Phone: STerling 3-4100
► f
INDEPENDENT TRACKING COORDINATION PROGRAM
824 Connecticut Avenue
Washington 6, D. C.
July 20, X96k
Dr* TlKMoas L, K. Smull
Director
Office of Research Grants and Coatracts
National Aeronautics and Space Administration
Wa&hington, D.C, 205lf6
Re: Research Grant tfep 35^60
Dear Dr« Sxaull:
Our last regular report was forwarded to you under date
of April 20, 1961f and covered the period from 1 Jfoveuiber 196^
to March 31, 1964*
ESiclosed please find report covering activities for the
period 1 April through 30 .June, 19&^.
Progr
rector
Enclosures :
Exhibits A tharough H
UNPUBLISHED
PRELIMINARY DA
ji. « X
SPSE
fitTfiHjS CONTROL N6,_
oSfK
Society of Photographic Scientists and Engineers ^
IHDEPEHDEHT TRACKHKJ COORDINATION PROGRAM -
SUMMARY OF PROGRESS AM) REPORT OF ACTIVITIES - 1 April to 30 June, 1961*
I. TRACKIWG and ACQUISITION DATA
A. Observations and Reports of Fix
1. 'PHOTOTRACK observations have been received at SPACON
during the period, as follows:
60 091 - 6 6k OOlfA - 13
63 0»t7A - 8 6k 005A - k
63 053A - 2
2. According to summary reports received at this office, a
number of visual reports of fix have been supplied to
independent research programs on satellites. At the
present time the majority of such reports are not being
relayed to SPACON .
3. During the period, the follow ng observations have been
received from independent tracking sources overseas and
forwarded to Croddard Space Flight Center:
60 091 - 15 63 038B
60 053 - 2 038C
61 Alpha 1 2 043A
62 A Ypsi k 053A
1
1
1
62 Kappa 1 5 05S - 1
2
35
3
fe B Kappa 1 3 Sk OOIA
63 03A - 5 OOlfA
63 Oll-A - 2 006A - ^
63 l^A - 2 OlOA - 1
63 27A - k OlOB - 1
63 3OA - 3 OllA - 8
6k 028B - k
B. Acquisition Data
1. Mean Orbital Elements
Reports of Fix are of primary interest to individuals or centers
conducting orbit studies on the particular satellites on which data is given
Individual observerations are of little use in satellite acquisition. Mean
orbital elements are the result of an analysis of a series of fixes and are
of use to anyone w.shing to acquire a satellite for observation: purposes.
An important element of the long-range goals of the Independent Tracking
Coordination Program has been to develop som-ces of acquisition data of this
type, not only from official tracking agencies but also from competent
ind viduals and groups.
- Continued on page 2
Sunanary - page 2 1 April - 30 June 196U
B» 1. (cont.)
Heretofore, the primary sources of mean orbital data, other than the
principal tracking centers, have been individuals with extraordinary interests
and/or computer resources, such as W* P. Overbeck, Director of the Savannah
River Laboratories and Herman Michielsen, Senior Staff Scientist, Lockheed
Missiles and Space Company. During the quarter, the ITCP received for the
first time sets of mean orbital data which were based on independent analysis
of independent observations carr ed out by a team of individuals with limited
computer resources and no background training in orbit analysis of this kind.
These results were reported in our Announcement Card issued 18 Jvne 196U, copy
of which is enclosed as Exhibit B,
It will be noted that the mean orbital elements supplied by Gregory
Roberts and Arthxir Arnold were based solely on observations made at Durban,
South Africa. The data obtained by Roberts and Arnold are of use to anyone
in the world having an interest in acquiring 1963-l^A or 1962 Kappa I during
a period from 30 to 60 days after issue < The analytical procedures were
carried out with the aid of a desk calculator, following methods suggested
in W. P. Overbeck 's, "A Letter to Gregory Roberts", which has been published
as part of the ITCP Program.
In the case of the more stable satellites, ways for describing the
orbit and mean motion in terms of "gear ratio elements" have been developed.
"Gear ratio elements "simplify long-term analysis of mean satellite motion •
They also supply data in a form which permits acquisition of the more stable
satellites from one to two years after the epoch of the elements. They will
be described in bulletins to be issued during the next quarter .
2. Daily Sattellite Ephemerides
An alternative method for communicating acquisition data of
particular interest in the shorter-lived or more erratic satellites is the
daily satellite ephemeris. Such an ephemeris, giving predicted orbital
arguments for OOh G.M.T. for each day of a 50-day period on six satellites
was issued dtiring the period. It is typical of the kind of daily ephemeris
that has been proposed for routine preparation at Goddard Space Flight Center,
and was, in this instance, prepared by W. P. Overbeck. A copy of the Daily
Ephemerides is attached hereto as Exhibit C. Ephemeris data on three of the
satellites (58 OOIA, 59 OOIA and 59 007A) were based on Smithsonian Astrophys-
ical Observatory mean orbital data. Data on the remaining three (60 OO6A,
60 OI3B, and 63 Okjh) were based on observations made by W* P. Overbeck. The
ephemeris contained on its reverse side tables of eccentricity functions for
the cxirrent value of eccentricity of each of the satellites listed on the
obverse side. True anomaly (PRV) and radius ratio (RAD) were given as
function/5of mean anomaly (PRM)* These elements were issued on April 5, 196U
in conjunction with a bulletin on "Work Sheets for Conversion of Satellite
Data to Rationalized Orbital Elements "(Exhibit D), Further details are given
in Section V of this Report.
- continued on Page 3
Sunnnary - P_ige 3 1 April - 30 June, 196U
II . Su pport of Inflation Studies .
There were no satellite inflations during the qu-rter. To date, none of
the photographic records showing traces of ECHO II (196U 004a) th<it have uome
to our attention give evidence of apparent brightness fluctuations attribu cable
to surface anomalies of the structure. Arrangements have been made to keep
the satellite under photographic surveillance to determine when and if bright-
ness fluctuations of this type becc»oe evident.
III. Satellite Trackers* Handbook .
Satellite tracking techniques and methods for orbit analysis continue
to develop at such a puce as to make it undesirable at this point to attempt
to "freeze" the material into the form of a handbook. Advances in trocking
methods, graphic forms and worksheets, and suggested procedures continue to
be issued in bulletin form. The individual bulletins are related to one
another and to the present literafore through common systems of notation and
terminology, and also through adherence to and systematic development of decimal
notation for describing angles, as well as times of events.
IV. Rationalized Tables of Trigonometri c Functions
During the quarter copies of "SEVEK PLACE COSHJES, SIHES AHD TAMGENTS
FOR EVERir TERTH MICBOTORS" were distributed to addressees and participants
in the Independent Tracking Coordination Program. A copy of these tables
of ^igonometric functions with rationalized arguments is attached hereto as
EXHIBIT F, The availability of these tables vastly simplifies desk calcu-
lator tracking methods.
V. Derivation of Rationalized Orbital Elements from Daily
S!^itii*».^^^^r"^^*' ^^^ ^°^^ Messages, and NOFAD/
SPADATS "4-line" Elenents.
eart^sa?2?ti?^-°''^'^\^^^^"*° ^""^^^^^ ^^^^ °" *^^ «^ti°" o^ ^^ artificial
earth satellite m an optimum form for Baking predict ons. The derivation of
rati^i?Sd oihf 1^ ^^ h ^^' examples are given of the derivation of
rationalized orbital elements from a variety of sources ^Exhibit D). Comnuta
^ZT/L "V^.^ ""l "^^ "" ^'^^^ ^°^ °'°*^^^^"^ rationalized ^rbitarSe!"
hS^ith Sork'Sjfr^'V"^^ ^^°^'^^' ^^^'^ ^°P^^ °^ ^"^-^^ i- -n^loS
r!Jf^i«?J A \^^^ C' " "^^ ^^ '^''^^ Exhibit D-1. Work Sheet B for obtaining
andT^S/Siibit'iT"^'"^'" "^Tt T'^ '^'^'^^' '^ attached h:^:^
sPADi?sTSnf^rnP-to i::oriL2t4?:ei 1^.^:^^^^^^
IZ T^ Ir'^lT^'l ¥ "°^" ''''' ' (F.hib.tT5) ro;:tf flrt^fva-
wMch L ^ ifirf ^ derivative of mean anomaly) from the anomalistic period
wnicn IS given in such elements to the nearest hundredth of a minute nniv
An improved Work Sheet pennitting derivation of me.n motiS vaLesThigher'*
el^Pnt°"- ^^" *^^ "Semi-Major Axis" values given in 1«)RAD/spM)A?S \-r n?
Summary - Page k 1 April - 30 June, 1961f
VI. Rules for Advancing the Epoch of Rationalized
Orbital Elements and Other Aids to Precise Computation.
Rationalized orbital elements may be routinely advanced to a subsequent
epoch without loss of precision. Rules for ccMnputing rationalized orbital
elements for a new epoch are described in ITCP Bulletin of 7 April, 1961^,
(EXHIBIT E) which gives an example of the necessary conq?utations in work
sheet form. Copies of blank Work Sheet C were supplied with the Bulletin
(exhibit E-l).
rrCP Bulletin of 7 April also gives rules for error-free combination
of polar angles and for obtaining the negative of an angle^i Drafting aids
to make accurate overlays, including a table for locating arc centers on
ITCP Chart #532 are supplied* The same bulletin briefly discusses the
availability of circular slide rules for five s gnificant f gures, the
relative merits of used desk calculators, and the availability of hand
calcxilators which permit COTiputation to eight significant figures*
VII. Digital Computer Program for Station Predictions (ZAYIK) >
The methods of prediction, observation and analysis which have been
consistently recommended by the Independent Tracking Coordination Program
have been based on obtaining fixes at or near the time of local culmination.
This is the instant when an artificial earth satellite transits the meridian
flx>m the mean orbit pole throijgh the observer's station. For radio observers,
this instant is practically undistiiiguishable from the instant of doppler
inflection. Satellite observations at local culmination i>ermit dealing
with the erfects of the earth's pear shape (third zonal harroonic) in a par-
ticxilarly efficient way and limit the problem of passing from mean to true
anomaly in making a prediction. From a computation point-of-view, a method
which requires conversion ft-om true to mean anomaly is much more efficient.
A method for predicting positions of artificial earth satellites at the
point of local culmination for desk calculatorjwas described in detail
in "A Letter to Gregory Roberts". These methods have been refined into a
digital computer program of great efficiency by W, P. Overbeck, and described
in a Bulletin dated May ik, 196^ entitled, 'ZAYIH: A Computer Program For
Predicting Positions of Artificial Satellites at the Point of Local Culmin*
ation", (EXHIBIT G), copy of which is attached hereto • ZAYIH uses rational-
ized orbital elements as input and accomplishes rejection of unobservable
or inacceptable passes with a minimum of non-productive computation. ZAYIN
computes the apparent positions of artificial earth satellites at the point
of culmination. It is designed for use by the optical observer who wishes
to make the type of observation that is most useftil in the determination of
orbital characteristics. It examines all revolutions of the satellite which
occur between any two selected dates. It rejects those passes which are below
the horizon, which occur while the observer is in daylight or for xfhich the
point of culmination is inside the Earth's shadow. For passes that are not
rejected, it prints out predictions in both alt-azimuth form and in celestial
coordinates, together with other data that is useful in setting the observing
instrument or in adjusting the predictions when observation indicates that
this is necessary. A number of observatories, including the Dominion
- continued on page 5
Sunanary - Page 5 1 April - 30 June, 196l^
Observatory, Ottava, Canada, are now uaing ZAYIH as a means for satisfying
requirements for station predictions on visually observable passes on
artificial earth satellites. Copies of the program in deck form are
being made available.
VIII. Long-term Tracking Techniques for Stable Satellites .
In the case of the more stable satellites, tracking procedures and
orbit analysis can be greatly simplified by expressing the motion of the
orbit and of perigee as functlonsof mean motion, rather than as functions
of time. Element sets of this kind have been named "Gear Ratio Elements"
by W. P. Overbeck, who is largely responsibile for their development. ITCP
Bulletin of June 11, 196^ reproduces Overbeck 's paper entitled, "dear Ratio
Orbital Elements for Tracking Artificial Earth Satellites". /(EXHIBIT H)
An important aspect of gear ratio elements is that they offer a
technique for orbit analysis which permits the casual observer to become
an authoritative source of long-range acquisition data on stable satellites.
It also permits the improvement of mean orbital data currently being supplied
by the official tracking agencies so as to be useful for satellite acquisition
over extended periods of time.
One of the prime objectives in the Independent Tracking Coordination
Program has been to reduce the amount of data on a given satellite required
for acquisition by an independent observer and to reduce the frequency of
communications on a specific satellite required for such purposes. It
appears that the gear ratio type of data package pennite use of a smaller
data package at less frequent intervals than any that have been advanced so
far. Although particularly appropriate for the longer-lived satellites,
gear ratio elements convey the essential information required for the shorter-
lived satellites and may, therefore, be of interest for general adoption for
the exchange of satellite acquisition information..-
^' Requirements for a Passive Geodetic Satellite .
The above topic was discussed by a panel on April 27, 196U at the
19W International Conference of the Society of Photographic Scientists
and Engineers held in Hew York City. Copies of the final program of the
Conference were furnished as EXHIBIT M to Report of April 20, 196J*. It is
anticipated that J. Hewitt's paper, "A 24-in. f/l Schmidt System for Pre-
cision Measurement of Satellite Positions" wiU appear in the forthcoming
issue of PHOKXSRAPHIC SCIEliCE AHD EHOHEERIIKJ, the SPSE Journal.
X. Matrix Methods
* -nnn/* ^^ ^^® meeting on April 2?, 196I^, Mr. Norton Goodwin, Director
of ITCP read a paper entitled, "Apparent Place Determination of Photographic
Star Images . A number of requests for preprints of this paper have been
received to date. It is anticipated that the subject matter will appear
in an ITCP Bulletin in the near future.
- continued on Page 6
SuBbary - Page 6 1 April - 30 June, 196^
XI. Anaouncaaents and Bulletins
A, Smnaary of Post Card Announcements on Radio -traasmittlng Satellites
Distributed during Period 1 April - 30 Jvme, 196'i-
Date Ifo. Diet * Identity of Satellites
V8 691 €k ookk 62 060A 63 OjifA
6k 005A 63 02lfA 6U 003A
V23 321 6h OOUa 62 O6OA 58 002B
6k 005A 6h OOIB 6k 015A
5/7 321 6k 00i^A 62 06QA 63 02ifA
6k 005A 63 038c 63 0U9C
5/21 678 6k 0(AA 62 06OA 64 006b
6ft OiOA 63 02l(A 62 015A
6/5 318 a OOftA
62 O6OA
6/18 627 6k 006B 6i; OlOA 61* 015A
63 02l*A 63 05ftA 62 015A
63 OlftA 62 OIQA 60 OO9A
6k QOkA 61* OO5A 63 O53A
B. Summary of Post Card Announcements on Brighter Satellites
V2 k65 €h QQkk 61* 005A 63 053A
60 OO9A 61 00l*A 58 OOIA
Vll 478 6k OOftA
16 U65 60 OO9A 61 OOftA 60 OO9A
63 Ol*TA 63 O53A 58 OOIA
5/1 1*63 61* OO9A 60 OO6A 58 OOIA
6h 005A 63 O53A 60 009B
5/15 1*63 60 OO9A 60 OO6A 58 OOIA
61* 005A 63 053A 60 009B
5/29 1*67 61* 00l*A 61* 005A 58 OOIA
60 009A 63 O53A 60 009B
6/11 1*65 61* 006B 61* 00l*A 60 009A
61* 005A 60 006a 63 O53A
6/25 1*65 60 009A 61* 005A 60 009B
58 OOIA 62 015A 59 007A
EKD OF REPORT
EXHIBITS ATTftCHED TO T3IS REPORT :
A* Financial Report, Quarter ending 30 June, 196l|- (Forms 1030-1031)
B. Modified Orbital Annonnc^nent Card dated June 18, 196'i' giving acquisi-
tion data supplied by independent observers in Durban, South Africa.
C» Daily Ephaaerides on Six Satellites vith Eccentricity Tables,
issued April 5, 196U
D. Bulletin dated April 5^ 196^ describing in detail Computation Work
Sheets for Conversion of Satellite Data to Rationalized Orbital Elements
D(l) Work Sheet A: For Obtaining Rationalized Orbital Eleanents from
Rationalised Daily Ephemerides
D(2) Work Sheet B: Rational Orbital Elements from Jtodified Orbital
Elements
D(3) Work Sheet C: Conversion of BORAD-SPADATS "if-line" Elements to
H,O.E.
E . Bulletin of April 7, 1^ containing Rules and information on
Advancing the Epoch and Drafting Aids to Making Accurate Overlays
E(1) Work Sheet D: For Advancing the Epoch of Rationalized Orbital Elements
F* Tables of Trigonometric Functions: "SEVEN PLACE COSIHES, SIBES ABB)
TAISSESTS FOR EVER? TEKEH MICROTURH"
G» Bulletin dated Mary ll^-, 196lf: ZAYIN: A Computer Program for
Predicting Positions of Artificial Satellites at the Point of
Local Culmination.
H. Bulletin dated June 11, 196k: "Gear Ratio" Elements for Tracking
Artificial Earth Satellites.
^BRIGHT
EXHIBIT ^
^■pBJECr
iPjame
64 006B
64 OlOA
64 015A
63 024A
63 054A
62 015A
Elek 2
Cosinus 25 Ariel 2
; Tiros 7
Tiros S
Ariel
;2 SOURCE
Norad
Norad
GSFC
Norad
Norad
GSFC
5 EPOCH of
14 Jun
16 Jun
15 Jun
13 Jun
13 Jun
12 Jun
§ perigee
OIH
08H
OOH
16H
14H
18H
SJ (UT)
14M64
56M53
49M10
53M36
33M38
39M96
^INCLIN.
60A20
49A02
51A66
58A23
58A50
53A86
tNODE W.
350A16
323A52
010A61
113A60
250A74
091A29
SmPD = ID
° PERIGEE
-04M24
-25M23
-20M06
-18M74
-17M97
-19M43
073A30
276A21
020A66
144A25
350A03
282A06
^ change/P
+A017
+A303
+A214
+A093
+A086
+-A172
5 A. PERIOD
156M386
91M434
101M195
97M439
99M370
100M55]
O change/P
-MOOOOO
-M00031
-M00009
-Moooo:
. -MOOOOl
-MOOOO]
^ECCEN.
U82847
U01347
U07331
U00256
U00235
U05417
P. RADIUS
4326#0
4121#5
4142 #2
4347#2
4405#3
42 09 #7
freq. X6/S
19T430
19T540
90T225
90T022
136T447
136T233
136T233
136T406
REMARKS
136T921
136T924
R. A. NODE
^J \J J_ ^^ 4aJ ^-f
290A87
075A27
265A00
041A77
229A53
089A82
1 BRIGHT
-28.8
-26,-30
-4
+ 3
-8r20
-16,-28
^OBJECT
63 014A
62 OlOA
60 009A
64 004A
64 005A
63 053A
5 NAME
Atl.Agena Midas 5
Echo 1
Echo 2
Saturn 5
Expl 19
i2 SOURCE
Durban*
Durban*
Norad
SAO
SAO
SAO
5 EPOCH of
19 Jun
19 Jun
13 Jun
20 Jun
20 Jun
20 Jun
^ perigee
OOH
lOH
09H
OIH
02H
OIH
^ (UT)
17M346
45M045
38M15
45M66
05M79
10M28
AiNCLIN.
87A310
86A705
47A30
81A47
31A45
78A62
^NODE W.
080A316
060A070
009A93
014A22
338A94
024A61
SmPD - ID
-04M3164 -04M5056 -17M15
-07xM18
-29M78
-07M78
^PERIGEE
292A181
116A440
061A89
135A79
135A65
161A05
ULJ change/P
-A11702
-A13042
-^A255
-A13272 +A66797
-A15244
^A. PERIOD
166M4257 152M9816 114M328 108M71494M237
115M594
LJ
O change/P
-MOOOOO
-MOOOOO
-M00030-M00010
-M00033
-M00019
^ECCEN.
U00660
U02906
U05459
U02347
U03343
U11288
P. RADIUS
6180#87
5716#51
4584#2
45 79 #3
41 18 #8
4333#6
freq. X6/S
Tumbling Tumbling
136T020
REMARKS
very slowpur.lS sec.
136T170
R. A. NODE
: 19rA1843 008A9299 036A34
280A32
320A64
261A06
*Elements supplied by Gregory Roberts and Arthur Arnold,
62 Draeonwvck, 7 St. Geore^es Street, Durban, South Africa.
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A
EXHIBIT D-1
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. -
NGR =
= t CG = . km CP =
. km
CG/CP =
Whole turns in PRM per day
= !o
@ JNL =
d
•
: TNR = \ NRP = 5
PRM =
t
•
-(@ JNL =
d
•
d
•
: TNR = \ NRP = \
PRM =
t
•
AJNL =
ATNR= ! ANRP= t
APRM =
t
•
divide each item by above value of
AJNL
TNR^= \ NRP J = \
PRM =
t
•
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. -
NGR =
\ CG = . km CP =
. km
CG/CP =
Whole turns in PRM per day
= io
@ JNL =
d
•
: TNR = J NRP - J
PRM =
t
•
-(@ JNL =
d
•
d
•
: TNR = \ NRP = !
PRM =
t
•
AJNL =
ATNR = * ANRP = \
APRM =
t
•
divide each item by above value of
AJNL
TNR^= ^ NRP J = \
PRM =
t
•
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. -
NGR =
= t
•
CG = . km CP =
. km
CG/CP =
•
Whole turns in PRM per day
- to
@ JNL =
d
•
•
•
TNR = * NRP = ^
PRM =
t
•
-(@ JNL =
d
•
d
•
•
•
TNR = \ NRP = \
PRM =
t
•
AJNL =
ATNR - ! ANRP = !
APRM =
t
•
divide each item by above value of
AJNL
TNR^= \ NRP J = *
PRM =
t
•
Additional covlz^ available, f^xom: I TCP, S2 4 Conn^ Av;e., ^Ja6k., P. C. 20006*
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. -
NGR = t
CG = . km CP =
. km
CG/CP = .
Whole turns in PRM per day
= \o
@ JNL =
d
• •
TNR = \ NRP = !
PRM =
t
•
-(@ JNL =
d
• •
d
•
TNR = \ NRP = \
PRM =
APRM =
t
•
AJNL =
ATNR = \ ANRP = ^
t
•
divide each item by above value of
AJNL
TNR^= \ NRP^ = ?
PRM =
t
•
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. -
NGR
= t
•
CG = . km CP =
. km
CG/CP
1
Whole turns in PRM per day
= to
@ JNL =
d
•
•
•
TNR = ! NRP - 5
PRM =
t
•
-(@ JNL =
d
•
d
•
•
•
TNR = ! NRP = t
PRM =
t
•
AJNL =
ATNR - \ ANRP = \
APRM =
t
•
divide each item by above value of
AJNL
TNR^ ! NRP = t
PRM =
t
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. -
@ JNL =
-(@ JNL =
AJNL =
NGR =
CG/CP --
d
•
d
•
d
•
= t CG = . km CP =
Whole turns in PRM per day
: TNR = t NRP = \
: TNR - \ NRP = \
. km
- to
PRM =
PRM =
APRM =
AJNL
PRM ^
t
•
t
•
t
•
t
•
)
ATNR = . ANRP = \
divide each item by above value of
TNR^= \ NRP^ = t
EXHIBIT D-2
WORKSHEET B: Rationalized Orbital Elements from Modified Orbital Elements
(items in square brackets are line No.'s of items listed in M.O.Els)
No. = [1]
JNE = epoch: from [4] = day mo.
+([5]/24 = h/24h/day
+([6]/1440 = "? /1440m/day
yr, obtain mjd
9 /360O/t
NGR = inclination = [7]/360°/t
CG/CP= eccentricity = [14]
CP = semi-major axis = [15] (1.60935 km/sm)/( 1.000000 - #3)
CG
TNR.
( . sm) (1.60935 km/sm)/(0. )
= semi-major axis times eccentricity = (#3)(#4)
= mean time of orbit pole at epoch
TNLq = fractional p
-(ANLq = [8]/360O/t
= TNAq
-(TNA
/3607t
TNR
o
TNR,
- 1^* time derivative of mean time of orbit pole
= [9]/ 1440 = ^ /1440m/t
Argument of perigee at epoch (from ascending node)
- (#10)/360O/t - ° /360O/t
NRP = argument of perigee at epoch (from north point of orbit)
= (#12) - ^250000
NRP J = ist time derivative of argument of perigee = (#16)[ ll]/360
= (#16)( 9 /360O/t) = ( t )(t )
PRM - mean anomaly at epoch
iSt
o
PRM, - l^*- time derivative of mean anomaly = 1440m/Cl2] •
= 1440m/ ^ /t
PRMg = I 2"*^ time derivative of mean anomaly = -(#16)2 [13 ]/2 880
= -( . )( "? )/2880 - . /2880
4oooooo
d
#0
_ t
#1
#2
#3
km #4
km #5
250000)
#9
000000
(ENL) #11
#12
#13
(ENL) #14
#15
(ENL) #16
Kddi.tA.onat copizi avaltabta ^fiom:
(ENL)2 #17
ITCP, S24 Conn. Avt., Waifi., 0.C. 20006.
WORKSHEET B: Rationalized Orbital Elements from Modified Orbital Elements
(items in square brackets are line No.'s of items listed in M.O.Els)
No.
JNE
[1]
epoch: from [4] =
+([5]/24 =
+([63/1440 =
#0
day mo.
h/24h/day
^ /1440m/day
yr, obtain mjd
NCR =
CG/CP =
CP
CG
TNRq =
9 /360O/t
inclination = [7]/360°/t
eccentricity = [14]
= semi- major axis = [15] (1.60935 km/sm)/( 1.000000 - #3)
= ( . sm) (1.60935 km/sm)/(0. )
= semi-major axis times eccentricity = (#3)(#4)
= mean time of orbit pole at epoch
TNL = fractional p
-(ANLq = [8]/360O/t
= TNAo
-(TNA
/ 3607t
TNR
o
TNR.
NRP
= 1^^ time derivative of mean time of orbit pole
= [93/1440 = ^ /1440m/t
Argument of perigee at epoch (from ascending node)
- (#10)/360O/t - ° /360O/t
= argument of perigee at epoch (from north point of orbit)
= (#12) - ^250000
1^^ time derivative of argument of perigee = (#16)[ 11 3/360
= (#16)( 9 /360O/t) = ( t )(t )
mean anomaly at epoch
PRM = 1® time derivative of mean anomaly = 1440m/[12 3 •
^ = 1440m/ ™ /t
PRM = I 2"^ time derivative of mean anomaly = -(#16)2[ 133/2880
"^ = -( . )( m )/2880 = . /2880
o
NRP. =
PRM
o
4oooooo
d
#1
#2
#3
km #4
km #5
t
•
t
•
)
t
•
t 2 50000)
t
•
t
t
•
(ENL)
#11
t
•
#12
#13
t
•
(ENL)
#14
! 000000
#15
t
•
(ENL)
#16
t
(ENL)2
#17
EXHIBIT D-3
WORKSHEET C: Conversion of NORAD-SPADATS "4-line Elements" to R.O.E.
(items in square brackets are line No. and item No. of items in NORAD elements)
No.
JNE
NGR
CG/CP
CP
CG
QNT
/360°/t
QNT,
TNR,
tnK
[0,3]
epoch = [ 1 ,3 ]
inclination = [2,3]/360O/t = °
eccentricity = [2,6]
semi- major axis = [1,4 ](a) = . (6378.17 km)
semi-major axis times eccentricity = (#3)(#4)
celestial longitude of midnight from pole of ecliptic at epoch
= t540608 + ^002738( #1 - 38400*^000000)
= ^540608 + *002738( ^ )
= t
^540608
1®^ time derivative of longitude of mean midnight
/360O/t
mean time of the orbit pole at epoch
QNRq = [2,5]/360°/t = °
-QNTq = #6
=TNR,
iSt
1 time derivative of mean time of orbit pole
QNRj = [3,5]/360Ot = ^ /SeO^/t
-(QNTj
= TNRj
Argument of perigee at epoch (from ascending node)
= L2,4]/360O/t = ° /360°/t
NRPq = Argument of perigee at epoch (from north point of orbit)
== #12 - *2 50000
NRP = 1^^ time derivative of argument of perigee
= [3,4]/360O/t = ° /3607t
PRM = mean anomaly at epoch = L3,3]/360°/t = ° /360°/t
- ( #12
- ((*)(#8) (*)=(-l) if#2^k5
o
km
km
_ t
= t0027379093(ENL) #7
(ENL)
_ t
!: 00273 8(ENL)
(ENL)
^ t
= t
= t
= t
(ENL)
PRM.
l®*- time derivative of mean anomaly = 1440/[2,7]
= 1440m/ "? A
»nd
= t
PRM = i 2^° time derivative of mean anomaly = -(#16)'^[ 3,7 ]/2 880
= -(
)( .
)/2880
/2880
(ENL)
(ENL)'
No.
JNE
NGR
CG/CP'
CP
CO
QNT
WORKSHEET C: Conversion of NORAD-SPADATS "4-line Elements" to R.O.E.
(items in square brackets are line No. and item No. of items in NORAD elements)
° /360O/t
(6378.17 km)
QNT,
TNR,
[0,3]
epoch = [1,3]
inclination = [2,3]/360O/t
eccentricity = [2,6]
semi- major axis = [l,4](a) =
semi-major axis times eccentricity = (#3)(#4)
celestial longitude of midnight from pole of ecliptic at epoch
= t540608 + t002738( #1 - 38400*^000000)
= {540608 + i002738( ? )
= ^540608 + \
1^^ time derivative of longitude of mean midnight
/360°/t
= t
mean time of the orbit pole at epoch
QNRq - [2,5]/360°/t = °
-QNTq = #6
=TNR
TNR^ =
o
l^*- time derivative of mean time of orbit pole
QNRj = [3,5]/360Ot = o /SeO^/t
-(QNTj
= TNRj
Argument of perigee at epoch (from ascending node)
= L2,4]/360O/t = ° /360°/t
Argument of perigee at epoch (from north point of orbit)
- #12 - ^2 50000
NRP, = l®*- time derivative of argument of perigee
= [3,4]/360O/t = ° /360°/t
PRM = mean anomaly at epoch = L3,3]/360O/t = ? /360°/t
- ( #12
- ((*)(#8) (*)=(-l)if #2^!25
NRP
o
o
PRMj^ = 1^^ time derivative of mean anomaly = 1440/[2,7]
= 1440m/ "? A
PRM = I 2^^ time derivative of mean anomaly = -(#16)^[3,7]/2880
km
km
0027379093(ENL)
(ENL)
00273 8(ENL)
(ENL)
(ENL)
(ENL)
2
= -( . )( . )/2880 - . /2880 = . (ENL)
Kddltlonait copizi avallablz {.fLom: ITCP, S24 Conn. Avz., n/as/i., V,C. 20006.
EXHIBIT D
^^hone: Slerling 3-4100
INDEPENDENT TRACKING COORDINATION PROGRAM
824 Connecticut Avenue
Washington 6, D. C.
BULLETIN
April 5, 1964
WORK SHEETS FOR CONVERSION OF SATELLITE DATA
TO RATIONALIZED ORBITAL ELEMENTS
COMPUTATION WORK SHEETS which may be used as guides for obtaining rationalized
orbital elements from various sources are discussed below. Blank copies are supplied
herewith. Additional work sheets may be obtained from this office upon request. Please
specify which work sheets are required.
Rationalized Orbital Elements from Daily Ephemeris: Example
Given
JNL
63047A rtPO 3^466
i"\joK«Qo43:jCo bi^l.i
CH 7b02.o 13T
_Kra ^^ ■Ml<P , , HKr,
■Mt^
^^^^^^g
b2? |.64b70 .p::;:.cjb* ,-'a4uZ
(Extract from Daily Ephemerides supplied by W. P. Overbeck 3 April 1964.)
WORK SHEET A: For Obtaining Rationalized Orbital Elements from
Rationalized Daily Ephemerides
No. G;i-(v;:-o
@ JNL =
-(@ JNL =
AJNL =
NGR = tOBi33 CG= 651.3 km CP =
CG/CP = .'
' '-" '■'■ * ■■*■.■■
... d.
li Km KSf = (
Whole turns in PRM per day =
J2.B km
tn
TNR = - *C6219
TNR = - *C4:j70
NRP =
NRP =
PRM = hi 932
•■'V;tib
PRM
= t-
?76462 )
,d
1 3^3 5470;
ATNR = -*0iC49 ANRP = ht2i69 APRM = 13*3 547 (
divide each item by above value of AJNL
TNR = - hv'A9 NRPj = !u21 i) PRM
Society of Photographic Scientists and Engineers
Decoding SATOR Messages: Description of Code
SATO R (Modified Orbital Elements for Prediction Purposes)
Code word : SATOR
Syinbolic form
SATOK
jkkkk
PPPPP
sssss
aabbc deeff ggggZ hhhhX NOWES iiiii
ARPER 111 11 mnnnX PERIOD ooooo
ECCEN qqqqq PERRA rrrrr RAFRE
{sssss repeated as necessary) RADEG ttttt
aa = last two digits of year satellite launched
bb = Greek letter designation, 01 = Alpha, 02 = Beta, etc.
c = component
d = reference time (epoch): last digit of numerical notation for month; i.e. 1 = January or November, 2 =
February ^r December, 3 = March, etc.
ee - reference time (epoch); date
ff = reference time (epoch); hour
gggg ' reference time (epoch): minutes and hundredths of minutes
Z a Universal tinne, Greenwich Mean Time
hhhh - inclination in degrees and hundredths of degrees. If the orbit inclination is negative (satellite fired west-
ward) group is preceded by NEGAT
X = always an X
NOWES = sub-indicator for geographical longitude of northbound node west of Greenwich at reference time
iiiii » longitude of northbound node in degrees and hundredths of degrees
j = 1 if plus: when the "prime sweep interval" is one day plus a certain number of minutes
2 if minus: when the "prime sweep interval" is one day minus a certain number of minutes
This is equivalent to saying that the same portion of the orbit plane will reappear at the same
location a certain number of minutes earlier each day.
kkkk = number of minutes and hundredths of minutes by which "prime sweep interval" differs from one day or 1440
minutes. This is another way of expressing the relative "westward motion" of the orbit plane.
ARPER = sub-indicator (argument of perigee) angular distance of perigee from node at reference time. For modified
orbital elements, this is also the position of the satellite in the ellipse at reference time (mean anomaly
at epoch is always equal to zero in this system)
mil - angular distance of perigee and satellite from northbound node, measured in the direction of satellite travel
in degrees and hundredths of degrees
m = 1 for plus, if perigee moves in the same direction as satellite travel
2 for minus, if perigee moves in the direction opposite to satellite travel
nnn = average decimal fraction of a degree which perigee moves per period, measured in thousandths of a degree
X = always an X
PERIOD = sub-indicator for perigee-to-perigee period (anomalistic period)
ooooo = perigee-to-perigee period (anomalistic period) in minutes and thousandths of a minute. If first two digits
are less than 85 it should be understood that 100 should be added in order to arrive at the correct period
(period cannot be less than about 88 minutes). Should the period be greater than 185 minutes a special no-
tation will be nnade in the message.
PPPPP ~ average per period change in perigee-to-perigee period, measured as a decimal fraction in one hundred
thousandths of a minute
ECCEN = sub-indicator for eccentricity
ggggg « eccentricity, measured as a decimal fraction in one hundred thousandths
PERRA » sub-indicator for radial distance of satellite from center of earth at perigee
rrrrr = radial distance of satellite from center of earth at perigee, measured in miles and tenths of miles
RAFRE ' sub-indicator for radio frequencies currently being transmitted from satellite
sssss - radio frequency in megacycles and hundredths of megacycles
RADEG = sub-indicator for right ascension of the ascending node expressed in degrees and hundredths of degrees in
order that this message may also serve the needs of those who prefer traditional orbital elennehts (Note that
this sub-indicator and the following code group represent a revision of the code appearing in the Fifth Sup-
plement to the Draft Manual)
ttttt = degrees and hundredths of degrees of right ascension (Note that right ascension is given in degrees and
hundredths of degrees rather than hours and minutes)
(From Satellite Report #7, National Academy of Sciences, National Research Council, p. 49-50)
Decoding SATOR Messages: Example
Given the following SATOR code message:
PART IV.
SATOR
29152
9936 9
RAFRE
6354A
217 98
00001
36.20
32716
ARPER
ECCEN
00000
5450 Z
24051
00268
RADEG
58 50X
lassx
PERRA
14726
NOWES
PERIOD
44038
(From Bulletin 9 1963-54A 716 Part IV, from NASA Goddard Space Flight Center.
Data Source NORAD)
Modified Orbital Elements from above SATOR code message:
^BRIGHT
m^'^7':
-4
< OBJECT
mm4^
63-054A
•^^NAME
■mm--
Tiros 8
« SOURCE
mmm.:
Nor ad
2 EPOCH of
m:0m-
27 Mar
S perigee
mmn,:
16H
ui (UT)
^1$^
54M50
^INCLIN.
tNODE W.
58A50
291A52
Smpd — id
.I*?|»^Moy|^:i
-17M98
^ PERIGEE
'^w
240A51
Lu change/P
+A086
5 A. PERIOD
AmA
99M369
O change/P
'^t'^tt^^ffi
-MOOOOl
^ ECCEN.
U00268
-8 P. RADIUS
^';:^^^^Hp^C^
4403#8
vSfreq. X6/S
^/l^^^^
136T233
Sreaaarks
fii^^R 136T924
"^R. A. NODtPiM^:
147A26
Rationalized Orbital Elements from Modified Orbital Elements: Example
Given above Modified Orbital Elements:
WORKSHEET B: Rationalized Orbital Elements from Modified Orbital Elements
(items in square brackets are line No.*s of items listed in M.O.Els)
No.
= [1]
JNE
= epoch: from [4] = 2" day 03 mo. 1964 yr, obtain mjd
+([5]/24 = h/24h/day
+([6]/1440 = : ."? 0/1440m/day
NGR
- inclination - [7 ]/360Vt = 58950 /360O/t
CG/CP= eccentricity = [14]
CP = semi-major axis = [15] (1.60935 km/sm)/( 1.000000
CG
TNR^
( :;;.: sm) (1.60935 km/sm)/(0.99732 )
semi-major axis times eccentricity = (#3)(#4)
mean time of orbit pole at epoch
TNLq = fractional part of #1
-(ANLq = [8]/360O/t
#3)
^/360Vt
TNR,
NRP
NRP,
-TNAq
-(TNA
™Ro
- 1^^ time derivative of mean time of orbit pole
= [91/1440 - -~17n?98/1440m/t
Argument of perigee at epoch (from ascending node)
= (#10)/360O/t = 240*?5i/360O/t
= argument of perigee at epoch (from north point of orbit)
= (#12) - ^250000
= 1^* time derivative of argument of perigee = (#16)[ll]/360
= (#16)( 9ra.(./360O/t) = (■I4'49l4)( t00023S9l
PRM =
mean anomaly at epoch
PRM, = 1®^. time derivative of mean anomaly = 1440m/[12]
= 1440m/ ^ /t
I 2^^*^ time derivative of mean anomaly = -(#16)^[13]/2880
1
PRM« =
63-054-01
#0
38481^000000
^666667)
^037847)
384814704514
#1
h62500
#2
.00268
#3
7106.30 km
#4
19.04 km
#5
t704514
^809778)
t 894736
t 2 50000)
^644736
#9
^012486(ENL)
#11
^668083
#12
^418083
#13
^003462(ENL)
#14
^000000
#15
= -(
)(
m
t)/2880 =
.030432/2880
= 14.49144 (ENL) #16
= h06-4 (ENL)2 #17
Rationalized Orbital Elements from NORAD-SPADATS ^^4-line^^ Elements: Example
Given:
(((((
716 009 1963-54A US 64 03 27 9 03 28 261
1 716 009 38481.72737050 01.11416577 -.953515-06 -.240759-10 178
2 716 009 058.4979 240.5085 147.2553 .0026835 0099.36 000721 179
3 716 009 147.0132 01.2459 -03.5655 -.853-06 -.127-3 001411 136
)))))
(From Element sets for NASA issued 29 March 64, Data Source NORAD)
WORKSHEET C: Conversion of NORAD-SPADATS "4-line Elements" to R.O.E.
(items in square brackets are line No. and item No. of items in NORAD elements)
No.
JNE =
NGR
CG/CP =
CP
CG
QNT
QNT,
TNR„
(6378.17 km)
[0,3]
epoch = [1,3]
inclination- [2,3]/360O/t
eccentricity = [2,6]
semi- major axis - [1,4 ](a) =
semi-major axis times eccentricity ^ (#3)(#4)
celestial longitude of midnight from pole of ecliptic at epoch
= t540608 + ^002738( #1 - 38400^000000)
- ^540608 +t002738( ^' : )
= ^40608 +!
1^^ time derivative of longitude of mean midnight
mean time of the orbit pole at epoch
QNR_ - [2,5]/360^/t
/360^/t
-QNTq
#6
=TNR^
TNR,
= ist
NRP„ =
1^^ time derivative of mean time of orbit pole
QNRj = [3,5]/360Ot = - ;0 /360^/t
-(QNTi
-TNRj
Argument of perigee at epoch (from ascending node)
-L2,4j/360O/t = H ^ /3607t
Argument of perigee at epoch (from north point of orbit)
- #12 - ::250000
NRP = l^t time derivative of argument of perigee
= [3,4]/360Vt = ^, /360°/t
PRM = mean anomaly at epoch - L3,3]/3607t = '° y360°/t
- ( #12
- ((*)(#8) (*)-(-l)if#2>j25
PRM. =
1
tst
PRM„
1**^ time derivative of mean anomaly = 1440/[2,7]
^2r
1440m/ ^ A
^ 2^ time derivative of mean anomaly = -(#16)'^[3,7]/2880
= -(• .' )(-.^ )/2880 - . /2880
#0
#1
#2
#3
km
#4
km
#5
(ENL)
^ t
!:002738(ENL)
(ENL)
(ENL)
t
i:
•
t
- )
)
t
t
•
(ENL)
t
/tPXTT \2
#6
- ^0027379093(ENL) #7^
#8
#9
#10
#11
#12
#13
#14
#15
#16
#17
} RAFTING AIDS TO MAKING ACCURATE OVERLAYS
In ITCP Bulletin of January 15, 1964 (1. 11) mention was made of an
"inexpensive yardstick compass such as #978 of Eugene Dietzgen Co.,
2425 Sheffield, Chicago, Illinois 60614, which could be purchased for
less than $15 and which is useful in preparing accurate overlays for
meridional stereographic nets."
Our attention has been drawn to the Keuffel and Esser Mark 1 Beam Compass, Item No.
55-1806, which is also available at less than $15. It is available at most drafting and sur-
veying supply houses and is distributed by Keuffel and Esser, Hoboken, New Jersey. Three
8" beams are supplied with this unit and additional beams are available.
ACCURATE LOCI for the centers of arcs to be swung with a beam compass can rapidly be
foxmd with the table given below. Distances in millimeters from the net center are given.
Location of Arc Centers for Preparing Accurate Overlays of Mean
Orbit Plane and of Observer's Parallel
On ITCP Chart #532
ORBIT PLANE ARC CENTER
OBSERVER CIRCLE CENTER
NGR
Center On
NGR
(turns)
X Axis (mm)
(turns)
.00
00.0
.50
.01
10.1
.49
.02
20.3
.48
.03
30.6
.47
.04
41.2
.46
.05
52.1
.45
.06
63.5
.44
.07
75.4
.43
.08
88.1
.42
.09
101.7
.41
.10
116.5
.40
.11
132.6
.39
.12
150.5
.38
.13
170.7
.37
.14
193.8
.36
.15
220.6
.35
.16
252.6
.34
.17
291.6
.33
.18
340.7
.32
.19
404.9
.31
.20
493.4
.30
.21
624.3
.29
.22
840.3
.28
.23
1268.9
.27
.24
2547.9
.26
.25
.25
NGO
Center On
NGO
(turns)
Y Axis (mm)
(turns)
.00
160.3
.50
.01
160.6
.49
.02
161.6
.48
.03
163.2
.47
.04
165.5
.46
.05
168.5
.45
.06
172.4
.44
.07
177.2
.43
.08
182.9
.42
.09
189.9
.41
.10
198.1
.40
.11
208.0
.39
.12
219.9
.38
.13
234.2
.37
.14
251.5
.36
.15
272.7
.35
.16
299.2
.34
.17
332.7
.33
.18
376.5
.32
.19
435.5
.31
.20
518.7
.30
.21
644.6
.29
.22
855.5
.28
.23
1279.0
.27
.24
2552.9
.26
.25
.25
ACUARC RULER
It is obvious from the above table that many of the arcs of circles which one would
like to draw on a stereographic net overlay are too large for beam compasses of prac-
tical radius. The ACUARC ruler is a flexible template that can be adjusted to approxi-
mate curves of any radius from about 7" to infinity. It is extremely useful in fitting
arcs of circles through three or four points plotted on a net overlay. It is sold by many
drafting and mapping supply houses and is manufactured by Hoyle Engineering Company,
Barstow, California, U.S.A. The list price is $10.00.
SLIDE-RULE MULTIPLICATION AND DIVISION TO FIVE SIGNIFICANT FIGURES
The Atlas slide rule is an ingenious device for multiplying and dividing to five
significant f^res. In addition to a circular scale around the periphery of the disc,
the slide rule contains a spiral scale of 25 coils occupying most of the face of the slide
rule, which is about 21 cm in diameter. For those who do not have access to a desk cal-
culator or a hand calculator, such as the CURTA described below, the Atlas Slide Rule
will prove a veiy useful aid to computation. It is distributed by Eugene Dietzgen Co.,
2425 Sheffield, Chicago, Illinois, 60614, U.S.A., and is available from most drafting and
surveying supply houses. It is listed as Dietzgen Part No. 1797A at a $13.50 list price.
USED DESK CALCULATORS
Greater than 5-place accuracy requires access to a desk calculator or at least a
hand calculator. Used and/or reconditioned desk calculators with automatic division
features and sufficient register dials to compute to 8 significant figures are available
through most stationery and office supply sales outlets, including those of the larger
department stores (e.g. Macy's in New York City). In general, a used desk calculator
with few features, but in good condition, will prove to be a better buy than a used desk
calculator in fair condition with many special features at the same price,
CURTA HAND CALCULATOR
The CURTA hand calculator, while not as convenient as a good electric-powered
desk calculator, can be used to solve problems to 8 significant figures. One version
with more limited registers is offered but is not considered to be as good a buy^ These
items are not toys, they are precision equipment and command a substantial price -
about $165. They are available through some of the larger stationery stores and are
listed in the Montgomery Ward catalog under Automobile Rally Accessories.
^ c^
Phone: STefling 3-4100
INDEPENDENT TRACKING COORDINATION PROGRAM
824 Connecticut Awnut
Washington 6, D. C.
BULLETIN
ZAYIN: A COMPUTER PROGRAM FOR PREDICTING POSITIONS
OF ARTIFICIAL SATELLITES AT THE POINT OF LOCAL CULMINATION
\i. P. Overbeck
May 14, 1964
Introduction
This paper describes an automatic prediction program, ZAYIN, which computes
the apparent positions of artificial earth satellites at the point of local
culmination. It is designed for use by the optical observer who wishes to make
the type of observation that is most useful in the determination of orbital
characteristics. It examines all revolutions of the satellite which occur
between any two selected dates» It rejects those passes which are below the
horizon, which occur while the observer is in daylight or for which the point
of culmination is inside the Earth's shadow. For passes that are not rejected,
it prints out predictions in both alt-asimuth form and in celestial coordinates,
together with other data that is useful in setting the observing instrument or
in adjusting the predictions when observation indicates that this is necessary.
ZAYIN uses Rationalized Orbital Elements as input. These were initially
described in an ITCP publication, **A Letter to Gregory Roberts", and have been
further discussed in subsequent ITCP Bulletins. A Bulletin of April 5> ^96h,
tells how to derive such elements from the information available from a variety
of sources. The output of ZAYIH is also, primarily, in rationalized format and
is arranged to facilitate mailing of predictions from a central computing facility
to distant observing stations.
ZAYIN is one of a group of programs which, together, comprise a complete
data processing system for satellite tracking, including such functions as;
computation of perturbations, preparation of tabular aids for desk calculator
computation and the reduction and analysis of observations. The data card format
and the subroutines used in ZAYIN are designed to be applicable to all programs
in this system.
The unique feature of ZAYIN is that it accomplishes its rejection of
unacceptable passes, with a minimum of non-productive computation, in the
coordinate system of the orbit pole, rather than that of the Earth's North Pole.
A fast subroutine for coordinate transformation, POLO, makes it possible to do
this expeditiously. ZAYIN also includes an "internal counter" system that permits
the program to make its own decisions as to whether certain steps in the compu-
tation are necessary.
Society of Photographic Scientists and Engineers
Symbols^ Units and Fortran Names
With few exceptions, the Fortran names used in ZAYIN are derived from
the three-letter symbols which have been consistently used in the ITCP
reference material • To maintain this consistency, we will use the same
three-letter symbols in the ensuing description of ZAYIN, even though they
differ from the Fortran names • It is believed that the reader can learn to
recognize the differences without confusion^ ^i/here it is necessary to use
Fortran names in the discussion, they will be xinderlined and will also include
the Fortran convention of writing the letter *'C* with a slash, as V*» "to
distinguish it from the numeral, zero*
The Fortran names require a fourth, prefix letter to differentiate between
fixed-point and floating-point variables* We have given this prefix letter an
added significance as follows:
I The letter "I* is used to designate the integral portion of a number*
For example, the mean anomaly, PRM, may include both full and fractional
turns, as in PRM « 1179-2835492 In this case, the name, IPRM, would have
the value, 1179> representing only the full turns.
A The fractional portion of a nvunber such as that above, would have the
prefix "A** In the above example, APRM would have the value, •26^k92
In ZAYIN, names that begin with "A", such as APRM, ANGR, ANRP , etc*,
represent the amplitude of the angle that is represented by the last
three letters in the name*
S The prefix "S* will represent the sine of an angle* Thus, when the angle
is named ANGR its sine will be named 3NGR .
C Similarly, *C* designates the cosine, so that CNGrR is the cosine of ANGR.
D The prefix *D" designates a difference or a derivative* Such a name
will usually require further definition*
The units of angular measure in ZAYIN are decimal turns j units of time are
decimal days and units of distance are kilometers* Througihout the program and
subroutines, the letter "Z* always represents the constant, 2 pi, needed as a
conversion factor between turns and radians wherever the standard trigonometric
Library Functions are used*
Input
The input data for ZAYIN is arranged on a series of punched cards as
described below. For each card, we give the full 72 column format, in which
blank spaces are indicated by the letter "b". The field assigned to each
Fortran variable is underlined* The numbers used in these examples correspond
to an actual case and, in a succeeding description of the output, the same case
is used so that the niambers may be directly compared*
Control Card
' . > t 1 t I I t
b^8475b5^55bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
JNL1 JNL2
This card gives the Modified Julian Dates, JNL, for starting, JNL1 , and
ending, JNL2, of the series of predictions.
C Card
I ( J I I I I t
0.1572^0 b -0.227057 b 6372.07 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb gfVERBECKbb bb
ANG0 ALN^ G^ IDEN IDEM1
The G card gives the observer's position, including his polar distance, NGO,
his longitude, LNO, and his radius, GO, from the Earth's center. It also
provides ^ spaces for alphanumeric data that may be needed for identification
or as an aid in addressing and mailing predictions. This information is
transferred from input to output in blocks of 5 characters, such as IDEK
and IDSK1 . As written, 2AYIN transfers only two such blocks but it may
j be easily changed to transfer more information.
A Card
j ' ?8^6.024^93^7 b .0917?9 b 06862.26 b OQl6.7^ b 211$7>^728U b 1^.266d^^ .66^E^3
' JNE AJKE ANGR CP CG IFRMO AFRMO k?m^ AFmiZ
This card contains a portion of the orbital elements for one satellite and
includes all of the information needed in calculating the principal
perturbations of its motion. Thus, for some programs, this is the only
card needed. The first item is the epoch of the elements, JNE, divided
into an integral portion, JKE, and a fractional portion, AJI\E , (The "l"
I prefix is not used for the integral portion because the letter "j" already
' defines it as a fixed point variable.) The next three items include the
inclination, NGR, the senimajor axis, CP, and the displacement of the orbit
center, CG. The remaining items are coefficients of the equation for the
mean anomaly, PRM, which, with the above values, would be written:
PRI^I « 21157.572844 + 15.268COIII (©iL) + .665E-05(ei.jl)^
For some programs, the second term coefficient, APRI-11 , is split into
integral and fractional portions but ZAYIK does not require this.
B Card
'-.7g6253b -.02071577 b -.785E-08 b 4..4e0355 b i-.02694540 b -i..117E-07 b .00E-05-^.5$
ATNRO ATNR1 ATNR2 AI^RPO A^RPI ANRP2 ROW GD
The B card contains the remaining orbital element data, starting with
coefficients for the two equations:
TNR » - 0.766255 - 0.02071577(ENL) - .785E-08(ENL)^
NRP - + 0.480555 + 0.02694540(ENL) + .117E"07(E1.'L)^
It also includes the values necessary in correcting for the effects of the
Earth's pear shape; RGW, which happens to be insignificant for this case,
and GD.
The full stack of data cards for a run of ZAYIN starts with the Control
Card and G Card and may then include any number of pairs of A and B Cards, one
pair for each satellite for which predictions are desired. A single blank card
is then added at the end of the stack.
2:
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Output
An example of the output of ZAYIN, obtained with the above input data, is
shown in Table I. The table will extend through as many pages as are needed to
reach the ending date, JNL2 « For each satellite, a fresh page is started.
The format is arranged so that the most important data is to the left of the
vertical line, where the sheet can be trimmed to fit the standard &q^ mailing
envelope. The alphanumeric identification data is printed at the upper left,
where it will match a transparent window in the envelope #
The input orbital element data is reproduced at the upper rigjht. Here, it
may be seen that, in transferring the coefficient, APRM1 , as a single, floating-
point number, we lose the identity of the last digit. This is not important to
ZAYIN.
Immediately below the identification, we print the starting date, JNL1 .
The purpose of this is to eliminate need for printing the first two digits in
the succeeding table of predictions.
Each prediction occupies two lines and the numbers are arranged in pairs,
upper and lower, which correspond to the upper and lower column headings* In
most cases, the numbers in each pair have a functional relationship as indicated
in the following discussion:
JNL 3oth niambers represent the predicted time of culmination transit.
UT The upper is expressed as the integral and fractional Modified
Julian Date (with the first two digits removed) and the lower is the
Universal Time in hours, minutes and seconds.
Qt;X This is the predicted apparent position in rationalized celestial
NaX coordinates, including the polar angle, QNX, and polar distance, NGX.
RA This is the same predicted position as above but is expressed in star
DECL chart coordinates of Right Ascension and Declination for the epoch of
the prediction. For precise work, these must be corrected for
precession of the coordinate system from the epoch of the charts.
NOR This is equivalent to an alt-azimuth prediction for the point of
OGX culmination but is expressed in rationalized coordinates with the
observer's geocentric zenith (obtained by projection from G through O)
as the pole» NOR is the polar angle of the orbit pole and OGX is the
zenith distance. These values are not corrected for effects of atmos-
pheric refraction.
NRO These represent the coordinates of the observer relative to the orbit
RGC pole. They are of interest to the observer who wishes to select those
observations which will be most useful in determining the location of
the orbit pole.
GV These numbers give the radius and mean anomaly of the satellite at the
PRM point of culmination. They are useful in selecting the most meaningful
observations. They may be used in combination with other data to
determine brightness and apparent rates of motion.
6
ONX
These values are used in setting an instmrnent that employs an
NXR equatorial mounting. ONX corresponds to the "local hour angle" of the
point of culmination and NXR determines the apparent "bearing angle**
or direction of travel of the satellite, as viewed in celestial
coordinates •
DMDT
DRGX
^^'^^ is the rate of change of mean anomaly of the satellite, with
respect to time, at the point of culmination. DRGX is the rate at
which the angle, RGX, changes with respect to time. These values
may be used in setting the proper rate of motion for a camera which,
like the Baker-I^nn camera, follows the motion of the satellite. They
may also be used in calculating corrections which should be applied
to the predicted positions when the predicted timing is found to be
in error.
SLA^J SLAN is a Fortran variable that corresponds to the ratio of slant
COGS range to radius. Ivtultiplied by GV, it gives the slant range and
aids the observer in estimating brightness. C0GS is the cosine of
the angle, OGS, between the observer and the Earth's shadow center.
For predictions that fall in the twilight zone, it is useful in
estimating darkness of the sky so that the observer can decide
whether observation of a faint satellite is feasible.
The remaining three columns contain numbers that are not related to the
prediction. However, they are pertinent to a later discussion of the performance
of the computer program. As described below, they indicate what the program has
done in the interval between predictions.
I/J !_ represents the nximber of passes rejected, after the preceeding
prediction, because they were below the observer's horizon* J
represents the number rejected because they occurred while the
observer was in daylight, even though they were above the horizon.
K/L K represents the nvimber of passes rejected because the culmination
point was inside the Earth's shadow although they were above the
horizon and occurred while the observer was in darkness. L represents
passes that were not rejected but which had to be returned for more
precise computation because the estimated mean anomaly was substantial]
different from that predicted by the equation for mean anomaly, PK-I.
M/N M is the number of returns for recomputation needed in refining the
prediction to six-digit precision. N is the total number of synodic
revolutions tested, up to the current prediction and including
previous predictions. It is a cumulative count so that, as indicated
by the final entry, the entire table involves the examination of
847 synodic revolutions, plus a few more that would have occurred
between the final prediction and the ending date, JKL2 .
In discussion of the Main Program, we will again refer to these coxjints to
explain how they control the routing of the computation and how they provide a
measure of the performance of the program.
Subroutines
The following discussion represents a functional description of each of the
subroutines, starting with an example of the necessary CALL statement • Internal
detail of some of these subroutines is given in separate Appendices. In each
CALL statement, the underlined arguments are the "input* arguments, whose values
must be supplied by the Main Program. The remaining arguments are the output
arguments, for which values are supplied by the subroutine.
CALL FRACT (A, ARN0)
PRACT is a very brief subroutine that usually proceeds an entry into POLO
or other subroutines and operations that involve trigonometric functions.
It extracts the fractional portion of a decimal number and expresses it in
the range from - ot:5 to + 0V5 In 2AYIN, the input argument for FRACT
usually has a temporary name, such as A or B. The CALL statement defines
the permanent name for storage (in this case, it is ARN0 ). As an example
of the operation of this subroutine, we might start with the number:
A . 2412.849256
FRACT would extract the fractional portion, .849256, and, because this is
outside the required range, it would subtract 1.0 to obtain - 0.150764,
which would be returned to the Main Program as the value for ARN0 . Use
of this subroutine permits us to combine several angles without concern
as to whether or not they acciimulate to more than a full revolution.
CALL PjfL0 (ANG0, SNG0, CNG0, AKGR, SNGR, CKGR, ARNtf , SRII0, CRK0, ARG0, SRG0,
CRG^, A0mu S{ZfMI, '^m, ANJ^R, SN0R, C0R)
POLO is the "workhorse" of ZAYIN, solving all problems in spherical
trigonometry throu^ coordinate transformation. The three underlined
input arguments are supplied by the Main Program and POLO supplies the
remaining 15 arguments. The chart on page 8 gives the rules for naming
the arguments for POLO and also helps to explain what POLO does. The
above CALL statement represents a case in which we wish to transform the
coordinates of an observer, 0, from the Earth's North pole, N, to the
orbit pole, R, with the Earth's center, G, as the center of both coordinate
systems. Examining the definitions given in the upper right portion of
the chart, it may be seen that this requires the following substitutions:
3 m 1«N 2«R Ox.G
These substitutions are then made in the list of "Call Names" given in the
lower portion of the chart to obtain the arguments as listed above in the
CALL statement. The order of letters obtained from this substitution
automatically defines the direction of measurement of each polar angle.
The subroutine contains its own switching system to route each computation
through the shortest path. A complete description, flow diagram and Fortran
listing for POLO are included in an ITCP Bulletin of November 19, 1962, on
"Fortran Programs for Preparation of Tabular Aids to Satellite Tracking".
CALL SHAD0 (JNL, AJNL, ANGS, ATNS, AC^T)
SHADO also requires FRACT and POLO as internal subroutines. From a given
input date, for which the integral portion is JNL and the fractional
portion is AJNL, SHADO computes the Earth's true anomaly and transforms it
from the pole of the ecliptic, Q, to the North pole, N, providing the
coordinates of the Earth's shadow center, NGS and TNS. It also supplies
the fractional value of the Tropical Year, QKT. Details of SHADO are given
in Appendix B. An understanding of the problem with which it deals can be
obtained from an ITCP Bulletin of March 28, 1964, on "Time as a Measure of
Direction in Space".
CALL I<EPLR (APRV, CP, CG, APRiM, DrPJ-l, R\DR, DRAD)
KEPLR contains Kepler's equations and computes the mean anomaly, PRI^I, and
the ratio of radius to semiraajor axis, RADR, from given input values of
the true anonaly, PRV, and eccentricity, as derived from CG and CP» It
also computes the rate of change of mean anomaly, DPRM , and rate of change
of radius ratio, DRAD, per unit change in true anomaly.
Naming of Variables used as Arguments in POLO
Definitions :
5 A point whose coordinates are to
be transformed*
1 Pole of coordinate system in which
coordinates of 5 Q^re known.
2 Pole of coordinate system in which
coordinates of 5 ^^^ desired.
Common center of both coordinate
systems •
Note : All angles must be expressed in
decimal revolutions (turns) and in the
range from - 0.5 to + 0.5
Call
Name
Dummy
Name
Description:
AIO^
SI 05
CI 05
A
B
C
Polar distance of point 5 fTom pole 1
Sine of polar distance, IO5
Cosine of polar distance, IO5
A102
S102
CI 02
D
E
F
Polar distance of pole 2 from pole 1
Sine of polar distance, 102
Cosine of polar distance, 102
A21^
3215
C215
G
H
Polar angle of point 5 from pole 2, measured about pole 1
Sine of polar angle, 215
Cosine of polar angle, 215
A205
S205
G205
P
R
Polar distance of point 5 from pole 2
Sine of polar distance, 205
Cosine of polar distance, 205
A52I
S52I
C52I
S
T
U
Polar angle of pole 1 from point 5, measured about pole 2
Sine of polar angle, 521
Cosine of polar angle, 521
AI52
SI 52
CI 52
V
w
X
Polar angle of pole 2 from pole 1, measured about point 5
Sine of polar angle, 152
Cosine of polar angle, 152
Note ; The underlined variables; AIO5, A102 and A215 represent the input data,
which must be supplied from the main program. All other values are computed by
POLO. In the CALL statement, the variables must be listed in the above order.
Main Program
The follov/ins description is based on the Flow Diagram of FifOire 1 but
also requires reference to the Fortran statement list in Table II. Numbers in
the Flow Diagram boxes correspond to the statement numbers. Although the
program is a continuously flov/ing sequence of operations, it is written in six
"Parts", each of v/hich completes a major portion of the logic. V/ithin each
Part, the statements are numbered consecutively.
PART I reads the input data and establishes the output format. It also estab-
lishes the control pattern as that of a "one station - many satellite"
program, predicting for any number of satellites in one computer run but
for only one observing location. Minor changes in this part of the program
will convert it to a "one satellite - many station" or "many satellite -
many station" program*
As written, 2AYIN starts by reading the Control Card and C Card (statements
10 - 15) to find the rajige of prediction dates and the observer's location.
It then reads the A Card for the first satellite. As indicated in the
Flow Diagram, the program returns to this point for additional satellites
and, if there are none, it is routed to the ETID through statement 16.
Continuing with a given satellite, 2AYIN reads the B Card (statements 17
and 18) and then proceeds to print the headings for the output table of
predictions (statements 19 - 52).
PART II establishes initial values of program variables and the value of one
constant, Z, (statement 40).
The independent variable in ZAYIN is Er\^L, the time elapsed since the epoch
of the orbital elements. Its initial value is determined (statements k^
and 42) from the starting date, JNL1 . Statement 4^ then finds the maximum
value, EITLM, from the ending date, JNL2 .
The average interval between times of local culmination is equal to the
"synodic period", SYNP. This is the reciprocal of the mean rate of
revolution of the satellite relative to the observer, measured in turns
per day. Statement 44 adds all of the rates involved: APRI^!1 , the mean
anomalistic motion of the satellite; ANRP1 , the notion of the perigee;
ATNR1 , the motion of the orbit pole and - itfO/day for the motion of the
observer. Statement 45 then find the reciprocal to define the value of
SYTJP.
The internal counters; I, J, K, L, M and N, are all set to zero in this
part of the program. The necessary statements are placed in an order that
depends on the points at which various return loops are to enter.
One of the time-saving features of ZAYIN is that it computes the coordinates
of the Earth's shadow center no more frequently than necessary, once at
the start of the program and once as a part of the completion of each
prediction. The initial values are supplied, in statements 48 and 49, for
the starting date, JNL1 ♦
Considerable time is saved in ZAYIN by rejecting "impossible" passes prior
to a final computation of the satellite radius, GV. Statement 55 allows
the program to start with the initial assumption that the satellite is at
its apogee radius, CP -»- CG.
10
FORTRAN Listing for ZAYIN Main Program Parts I and II
C ZAYIN
PART I
10 READ 11,JNL1,JNL2
11 P0RHAT(2I6)
12 READ 1?,ANQ0,ALN$2(,G0,IDEN,IDEN1
15 P0RMAT(F8.6,F1O.6,P8.2,54X,2A5)
14 READ 15,JNE,AJNE,ANGR,0P,CG,IPRH,APRM0,APRM1, APRM2
15 P(JfRMAT(l5,P9.8,F8.6,F9.2,F8.2,l6,P7.6,P12.6,E8.5)
16 IF(JNE)172,172,17
17 READ 18,ATNR0,ATNR1,ATNR2,ANRP0,ANRP1,ANRP2,RGW,QD
18 FORMAT (F8.6,F11 .8,E10.5f F9.<5,F11 .8,E10.5,E8.2,F5.2)
19 WRITE 0UTPUT TAPE 6,20,IDEN,IDEN1 ,JNE,AJNE,CP,CG
20 F0RMAT(1H1,5X,2A5,7X,I5,F7.6,4H CP-,F8.2,4h CG-,F7.2)
21 WRITE JZfUTPUT TAPE 6,22,ANGR,RGW,GD
22 F0RMAT(25X,4HNGR»,F7.6,5H RGW»,E8.2,4h GD«,P5,2//)
25 WRITE pfUTPUT TAPE 6,24,IPRM,APRM0,APRM1 ,APRM2
24 F0RMAT(25X,I5,F7.6,F12.8,E1O.5)
25 WRITE JZfUTPUT TAPE 6,26,ANRP0,ANRF1 ,ANRP2
26 F0RMAT(25X,5HNRP,F9.6,F12.8,E1O.5)
27 WRITE jifUTPUT TAPE 6,28,JNL1 ,ATNR0,ATNR1 ,ATNR2
28 F0RMAT(2X,5HJNL1-,I5,15X,5HTNR,F9.6,F12.8,E1O.V/)
29 WRITE jZfUTPUT TAPE 6,50
50 F0RMT(4X,61HJNL QKX RA NjZfR NR0 GV jZfNX DMD
IT SLAN)
51 WRITE JZfUTPUT TAPE 6,52
52 FJ2JRI.IAT(5X,72HUT NGX DECL 0GX RGJ? PRM NXR DRGX
1 C0GS I/J K/L M/N)
PART II
40 2-6.28518551
41 A-JNL1-AJNE
42 ENL.A+AJNE
45 ENLM-JNL2-JNE
44 A-APRM1 +ANRP1 +ATNR1 -1 .
45 SYNP-I ./A
47 N-0
48 AJNL-0.
49 CALL SHADj?(JNL1,AJNL,ANGS,ATNS,AQNT)
50 I-O
51 J-0
52 K-O
55 L.0
54 M-O
55 GV-CP+CQ
11
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PART EZ"
PART III starts with statement ^0, which sends ZAYIN back for &.nother satellite
if E^lh exceeds the maximum value, £I'JU! » It then includes the principal
components of a ''search loop", which searches for acceptable passes* The
action of this loop can be better understood vdth the aid of a preliminary
review of the geometry for the general situation, Figure 2, and for the
rejection criteria, Figure 5»
Figure 2 is a view of the Earth from the direction of the orbit pole, R«
The orbit then lies in the plane of the paper and its inclination is
represented by the polar distance, NGR, of the Earth's North pole, N, from
the orbit pole» In this view, the observer's circular path is tilted and
his relationship to the orbit is described by the direction coordinates,
NBO and RGO, and his radius, GO.
In Figure 5(a), it may be seen that, if the sine of the angle, RGO, is
greater than GO/GV, the satellite will be above the observer's horizon.
For preliminary screening, we can apply this criterion without knowing
whether the satellite is actually at the point, V, and without knowing
its exact radius, GV» To avoid unnecessary rejections, we can assume that
GV has its maximiom value, CP + GG*
R
////■'//' ■' / ' ' / / / ////// / / I
I i ! I : ! I I i
X
Fig. 2. Requirements for Optical Observation
at the Point of Local Culmination
In this diagram, the Eartli is visible from the orbit pole, R, and Uie
Eartli's center, G, is the center of the coordinate system. The orbit is
iji the plane of tlie paper and its inclination is represented by the p<)lar
distance, NGR, of tlie Earth's Nortli Pole, N. The polar angle, NRG, of
the observer, O, must equal tlie polar angle, NRV, of tlie satellite, V.
For successful optical observation, the observer's polar distance, RGO,
must be such that the satellite will be above his horizon (see Fig. 3(a) ).
The position of the Earth's shadow center, S, must be such tliat tlie ob-
server is on tlie shadowed side of the terminator (see Fig. 3(b)) ajid tJie
satellite is outside the cylmdrical region defined by the Eartli's shadow
(see Fig. 3(c)).
Fig, 3 (a) Criterion for Relection
For the limiting case, the satellite, V,
is on the horizon of the observer, O. The
angle, OVG, is equal to the angle, RGO,
and the sine of either OVG or RGO is GO/
GV. Thus, if the sine of RGO is less than
GO/GV, the satellite will be below the
observer's horizon.
14
Returning to Figure 2, the position of the Earth's shadow center, S, may
be defined by the direction coordinates, NRS and BjS. The question as to
whether the observer is in daylight depends on the angle, 3G0.
As illustrated in Figure 5(b), the cosine of SGO must be positive and, if
we wish to insure that the observer is well inside the twilight zone, we
can specify a minimum value. As written, ZAYIH requires that the observer
be at least 0i:028 away from the "terminator**, or edge of the shadow. This
criterion is independent of the exact location of the satellite.
Looking, again, at Figure 2, it may be seen that the coordinates of the
satellite would be: NRV, RGV and GV. RGV is always equal to ot:25. At the
point of culmination, NRV will be equal to NBO and, again, we may assume
that GV has the maximum value, GP + OG.
As shown in Figure 5(c), the satellite must be outside the cylinder
defined by the Earth's shadow, regardless of whether the angles SGV, is
greater or less than 0^:25. This requires that the sine of SGV be greater
than GO/GV. Again, the assiomption that GV » CP + CG is one that avoids
unnecessary rejections.
Fig, 3 (b) Criterion for Rejection
If the angle, SGO, is greater than 0^25,
the observer will be in sunlight. To en-
sure that his sky is reasonably dark, we
require that he be about 0J028 away from
the edge of the shadow. Thus, if the cosine
of SGO is negative, or less than 0.174, the
circumstances are considered to be un-
satisfactory.
Fig. 3 (c) Criterion for Rejection
If the sine of the angle, SGV, is less
than GO/GV, the satellite will be inside
the cylinder defined by the Earth^s shadow.
On the side towards the Sun, it will then be
below the horizon for any observer who is
inside the shadow. On the side away from
the Sun, it will be inside the shadow and
will not be illuminated.
15
Returning; to the Flow Diagra:n and Fortran listing, we should, temporarily,
ignore the "bypass" lines at statements 65 and 72 because these are best
related to the* functions of Part IV of the iprogr&xn. The first task is
then to locate the position of the observer, 0, in the coordinates of the
orbit pole, R» This is done in four steps:
a. Evaluate TNR from the TIIR equation (statement 61 )•
b. Sum the necessary polar angles to find NH) (statement 62).
c. Call FRAGT to express NRO in the proper range (statement 6^).
d. Call POLO to transform from N to R (statement 64). This
statement is the example that was used in the functional
description of POLO, above.
The criterion of Figure 5(a) is then applied in statements 66 and 67.
SRGt^ - G0/GY is given the name, P0GX , only because its value is useful
at a later point in the program. If the pass is rejected (below the
horizon), we add 1 to the "l" counter (statement 68). Returning through
the search loop, we add 1 to the "l-l" counter and advance El^h by one synodic
period (statements 69 and 70 )•
For the passes that are above the horizon, we then find the coordinates
of the shadow center, S, in the same manner as used above; summing the
polar angles (statement 75), placing in the proper range (statement 7^)
and transforming coordinates (statement 75 )• Statements 76 and 77 then
represent two applications of the Law of Cosines, which are written in an
order that economizes on computing time. V/ritten in full, statement 76
v/ould be:
cos SGV = cos RaS cos RGV + sin RG3 sin RGV cos SRV
but RGV is always ot:25 and, at the point of culmination, SRV will equal
3R0 so that the equation reduces to:
cos SGV « sin RGS cos SRO
Similarly, statement 77 could be written in full as:
cos SGO = cos RGS cos RGO + sin RGS sin RGO cos SRO
Having already evaluated the product, sin RGS cos SRO, we can v;rite:
cos SGO « cos RGS cos RGO + sin RGO cos SGV
The criterion of Figure 5(b) is then applied in statement 78 and, if the
pass is rejected (in daylight), v/e add 1 to the "j" counter (statement 79)
and, in returning through the search loop, add 1 to "ll" and SYIIP to EKL.
For surviving passes, the criterion of Figure 5(c) is applied in statement
£-1. For those rejected here (in shadow), we add 1 to the "K" counter, 1 to
"N* and SYT'IP to ETIL.
Much of the speed of ZAYIi: is due to the effectiveness of this search loop.
On the average, half of the passes v/ill be rejected because they are below
the horizon. (In Table I, there are a total of 480 "l" rejections in the
847 passes examined.) For this reason, the "I" reject path has been kept
16
as short as possible. Of the remaining passes, approximately half (240
in Table l) will occur with the observer in daylight. Thus, the "j" path
is just outside the "I" path. For a low-flying satellite, as was selected
for our example in Table I, many passes that are otherwise acceptable will
lie inside the Earth's shadow (106 in Table l). The fact that ZAYIN rejects
these prior to final computation of the satellite position is an important
time-saving feature. It should be noted that the assumption that GV » CP 4 CG
is one that is designed to save all possible passes* This leads to some
tinnecessary computer work in that a few of these may be, later, rejected in
Part IV. An alternative assumption would be that the satellite is at
perigee radius where GV - CP - CG. Such an assumption would minimize
computer work and would select only those passes that are easiest to observe.
If ZAYIN were to be used in predicting for a large number of observing
stations, such an assiunption might be preferred.
FORTRAN Listing for ZAYIN Main Program Part m
60 IF(ENL-ENLM)6l,6l,l4
61 ATNR-ATNR0+ATNR1 *ENL4ATNR2*ENL**2
62 A>c-ATNR.fAJNE<t-ENL<»ALN(2f
65 CALL FRACT(A,ARNj2f)
64 CALL P(ifL(2((ANG{;(,SNG0,CNG0,ANGR,SNGk,
:GNGR,ABN0,SRN0,CHN0,ARG0,SRG$f,
1 CRQ0,i^RN,S0RN .Ci^RN ,AN0R,SN0R,CN0R}
65 1F(L^I+N)90,90,66
66 Pj2fGX-SRG0-G0/GV
67 IP(P0GX)68,68,72
68 I-I+1
69 N.JI+1
70 ENL-ENL't'SYNF
71 G0 T0 55
72 IP(L4M)75,75,76
75 A-ATNS-ATNR
74 CALL FRACT(A,ABNS}
75 CALL P0L0(ANGS,SNGS,CNGS,ANGR,SNGR,
CNGR , ARNS , SRNS ,CRNS ,ARGS , SRGS ,
1 CRGS ,ASRN ,SSRN ,CSRN,AHSR,SNSR,(»ISR)
76 CS6V-SRGS*C0SP( (ASRN-AORN ) ♦Z )
77 CSG0.CRG0*CRGS-«-SRG0*CSGV
78 IF(CSG0-.175648)79,79,81
79 J-J+1
eo 00 70 69
81 IP(SQRTP(1 .-CSGV**2)-G0/GV)82, 82,90
82 K-K-fl
85 G0 Tpf 69
17
PART IV is concerned with matching the polar angle, IIRV, of the satellite with
that of the observer, NRG (see Figure 2), It starts by evaluating NRP
(statement 90) and PRM (statenent 91) from the orbital element equations.
Ignoring the bypass at statement 92, the sum of NRP and PR*M is compared
with IIRD in statements 95> 9^ QJ^d 95* If there is an appreciable difference
(the value, olflj, was selected arbitrarily), the value of EIIL is adjusted
according to the formula in statement 97 and, adding 1 to the "L** counter,
we return to statement 6^ to try again. This returns the case to the
search loop to redetennine the coordinates of the observer, 0. The pass
must again pass the rejection criteria but, this time, it is not necessary
to redetermine the coordinates of S, which change rather slowly. Statement
72 permits us to bypass statements 75 through 75 •
As indicated by Table I, very few passes are returned through "L". Such
returns occur occasionally if the satellite has very high orbital eccentri-
city. ::ost cases advance to statements 99> 100 and 101, v;here v;e determine
the mean anomaly, APRI-!D , that is equivalent to the polar angle, PRO, betv;een
the perigee, F, and observer, 0, (see Figure 2). This is done by using PRO
as the input value of true anomaly for KEPLR.
Statement 102 is the first determination of the r.ctLU,! radius, GV, and
includes a small adjustment, GD cos NRO, for efioci:,,- of the pear shape.
Statement 10^ detennines the rate of change, DMDT , oi' the computed value
of mean anomaly as a function of time. It includes effects of motion of
the satellite, the perigee, the orbit pole and the observer.
The desired value of mean anomaly, APPXID , is then compared with that
predicted by the orbital element equation in statement 91 j APRi4 . This
requires three statements; 104, IO5 and IO6, using FRACT because we are
interested in only the fractional portion. If the difference represents
more than 0^000001 in timing, we add 1 to the "M" counter, adjust EITL
by an amount determined from the difference in mean anomaly and its rate
of change, DMDT , and return to statement 61 to try again* The accurate
determination of DIIDT in statement IO5 is an important factor in keeping
the number of these returns to a minimum. As indicated in Table I, at
least one return through "M" is usually necessary but the count is very
rarely higher than 2.
In returning through "M", it is necessary, again, to pass all rejection
criteria and, this time, v/ith the correct value of GV. It is hei^e that
an occasional pass will be rejected after having progressed all the way
through Part IV.
Use of the synodic period, SYNP , as a first approximation from each time
of local culmination to the next is valid only after finding one correct
time of culmination. This explains the bypass at statement 65, which
requires that there must have been at least one adjustment at "L" or "M"
or one complete prediction to bring the observer and satellite into
"synchronism" before the rejection criteria can be applied.
The by-pass at statement 92 insures that the rough adjustment of mean
anomaly in statement 97 is applied only once. If this adjustment is
made too closely, and if it is permitted to remain in effect, the program
can go into a state of sustained recycle when there is a substantial
difference between true anomaly and mean anomaly at the culmination
point.
On completing Part IV, ZAYIN has fully established the situation shown
in Figure 2, vdth NRV, for the satellite, substantially equal to IJRO,
for the observer. The observer, satellite and orbit pole are lined up
on the same great circle.
18
FORTRAN Listing for ZAYDSf Main Program Part IV
90 ANRP-ANRP0+AKRP1 *ENL+ANRP2*ENL*»2
91 APRM«APRM04APRM1 *ENL4APRM2*ENL**2
92 IP(M)95,95,99
95 A—APRM-ANRP-AORN
94 CALL FRACT(A,AMB0)
95 IP(ABSF(AMBfZf)-.15)99,99,96
96 L-UI
97 ENL-ENL4AMB(af/(APRM1 -jANJRPI -(1 .-ATNR1 )*SNQ0*CN0R/SRa0)
98 GJZf 0^61 ^ rv ^,
99 A»-ANRP-A0HN
100 CALL PRACT(A,APR0)
101 CALL KEPLR(APR0,OP,Ca,APiM3,DPRM,RADR,DRAD)
102 GV-CP*RADR4GD»C0RN
105 DMDr«APRMl4ANRP1-DPBM*(l,-ATNRl)*SNajZf*CN0VSRG0
104 A«APRMI>-APRM
105 CALL FRACT(A,B)
106 IP(ABSP(B)-IMDr*.1E-05)l20,120,107
107 MmM+1
108 mLmWUB/JMDT
109 00 TJlf 61
PART V finishes the prediction after the exact time of culmination has been
found in Part 17. The order of the statements in Part V is not as
logical as it could be and, although this does not affect the computer,
we will discuss them in an order that should be easier for the reader to
understand.
The alt-azimuth prediction in column 4 of Table I is already partly
completed. The value of NOR has been stored as a part of the coordinate
transformation in statement 64. Statement 120 completes the calculation
of OGX, which was partially done in statement 66 •
The next task is the conversion to celestial coordinates. These are
first determined with relation to the orbit pole, R. In Figure 2, it
may be seen that IJRX is equal to l^BD (already determined) and, in Figure
5(a), it may be seen that RGX will be equal to RGO + OGX, as done in
statement 121. These coordinates, NRX and RGX, must then be transformed
to the N pole, as is done in statement I50, providing values of RKX and
NGX. To express the polar angle as (JRX, we need a value of QiNT, obtained
with statements 124 through 129 • We make the necessary summation of angles
and adjust to the proper range in statements I5I through 154. The call of
SHADO, in statement 129 not only provides the required value of QNT but
also provides revised values of TNS and NGS for use with the next pred-
iction.
Miscellaneous computations are included in statement 122, which computes
the ratio of slant retnge to radius, SLAN ; statement 125> which computes
the rate of change of RGX with time; statement 155> which computes the
local hour single, ONX, and statement 156, which simply reverses the sign
of CRN so that it can be printed out as NRO.
19
On completion of Part V, all of the real work of prediction is done
and values of 9S variables that are involved in the prediction are in
storage. At this point, we can make any decision that we like as to which
values we wish to have in the output record •
PART 71 is then concerned only with the conversion of data to the particular
form desired for the printout of Table I. Statements 1A0 through lA6
convert decimal days to Universal Time in hours, minutes and seconds.
Statements l47 through 155 convert the value of Q^IX to hours and minutes
in Right Ascension. Statement 154 converts NGX to degrees of Declination.
Statements 154 through l64 cause all negative decimal fractions to be
expressed as positive decimal fractions, simply to save space in the
output format by eliminating minus signs. Statements I65 through I68 do
the printing.
Finally, statement I69 adds one synodic period, statement 170 adds 1 to
the **N" count and statement 171 sends 2AYIN back to look for the next
observable pass.
Program Performance
ZAYIN has been tested with a variety of satellites, covering the full
range of values of inclination and eccentricity. It works equally well for
both northern hemisphere and southern hemisphere observing locations. The
example used for Table I was chosen to illustrate the performance through a
full revolution in TNR, the Local Mean Time of the Orbit Pole, for a low-
flying satellite, which makes relatively few observable passes (60 06 A),
The nature of the satellite is apparent from the rather large number of passes
rejected because they are below the horizon (l) or inside the shadow (K).
The only known limitation of the program is that it will not operate depend-
ably if one attempts to predict for a period much beyond 100 days from the
epoch of the orbital elements. This is a result of handling the mean motion,
PRI-'II , as a single floating point number, rather than breaking it into integral
and fractional parts and using double precision in multiplying it by EIJL.
The multiplication then results in uncertainty in the sixth decimal digit
so that, in attempting to balance to Oi 000001 , the program can fall into a
sustained loop. This is not a serious limitation but the user should be aware
of its existence.
Tracing through the rejects, as well as the predictions in Table I, one
finds that the entire computation includes:
1261 coordinate transformations using POLO
45 orbital computations using KPLR
22 detenninations of the Earth's shadow position v;ith SHADO
plus the computations called for in many cycles through the Main Program.
Using the IBM 704, which is now regarded as a slow machine, the entire job,
plus printout on magnetic tape, requires about 95 seconds. A faster machine
could do it in about I6 seconds. This proves, at least, that there are a great
many microseconds in one second.
20
FORTRAN Listii^ for ZAYIN Main Program Parts V and VT
C PART V
20 AJ2(GX-ATANF(CRGff/P0GX)/Z
21 ARSX«ARGI^U0GX
22 SLAN-ABSP(CRGJf/SINF(A(2fGX*Z))
25 DRGX— (l.-ATNRl)*SNG^*aNffR/SLAN
24 A.EIIUAJNE
2^ IB.A
26 JNL-JN&t-IB
27 B-IB
28 AJNL-A-B
29 GALL SHAS{ir(JKL,AJNL,ANGS,ATNS,AQNT)
50 CALL F0t^(ARGX,SRGX,CRGX,ANGR,SNGR,CNGR,A0RK,S^RN,C0RN,ANGX,SNGX,
1 CNGX»ARNX,SRNX,CRNX,ANXR, SNXR,CMXR)
51 A«AQNT<»ATI«R+ARia
52 CALL PRACT(A,AQNX)
55 IP(AQNX)154,155,155
54 AQtIXi^QN<X4-1 .
5^ A0ini^-Am^*Amx
56 AI?R^»-A0RN
Continuation of PART VI
169 ENLi^ENL-fSYNP
170 NiJI+1
171 QfS 10 ^
172 END FILE 6
REWIND 6
ST0t> 77777
EKD(2,0,1,0,1)
PART VI
40 A-24.4-AJNL
41 lUTH-A
42 B-IUTH
45 3-60,*(A-B)
44 lUTM-B
45 G«IUTM
46 IUTS-60.*(B-C)
47 B-ACJNX-.25
48 IF( 8)149,150, 150
49 B.B<f1.
50 B»24.*B
51 IRAH-B
52 C-IRAH
55 RAM=60.*(B-G)
54 DECL-560.*(.25-ANGX)
55 IF(AN(!(R)156,157,157
56 AN^R.JU$;0R4>1 .
57 IP(A0GX)158,159,159
58 A0GX»A0GXt.1 .
59 IP(AJIfl(!f)l60,l6l,l6l
60 A^^SmAN^f^i .
61 lP(Aj»a)l62,l65,i65
62 A0^X«A0NX4l .
65 IP(ANXR)164,165,165
64 ANXRsANXR^I .
65 raiTE 0UTPUT TAPE 6,l66,JNL,AJNL,Aqia,IRAH,RAl.!,AN0R,ANI^,GV,A0KX,
1DMDT,SLAK,I,K,M
66 P^RMAT(1X,I5,P6.5,1X,P6.5,I5,P5.1,1X,F6.5,1X,F6.5,F7.1,1X,F5.4,
1P7.5,1X,P5.4,5l4)
67 WRITE 0UTHJT TAPE 6,168,IUTH,IUTM,IUTS,ANGX,DECL,A0GX,ARG0,APRM,
1 ANXR, DRGX, CSG(lf , J , L , N
68 F0RI1AT(1X,5I5i1X,F6.5,F8.5.1X,F6.5,1X,F6.5,1X,F6.5,1X,F5.4,F7.5.
11X,P5.4,5I4//)
(See continuation, above.)
21
Appendix A. FORTRAN Listing for FRACT
FRAOT
SUBR0UTINE PRAGT(A,B)
10 I=A
11 0»I
12 D«A-C
15 IF(ABSP(D)-.5)19,19,U
14 IP(D)15,19,17
15 B«1.+D
16 G0 T0 20
17 B»D-U
18 GJZf TJZf 20
19 B«D
20 RETURN
Comments
"a" ia the input argument and may be any decimal nvimber that includes both
integral and fractional parts. The output argument, "B", will correspond to the
fractional part and, if this is greater thaji 0,5 in magnitude (statement Ij), 1 #0
is subtracted (statement I?)* Thus, the output argument is always in the range
from -0,5 to +0.5, The object deck for FRACT will consist of only two cards.
22
Appendix B, FORTRAN Listing for SHADO
SHAD0
SUBRjZSUTINE SHADpf(I,A,B,C,D)
10 G«I -58200
11 H.G4-A
12 ANQM«0.495026+.275791E-02*H
15 AClNT-ANQM+0.5
14 ANQP.0. 054201 +*15E-06*H
15 E-ANQM-ANQP
16 CALL PRACT(E,APqjM)
17 2«6.28518551
18 APQM«APQM*Z
19 APQE-APQM+.01675572*SINP(APQM)+.14004E-05*SIKF(2.*APQM)
20 APQM2«APQE-.01675572*SINP(APQE)
21 APQE«APQE+(APQM-APQK2)/(1 .-•O1675572*C0SF(APQE))
22 F=C(^SP(APaE/2.)
25 IF(F)26, 24,26
24 APQS-.5
25 00 TJ2f 27
26 APQ3«.2.*ATANP(1 .01687815*SINF(APQP/20/F)/Z
27 E-ANQP+APQS
28 CALL FRA0T(E,A1TQS)
29 AQGN-r ,0651 5206
50 AQGS«,25
51 CALL P0L0(AQGS,SQGS,CQGS,AQGN,SQGN,CQGN,AJIQS,SNQS,CNQS,B,SNGS,
1CNGS,ASNQ,SSNQ,0SNa,AQSN,S(lSN,CQSN)
52 E—AQJTT-ASNQ
55 CALL FRACT(E,C)
54 J-AQNT
55 P=J
56 D«AQrrr-p
RETURN
Discussion
Equations in SHADO use the Modified Julian Date, 58200, as the epoch.
Thus, they should be periodically revised. These include statement 12, which
computes the Earth's mean polar angle, NQM, and statement l4, which computes
the perigee position.
Statement I9 makes a first approximation of the eccentric anomaly, using
equation (45) from page I6I of Moulton's "Celestial Mechanics". Statement 20
computes the corresponding mean anomaly. Statement 21 then uses Moulton*s
equation (47), page I62, to obtain a closer approximation of the eccentric
anomaly. For the Earth's orbital eccentricity, this second approximation is
sufficient.
SHADO then computes the true anomaly, (statement 26) and then transforms
the Earth's position from the pole of the ecliptic, Q, to the North celestial
pole, N, using POLO. Statement 29 gives the value used for obliquity of the
ecliptic.
23
Appendix C. FORTRAN Listing for KEPLR
C KEPLR
SUBRJiJaTINE KEPLR(W,B,C,D,EiP,G)
9 A«W
10 H-C/B
11 0-1 .-H
12 P-1.+H
15 (i-S(lRTF(0/P)
14 R=SQRTP()2f*P)
15 IP(A)21 ,16,24
16 D-0.
17 F-0
18 E-F*0/R
19 GUO.
S-1.
20 C30 T0 44
21 S«-1.
22 A— A
25 Gj2f T0 25
24 S=1 .
25 IF(A-.5)54,26,51
26 D=.5
27 F-P
28 E-F*P/R
29 G=0.
50 G<2( T0 44
51 A=1 .-A
52 T=-1.
55- Gi? Tj2( 55
54 T-1 .
55 Z-6. 2851 8551
56 U-A*2/2.
57 V-2.*ATANF(Q*SINF(U)/CJZfSF(U))
58 D-(V-K*SINF(V))/Z
i}9 F-1 .-H*C0SF(V)
40 E-F**2/R
41 G-S*T*Z*E*H*SINF(2.*U)/R
42 IF(T)45,44,44
45 D-1 .-D
44 D-S*D
RETURN
Discussion
rJlFLR uses conventional orbital formulae, which are included in statements
57 through 40, The remaining statements are concerned with establishing constants
and with the routing of trivial solutions. The input true anomaly, V;, may have
any value in the range of - ito to + 1V0 •
24
Phone: STcrling 3-4100
^.^x^Xv^ j£
INDEPENDENT TRACKING COORDINATION PROGRAM
8S4 Conntctkut Avtnut
Wishlngton 6, D. C
June 11, 1964
BULLETIN
"Gear Ratio" Orbital Elements
for
Tracking Artificial Earth Satellites
W. P. Overbeck
June 8, 1964
CONTENTS:
Introduction
The Derivation of Gear Ratio Elements
a. Basic Orbital Elements
b. Treatment of Kepler's Laws
c. Cumulation Effects of Gravitational Perturbatiqps
d* Acceleration Effects
Comparison of Theoretical Gear Ratios with Those Observed
over Long Periods
I ^^ Applications of Gear Ratio Elements
a* Use of Gear Ratio Elements to Furnish Rationalized
Orbital Elements for Prediction
b. Use of Gear Ratio Elements in Simplified Satellite
Tracking for Casual as well as Meticulous Observers
c. Use of Gear Ratio Elements to Improve Tracking
Agency Data for Long Term Predictions
d. Gear Ratio Elements for Keeping Track of the ECHO
Satellites
Conclusions
INTRODUCTION
j
i One of the main problems facing those who attempt to supply satellite
I prediction data is the rapid rate at which this data becomes obsolete.
I The principal reason for this is the erratic nature of the accelerations due
to atmospheric drag, solar wind and radiation pressure. There is no way to
eliminate this problem completely but there is a way to isolate it arid
express it in a form such that it can be more easily controlled by the
independent observer.
In 1956> I used a tracking method that I called the "gear ratio method*,
because it treats the components of satellite motion as though they were
coupled to one another like the gears in a gear train. X discarded this method
because its performance seemed erratic. However, it was particularly useful
i in its application to long-term extrapolation from each observation to the
I next. Remembering this particular quality, I resurrected the Gear Ratio
Society of Photographic Scientists and Engineers
method for use in a recent effort to track ECHO II through daytime optical
observation. The results were surprisingly successful, particularly when the
method was coupled with refinements that have been developed during the
period since 1958» ^ i'^s modernized form, the Gear Ratio method is found to
have the following advantages:
1, With few exceptions, the characteristics of any satellite that Is
now in orbit can be expressed in a "permanent* set of Gear Ratio
Orbital Elements which can be easily kept up-to-date with no
further information other than that derived by the observer from
his own observations.
2. With careful measurement by an experienced observer, the Gear Ratio
Elements will give very precise prediction over periods as long as
2 to 5 years. Relatively little analytical effort is required in
the interpretation of observations.
5. Gear Ratio Elements can also be used and kept up-to-date by a
beginner who does not care for precision or who is not equipped for
precise measurement. Under such usage, the Gear Ratio Elements
do not deteriorate. The effects of measurement error are not
cumulative.
The ECHO balloon satellites are the principal exceptions to the above
comments in that additional refinements are needed for precise prediction.
However, the ability to obtain dependable, though rough, predictions is
retained. The Gear Ratio Elements appear to be the best way to provide a
long-term, though incomplete, description of the behavior of such satellites.
The following discussion is in two parts. The first of these reviews the
background of theory and experimental evidence from which the Gear Ratio
Elements are derived. The second part explains a few applications of the
Gear Ratio Elements to satellite tracking problems. The discussion assumes
that the reader is familiar with other recent publications of the Independent
Tracking Coordination Program.
THE DERIVATION OF GEAR RATIO ELEMENTS
a. Basic Orbital Elements
If the Earth were a perfect sphere, if it had no atmosphere and if there
were no other nearby massive objects, such as the Sun and Moon, the behavior
of an artificial satellite could be penzianently described by a set of six
nxombers, known as the "orbital elements". The nature of these elements is
illustrated in Figure 1 .
In this diagram, we view the Earth from the direction of the orbit pole, R,
so that the orbit lies in the plane of the paper, which also includes the
Earth's center, 0, (hidden under R). The inclination of the orbit is represented
by the polar distance, NGR, between the Earth's North pole, N, and the orbit
pole. Its value, expressed in turns, is one of the six elements.
Using the pole of the ecliptic, Q, as a reference direction, the location
of R is further defined by the polar angle, ONR, expressed in turns and
measured about the North pole. The value of QNR is a second orbital element.
The point in the orbit nearest to the Earth's center is the perigee, P,
and its position is defined by a third orbital element, the value of the polar
angle, NRP, also measured in turns.
The above three elements define the orientation of the orbit which,
according to Kepler's First Law, must then be drawn as an ellipse, with one of
its foci at the Earth's center. The size and shape of the ellipse can then be
defined in terms of its semimajor axis, CP, and the displacement, CO, of its
center from the Earth's center. The value of CP, in kilometers, is a fourth
orbital element and the eccentricity, CG/CP, is the fifth.
Finally, if we select an instant at which the satellite is at the perigee,
wo need supply only one more n^Amber, the corresponding epoch, Jl^, as a sixth
orbital element •
Such economy in description is seldom necessary and it is customary to
expand this group of numbers so that we can specify initial positions of the
satellite other than at the perigee • A more general situation would be that
in which the satellite is at a point, V. This may be defined in terms of a
polar angle, PRV, known as the "true anomaly". Kepler's Laws are then used
in providing a description of the variation of this angle as a function of time.
Fig. 1. Definition of Orbital Elements:
Points and directions in the plane
of the orbit are represented in
blade and are mapped in the plane
of the paper* Points ^lich lie above
the plane of the orbit, and the angles
between them, are shown in grey.
QNR: The right-handed polar angle measured from the pole of the ecliptic Q, around
the North Pole, N, to the mean pole of the orbit, R. This value, together with
the Polar distance, NGR, determines the orientation of orbit's coordinate sys-
tem relative to that in \^ich the observer's position is defined.
NGR: The polar distance from the North Pole, N, measured at the center of its earth,
G, to mean orbit pole, R.
NRP: "Argument of perigee from the North point of the orbit" = The right-handed
polar angle measured from the North Pole, N, around the mean orbit pole, R,
to perigee, P.
P: Perigee, the point in the orbit closest to G, the center of the earth.
V: The center of mass of the satellite.
PRV: "True anomaly" = The right-handed polar angle measured from perigee, P,
around the mean orbit pole, R, to the center of mass of the satellite.
C: Center of elliptical orbit
CG: Distance from center of orbit, C, to center of Earth, G.
CP: "Semi-major axis" = Distance from center of orbit, C, to perigee, P.
CQ/CP = Eccentricity of orbit.
b. Treatment of Kepler's Laws by Tabulation of Eccentricity Functions
As indicated above, Kepler's First Law defines the orbit as an ellipse,
with one focus at the Earth's center. This can be expressed in the equation:
GV 1 - e2
CP
1 + e cos(PRV) ^^^
in which e is the eccentricity, CG/CP. The ratio of satellite radius to semi-
major axis, GV/CP, is the "radius ratio*. For any given value of eccentricity,
we can compute values of the radius ratio as a function of true anomaly and
can tabulate these, as in the fourth column of Table I. Vte can also use a
derivative of equation (1) to calculate the rates of change of radius ratio,
as listed in the fifth column.
With the assumptions made earlier (spherical Earth, no atmosphere, no Sun
or Moon), the satellite would continue to travel around its orbit forever,
completing each revolution in exactly the same time interval. This can be
expressed, according to Kepler's Third Law, in the equation:
n2(CP)5 « .75^2E+l4 (2)
in which n is the "mean motion", in revolutions per day, and GP is measured in
kilometers. The constant, •75A02E+14 (electronic computer format for .75402x10''^)
applies only to earth satellites and would be different for satellites of other
planets.
The true angular motion of the satellite, d(PRV)/dt, will vary from the
mean motion, being faster at perigee and slower at apogee. To simplify the
description of this notion, it is compared with that of a fictitious object, M,
which travels around R at a constant rate, d(PRI-l)/dt, which is equal to the
mean motion, n. Kepler's Second Law permits us to define a "velocity ratio",
or ratio of true motion to mean motion, in the eqiiation:
gj} (Gv/cp)2 - y/irr^ (5)
This differential equation can be solved, uniquely, for PRM as a function of
PRV. A solution for PRV as a function of PRM requires successive approximations*
V/e have tabulated values of PRvi in the second column of Table I and have
inverted equation (5) to obtain inverse values of the velocity ratio, as listed
in the third column. The entire table is then called a table of "eccentricity
functions" because it applies to only one value of eccentricity. Now that we
have electronic computers to produce them, it is much more convenient to
interpolate in such tables than to use the basic equations.
Such equations, or tables, provide an artifice which permits us to
describe the motion of the satellite in the simple time equation:
PRM . PRMo + PRMi(ENL) (4)
in which PRMq is the initial value at any epoch, JNE. PRMi is equal to the
mean motion, n, and ENL is the elapsed time since epoch. Presiimably, one might
then determine the value of PRM at any time and then translate to PRV. A table
using PRM as the argument would therefore seem more useful. However, such a
table is more costly, in terms of computational time and effort. Instead, we
use a prediction procedure in which we, first, select a position, PRV. We
then translate to PRM and use equation (4) to determine the time.
TABLE I
Mean Anomaly (PRM) and Other Values
Tabulated as Functions of True Anomaly (PRV)
Where Eccentricity (CG/CP> = .019607
?m
PBM
0.00
0.000000
0.01
0.009614
0.02
0.019229
0.05
0.028847
0,04
0.058470
0.05
0,048098
0.06
0,057754
0.07
0,067578
0.08
0,077052
0.09
O.O86697
0.10
0.096575
0.11
0.106066
0.12
0.115775
0.15
0.125496
0.14
0.155256
0.15
0.144994
0.16
0.154772
0.17
0.164570
0.18
0,174588
0.19
0.184229
0.20
0,194092
0.21
0.205977
0.22
0.215887
0.25
0.225820
0.24
0.255///
0.25
0.245759
0.26
0.255766
0.27
0.265797
0.28
0.275855
0.29
0.285955
0.50
0.294058
0.51
0.504166
0.52
0.51 451 8
0.55
0.524492
0.54
0.55^89
0.55
0.544907
0.56
0.555146
0.57
0.565404
0.58
0.575682
0.59
0.585976
0.40
0.596288
0.41
0.406614
0.42
0.416954
0.45
0.427507
0.44
0.457671
0.45
0.448044
0.46
0.458426
0.47
0.468815
0.48
0.479206
0.49
0.489602
0.50
0.500000
0.961 556
0.961429
0.961647
0.962011
0,962518
0.965168
0.965957
0,964884
0.965945
0.967157
o.968'i56
0.969897
0.971456
0.975126
0.974905
O.97678O
0.978750
0.980807
0.982945
0.985151
0.987422
0.989748
0.992120
0.994550
0.996967
0.999425
1.001889
1 .004554
1 .006808
1 .009242
1,011645
1.014008
1.016521
1.01 8575
1 .020759
1 .022864
1 .024881
1 .026801
1,028617
1.050519
1.051900
1 .055554
1 .054672
1 .055851
1 .056885
1 .057765
1 .058492
1 .059062
1 .059470
1.059716
1 .059798
Radius
Ratio
0.980595
O.98O45O
O.98O542
O.98O727
0,980986
0.981 517
0.981 71 9
0.982191
0.982751
0.985557
0.984007
0.984759
0.985550
0.986577
0.987277
0.988227
0.989225
0.990262
0.991540
0.992452
0.995596
0.994765
0.995956
0.997165
0.998586
0,999616
1 .000848
1 .002078
1 .005502
1.004514
1.005709
1 .006885
1 .OO8051
1.009148
1.010229
1.011270
1.012267
1.015215
1.014110
1.014949
1.015727
1.016442
1.017091
1.017670
1.018177
1,018610
1.018966
1.0192^
1.019446
1.019566
1.019607
d(R.R,)
0.000000
0.007458
0.014851
0.022211
0.029494
0.056674
0.045724
0.050621
0.057559
0.065855
0.070140
0.076177
0.081940
0.087407
0.092557
0,097570
0,101824
0,105905
0.109588
0.112865
0.115712
0.118122
0.120080
0.121576
0.122600
0.125146
0,125206
0.122777
0,121858
0,120448
0,118551
0.116169
0.115509
0.109981
0.106195
0.101964
0.097502
0.092228
O.O8676I
O.O8O922
0.074755
0.068225
0.061 41 8
0.054544
0.047052
0.059514
0.051 822
0.025990
0,016055
0,008044
0,000000
With the aid of equation (4), we can write what are called \inperturbed
orbital elements" as follows, using values for a particular satellite, I960 Nu 2,
to provide a numerical examples
Unperturbed Orbital Elements.
1960 Nu 2
Jl^ « 5B442 .026710
HGR - .078485
cp - 7445.85
CG - 145.95
Vm « .654629 + 15.51296051 (K^TL)
QNR « .804751
NRP « .475764
For such elements, the epoch, JNE, may have any selected value. The
meticulous reader may find that, in the above elements, the value of n^(CP)^
does not agree exactly with equation (2). The above values include corrections
for the gravitational perturbations discussed below. However, equation (2) is
quite adequate for a first approximation.
c. Cumulative Effects of Gravitational Perturbations
The Earth is not a perfect sphere but resembles, more nearly, an oblate
spheroid, having an equatorial radius of 6578.17 1^» ^-rid polar radius of
6556.79 km. Through observation of artificial satellites, it has also been
found that the Earth is slightly pear-shaped, having; more mass to the South of
the equator than to the North. In addition, its equatorial cross -section is
elliptical, with the maxiimam radius (at about 0t;06l W) being about 0.17 km.
greater than the minimum. In theory, these differences from a spherical figure
may be treated as extra masses which exert extra "perturbing** forces on the
satellite, giving rise to perturbations in its motion.
The most important perturbations are those that are due to the oblateness.
In prediction, these must be considered. However, it is usually possible to
neglect effects of the elliptical equator and, except in precise prediction,
those of the pear shape. The most noticeable effect of the oblateness is the
precession of the orbit pole which, in a manner similar to the precession of
a gyroscope, moves in a circular path about the Earth's North pole. This
motion can be expressed in an equation:
QNR - (JNRq + QNRi(ENL) (5)
in which the value of the coefficient, QNRi , can be approximated from the
forrnula:
p2
QNRi - - ♦<SgO?7E-»? (PRMi ) C08(NGR) (6)
F
in which p is equal to CP(1 - o^).
For the Gear Ratio Elements, we regard the ratio, QMRi/PFM-j, as the first
of two gear ratios. Motion of the satellite is so coupled to that of the orbit
pole that, for each mean revolution of the satellite, the pole is moved by the
amount:
i^ ■ S! - - ^^^ -(-« o)
In equating this to AQNR/APHM, we indicate that it is applicebde to large,
as well as small changes in PHU*
A second effect of the oblateness is observed as a motion of the perigee
so that the polar angle, NRP, may be described by a time equation:
KRP » NRPq + NRPi(EIJL) (8)
For this equation, an approximate value of NRP^ may be obtained from:
NRP^ . >^<^?7E4^ (pj^j^j (2 - 2.5 sin2(KGR)) (9)
The second gear ratio is obtained by dividing equation (9) by equation (6) to
obtain:
ANRP . NRPi . 2 - 2.5 sin^C^GR) /^qv
AQNR QNRi cos(KGR) ^ '
which is also applicable to large, as well as small changes. It should be
noted that this ratio depends only on the inclination, NGR.
These equations have been described as yielding "approximate* values
and, in the proceeding section, we also indicated that equation (2) gives a
first approximation. In "Smithsonian Contributions to Astrophysics", Vol. 6,
p 67, Kozai gives more complete fonmilae than those of equations (2), (6) and
(9)» We have obtained excellent results by using the Kozai formulae to
compute the rates of notion and by deriving the ratios from these rates.
In writing orbital elements to include the effects of oblateness, the
normal practice is to express CJt^R and IvRP in terms of time equations, such as
equations (5) and (6). However, the Gear Ratio Elements would be written:
JNE » 56^2.028710
NGR = .078485
OP = 7445.85
CG - 145.95
PR^! » .654629 + APRM
qjIR = .804751 - .001052864( APRM)
KRP «.. 475764 - 1.6555966(AQrJR)
and we might then supply an auxiliary prediction equation which, at this stage,
would be written:
APRM « 15.51296051 (EtIL)
However, it should be noted that the prediction information can be expressed
in forms other than the above equation. It can be expressed as a table of
values of PRM as a function of time, as a graph or in the fona of periodic
announcements of observed values.
d. Acceleration Effects
The effects of atmospheric drag and radiation pressure make it necessary
to add another term to equation (4) so that it becomes:
PRM » PRMo + PRMl(EI^X) + FBMaCEKL)^ (11)
in which the acceleration coefficient, PRM2, is usually of a magnitude that
can be most conveniently expressed in microtums per day2.
When the perigee radius (CP - CG) is about 69OO km. or less, the accel-
eration will be primarily due to atmospheric drag, PRM2 will be positive and
may range from 10 to 1,000/ut/d2 for reasonably long-lived satellites. As a
satellite approaches the end of its lifetime, PRM2 ^^Y increase rapidly to
values as great as 100,000/it/d2,
If the perigee radius is greater than 69OO km., the effects of radiation
pressure become significant and may result in either positive or negative
values of PRM2, generally of the order of + 1 yut/d^. The ECHO satellites are
a notable exception, being abnormally sensitive to both radiation pressure and
atmospheric drag. For these satellites, typical values of ?m2 might range
from - 500 to + 2,000 ;ut/d2.
In any case, the acceleration coefficient is highly variable and can
seldom be predicted with an accuracy much better than + ^Ofo* So, in writing
an equation for long-term prediction, v/e usually use a mean value that has
been determined from observations over a substantial period. In addition to
yielding the best possible predictions, this value also serves as an indicator
of the length of time that the mean anomaly equation remains useful. For
example, if we expect the equation to give values of PRI*I that are accurate to
+ t:01 , its accuracy begins to become questionable when the acceleration term
approaches this value or when:
EI^Tj is equal to or greater than -/.Ol /PRM2
The acceleration affects other orbital elements because the mean motion,
n, is variable, as indicated by:
LtZMl - m:^ + 2PRi^2(E^) (^2)
l(EML)
v/hich may be derived from equation (11 ). As indicated by preceeding formulae,
this variation can affect the semimajor axis, the eccentricity and the rates
of motion of both orbit pole and perigee.
The resulting rate of change of the semimajor axis is usually quite small
so that the effects of acceleration are best treated by occasional recomputa-
tion or adjustment according to the equation:
A(OP) = -i^An (15)
The theory for estimating relationships between acceleration and eccen-
tricity is rather unsatisfactory, except for after-the-fact analysis.
However, in his book, "Satellites and Scientific Research", King-Hele presents
some useful formulae which apply to satellites having moderately high accel-
eration (about .5E-5 or more) and for which the initial eccentricity is
between 0.2 and 0.02. Expressed in our symbols, one of these formulae
estimates the lifetime, t^, of the satellite as:
ti,- ? eo (PRMi) (^ ) (14)
8 (PRM2)
in which ©o is the initial value of eccentricity. The variation of eccentricity
with time can then be expressed by:
e - eoVl - (ENL)/tL (15)
where ENL is the elapsed time since the epoch of the initial value.
8
For lesser acceleration, corresponding to a perigee radius greater than
about 6900 km«, the above formulae become meaningless. Experience indicates
that one may as well adopt the most convenient assxomption; that the eccentricity
remains constant until the observations indicate that a change is necessary.
(This excludes effects of the pear shape, which we handle through geometric
adjustment of predictions and observations, rather than as a cyclic variation
in eccentricity.) Again, the ECHO satellites are an important exception in
that they undergo large, cyclic variations in eccentricity as a result of
radiation pressure.
Theory becomes even less satisfactory in predicting the effects of
acceleration on the motion of the orbit pole eind perigee. Generally, the
theory requires that additional tenns, dependent on (EI^IL)^, be added to
equations (5) and (8). However, it is difficult to derive satisfactory values
for the necessary coefficients, QiriR2 and KRP2.
COMPARISON OF THEORETICAL GEAR RATIOS
WITH THOSE OBSERVED OVER LONG PERIODS
For the Gear Ratio Elements, \ie have made two simplifying assumptions.
First, the ratio between motion of the orbit pole and mean motion of the
satellite is assumed to follow the equation:
AQNR
APRM
= A + B( APRM)
(16)
in which A and B are constants to be determined empirically. Second, the
ratio between motion of the perigee and that of the pole is asstimed to remain
a single -valued constants To test these assumptions, we have studied records
for several satellites, with the results indicated in the following Table II.
TABLE n
Study of Observed Gear Ratios
Satellite
Perigee
Radius
64 005 A
6656.02
1956 Alpha
671 8.51
1959 Alpha 1
69^6. 6k
1959 Eta
6894.55
1960 Zeta 1
68V1.96
65 047 A
6851 .50
i960 Ku 2
7297.90
AQ^^R/APR-1
A B
AITRP/AQI'IR
-I.5458I5
-.001195^75 -.259E-8
-.001055650 -.659E-9 -1 .495750
-.000651741 negligible -1 .5055958
-.000825597 negligible -1 .4888692
-.001177486 negligible -1 .4988108
-.001 029629 negligible -1 .5771 661
-.001052884 negligible -1 .6555966
Revolutions
Examined
1800
15600
8700
8500
22000
1600
19000
Values of B less than •1E-12 were considered negligible. In no case v/as
there a significant departure from a single value for AQNR/AIiRP»
9
These results should not be viewed with surprise because they represent
an after-the-fact fit to data that is limited in precision and the implied
acceleration effects are not greatly different than those that would be
predicted by other means. The only conclusions that we wish to draw are:
1 . For satellites having a perigee radius of the order of 69OO km.
or more, constant values for both AQ^^/APK^i and ANRP/AQ?^ may
be used over periods of time as long as 2 to 5 years.
2# For satellites having a perigee radius less than 69OO km., the
ratio, ANRP/AQMR, remains constant and the ratio, AQNR/APRM,
can be closely approximated as a linear function of aPP^*-*
These conclusions apply to mean rates of motion. Periodic variations due to
the pear shape are treated separately.
Obviously, a long period of observation is needed to establish accurately
measured values of these ratios. However, we can use the Kozai formulae to
calculate initial values and, as indicated by the following Table III, these
values are adequate to serve for several hundred revolutions.
TABLE m
Comparison of Observed and Theoretical Gear Ratios
Satellite
i960 Zeta 1
i960 Ku 2
65 047 A
Observed
-.001177486
-.001052884
-.001029629
AQITR/ AFRKI
Theoretical Observed
-.00117750 -1.4988108
-.00105215 -1.6555974
-.00102906 -1 .5771 681
AnRP/Aq(N^R
Theoretical
-1.4988108
-1.6554171
-1.5771675
This tern is shown only
to indicate where it is
placed, when its value
is significant.
Thus the final form of the Gear Ratio Elements for I96O liu 2 might be
written:
JNE = 58442.028710
NGR « .078485
cp « 7445.85
CO » 145.95
PRM « 16624.654629 + APRM /
QNR « .804751 - .OO1O52884(APK«0 - 0.0( APRH)^
KRP .= .475674 - 1.6555966(AQNR)
This differs from the form written previously only in the addition of the
integral number of turns, since launching, to PRMq* This is useful in
coordinating different sets of elements that are derived at different times.
The auxiliary prediction equation may now be written as:
APRN! - 15.51296051 (ENL) + .127E-6(ENL)^
and it must be recognized that the values of CP and CG will require occasional
revision.
The usefulness of the Gear Ratio Elements will become more apparent in
the following description of various applications. However, at this point,
it should be noted that their basic characteristic is that they use PRM as
the independent variable, rather than ENL. In essence, the mean anomaly, PRM,
is the satellite's measure of time, just as our time, JNL, is based on mean
revolutions of the Earth about the Sun. The prediction equation is simply a
means of translating between the two systems of time.
10
APPLICATIONS OF GEAR RATIO ELEMENTS
a. Use of Gear Ratio Elements to Furnish Rationalized Orbital Elements for PredictiMi
It is possible to make predictions directly from the Gear Ratio Elements*
However, to take advantage of other ITCP publications on the subject of
predictiont the Gear Ratio Elements can be converted to the notmal foxa of
Rationalized Orbital Elements* Using the above I960 Nu 2 elements as an
example, we first add the APRM etjation to PRI^Iq to obtain:
PRK - 16624.654629 + 15.51296051 (EITL) + •127E-6(EriL)2 (1?)
The APRH equation is then multiplied by the AQCJI^/APRM ratio and added to
QNRq to obtain:
qjTR « .8O575I - .01422758(ENL) - .154E-.9(ENL)^ (18)
and the AQNR portion of this is multiplied by the ANRP/AQ^IR ratio and added
to NRPo to obtain:
I^RP . ,475764 + *02525928(EtIL) + .218E-9(ENL)^ (19)
It should be noted, here, that the coefficients of (Ei^)^ in equations
(18) and (19) are slightly less than we would have derived from a more
elaborate, but rather uncertain theory* Evidently, this difference from
past practice has little practical significance*
To make equation (18) easier to use, we usually combine it with an
equation for QJ^T* Using methods outlined in previous ITCP Bulletins, the
applicable equation would be:
q^n = .655678 + .00275791 (EI.!,) (20)
and, combining with equation (I8), we have:
t?:r x= .149075 - .01696549(ENX) - .154e-6(ekl)^ (21)
Equations (17 )» (19) and (21) are those normally used with the Rationalized
Orbital Elements* The values of JKE, NOR, CP and CG can be copied directly
from the Gear Ratio Elements to complete the full set of Rationalized Orbital
Elements*
b. Use of Gear Ratio Elements in Simplified Satellite Tracking
for Casual as well as Meticulous Observers
For tracking, we replace the PRM equation by a "Revolutions Log", a
device that was first used in satellite tracking by Arthur S* Leonard*
Equation (17) can be expanded into a table of values of PHM, with first and
second differences, as shown in Table IV*
TABLE IV
Tentative Revolutions Log for 1960 Nil 2
(based on equation (17), above)
JJIL Pmi 1st Diff. 2nd Diff.
58440 1 6597 .240752 .0000254
155.1296158
58^0 1 6752 .570568 .0000254
155.1296412
58^0 1 6867 .500009 .0000254
155*1296666
58470 1 7002 .629676 .0000254
155*1296920
58480 17157.759568 .0000254
155.1297174
56490 17272.889O85 .0000254
155 •1297428
58500 17408.018828 *0000254
The usual practice is to write these values in pencil so that they can, later,
be erased and replaced by observed values*
11
Using a 10 day interval, the values in the third colurui will be 10 tines
the mean motion, n, and those in the fourth column will be 200 times the
acceleration coefficient. As outlined in previous ITCP Bulletins, predictions
may be made by interpolation in such a table. Its form is comparable to that
of the Daily Satellite Ephenerides.
The prediction procedure involves the calculation of a time, JNL, at
which the satellite is expected to appear at a particular position such as
the point of local culmination. For this position, we will have calculated
a value of PH.. The observation then represents a measurement of the actual
tine at v;hich the satellite appeared at, or very near this position. As a
meticulous observer, I will have measured this time to + 0^0000001 . I will
then use this actual time to recalculate the position and will correct for
differences between the calculated and actual point of observation. I will
also make corrections for effects of the pear shape and the ellipticity of
the equator. All of this is done with the objective of obtaining a measured
value of PRM that is accurate to about + 0l;00001 .
Another observer, Mr. X, may either be less meticulous or may lack the
means for making precise obseirvations. I!e may simply use the predicted value
of PH'I as a measured value to correspond with his measurement of the time.
\;e cG.n assume that his accuracy is + o4o0001 in timing &Jnd 4 OiOOl in PRI'I.
Let UG then assume that ::r. X and I both start tracking 19^0 >Iu 2, using
the above table and the Gear Ratio Elements. For convenience, we will also
assume that his location is the same as mine and that he makes each observation
at the same time that I do. After about 60 days, our records of observations
might compare as shovm in Table V.
TABLE V
RECORD OF OBSERVATIONS
Comparing Log of a Metlculods Observer (W.P.O)
with Log of a Casual Observer (Mr. X)
Predicted
PK-I
u. p. C
).
JML
?m
Reside
58462.40115515
16699.946458
16899.94651 8
+80
58482.05086529
17165.472894
17165.475597
+505
58484.04727861
17192.450126
17192.450724
+598
58485.08494712
17206.472112
17206.472747
+655
58486.04569964
17219.427707
17219.428590
+665
58494.02957450
17527.5^616
17527.5^1716
+1102
58495*06725567
175^1 .562460
175^1.565655
+1175
58497.06565582
17568.540066
17568.541567
+1299
Mr. X
PR-:
16899.9469
17165.4752
17192.4512
17206.4728
17219.4260
17527.5^16
175^1.5642
17568.5410
Re 3 id .
+400
+500
+1100
+700
+500
+1000
+1700
+900
V/e can asstime that Mr. X's values of JIJL will be the same as mine (in the first
column) except that they will include only five digits to the right of the
decimal.
In the above table, each "residual" column represents the difference
between the observed valua and the predicted values, which are in the second
column. As we proceed, Mr. X and I will both plot these residuals against time,
as shovm in Figure 2.
12
JNL
Fig. 2. Plot of residuals in NRM obtained by a meticulous observer (O)
as compared with those of a casual observer (X). Data from Table V.
On JNL - 58500, Mr. X and I both decide to review the situation and
bring our orbital elements up to date* V/e both draw a smooth curve through
the data points* Ky curve ia represented by the solid line and Mr. X's curve
will probably fall somewhere within the shaded area.
Our next step is to correct the values of PK^I in the Revolutions Log by
amounts equal to the ordinate s of this curve for the dates used in the Log*
A comparision of the results that we obtain might be as shown in Table VI.
TABLE VI
Corrected Revolutions Logs of
Meticulous and Casual Observer Compared
w.
P. 0. Log
■:r.
X's Log
JITL
PRK
1st Diff.
2nd
PKM
1st
2nd
58440
16597.240752
.000056
16597. 2408
.0000
155.129626
155.1296
58450
1^752.570578
155.129711
.000085
16752.5704
155.1297
.0001
58460
16867 .500089
155.129782
.000068
16867.5001
155.1298
.0001
58470
17002.629871
155.129959
.000157
17002.6299
155.1299
.0001
58480
17157.759810
155.150155
.000216
17157.7598
155.1502
.0005
58490
17272.889965
.000228
17272.8900
.0001
155.150585
155-1505
..._.-.«.
58500
17408.020548
155.150585
.000200
17408.0205
155.1505
.0002
58510
17545.150951
155.150785
.000200
175^5.1508
155.1507
.0002
58520
17678.281714
17678.2815
In both cases, we have extrapolated the table ahead by selecting a mean value
for the second difference and adding it in for two more 10 day intervals.
Thus, the portion of each Log below the dashed line represents a new "tentative"
Log, with which we continue to predict for subsequent observations.
13
At the sane time that we revise the Log, Mr. X and I vdll both derive new
orbital elements and we will assume that we select the date, JJJL « ^B^^0^ As
compared with the previous Gear Ratio Elements, the data that we will work from
will be:
Data Required for Revision of Elements
W. P> 0. Mr. X
JITL 56510.0 58510.0
PRM 1 7545 . 1 50951 1 75^5 . 1 5O8
APRM 918.496502 9I8.4962
The new Gear Ratio Elements that we derive will be:
New Gear Ratio Elements Derived from
Casual and Meticulous Observations Compared
W. P. . Mr. X
Jlffi 58510.0 58510.0
NGR .078485 .078485
cp 7445.46 7445.46
CG 145.95 145.95
PR>i 17545.150951 + aprn: 175^5*1508 + Avm
q^m .85768I - .001052864( API^O .857681 - .001 052884 ( a PH-)
MRP .055575 - 1.6555966( AOrm) .055575 - 1 .6555966 (a QNR)
If I were to v;rite an auxiliary prediction equation, it would be;
APR-I « 15.5150685(ENL) + .100E-5(E^IL)^
tihereas Mr. X's prediction equation would be:
APRl^I « 15.51506(EML) + .1E-5(ENL)^
Tlaus, it nay be seen that the nain difference betv/een my results and those
of i:r. X is that I have more quantitative information as to the acceleration
that has taken place and my prediction data will be more accurate in timing.
However, the new Gear Ratio Elements that we derive will not be significantly
different. Insofar as tracking is concerned, I could have abandoned the
satellite for 70 days, leaving it to Mr. X. At the end of that time, I would
have found his elements acceptable in recovering it. So, Mr. X easily qualifies
as an Independent Satellite Tracker. {\le once defined a "tracker" as a man who
can keep track of satellites by himself.)
It should be noted that the above example is fictitious in that it was
necessary to show a rather sudden increase in acceleration in order to make the
residuals large enough to clearly illustrate the method of correction. I960 llyx 2
would never show such a drastic change. If actual data had been used, the curve
of Figure 2 would have been very close to the time axis. I would have made very
small corrections, plotting the curve on a larger scale, and Mr. X would have
concluded that there was no need to make corrections.
14
c. Use of Gear Ratio Elements to Impr ove Tracking Agency Data for Long Term Predicticmfl
In using data issued by the official tracking agencies, one should recog-
nize its limitations. The principal objective of the agency is to provide good
short-term prediction. Observations are analyzed in batches to derive the best
orbital elements to fit current observations and these are then extrapolated one
or two weeks ahead. Before these elements begin to deteriorate, the agency will
have a new batch of observations, a new analysis and a fresh set of elements.
Thus, there is little reason to aim for long-term accuracy of any one set of
elements.
There are a number of methods by which tracking agency data can be
"smoothed* to obtain more uniform and better long-term performance. However,
in smoothing the data, one should understand which items are likely to be
most accurate and which should be regarded with suspicion. First, the rates
of motion given for orbit pole and perigee are usually theoretical, rather than
measured values. As a result, one sometimes obtains better values by comparing
successive sets of elements, rather than by using those stated in the official
bulletins. The remaining items are then comparable to those that we have
described as "unperturbed* orbital elements and can be discussed, in order of
decreasing accuracy, as follows:
JNE The tracking agencies measure time very precisely so that there need
never be any doubt as to the precision of the stated epocli.
CP The semimajor axis is derived from measurement of the mean motion and
this value should always be quite precise.
N^ The inclination is relatively easy to measure and does not change
rapidly. After the first week or two that a satellite is in orbit,
tracking agency values for NGR will stabilize to within about + 50
microtums of the correct value.
QjN^R This value is also relatively easy to measure but more observations are
needed in finding a correct value. So, the accuracy will generally be
about + 100 microtums for SAO elements and not quite this good for
NORAD elements.
CG This depends on eccentricity and the ability to obtain a good measurement
varies with the orientation of the orbit. In prediction elements, the
values seem to have a "scatter" of about + 0.0005.
NRP The perigee position is the most difficult to measure, particularly if
the eccentricity is low. Errors can be of the order of + I5OO microtums.
Tracking agency data will also contain the variations due to the pear
shape. For long-term, smoothed elements, ve must separate these variations.
FRM Accuracy in measurement of mean anomaly is affected by the accuracy of the
perigee position, which serves as the reference point. It should be
realized that, by adding NHP and PRM to obtain NRM, we can usually
obtain a more accurate measure of the satellite position because NRM is
measured from a fixed reference point.
Table VII then illustrates how we can use the Gear Ratio Elements to improve
tracking agency data. In this case, we have used Ephemeris VI data, issued by
the Smithsonian Astrophysical Observatory. Once we have established values for
the gear ratios, the only data that we need accept from the SAO epheme rides are
the values of JNE, NGR and KBM. In effect, we use SAO just as we used Mr. X,
to keep track of how many times the satellite goes by and when. Thus, the first
two columns of the table are similar to the Revolutions Log, except that, for
convenience, we have used the ^k day interval that corresponds to the dates for
which the Ephemeris VI is issued.
15
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To start the table, we initially used the SAO values of NRM, to find
temporary ratios, AQNR/ANRM and ANIff/AQNR, that would give a reasonably
good fit to the SAO values for QNR and NRP. This made it possible to
establish a few of the calculated values, QNR(ViTO) and NRP(WPO)» The cal-
culated KRP(V/PO) values could then be subtracted from the NRM(SAO) values to
obtain some starting values for PRM(WPO). With these, we derived the equations
shown at the upper left. For convenience, these were based on an initial value
of zero for PBM. Based on this start, we now fill in new values in the table
according to the following procedure:
Procedure Used in Preparing Table VII
1. From each new issue of Ephemeris VI, we enter the values of JNL and
Nia^l(SAO) in colisnns 1 and 2»
2. We extrapolate the value of KRP(WFO) forward, using the tabulated
1st and 2nd difference to obtain a tentative value for. NRP.
5» The tentative value of NRP is subtracted from NRM(SAO) to obtain a
tentative value for PRM(^VPO).
hm The tentative value of PRM(V/PO) is then used to calculate final
values of QrTR(WPO) and NRP(lTffO) that are entered in columns 7 and 11.
5. The final NRP(WPO) is subtracted from NRM(SAO) to obtain the final
value for PRM(WPO). This final value seldom varies from the tentative
value by more than a few mi c returns so that it is not necessary to
recalculate CiiCR(WPO) and NRPCV.TO).
6» The first and second differences are then filled in for PHM (colximns
4 and 5), (3J:R (columns 8 and 9) and NRP (coliimns 12 and I5).
7. The SAO values of QNH(SAO) and NRP(SAO) are entered in columns 6
and 10 for comparision.
Columns lA and 15 illustrate how we can keep track of the variation in
semimajor axis. The first entry in this column was calculated from the mean
motion, corresponding to 1/l4 of the value entered in column 4 of the same row.
Succeeding values are obtained by subtracting the numbers listed as A CP in
column l4. The AGP values are obtained with the equation shown above the table,
in which AAPRM represents the 2nd difference in PEM. The constant in this
equation, - 21 .56OI , is 1/l4 of the value obtained vdth equation (15)»
Coli-imns 16 and 17 compare the SAO values for eccentricity v;ith those
calculated from the King-Hele equations (l4) and (I5). Equation (l4) gived a
lifetime, t^, of 629 days from JNL = ^Sh^6f indicating that this satellite
should reenter the Earth's atmosphere on or about November 1, 1965« It should
be a very spectacular sight because this is the rocket that is filled with
sand*
As indicated, the QNR(SAO) values agree with the calculated QNR(WPO)
values quite well and, as anticipated, the NRP(SAO) values do not show as
good agreement. The calculated values are definitely better for long-term
prediction and have given more accurate predictions of position angles.
17
V/ith the 14 day interval, one must divide first differences by 14 to
obtain daily motion and must divide 2nd differences by 2 x (l4)^ or ^92 to
obtain the acceleration coefficients. For example, if v;e v/ish to write
Rationalized Orbital Elements for the epoch, JITE = 5^55^? we first interpolate
in the table to obtain:
Base Value
1st Diff.
2nd Diff.
PRM
1680.019550
215.27^191
.090684
(JL^R
-.520565
-.256775
-.000528
NRP
.215694
•596927
.000507
V;e then divide 1st differences by l4 and 2nd differences by 592 to write:
PRM = I68O.OI955O + 15.2625851 (Elsli) + .251E-5(ENL)^
ONR = -.520565 - .0185411 (ENL) - .857E-6(ENL)^
NRP = .215694 + .0285519(ENli) + .129E-5(ENL)^
For comparision, the SAO equations for the same epoch may be written:
PRM « 1680.02241 + 15.265657 (ENL) + .259E-5(ENL)^
QNR - -.520569 - .018524 (ENL)
NRP « .210816 + .027669 (ENL)
This comparision helps to illustrate several points:
1 . The two sets of elements describe the same initial position because
the NRM values for each are nearly equal, .255224 vs .255226.
2. The principal difference in long-term prediction v/ill be due to
the difference in meaji motion. The value obtained from the Gear
Ratio Elements will usually be better because it must fit the past
history.
\jhile the rates of notion of the orbit pole agree quite well, the
rates of motion of the perigee differ substantially. This may be
ascribed to possible effects of the pear shape on the SAO analysis.
iin incidental benefit of the table of Gear F^tio Elements is that v/e begin to
see the pc.ttern of acceleration. The early period, between ^6h^6 and 58492, is
one in which the perigee is in sunlight. From 5^506 through ^^h8 it is in
darkness, where the atmospheric drag is reduced. The entire cycle should
extend through about 157 days.
The values in the first row of Table VII represent my extrapolation back
to the time of launching, JNL = 58425.686866^ These values place the orbit
directly over Gape Canaveral and allow the real satellite a few minutes to
rise from the launching pad to meet its mathematical model.
18
d. Gear Ratio Elements for Keeping Track of the ECHO Satellites
, Radiation pressure causes large, cyclic changes in eccentricity for the
ECHO satellites, ranging up to 0,05 for ECHO I and 0.O25 for ECHO II. This
phenomenon leads to apparent irregularities in the motion of the perigee.
For example, as the eccentricity passes through zero, the perigee position
must jtimp from one side of the orbit to the other. These variations also
lead to cyclic changes in the rate of motion of the orbit pole because, as
indicated by equation (6), this rate is affected by the eccentricity. These
effects of radiation pressure are coupled with large, irregular variations in
acceleration. Thus, it becomes virtually impossible to describe the overall
behavior of the ECHO satellites with a simple set of long-term equations.
One can devise means for accurate prediction for these satellites over
short periods of time. However, the most useful application of the Gear Itatio
Elements to this type of satellite appears to be that of providing a means for
rough prediction that remains valid over long periods of time. !ath this
objective in mind, we can make the Gear Ratio Elements simpler, rather than
more complicated.
We start with the basic assiomption that the eccentricity is zero and that
there is no perigee position. Vfe then need only one gear ratio, that between the
orbit pole position, QMR, and the mean polar angle, NRM, of the satellite.
With these assumptions, we can write Gear Ratio Elements for the two ECHO
satellites as follows:
Gear Ratio Elements for Tracking ECHO Satellites
ECHO , I ECHO II
JNE = 58541 .0 JNE = 585^.0
NGR = .151581 NGR = .226^»00
CP « 7806.94 OP =» 7547.40
NRM = 5704.0997 + ANRM NRM = 1697.8^2 ^ AKRU
QNR = .272561 - .000755181(ANRM) QNR = .819782 - .00017l478( ANIW)
For ECHO I, the integral number of turns was taken from an arbitrary starting
point. For ECHO II, the count is taken from the time of launching.
Auxiliary prediction equations, using the same epoch as for each set of
elements, would be:
For ECHO I: AWRM - 1 2.595787 (ENL) + .174E-4(ENL)^
For ECHO II: ANRM - 1 5. 258029 ( ENL) + .895E-4(ENL)^
These will be somewhat le&a dependable than for other satellites, due to the
rather rapid changes in acceleration.
19
Such IJear Ratio Elements can be kept up to date partly through observation
and partly through the use of data from any other source. As an example, the
ITCP issued Modified Orbital Elements on June 4, 1964, including the following
data for ECHO II:
OBJECT 64 004a
NAME ECHO II
SOURCE SAO
EPOCH of 06 Junl
perigee 02 H > »- JNL « 58552.104410
(UT) 50M55 J
INOLIN 8IA5O
NODE W OOOAI7
MPD=1D -07M17
PERIGEE 155A79 *NRP « .182750
change/P -AI659I
A Period 108K720
change/P -M00012
ECCEN UO25I7
P RADIUS 4580#8
R A NODE 291 A77
Of this data, we need only the two values indicated for JNL and NRP. (The NRP
value is obtained by dividing the argument of perigee, 155*79» by 56O and
subtracting ot:25).
Modified Orbital Elements are always written for an epoch at which PRM is
zero so that the above value of NRP is also equal to NRM. Using the ECHO II
prediction equation, the full value of NRM can readily be identified as:
NRM = I752.I8275O for JNL = 58552.104410
and these values can be regarded as equivalent to any observation that the
tracker might obtain through his own effort.
As a check on the effectiveness of the Gear Ratio Elements, we can use
the above data to compute QNR in:
QNR = .819782 - .000171478(1752.182750 - 1697.846200)
from which the result (0^810464) should be numerically equal to the value given
in the Modified Orbital Elements for the Right Ascension of the Node (291^77)
Based on past history, the above Gear Ratio Elements for the ECHO satellites
should give satisfactory prediction for a year or more and should thus provide a
good beginner's exercise in tracking, when coupled with graphical methods of
prediction such as those involving the Rationalized V/ulff Net.
CONCLUSIONS
Obviously, Gear Ratio Elements are not truly "permanent" in that they
must ultimately deteriorate due to lack of arithmetical precision. However,
they are correct in principle in that motion of the orbit pole and perigee is
more properly a function of motion of the satellite, rather than of time.
It is evident that they pl-ovide a more enduring model of satellite behavior,
permitting the tracker to continue for a much longer period before adjustments
become necessary. Since starting this investigation of their perfo nuance, I
have continued tracking three satellites, each for about 200 days, and have
continued to obtain accurate predictions for each from a single set of Gear
Ratio Elements. During the entire period, the process of bringing the elements
up to date and making new predictions has been completely automatic. It now
appears that the same elements will continue to be satisfactory for at least
one year.
20
Phone: STerlins 3-4100 EXHIBIT E
INDEPENDENT TRACKING COORDINATION PROGRAM
824 Connscticut Avcnut
Waihington 6, D. C
BULLETIN
7 April 1964
CONTENTS: Rules for Combining Polar Angles
Rules for the Negative of an Angle
Worksheet for Advancing Epoch of Rationalized Orbital Elements
Drafting Aids to Making Accurate Overlays
Circular Slide Rule for Five Significant Figures
More Significant Figures with Used Desk Calculators
More Significant Figures with Curta Hand Calculator
Rules for Combining Polar Angles
The predictable way in which the 3-letter "names" of polar angles change under
addition and subtraction is illustrated in the work sheets.
IN GENERAL: Polar angles may properly be combined if, and only if, they have a
common pole (common middle letter- identity) and another identity in common.
If the INITIAL identity of one is the san>e as the TERMINAL identity
of the other, the SUM ANGLE will have the remaining initial and terminal
identities; for example:
ABC + CBD = ABD
If the INITIAL identities are the same, the INITIAL identity of the
REMAINDER ANGLE will be the remaining identity of the subtrahend;
for example:
ABD - ABC = CBD
U the TERMINAL identities are the same, the TERMINAL identity
of the REMAINDER ANGLE will be the remaining identity of the subtrahend;
for example:
i ABD - CBD = ABC
I Rule for the Name of the Negative of an Angle
If an angle is defined by the three-letter symbols, ABC, then the angle, CBA
I is the negative of ABC, for example:
I
-ABC = CBA
Society of Photograpbic Scientists and Engineers
WORKSHEET FOR ADVANCING EPOCH: Example
:uSec
Data used in the example given below was taken from the Work Sheet B example discuSSed
in ITCP Bulletin 5 April 1964. Additional copies of Work Sheets are available on request to
this office. Please specify which Work Sheets are desired.
WORK SHEET D: For Advancing the Epoch of Rationalized Orbital Elements
No.
JNE
-(JN?!
new epoch
old epoch
^ S!NE = interval
d
i
(^NE)
2 ^
Transmits:
mc/s
mc/s
NCR =
t
CO == . km CP
km CG/CP =
TNRw -
+ TNR^(^NE)
+ TNRw(5;nE)''
TNR,
_ t
TNRv =
+2 TNR^(?NE)
TNR, =
(ENL) tnr„
(ENL) tnr^
(ENL)^
(ENL)i
NRP
NPR .(?;ne)
NRP^C^lNE)''
NRP^= i
NRP. = :
+2NRP^(?:NE) = \
(ENL) NRPjj
::, t
(ENL)^
NRP^ = ^ (ENL) NRP2 = o (ENL)2
PRM . -
+ prm^(?;ne)
+ PRM
:^^NE)-
PRM,
= t
PRM^
+2PRM^(?;NE)
PRM,
(ENL) PRM^
= t
(ENL) PRM^
(ENL)2
(ENL)2
EXHIBIT E-1
^WORK SHEET D: For Advancing the Epoch of Rationalized Orbital Elements
No.
JNE
■{JNt
= new epoch
= old epoch
= :^NE = interval =
d
(|JNE)
2 =
Transmits:
mc/s
mc/s
NGR =!
CG = . km CP = .km CG/CP = .
TNRm =
+ TNR^(?lNE)
+ TNRw(?iNE)
3 _
t
O
t
t
TNR« =
_ t
TNR,
+2 TNRw(?lNE) =
TNR^ = .
(ENL) TNR„
= t
(ENL) TNR,
= t
(ENL)'
(ENL)'
NRP^ =
+ NPR .(?INE)
+ NRP^(?lNE)^
t
t
t
NRP
= t
NRP^
+2 NRPw(?lNE)
t
t
NRP.
= t
(ENL) NRP^
= t
(ENL) NRPg
(ENL)'
(ENL)'
PRM^ =
+ PRM^(?INE)
+ PRM,
j(^NE)
2 _
t
t
PRM
= t
PRM^
+2PRM^(?lNE)
t
t
_ t
(ENL) PRMw
- t
PRMj = . (ENL) PRMg
= t
(ENL)2
(ENL)2
Additional coplz& available. ^Kom: ITCP, S24 Conn. Ave., Wa-S^,, 0,C. 20006.
WORK SHEET D: For Advancing the Epoch of Rationalized Orbital Elements
No.
JNE
-(JN?I
= new epoch =
= old epoch =
d
•
d
•
)
= ?INE
NGR
= interval =
= ! CG =
a
•
(?JNE)2 =
. km CP =
Transmits:
. km CG/CP = .
mc/s
mc/s
TNRrf -=
+ TNR^(fJNE)
+ TNR2(?!NE)^
t
e
t
t
TNRq =
_ t
TNR,
+2 TNRw(?!NE)
TNRj = .
(ENL) TNR^ = I
(ENL) TNR = i
(ENL)'
(ENL)'
NRP^ =
+ NPR .(?!NE)
+ NRP^(?lNE)^
t
t
t
NRPq =
= t
NRP.
_ t
+2NRP^(?!NE) = \
NRP.
(ENL) NRP^ = \
(ENL) NRPg
= t
(ENL)'
(ENL)'
PRM^ =
+ PRM^(?lNE)
+ PRM,
■^t^n
t
t
PRM
= t
PRM^ = .
+2PRM^(|2NE) = \
PRMj = \
(ENL) PRM^
- t
(ENL) PRMg
= t
(ENL)'
(ENL)'
SEVEN PLACE
COSINES, SINES
AND TANGENTS
FOR EVERY TENTH MICROTURN
EXHIBIT F
Seven Place
Cosines, Sines and Tangents
For Every Tenth Microturn
Norton Goodwin, Director
Independent Tracking Coordination Program
Society of Photographic Scientists and Engineers
Photographically composed and printed m the United States of Annerica by
H. G. Roebuck & Son, Inc. 2140 Aisquith Street, Baltinnore, Maryland 21218
NOTE ON PHOTOGRAPHIC TYPOGRAPHY
These tables were photographically composed from digital
computer tape records. The particular typefaces in which these
tables are set are Spartan Book Condensed Large and Spartan
Heavy Condensed Large, The selection was made from trial
copy composed in a variety of fonts.
Typography was prepared by a commercial printer on a
conventional photocomposition unit controlled by perforated
paper tapes. The control tapes were produced by a converter
unit designed to process magnetic tape records into a lorin suit-
able for general-purpose phototypesetting machines.
Acknowledgment is made of the assistance of Robert H.
Blechen, Computer Sciences Department, The Rand Corpora-
tion, in securing a magnetic tape record of the tabular values,
edited in the specified page format, and to Donald Rollert and
Carl Rosencrown, Graphic Systems Engineering Department,
Mergenthaler Linotype Company for their concern in convert-
ing the magnetic tape record to Linofilm tape.
©1964, Society of Photographic Scientists and Engineers.
PREFACE
THESE TABLES differ from trigonometric tables now avail-
able in that the sine and cosine values of a given argument ap-
pear on opposite pages, and in that decimal fractions of the
period of these functions are the arguments. jThey were designed
to facilitate routine desk-calculator transformations of the co-
ordinates of artificial earth satellites in particular. [They should
prove generally advantageous in any area, such as"space naviga-
tion or electrical engineering, involving cyclical coordinate
changes.
The prime source of these tables is a subtabulation per-
formed by Dr. E. C. Bower based on key values obtained from
Francois Callet's "Tables Portatives de Logarithmes." Values
have been correctly rounded from an accuracy of one unit in the
fifteenth decimal place.
The arrangement of values in the one customary in logarith-
mic tables. Each page lists five hundred arguments at ten micro-
turn intervals. The first two significant figures of the argument
identify particular pages, the next two identify particular rows,
and the last significant figure identifies a particular column. To
facilitate "reading up," a tenth column is provided which gives
the same value as the zeroth column in the succeeding row.
Complete values are given only in the zeroth column. In the
remaining columns, unless the value is printed in boldface, the
missing first two significant figures are those of the first complete
value in the same row, or higher. If the value is printed in bold-
face, the missing first two significant figures are those of the first
value in the succeeding row.
Tabular values of cosines and sines of from tOOOOO to
t25000 are given. Since arctan alb = arccot b/a, tangents are
only given for fiom tOOOOO to tl2500 and cotangents for from
tl2500 to t25000. All tabulated values are positive. Rules for
determining the senses and magnitudes of the various functions
in the remaining three quadrants are given in an Appendix.
Washington, March, 1964
Norton Goodwin
COS too--
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
1.00 00000
.99 99998
99992
00000
99998
99991
00000
99997
99990
00000
99997
99990
00000
99996
99989
00000
99996
99988
99999
99995
99987
99999
99994
99986
99999
99994
99985
99998
99993
99983
99998
99992
99982
03
04
05
06
99982
99968
99951
99929
99981
99967
99949
99927
99980
99965
99947
99924
99979
99964
99945
99922
99977
99962
99942
99919
99976
99960
99940
99917
99974
99958
99938
99914
99973
99956
99936
99911
99971
99955
99934
99909
99970
99953
99931
99906
99968
99951
99929
99903
96
95
94
93
07
08
09
10
11
12
99903
99874
99840
99900
99870
99837
99898
99867
99833
99895
99864
99829
99892
99861
99826
99889
99857
99822
99886
99854
99818
99883
99851
99814
99880
99847
99810
99877
99844
99807
99874
99840
99803
92
91
90
89
88
87
99803
99761
99716
99799
99757
99711
99795
99752
99706
99791
99748
99701
99787
99743
99696
99782
99739
99692
99778
99734
99687
99774
99730
99682
99770
99725
99677
99765
99720
99672
99761
99716
99666
13
14
15
16
99666
99613
99556
99495
99661
99608
99550
99488
99656
99602
99544
99482
99651
99596
99538
99476
99646
99591
99532
99469
99640
99585
99526
99463
99635
99579
99520
99456
99630
99573
99513
99449
99624
99568
99507
99443
99619
99562
99501
99436
99613
99556
99495
99430
86
85
84
83
17
18
19
20
21
22
99430
99360
99287
99423
99353
99280
99416
99346
99272
99409
99339
99265
99402
99332
99257
99395
99324
99249
99389
99317
99242
99382
99310
99234
99375
99302
99226
99368
99295
99218
99360
99287
99210
82
81
80
79
78
77
99210
99130
99045
99203
99121
99036
99195
99113
99027
99187
99104
99018
99179
99096
99010
99170
99088
99001
99162
99079
98992
99154
99071
98983
99146
99062
98974
99138
99053
98965
99130
99045
98956
23
24
25
26
98956
98863
98766
98666
98947
98854
98756
98655
98938
98844
98747
98645
98928
98834
98737
98635
98919
98825
98727
98624
98910
98815
98716
98614
98901
98805
98706
98603
98891
98796
98696
98593
98882
98786
98686
98582
98872
98776
98676
98572
98863
98766
98666
98561
76
75
74
73
27
28
29
30
31
32
98561
98452
98340
98550
98441
98329
98540
98430
98317
98529
98419
98305
98518
98408
98294
98507
98397
98282
98496
98385
98271
98485
98374
98259
98475
98363
98247
98464
98351
98235
98452
98340
98224
72
71
70
98224
98103
97979
98212
98091
97966
98200
98079
97953
98188
98066
97941
98176
98054
97928
98164
98041
97915
98152
98029
97902
98140
98016
97889
98128
98004
97876
98115
97991
97863
98103
97979
97850
69
68
67
33
34
35
36
97850
97718
97582
97442
97837
97705
97568
97428
97824
97691
97554
97413
97811
97678
97540
97399
97798
97664
97526
97385
97785
97651
97512
97370
97772
97637
97498
97356
97758
97623
97484
97341
97745
97610
97470
97327
97732
97596
97456
97312
97718
97582
97442
97298
66
65
64
63
37
38
39
40
41
42
97298
97150
96998
97283
97135
96982
97269
97120
96967
97254
97105
96951
97239
97089
96936
97224
97074
96920
97209
97059
96905
97195
97044
96889
97180
97029
96873
97165
97013
96858
97150
96998
96842
62
61
60
96842
96682
96518
96826
96666
96502
96810
96650
96485
96794
96633
96468
96778
96617
96452
96762
96601
96435
96746
96584
96418
96730
96568
96401
96714
96551
96384
96698
96535
96367
96682
96518
96350
59
58
57
43
44
45
46
96350
96179
96003
95823
96333
96161
95985
95805
96316
96144
95967
95787
96299
96126
95950
95769
96282
96109
95932
95751
96265
96091
95914
95732
96248
96074
95896
95714
96231
96056
95878
95695
96213
96039
95860
95677
96196
96021
95842
95658
96179
96003
95823
95640
56
55
54
53
47
48
49
95640
95452
95261
95621
95433
95242
95603
95414
95222
95584
95395
95203
95565
95376
95183
95547
95357
95164
95528
95338
95144
95509
95319
95125
95490
95300
95105
95471
95280
95085
95452
95261
95066
52
51
50
10
9
8
7
6
5
4
3
2
1
SIN t24-
SIN tOO--
1
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.00 00000
06283
12566
00628
06912
13195
01257
07540
13823
01885
08168
14451
02513
08796
15080
03142
09425
15708
03770
10053
16336
04398
10681
16965
05027
11310
17593
05655
11938
18221
06283
12566
18850
03
04
05
06
18850
25133
31416
37699
19478
25761
32044
38327
20106
26389
32673
38956
20734
27018
33301
39584
21363
27646
33929
40212
21991
28274
34557
40841
22619
28903
35186
41469
23248
29531
35814
42097
23876
30159
36442
42726
24504
30788
37071
43354
25133
31416
37699
43982
96
95
94
93
07
08
09
10
11
12
43982
50265
56548
44610
50894
57177
45239
51522
57805
45867
52150
58433
46495
52779
59062
47124
53407
59690
47752
54035
60318
48380
54663
60947
49009
55292
61575
49637
55920
62203
50265
56548
62831
92
91
90
89
88
87
62831
69114
75398
63460
69743
76026
64088
70371
76654
64716
70999
77282
65345
71628
77911
65973
72256
78539
66601
72884
79167
67230
73513
79796
67858
74141
80424
68486
74769
81052
69114
75398
81681
13
14
15
16
81681
87963
94246
.01 00529
82309
88592
94875
01158
82937
89220
95503
01786
83565
89848
96131
02414
84194
90477
96760
03042
84822
91105
97388
03671
85450
91733
98016
04299
86079
92362
98644
04927
86707
92990
99273
05556
87335
93618
99901
06184
87963
94246
00529
06812
86
85
84
83
17
18
19
20
21
22
06812
13095
19378
07440
13723
20006
08069
14351
20634
08697
14980
21263
09325
15608
21891
09954
16236
22519
10582
16865
23147
11210
17493
23776
11838
18121
24404
12467
18749
25032
13095
19378
25660
82
81
80
79
78
77
25660
31943
38226
26289
32571
38854
26917
33200
39482
27545
33828
40110
28173
34456
40739
28802
35084
41367
29430
35713
41995
30058
36341
42623
30687
36969
43252
31315
37597
43880
31943
38226
44508
23
24
25
26
44508
50791
57073
63356
45136
51419
57701
63984
45765
52047
58330
64612
46393
52675
58958
65240
47021
53304
59586
65868
47649
53932
60214
66497
48278
54560
60843
67125
48906
55188
61471
67753
49534
55817
62099
68381
50162
56445
62727
69010
50791
57073
63356
69638
76
75
74
73
27
28
29
30
31
32
69638
75920
82202
70266
76548
82831
70894
77177
83459
71523
77805
84087
72151
78433
84715
72779
79061
85343
73407
79689
85972
74035
80318
86600
74664
80946
87228
75292
81574
87856
75920
82202
88484
72
71
70
69
68
67
88484
94766
.02 01048
89113
95395
01677
89741
96023
02305
90369
96651
02933
90997
97279
03561
91625
97907
04189
92254
98536
04818
92882
99164
05446
93510
99792
06074
94138
00420
06702
94766
01048
07330
33
34
35
36
07330
13612
19894
26175
07958
14240
20522
26804
08587
14868
21150
27432
09215
15497
21778
28060
09843
16125
22406
28688
10471
16753
23035
29316
11099
17381
23663
29944
11728
18009
24291
30572
12356
18637
24919
31201
12984
19266
25547
31829
13612
19894
26175
32457
66
65
64
63
37
38
39
40
41
42
32457
38738
45020
33085
39366
45648
33713
39995
46276
34341
40623
46904
34970
41251
47532
35598
41879
48160
36226
42507
48788
36854
43135
49417
37482
43763
50045
38110
44392
50673
38738
45020
51301
62
61
60
59
58
57
51301
57582
63863
51929
58210
64491
52557
58838
65119
53185
59466
65747
53813
60095
66376
54442
60723
67004
55070
61351
67632
55698
61979
68260
56326
62607
68888
56954
63235
69516
57582
63863
70144
43
44
45
46
70144
76425
82706
88986
70772
77053
83334
89614
71400
77681
83962
90242
72028
78309
84590
90870
72656
78937
85218
91499
73285
79565
85846
92127
73913
80193
86474
92755
74541
80821
87102
93383
75169
81450
87730
94011
75797
82078
88358
94639
76425
82706
88986
95267
56
55
54
53
47
48
49
95267
.03 01547
07827
95895
02175
08455
96523
02803
09083
97151
03431
09712
97779
04059
10340
98407
04687
10968
99035
05315
11596
99663
05943
12224
00291
06571
12852
00919
07199
13480
01547
07827
14108
52
51
50
10
9
8
7
6
5
4
3
2
1
COS t24-
COS tOO"
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.99 95066
94866
94663
95046
94846
94642
95026
94826
94622
95006
94806
94601
94986
94785
94581
94966
94765
94560
94946
94745
94539
94926
94724
94518
94906
94704
94498
94886
94683
94477
94866
94663
94456
53
54
55
56
94456
94245
94029
93810
94435
94223
94008
93788
94414
94202
93986
93766
94393
94180
93964
93744
94372
94159
93942
93722
94351
94138
93920
93699
94330
94116
93899
93677
94308
94094
93877
93655
94287
94073
93855
93632
94266
94051
93833
93610
94245
94029
93810
93587
46
45
44
43
57
58
59
60
61
62
93587
93360
93130
93565
93338
93106
93542
93315
93083
93520
93292
93060
93497
93269
93036
93474
93246
93013
93452
93222
92989
93429
93199
92966
93406
93176
92942
93383
93153
92918
93360
93130
92895
42
41
40
39
38
37
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30057
25715
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86
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83
17
18
19
20
21
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80
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77327
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76
75
74
73
27
28
29
30
31
32
72512
68133
63750
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63312
71637
67256
62873
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66818
62435
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61996
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61558
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72
71
70
69
68
67
59365
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50586
58926
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33
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66
65
64
63
37
38
39
40
41
42
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19770
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21094
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62
61
60
59
58
57
15357
10940
06521
14915
10499
06079
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10057
05637
14032
09615
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11382
06963
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10940
06521
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43
44
45
46
02099
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01657
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88374
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92361
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86601
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98560
94133
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85271
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93690
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56
55
54
53
47
48
49
84384
79948
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75065
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83053
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72845
81279
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72401
80835
76397
71956
80392
75953
71512
79948
75509
71068
52
51
50
10
9
8
7
6
5
4
3
2
1
SIN tl2-
SIN til-
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.66 13119
17830
22540
13590
18301
23010
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18772
23481
14532
19243
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15004
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53
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46
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57
58
59
60
61
62
46046
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50270
54962
59650
50740
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42
41
40
39
38
37
60119
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60587
65273
69955
61056
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70423
61525
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70892
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72296
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63
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65
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36
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34
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67
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32
31
30
29
28
27
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17102
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12912
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73
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26
25
24
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77
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80
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22
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18
17
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83
84
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16
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87
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12
11
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93
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06
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02
01
00
10
9
8
7
6
5
4
3
2
1
COS tl3--
COS tll-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
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84
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93
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96
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49
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43
42
41
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32
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30
29
28
27
26
25
24
23
22
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20
19
18
17
16
15
14
13
12
11
10
09
08
07
06
05
04
03
02
01
00
1
SIN tl3-
SIN til--
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.63 74240
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03
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96
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94
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07
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92
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13
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15
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86
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17
18
19
20
21
22
56176
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82
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98253
04909
01408
97903
04559
01057
97552
04209
00707
97201
03859
00356
96851
03509
00006
96500
03159
99656
96149
02809
99305
95798
02458
98955
95447
43
44
45
46
95447
91937
88423
84906
95096
91586
88072
84554
94746
91234
87720
84203
94395
90883
87369
83851
94044
90532
87017
83499
93693
90181
86665
83147
93342
89829
86313
82795
92990
89478
85962
82442
92639
89126
85610
82090
92288
88775
85258
81738
91937
88423
84906
81386
56
55
54
53
47
48
49
81386
77863
74336
81034
77510
73983
80682
77157
73630
80329
76805
73277
79977
76452
72924
79625
76100
72571
79272
75747
72218
78920
75394
71865
78568
75041
71512
78215
74689
71159
77863
74336
70806
52
51
50
10
9
8
7
6
5
4
3
2
1
SIN tl5-
SIN t08--
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.50 90414
95821
.51 01227
90955
96362
01767
91496
96903
02307
92037
97443
02848
92577
97984
03388
93118
98524
03928
93659
99065
04469
94199
99605
05009
94740
00146
05549
95281
00686
06089
95821
01227
06630
53
54
55
56
06630
12031
17430
22827
07170
12571
17970
23367
07710
13111
18510
23906
08250
13651
19049
24446
08790
14191
19589
24985
09331
14731
20129
25525
09871
15271
20669
26064
10411
15810
21208
26604
10951
16350
21748
27143
11491
16890
22287
27683
12031
17430
22827
28222
46
45
44
43
57
58
59
60
61
62
28222
33615
39006
28762
34154
39545
29301
34694
40084
29840
35233
40623
30380
35772
41162
30919
36311
41701
31458
36850
42240
31998
37389
42779
32537
37928
43318
33076
38467
43857
33615
39006
44395
42
41
40
39
38
37
44395
49782
55167
44934
50321
55706
45473
50859
56244
46012
51398
56782
46550
51937
57321
47089
52475
57859
47628
53014
58397
48166
53552
58936
48705
54090
59474
49244
54629
60012
49782
55167
60550
63
64
65
66
60550
65931
71310
76687
61088
66469
71848
77224
61627
67007
72385
77762
62165
67545
72923
78299
62703
68083
73461
78837
63241
68621
73999
79374
63779
69159
74536
79912
64317
69696
75074
80449
64855
70234
75612
80987
65393
70772
76149
81524
65931
71310
76687
82061
36
35
34
33
67
68
69
70
71
72
82061
87434
92805
82599
87971
93342
83136
88508
93879
83673
89046
94416
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89583
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95489
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90657
96026
85823
91194
96563
86360
91731
97100
86897
92268
97637
87434
92805
98173
32
31
30
29
28
27
98173
.52 03540
08904
98710
04077
09441
99247
04613
09977
99784
05150
10513
00320
05686
11050
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11586
01394
06759
12122
01930
07295
12658
02467
07832
13195
03003
08368
13731
03540
08904
14267
73
74
75
76
14267
19627
24986
30342
14803
20163
25521
30877
15339
20699
26057
31413
15875
21235
26593
31948
16411
21771
27128
32484
16947
22307
27664
33019
17483
22843
28200
33555
18019
23378
28735
34090
18555
23914
29271
34625
19091
24450
29806
35161
19627
24986
30342
35696
26
25
24
23
77
78
79
80
81
82
35696
41048
46398
36231
41583
46933
36767
42118
47468
37302
42653
48003
37837
43189
48538
38372
43724
49073
38908
44259
49607
39443
44794
50142
39978
45328
50677
40513
45863
51212
41048
46398
51746
22
21
20
19
18
17
51746
57092
62436
52281
57627
62970
52816
58161
63505
53350
58696
64039
53885
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64573
54420
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54954
60299
65641
55489
60833
66176
56023
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56558
61902
67244
57092
62436
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83
84
85
86
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73118
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83791
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73651
78989
84324
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84857
69380
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69914
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80590
85924
70448
75787
81123
86458
70982
76320
81657
86991
71516
76854
82190
87524
72050
77388
82724
88058
72584
77921
83257
88591
73118
78455
83791
89124
16
15
14
13
87
88
89
90
91
92
89124
94455
99785
89657
94988
00317
90191
95521
00850
90724
96054
01383
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96587
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91790
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02449
92323
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02981
92856
98186
03514
93389
98719
04047
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99252
04579
94455
99785
05112
12
11
10
09
08
07
.53 05112
10437
15760
05644
10969
16292
06177
11502
16824
06710
12034
17356
07242
12566
17888
07775
13099
18421
08307
13631
18953
08840
14163
19485
09372
14695
20017
09905
15228
20549
10437
15760
21081
93
94
95
96
21081
26400
31716
37031
21613
26931
32248
37562
22145
27463
32779
38093
22677
27995
33311
38625
23209
28526
33842
39156
23740
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34374
39687
24272
29590
34905
40219
24804
30121
35437
40750
25336
30653
35968
41281
25868
31185
36499
41812
26400
31716
37031
42343
06
05
04
03
97
98
99
42343
47654
52962
42874
48185
53493
43405
48715
54023
43937
49246
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44468
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55615
45530
50839
56146
46061
51370
56676
46592
51900
57207
47123
52431
57737
47654
52962
58268
02
01
00
10
9
8
7
6
5
4
3
2
1
COS tl6--
COS t08-
♦
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.86 07420
04220
01017
07100
03900
00696
06781
03580
00376
06461
03259
00055
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02939
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53
54
55
56
.85 97810
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94921
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84948
46
45
44
43
57
58
59
60
61
62
84948
81724
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81402
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42
41
40
39
38
37
75267
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63
64
65
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63933
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62959
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56462
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36
35
34
33
67
68
69
70
71
72
52557
49300
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48974
45712
51906
48648
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51581
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43754
49952
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43427
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43101
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42774
32
31
30
29
28
27
42774
39507
36235
42448
39180
35908
42121
38853
35581
41794
38526
35253
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34926
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34271
40487
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73
74
75
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26
25
24
23
77
78
79
80
81
82
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22
20
19
18
17
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03339
09615
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97050
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96719
93404
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16
15
14
13
87
88
89
90
91
92
86764
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80111
86432
83107
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83772
80444
77113
83439
80111
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12
11
10
09
08
07
76779
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93
94
95
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06
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97
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02
01
00
10
9
8
7
6
5
4
3
2
1
SIN tl6-
SIN t08-
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.48 17537
23042
28545
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03
04
05
06
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96
95
94
93
07
08
09
10
11
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92
91
90
89
88
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13
14
15
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86
85
84
83
17
18
19
20
21
22
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82
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80
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23
24
25
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4^128
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76
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27
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33
34
35
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66
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37
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40
41
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62
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52
51
50
10
9
8
7
6
5
4
3
2
1
COS tl6-
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.
1
2
3
4
5
6
7
8
9
10
99
98
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01
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03
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11
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34767
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92
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13
14
15
16
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11110
86
85
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17
18
19
20
21
22
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82
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23
24
25
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76
75
74
73
27
28
29
30
31
32
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73835
79770
76648
73523
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76336
73210
79146
76023
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77897
74773
71646
77585
74461
71333
77273
74148
71020
76961
73835
70707
72
71
70
69
68
67
70707
67575
64440
70394
67262
64126
70081
66948
63812
69768
66635
63499
69455
66321
63185
69141
66008
62871
68828
65694
62557
68515
65381
62243
68202
65067
61929
67888
64753
61615
67575
64440
61301
33
34
35
36
61301
58159
55013
51864
60987
57844
54699
51549
60673
57530
54384
51234
60359
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60045
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53754
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59730
56587
53439
50289
59416
56272
53124
49973
59102
55957
52809
49658
58788
55643
52494
49343
58473
55328
52179
49027
58159
55013
51864
48712
66
65
64
63
37
38
39
40
41
42
48712
45556
42397
48396
45240
42081
48081
44924
41765
47766
44609
41448
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44293
41132
47134
43977
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46819
43661
40500
46503
43345
40183
46188
43029
39867
45872
42713
39551
45556
42397
39234
62
61
60
59
58
57
39234
36068
32899
38918
35751
32582
38601
35434
32264
38285
35118
31947
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34801
31630
37652
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31313
37335
34167
30995
37018
33850
30678
36702
33533
30361
36385
33216
30043
36068
32899
29726
43
44
45
46
29726
26549
23370
20187
29408
26232
23052
19868
29091
25914
22733
19550
28773
25596
22415
19231
28456
25278
22097
18912
28138
24960
21779
18594
27820
24642
21460
18275
27503
24324
21142
17956
27185
24006
20824
17638
26867
23688
20505
17319
26549
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20187
17000
56
55
54
53
47
48
49
17000
13810
10617
16681
13491
10297
16362
13172
09978
16044
12853
09658
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14768
11575
08380
14449
11256
08060
14129
10936
07740
13810
10617
07420
52
51
50
10
9
8
7
6
5
4
3
2
1
SIN tl6-
SIN t07-
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.45 39905
45502
51098
40465
46062
51658
41025
46622
52217
41584
47181
52776
42144
47741
53336
42704
48301
53895
43264
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54455
43823
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55014
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49979
55573
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56133
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51098
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53
54
55
56
56692
62284
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73463
57251
62843
68433
74022
57811
63402
68992
74580
58370
63961
69551
75139
58929
64520
70110
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59488
65079
70669
76256
60047
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71228
76815
60607
66197
71786
77374
61166
66756
72345
77932
61725
67315
72904
78491
62284
67874
73463
79049
46
45
44
43
57
58
59
60
61
62
79049
84634
90217
79608
85193
90776
80167
85751
91334
80725
86309
91892
81284
86868
92450
81842
87426
93008
82401
87984
93566
82959
88543
94124
83518
89101
94683
84076
89659
95241
84634
90217
95799
42
41
40
39
38
37
95799
.46 01378
06956
96357
01936
07513
96915
02494
08071
97473
03051
08629
98031
03609
09186
98589
04167
09744
99146
04725
10301
99704
05283
10859
00262
05840
11416
00820
06398
11974
01378
06956
12531
63
64
65
66
12531
18105
23678
29248
13089
18663
24235
29805
13646
19220
24792
30362
14204
19777
25349
30919
14761
20334
25906
31475
15319
20892
26463
32032
15876
21449
27020
32589
16433
22006
27577
33146
16991
22563
28134
33703
17548
23120
28691
34260
18105
23678
29248
34816
36
35
34
33
67
68
69
70
71
72
34816
40383
45948
35373
40940
46504
35930
41496
47061
36486
42053
47617
37043
42609
48173
37600
43166
48730
38157
43722
49286
38713
44279
49842
39270
44835
50398
39826
45391
50955
40383
45948
51511
32
31
30
29
28
27
51511
57072
62631
52067
57628
63187
52623
58184
63743
53179
58740
64299
53735
59296
64854
54292
59852
65410
54848
60408
65966
55404
60964
66522
55960
61520
67077
56516
62075
67633
57072
62631
68189
73
74
75
76
68189
73744
79298
84850
68744
74300
79853
85405
69300
74855
80409
85960
69856
75411
80964
86515
70411
75966
81519
87070
70967
76521
82074
87625
71522
77077
82630
88180
72078
77632
83185
88735
72633
78188
83740
89290
73189
78743
84295
89845
73744
79298
84850
90400
26
25
24
23
77
78
79
80
81
82
90400
95948
.47 01495
90955
96503
02049
91510
97058
02604
92065
97613
03158
92620
98167
03713
93175
98722
04267
93729
99276
04822
94284
99831
05376
94839
00386
05931
95394
00940
06485
95948
01495
07039
22
21
20
19
18
17
07039
12582
18123
07594
13136
18677
08148
13690
19231
08702
14244
19785
09257
14799
20339
09811
15353
20893
10365
15907
21446
10919
16461
22000
11474
17015
22554
12028
17569
23108
12582
18123
23662
83
84
85
86
23662
29199
34734
40267
24216
29752
35287
40821
24769
30306
35841
41374
25323
30860
36394
41927
25877
31413
36948
42480
26431
31967
37501
43033
26984
32520
38054
43586
27538
33074
38608
44140
28092
33627
39161
44693
28645
34181
39714
45246
29199
34734
40267
45799
16
15
14
13
87
88
89
90
91
92
45799
51328
56856
46352
51881
57409
46905
52434
57962
47458
52987
58514
48011
53540
59067
48564
54093
59619
49117
54645
60172
49670
55198
60724
50223
55751
61277
50776
56304
61830
51328
56856
62382
12
11
10
09
08
07
62382
67906
73428
62935
68458
73980
63487
69011
74532
64039
69563
75084
64592
70115
75636
65144
70667
76188
65697
71219
76740
66249
71772
77292
66801
72324
77844
67354
72876
78396
67906
73428
78948
93
94
95
96
78948
84467
89983
95498
79500
85018
90535
96049
80052
85570
91086
96600
80604
86122
91638
97152
81156
86673
92189
97703
81708
87225
92741
98254
82260
87777
93292
98805
82811
88328
93843
99357
83363
88880
94395
99908
83915
89431
94946
00459
84467
89983
95498
01010
06
05
04
03
97
98
99
.48 01010
06521
12030
01561
07072
12581
02113
07623
13131
02664
08174
13682
03215
08725
14233
03766
09276
14784
04317
09826
15334
04868
10377
15885
05419
10928
16436
05970
11479
16986
06521
12030
17537
02
01
00
10
9
8
7
6
5
4
3
2
1
COS tl7-
COS t07-
»
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.89 10065
07211
04353
09780
06925
04067
09495
06640
03781
09209
06354
03495
08924
06068
03209
08639
05783
02923
08353
05497
02637
08068
05211
02351
07782
04925
02064
07497
04639
01778
07211
04353
01492
53
54
55
56
01492
,88 98627
95759
92887
01206
98340
95472
92600
00919
98054
95185
92312
00633
97767
94898
92025
00346
97480
94610
91737
00060
97193
94323
91450
99773
96907
94036
91162
99487
96620
93749
90875
99200
96333
93462
90587
98914
96046
93174
90299
98627
95759
92887
90012
46
45
44
43
57
58
59
60
61
62
90012
87133
84250
89724
86845
83962
89436
86557
83673
89148
86268
83385
88860
85980
83096
88573
85692
82808
88285
85404
82519
87997
85115
82231
87709
84827
81942
87421
84539
81653
87133
84250
81364
42
41
40
39
38
37
81364
78475
75582
81076
78186
75293
80787
77897
75003
80498
77608
74714
80209
77318
74424
79920
77029
74134
79631
76740
73845
79342
76450
73555
79053
76161
73265
78764
75872
72976
78475
75582
72686
63
64
65
66
72686
69786
66883
63976
72396
69496
66592
63685
72106
69206
66301
63394
71816
68915
66011
63103
71526
68625
65720
62812
71236
68335
65430
62521
70946
68044
65139
62230
70656
67754
64848
61939
70366
67464
64557
61648
70076
67173
64267
61356
69786
66883
63976
61065
36
35
34
33
67
68
69
70
71
72
61065
58151
55234
60774
57860
54942
60483
57568
54650
60191
57277
54358
59900
56985
54066
59609
56693
53774
59317
56401
53482
59026
56110
53190
58734
55818
52898
58443
55526
52605
58151
55234
52313
32
31
30
29
28
27
52313
49389
46461
52021
49096
46168
51729
48803
45875
51436
48511
45582
51144
48218
45289
50851
47925
44996
50559
47632
44702
50266
47340
44409
49974
47047
44116
49681
46754
43823
49389
46461
43530
73
74
75
76
43530
40595
37656
34714
43236
40301
37362
34420
42943
40007
37068
34126
42649
39714
36774
33831
42356
39420
36480
33537
42063
39126
36186
33242
41769
38832
35892
32948
41475
38538
35597
32653
41182
38244
35303
32358
40888
37950
35009
32064
40595
37656
34714
31769
26
25
24
23
77
78
79
80
81
82
31769
28820
25868
31474
28525
25573
31180
28230
25277
30885
27935
24982
30590
27640
24686
30295
27345
24391
30000
27049
24095
29705
26754
23799
29410
26459
23504
29115
26163
23208
28820
25868
22912
22
21
20
19
18
17
22912
19953
16990
22616
19657
16694
22321
19361
16397
22025
19065
16101
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18768
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18472
15508
21137
18176
15211
20841
17879
14914
20545
17583
14618
20249
17287
14321
19953
16990
14024
83
84
85
86
14024
11054
08081
05104
13727
10757
07784
04807
13430
10460
07486
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10163
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11946
08973
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03019
11649
08676
05700
02721
11351
08379
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02423
11054
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02124
16
15
14
13
87
88
89
90
91
92
02124
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96154
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98842
95855
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98544
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97946
94958
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00335
97349
94360
00036
97050
94061
99738
96751
93761
99439
96453
93462
99141
96154
93163
]1
11
10
09
08
07
93163
90169
87172
92864
89869
86872
92565
89570
86572
92265
89270
86272
91966
88970
85972
91667
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85672
91367
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85371
91068
88071
85071
90768
87771
84771
90469
87471
84471
90169
87172
84171
93
94
95
96
84171
81166
78158
75147
83870
80866
77857
74846
83570
80565
77556
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83270
80264
77255
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82969
79963
76954
73941
82669
79663
76653
73640
82368
79362
76352
73338
82068
79061
76051
73037
81767
78760
75749
72735
81467
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72434
81166
78158
75147
72132
06
05
04
03
97
98
99
72132
69114
66092
71830
68812
65790
71529
68510
65487
71227
68208
65185
70925
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67603
64580
70322
67301
64277
70020
66999
63975
69718
66697
63672
69416
66394
63369
69114
66092
63067
02
01
00
10
9
8
7
6
5
4
3
2
1
SIN tl7-
SIN t07--
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.42 57793
63477
69160
58361
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59990
46
45
44
43
57
58
59
60
61
62
59990
57468
54941
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57215
54689
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56963
54436
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56710
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53424
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53171
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52918
57720
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52665
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42
41
40
39
38
37
52412
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52159
49625
47088
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46834
51652
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46580
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48864
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63
64
65
66
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42002
39454
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41747
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39964
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36
35
34
33
67
68
69
70
71
72
34603
32044
29482
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31788
29225
34091
31532
28969
33835
31276
28712
33579
31019
28456
33324
30763
28199
33068
30507
27943
32812
30251
27686
32556
29994
27429
32300
29738
27173
32044
29482
26916
32
31
30
29
28
27
26916
24346
21773
26659
24089
21516
26402
23832
21258
26145
23575
21001
25889
23318
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25632
23060
20486
25375
22803
20228
25118
22546
19970
24861
22288
19712
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22031
19455
24346
21773
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73
74
75
76
19197
16617
14033
11445
18939
16358
13774
11186
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16100
13516
10927
18423
15842
13257
10668
18165
15583
12998
10409
17907
15325
12740
10150
17649
15067
12481
09891
17391
14808
12222
09632
17133
14550
11963
09373
16875
14291
11704
09114
16617
14033
11445
08854
26
25
24
23
77
78
79
80
81
82
08854
06260
03662
08595
06000
03401
08336
05740
03141
08076
05481
02881
07817
05221
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07557
04961
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07298
04701
02101
07038
04441
01841
06779
04181
01580
06519
03921
01320
06260
03662
01060
22
21
20
19
18
17
01060
•90 98454
95845
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95584
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97933
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00278
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95062
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98976
96367
93756
98715
96106
93494
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95845
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83
84
85
86
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90617
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86161
83535
91140
88521
85898
83272
90878
88259
85636
83009
90617
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85373
82747
16
15
14
13
87
88
89
90
91
92
82747
80116
77482
82484
79853
77218
82221
79589
76955
81958
79326
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81169
78536
75900
80906
78273
75636
80642
78009
75372
80379
77746
75108
80116
77482
74844
12
11
10
09
08
07
74844
72203
69558
74580
71939
69293
74316
71674
69029
74052
71410
68764
73788
71145
68499
73524
70881
68234
73260
70616
67970
72996
70352
67705
72732
70087
67440
72467
69823
67175
72203
69558
66910
93
94
95
96
66910
64258
61602
58943
66645
63992
61336
58677
66380
63727
61071
58411
66114
63461
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60539
57878
65584
62930
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57612
65319
62665
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62399
59741
57079
64788
62134
59475
56813
64523
61868
59209
56547
64258
61602
58943
56280
06
05
04
03
97
98
99
56280
53614
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53347
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49340
54414
51745
49073
54147
51478
48806
53881
51211
48538
53614
50944
48271
02
01
00
10
9
8
7
6
5
4
3
2
1
SIN tl8--
SIN t06--
1
2
3
4
5
6
7
8
9
10
■
00
01
02
.36 81246
87087
92927
81830
87671
93510
82414
88255
94094
82998
88839
94678
83582
89423
95262
84166
90007
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91175
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85919
91759
97597
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92343
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87087
92927
98765
99
98
97
03
04
05
06
98765
.37 04602
10437
16271
99349
05185
11021
16854
99932
05769
11604
17438
00516
06352
12187
18021
01100
06936
12771
18604
01683
07520
13354
19187
02267
08103
13938
19771
02851
08687
14521
20354
03434
09270
15104
20937
04018
09854
15688
21520
04602
10437
16271
22103
96
95
94
93
07
08
09
10
11
12
22103
27934
33764
22687
28518
34347
23270
29101
34930
23853
29683
35513
24436
30266
36095
25019
30849
36678
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31432
37261
26185
32015
37844
26768
32598
38427
27351
33181
39009
27934
33764
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92
91
90
89
88
87
39592
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51244
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46001
51826
40757
46584
52409
41340
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52991
41923
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53573
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54156
43088
48914
54738
43671
49496
55320
44253
50079
55903
44836
50661
56485
45419
51244
57067
13
14
15
16
57067
62889
68710
74529
57650
63472
69292
75111
58232
64054
69874
75693
58814
64636
70456
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59979
65800
71620
77438
60561
66382
72202
78020
61143
66964
72784
78602
61725
67546
73366
79184
62307
68128
73947
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62889
68710
74529
80347
86
85
84
83
17
18
19
20
21
22
80347
86163
91978
80929
86745
92559
81510
87326
93141
82092
87908
93722
82674
88489
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83837
89652
95466
84418
90234
96047
85000
90815
96628
85582
91396
97210
86163
91978
97791
82
81
80
79
78
77
97791
.38 03603
09413
98372
04184
09994
98953
04765
10575
99535
05346
11156
00116
05927
11736
00697
06508
12317
01278
07089
12898
01859
07670
13479
02440
08251
14060
03022
08832
14641
03603
09413
15221
23
24
25
26
15221
21029
26834
32638
15802
21609
27415
33219
16383
22190
27995
33799
16964
22771
28576
34379
17545
23351
29156
34960
18125
23932
29737
35540
18706
24512
30317
36120
19287
25093
30897
36700
19867
25673
31478
37281
20448
26254
32058
37861
21029
26834
32638
38441
76
75
74
73
27
28
29
30
31
32
38441
44242
50042
39021
44822
50622
39601
45402
51202
40182
45982
51781
40762
46562
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41342
47142
52941
41922
47722
53521
42502
48302
54101
43082
48882
54680
43662
49462
55260
44242
50042
55840
72
71
70
69
68
67
55840
61636
67432
56420
62216
68011
56999
62796
68590
57579
63375
69170
58159
63955
69749
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64534
70328
59318
65114
70908
59898
65693
71487
60477
66273
72066
61057
66852
72646
61636
67432
73225
33
34
35
36
73225
79017
84807
90596
73804
79596
85386
91175
74384
80175
85965
91754
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80754
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92333
75542
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93490
76700
82491
88281
94069
77280
83070
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94648
77859
83649
89439
95226
78438
84228
90018
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79017
84807
90596
96384
66
65
64
63
37
38
39
40
41
42
96384
.39 02170
07954
96962
02748
08532
97541
03327
09111
98120
03905
09689
98698
04484
10267
99277
05062
10845
99855
05640
11424
00434
06219
12002
01013
06797
12580
01591
07376
13158
02170
07954
13737
62
61
60
59
58
57
13737
19518
25298
14315
20096
25875
14893
20674
26453
15471
21252
27031
16049
21830
27609
16627
22408
28187
17206
22986
28765
17784
23564
29342
18362
24142
29920
18940
24720
30498
19518
25298
31076
43
44
45
46
31076
36852
42627
48401
31653
37430
43205
48978
32231
38007
43782
49555
32809
38585
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39162
44937
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51864
35119
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52441
35697
41472
47246
53018
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53595
36852
42627
48401
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56
55
54
53
47
48
49
54173
59943
65712
54750
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66288
55327
61097
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61674
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56481
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69172
58212
63981
69749
58789
64558
70326
59366
65135
70902
59943
65712
71479
52
51
50
10
9
8
7
6
5
4
3
2
1
COS tl8-
COS t06-
00
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
,92 97765
95450
93132
90809
88484
86154
83821
81484
79144
76799
1
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81250
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78675
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87785
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83120
80782
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76095
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72100
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65024
62658
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57914
55537
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74217
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67150
64787
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57677
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52918
73982
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66914
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36148
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11881
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06984
04530
26243
23818
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18956
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14080
11637
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06739
04285
26000
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13836
11392
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04039
02318
.91 99858
97393
94925
92453
89978
87499
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82530
80040
10
02073
99611
97146
94678
92206
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8
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66442
64078
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56964
54585
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45035
42638
40237
37833
35425
33014
30599
28180
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16032
13592
11148
08700
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84022
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82653
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34944
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18226
15788
13348
10903
08455
06003
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25273
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13103
10658
08210
05758
03302
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63368
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58627
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12615
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90721
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00350
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90473
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85513
83027
80538
78045
00104
97640
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92700
90225
87747
85264
82779
80289
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1
10
95450
93132
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83821
81484
79144
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62658
60288
57914
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14568
12126
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07229
04776
02318
99
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
70
99858
97393
94925
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89978
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85016
82530
80040
77546
69
68
67
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
SIN tl8--
SIN t05-
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.33 87379
93290
99200
87970
93881
99791
88562
94472
00382
89153
95063
00973
89744
95654
01563
90335
96245
02154
90926
96836
02745
91517
97427
03336
92108
98018
03927
92699
98609
04518
93290
99200
05108
53
54
55
56
•34 05108
11015
16921
22825
05699
11606
17512
23416
06290
12197
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06881
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19283
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14559
20464
26367
09243
15150
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26958
09834
15740
21645
27548
10425
16331
22235
28138
11015
16921
22825
28728
46
45
44
43
57
58
59
60
61
62
28728
34630
40530
29319
35220
41120
29909
35810
41710
30499
36400
42300
31089
36990
42890
31679
37580
43480
32270
38170
44070
32860
38760
44660
33450
39350
45250
34040
39940
45839
34630
40530
46429
42
41
40
39
38
37
46429
52327
58223
47019
52916
58813
47609
53506
59402
48199
54096
59992
48788
54685
60581
49378
55275
61171
49968
55865
61760
50558
56454
62350
51147
57044
62939
51737
57633
63528
52327
58223
64118
63
64
65
66
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76013
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82
81
80
79
78
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23
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25
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63293
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76
75
74
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27
28
29
30
31
32
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33
34
35
36
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39900
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66
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30114
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62
61
60
59
58
57
29905
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29278
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25086
29069
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24876
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43
44
45
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23615
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17291
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21300
19192
17080
23195
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16869
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56
55
54
53
47
48
49
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10297
14329
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52
51
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8
7
6
5
4
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2
1
SIN tl9--
SIN t04--
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.27 89911
95944
.28 01976
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53
54
55
56
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08610
14640
20669
26696
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09816
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30312
12831
18860
24888
30915
13434
19463
25491
31517
14037
20066
26093
32120
46
45
44
43
57
58
59
60
61
62
32120
38145
44169
32722
38748
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33325
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33928
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42
41
40
39
38
37
50193
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51397
57419
63440
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59827
65847
54408
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66449
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61633
67653
56215
62236
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63
64
65
66
68255
74274
80291
86308
68857
74876
80893
86909
69459
75477
81495
87511
70061
76079
82096
88112
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71265
77283
83300
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71867
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91721
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36
35
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67
68
69
70
71
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92323
98337
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32
31
30
29
28
27
10362
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241B4
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33196
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26
25
24
23
77
78
79
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54213
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16
15
14
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87
88
89
90
91
92
12384
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27958
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29155
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29754
18375
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12
11
10
09
08
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93
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02
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10
9
8
7
6
5
4
3
2
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1
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5
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7
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46
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59
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42
41
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39
38
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63
64
65
66
79828
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36
35
34
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67
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69
70
71
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67486
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32
31
30
29
28
27
67121
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65107
63272
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63088
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73
74
75
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22
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20
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18
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16
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87
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89
90
91
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08
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93
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01
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10
9
8
7
6
5
4
3
2
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SIN t20-
SIN t04--
1
2
3
4
5
6
7
8
9
10
99
98
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01
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92
91
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79
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23
24
25
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30245
36306
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49032
31457
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43579
49638
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44185
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32669
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76
75
74
73
27
28
29
30
31
32
50849
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55090
61147
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56302
62358
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62964
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72
71
70
69
68
67
69020
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69625
75680
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76891
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71442
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84155
72653
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84760
73258
79312
85365
73864
79918
85970
74469
80523
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75075
81128
87181
33
34
35
36
87181
93232
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87786
93838
99888
05937
88391
94443
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06542
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95048
01098
07147
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08962
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09566
92022
98073
04123
10171
92627
98678
04727
10776
93232
99283
05332
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66
65
64
63
37
38
39
40
41
42
11381
17428
23474
11985
18033
24079
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24683
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15009
21056
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15614
21660
27706
16219
22265
28310
16823
22870
28915
17428
23474
29519
62
61
60
59
58
57
29519
35563
41606
30124
36168
42211
30728
36772
42815
31333
37376
43419
31937
37981
44023
32542
38585
44627
33146
39189
45232
33750
39794
45836
34355
40398
46440
34959
41002
47044
35563
41606
47648
43
44
45
46
47648
53689
59729
65767
48252
54293
60333
66371
48857
54897
60937
66975
49461
55501
61541
67579
50065
56105
62144
68183
50669
56709
62748
68786
51273
57313
63352
69390
51877
57917
63956
69994
52481
58521
64560
70598
53085
59125
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53689
59729
65767
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56
55
54
53
47
48
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71805
77841
83877
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83273
89308
77841
83877
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52
51
50
10
9
8
7
6
5
4
3
2
1
COS t20--
COS t04-
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.96 85832
84267
82699
85675
84110
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81756
84737
83170
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83013
81441
84424
82856
81284
84267
82699
81127
03
04
05
06
81127
79551
77971
76387
80969
79393
77813
76229
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79235
77655
76070
80654
79077
77496
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80497
78919
77338
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78761
77180
75594
80182
78603
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75435
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78445
76863
75277
79866
78287
76704
75118
79709
78129
76546
74959
79551
77971
76387
74800
96
95
94
93
07
08
09
10
n
12
74800
73209
71614
74641
73049
71454
74482
72890
71294
74323
72731
71134
74164
72571
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72412
70815
73846
72252
70655
73687
72093
70495
73527
71933
70335
73368
71773
70175
73209
71614
70015
92
91
90
89
88
87
70015
68412
66806
69855
68252
66645
69695
68091
66484
69534
67931
66323
69374
67770
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69214
67609
66001
69054
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65840
68893
67288
65679
68733
67127
65518
68573
66967
65357
68412
66806
65195
13
14
15
16
65195
63581
61963
60342
65034
63420
61801
60179
64873
63258
61639
60017
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63096
61477
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86
85
84
83
17
18
19
20
21
22
58716
57087
55453
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82
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78
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53816
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23
24
25
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76
75
74
73
27
28
29
30
31
32
42251
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72
71
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69
68
67
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33
34
35
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32189
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28635
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30668
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66
65
64
63
37
38
39
40
41
42
25405
23699
21990
25234
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22161
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23699
21990
20277
62
61
60
59
58
57
20277
18560
16839
20105
18388
16667
19934
18216
16495
19762
18044
16322
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17872
16150
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15977
19247
17528
15805
19075
17356
15632
18903
17184
15460
18732
17011
15287
18560
16839
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43
44
45
46
15115
13386
11654
09918
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11481
09745
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09571
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55
54
53
47
48
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04688
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52
51
50
10
9
8
7
6
5
4
3
2
1
SIN t20-
SIN t03--
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.21 81432
87564
93694
82046
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94307
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53
54
55
56
99824
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10243
16370
22496
04727
10855
16983
23109
05340
11468
17595
23722
05953
12081
18208
24334
46
45
44
43
57
58
59
60
61
62
24334
30460
36584
24947
31072
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25559
31685
37809
26172
32297
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26784
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42
41
40
39
38
37
42708
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63
64
65
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36
35
34
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67
68
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32
31
30
29
28
27
.23 03894
10008
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26
25
24
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77
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81
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22
21
20
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18
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84
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16
15
14
13
87
88
89
90
91
92
07702
13799
19896
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11
10
09
08
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06
05
04
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98
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68637
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02
01
00
10
9
a
7
6
5
4
3
2
1
COS t21-
COS t03-
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.97 59168
57795
56419
59031
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56281
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57520
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53
54
55
56
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46
45
44
43
57
58
59
60
61
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48079
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46535
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41
40
39
38
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36
35
34
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67
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69
70
71
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32
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28
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73
74
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26
25
24
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77
78
79
80
81
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14233
12740
15574
14084
12590
15425
13935
12441
15276
13786
12291
15127
13636
12142
14978
13487
11992
14829
13338
11842
83
84
85
86
11842
10343
08840
07333
11692
10193
08689
07182
11543
10042
08538
07031
11393
09892
08388
06880
11243
09742
08237
06729
11093
09592
08087
06578
10943
09441
07936
06426
10793
09291
07785
06275
10643
09141
07634
06124
10493
08990
07483
05973
10343
08840
07333
05822
16
15
14
13
87
88
89
90
91
92
05822
04307
02788
05670
04155
02636
05519
04003
02484
05368
03852
02332
05216
03700
02180
05065
03548
02028
04913
03396
01875
04762
03244
01723
04610
03092
01571
04459
02940
01418
04307
02788
01266
12
11
10
09
08
07
01266
,96 99740
98210
01114
99587
98056
00961
99434
97903
00809
99281
97750
00656
99128
97597
00503
98975
97443
00351
98822
97290
00198
98669
97136
00045
98516
96983
99893
98363
96829
99740
98210
96676
93
94
95
96
96676
95138
93597
92051
96522
94984
93442
91897
96369
94830
93288
91742
96215
94676
93133
91587
96061
94522
92979
91432
95907
94368
92824
91277
95754
94214
92670
91122
95600
94059
92515
90967
95446
93905
92361
90812
95292
93751
92206
90657
95138
93597
92051
90502
06
05
04
03
97
98
99
90502
88949
87392
90347
88794
87236
90192
88638
87080
90037
88482
86924
89881
88327
86768
89726
88171
86612
89571
88015
86456
89415
87860
86300
89260
87704
86144
89105
87548
85988
88949
87392
85832
02
01
00
10
9
8
7
6
5
4
3
2
1
SIN t21-
SIN t03--
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.18 73813
79985
86155
74430
80602
86772
75048
81219
87390
75665
81836
88007
76282
82453
88624
76899
83070
89241
77516
83687
89858
78133
84304
90475
78750
84921
91092
79368
85538
91709
79985
86155
92325
03
04
05
06
92325
98495
.19 04663
10831
92942
99112
05280
11448
93559
99729
05897
12065
94176
00345
06514
12681
94793
00962
07131
13298
95410
01579
07747
13915
96027
02196
08364
14531
96644
02813
08981
15148
97261
03430
09598
15765
97878
04046
10214
16381
98495
04663
10831
16998
96
95
94
93
07
08
09
10
11
12
16998
23164
29330
17615
23781
29946
18231
24398
30563
18848
25014
31179
19465
25631
31796
20081
26247
32412
20698
26864
33029
21315
27480
33645
21931
28097
34262
22548
28713
34878
23164
29330
35495
92
91
90
89
88
87
35495
41659
47822
36111
42275
48438
36728
42891
49054
37344
43508
49671
37960
44124
50287
38577
44740
50903
39193
45357
51519
39810
45973
52136
40426
46589
52752
41042
47206
53368
41659
47822
53984
13
14
15
16
53984
60146
66307
72467
54601
60762
66923
73083
55217
61378
67539
73699
55833
61994
68155
74315
56449
62610
68771
74931
57065
63227
69387
75547
57681
63843
70003
76163
58298
64459
70619
76779
58914
65075
71235
77395
59530
65691
71851
78011
60146
66307
72467
78626
86
85
84
83
17
18
19
20
21
22
78626
84785
90943
79242
85401
91559
79858
86017
92174
80474
86632
92790
81090
87248
93406
81706
87864
94021
82322
88480
94637
82938
89096
95253
83553
89711
95868
84169
90327
96484
84785
90943
97100
82
81
80
79
78
77
97100
.20 03256
09411
97715
03872
10027
98331
04487
10642
98947
05103
11258
99562
05718
11873
00178
06334
12489
00794
06949
13104
01409
07565
13720
02025
08180
14335
02640
08796
14951
03256
09411
15566
23
24
25
26
15566
21720
27873
34025
16181
22335
28488
34640
16797
22951
29103
35256
17412
23566
29719
35871
18028
24181
30334
36486
18643
24797
30949
37101
19258
25412
31564
37716
19874
26027
32180
38331
20489
26642
32795
38946
21105
27258
33410
39562
21720
27873
34025
40177
76
75
74
73
27
28
29
30
31
32
40177
46327
52477
40792
46942
53092
41407
47557
53707
42022
48172
54322
42637
48787
54937
43252
49402
55552
43867
50017
56167
44482
50632
56781
45097
51247
57396
45712
51862
58011
46327
52477
58626
72
71
70
69
68
67
58626
64774
70922
59241
65389
71536
59856
66004
72151
60471
66619
72766
61085
67233
73380
61700
67848
73995
62315
68463
74610
62930
69078
75224
63545
69692
75839
64160
70307
76454
64774
70922
77068
33
34
35
36
77068
83214
89359
95503
77683
83829
89973
96117
78297
84443
90588
96732
78912
85058
91202
97346
79527
85672
91817
97960
80141
86287
92431
98575
80756
86901
93045
99189
81370
87516
93660
99803
81985
88130
94274
00418
82599
88744
94889
01032
83214
89359
95503
01646
66
65
64
63
37
38
39
40
41
42
.21 01646
07789
13930
02261
08403
14544
02875
09017
15159
03489
09631
15773
04103
10245
16387
04718
10860
17001
05332
11474
17615
05946
12088
18229
06560
12702
18843
07175
13316
19457
07789
13930
20071
62
61
60
59
58
57
20071
26211
32350
20685
26825
32964
21299
27439
33578
21913
28053
34192
22527
28667
34806
23141
29281
35419
23755
29895
36033
24369
30508
36647
24983
31122
37261
25597
31736
37875
26211
32350
38488
43
44
45
46
38488
44626
50762
56898
39102
45239
51376
57512
39716
45853
51990
58125
40330
46467
52603
58739
40943
47081
53217
59352
41557
47694
53830
59966
42171
48308
54444
60579
42785
48921
55057
61193
43398
49535
55671
61806
44012
50149
56285
62420
44626
50762
56898
63033
56
55
54
53
47
48
49
63033
69167
75300
63646
69780
75913
64260
70394
76527
64873
71007
77140
65487
71620
77753
66100
72234
78366
66713
72847
78980
67327
73460
79593
67940
74074
80206
68554
74687
80819
69167
75300
81432
52
51
50
10
9
8
7
6
5
4
3
2
1
COS t21-
COS t03-
*
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
02
.98 22873
21693
20510
22755
21575
20392
22637
21457
20273
22519
21339
20154
22401
21220
20036
22283
21102
19917
22165
20984
19798
22047
20865
19680
21929
20747
19561
21811
20629
19442
21693
20510
19323
03
04
05
06
19323
18132
16937
15739
19204
18013
16818
15619
19085
17893
16698
15498
18966
17774
16578
15378
18847
17655
16458
15258
18728
17535
16338
15138
18609
17416
16219
15018
18490
17296
16099
14897
18371
17177
15979
14777
18251
17057
15859
14656
18132
16937
15739
14536
96
95
94
93
07
08
09
10
11
12
14536
13330
12119
14416
13209
11998
14295
13088
11877
14175
12967
11756
14054
12846
11634
13933
12725
11513
13813
12604
11391
13692
12483
11270
13571
12362
11148
13450
12241
11027
13330
12119
10905
92
91
90
89
88
87
10905
09687
08465
10784
09565
08343
10662
09443
08220
10540
09321
08098
10418
09199
07975
10297
09077
07853
10175
08954
07730
10053
08832
07608
09931
08710
07485
09809
08588
07362
09687
08465
07239
13
14
15
16
07239
06010
04776
03539
07117
05887
04653
03415
06994
05763
04529
03291
06871
05640
04405
03167
06748
05517
04282
03043
06625
05393
04158
02919
06502
05270
04034
02795
06379
05147
03910
02670
06256
05023
03787
02546
06133
04900
03663
02422
06010
04776
03539
02298
86
85
84
83
17
18
19
20
21
22
02298
01052
.97 99803
02173
00928
99678
02049
00803
99553
01924
00678
99428
01800
00553
99303
01675
00428
99177
01551
00303
99052
01426
00179
98927
01302
00054
98801
01177
99928
98676
01052
99803
98551
82
81
80
79
78
77
98551
97294
96033
98425
97168
95907
98299
97042
95781
98174
96916
95654
98048
96790
95528
97923
96664
95401
97797
96538
95275
97671
96412
95148
97545
96286
95022
97420
96159
94895
97294
96033
94769
23
24
25
26
94769
93500
92228
90952
94642
93373
92101
90824
94515
93246
91973
90696
94389
93119
91846
90568
94262
92992
91718
90441
94135
92865
91591
90313
94008
92737
91463
90185
93881
92610
91335
90056
93754
92483
91208
89928
93627
92356
91080
89800
93500
92228
90952
89672
76
75
74
73
27
28
29
30
31
32
89672
88388
87101
89544
88260
86972
89416
88131
86843
89287
88002
86714
89159
87874
86584
89031
87745
86455
88902
87616
86326
88774
87487
86197
88645
87358
86068
88517
87230
85938
88388
87101
85809
72
71
70
69
68
67
85809
84514
83214
85680
84384
83084
85550
84254
82954
85421
84124
82824
85291
83994
82694
85162
83864
82563
85032
83735
82433
84903
83605
82303
84773
83475
82172
84643
83344
82042
84514
83214
81911
33
34
35
36
81911
80604
79293
77979
81781
80473
79162
77847
81650
80342
79031
77715
81520
80211
78899
77584
81389
80080
78768
77452
81258
79949
78637
77320
81128
79818
78505
77188
80997
79687
78374
77056
80866
79556
78242
76924
80735
79425
78110
76792
80604
79293
77979
76660
66
65
64
63
37
38
39
40
41
42
76660
75338
74011
76528
75205
73879
76396
75073
73746
76264
74940
73613
76132
74808
73480
75999
74675
73347
75867
74542
73214
75735
74410
73081
75602
74277
72948
75470
74144
72814
75338
74011
72681
62
61
60
59
58
57
72681
71347
70009
72548
71214
69875
72415
71080
69741
72281
70946
69607
72148
70813
69473
72015
70679
69339
71881
70545
69205
71748
70411
69071
71614
70277
68936
71481
70143
68802
71347
70009
68668
43
44
45
46
68668
67322
65973
64619
68533
67187
65837
64484
68399
67052
65702
64348
68264
66918
65567
64213
68130
66783
65432
64077
67995
66648
65296
63941
67861
66513
65161
63806
67726
66378
65026
63670
67591
66243
64890
63534
67457
66108
64755
63398
67322
65973
64619
63262
56
55
54
53
47
48
49
63262
61901
60536
63126
61765
60400
62990
61629
60263
62854
61492
60126
62718
61356
59989
62582
61219
59852
62446
61083
59716
62310
60946
59579
62174
60810
59442
62037
60673
59305
61901
60536
59168
52
51
50
10
9
8
7
6
5
4
3
2
1
SIN t21--
SIN t02--
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.15 64345
70550
76755
64965
71171
77376
65586
71791
77996
66206
72412
78616
66827
73032
79237
67447
73653
79857
68068
74273
80478
68689
74894
81098
69309
75514
81719
69930
76135
82339
70550
76755
82959
53
54
55
56
82959
89163
95366
.16 01568
83580
89783
95986
02189
84200
90404
96607
02809
84821
91024
97227
03429
85441
91644
97847
04049
86061
92265
98467
04669
86682
92885
99088
05290
87302
93505
99708
05910
87922
94125
00328
06530
88543
94746
00948
07150
89163
95366
01568
07770
46
45
44
43
57
58
59
60
61
62
07770
13971
20172
08390
14591
20792
09010
15211
21412
09631
15832
22032
10251
16452
22652
10871
17072
23272
11491
17692
23892
12111
18312
24512
12731
18932
25132
13351
19552
25752
13971
20172
26372
42
41
40
39
38
37
26372
32571
38769
26992
33191
39389
27612
33811
40009
28231
34430
40629
28851
35050
41249
29471
35670
41868
30091
36290
42488
30711
36910
43108
31331
37530
43728
31951
38150
44348
32571
38769
44967
63
64
65
66
44967
51165
57361
63557
45587
51784
57981
64177
46207
52404
58600
64796
46827
53024
59220
65416
47446
53643
59840
66035
48066
54263
60459
66655
48686
54883
61079
67274
49305
55502
61698
67894
49925
56122
62318
68513
50545
56742
62938
69133
51165
57361
63557
69752
36
35
34
33
67
68
69
70
71
72
69752
75947
82141
70372
76567
82760
70991
77186
83380
71611
77805
83999
72230
78425
84619
72850
79044
85238
73469
79664
85857
74089
80283
86477
74708
80902
87096
75328
81522
87715
75947
82141
88334
32
31
30
29
28
27
88334
94527
.17 00719
88954
95146
01338
89573
95766
01957
90192
96385
02577
90812
97004
03196
91431
97623
03815
92050
98242
04434
92669
98862
05053
93289
99481
05672
93908
00100
06291
94527
00719
06910
73
74
75
76
06910
13101
19291
25480
07529
13720
19910
26099
08149
14339
20529
26718
08768
14958
21148
27337
09387
15577
21767
27956
10006
16196
22386
28575
10625
16815
23005
29194
11244
17434
23624
29812
11863
18053
24242
30431
12482
18672
24861
31050
13101
19291
25480
31669
26
25
24
23
77
78
79
80
81
82
31669
37857
44044
32288
38476
44663
32907
39094
45281
33525
39713
45900
34144
40332
46519
34763
40951
47137
35382
41569
47756
36001
42188
48375
36619
42807
48993
37238
43425
49612
37857
44044
50231
22
21
20
19
18
17
50231
56416
62602
50849
57035
63220
51468
57654
63839
52086
58272
64457
52705
58891
65075
53324
59509
65694
53942
60128
66312
54561
60746
66931
55179
61365
67549
55798
61983
68168
56416
62602
68786
83
84
85
86
68786
74970
81153
87335
69404
75588
81771
87953
70023
76207
82389
88572
70641
76825
83008
89190
71260
77443
83626
89808
71878
78061
84244
90426
72496
78680
84862
91044
73115
79298
85481
91662
73733
79916
86099
92281
74351
80535
86717
92899
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16383
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06
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67024
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02
01
00
10
9
8
7
6
5
4
3
2
1
COS t22-
COS t02-
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.98 76883
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46
45
44
43
57
58
59
60
61
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67676
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42
41
40
39
38
37
66859
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63
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36
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67
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32
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26
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22
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20
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18
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16
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12
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02
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6
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3
2
1
SIN t22--
SIN t02-
1
2
3
4
5
6
7
8
9
10
99
98
97
00
01
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.12 53332
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6
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4
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COS t22-
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1
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3
4
5
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9
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99
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76
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29
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66
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37
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62
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84717
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85572
84622
83668
43
44
45
46
83668
82711
81749
80784
83573
82615
81653
80687
83477
82519
81556
80590
83381
82423
81460
80493
83286
82327
81364
80397
83190
82230
81267
80300
83094
82134
81170
80203
82998
82038
81074
80106
82902
81942
80977
80009
82807
81846
80881
79912
82711
81749
80784
79815
56
55
54
53
47
48
49
79815
78841
77864
79717
78744
77766
79620
78646
77668
79523
78549
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79426
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79328
78353
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79231
78256
77276
79134
78158
77178
79036
78060
77080
78939
77962
76982
78841
77864
76883
52
51
50
10
9
8
7
6
5
4
3
2
1
SIN t22-
SIN tOl--
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
52
.09 41083
47338
53593
41709
47964
54218
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53
54
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66101
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46
45
44
43
57
58
59
60
61
62
84861
91113
97365
85486
91739
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86112
92364
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42
41
40
39
38
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09868
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63
64
65
66
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24245
30495
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37369
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44243
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26745
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45493
27370
33620
39869
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34245
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41119
47368
36
35
34
33
67
68
69
70
71
72
47368
53616
59864
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54241
60489
48617
54866
61114
49242
55490
61738
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63613
51741
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64237
52366
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64862
52991
59239
65487
53616
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32
31
30
29
28
27
66112
72359
78605
66736
72983
79230
67361
73608
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74233
80479
68610
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69235
75482
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69860
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77356
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73
74
75
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97343
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26
25
24
23
77
78
79
80
81
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16077
22321
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23569
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16077
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22
21
20
19
18
17
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83
84
85
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53532
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70382
52284
58525
64766
71006
52908
59149
65390
71630
53532
59773
66014
72254
16
15
14
13
87
88
89
90
91
92
72254
78494
84733
72878
79118
85357
73502
79742
85981
74126
80366
86605
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82237
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76622
82861
89100
77246
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77870
84109
90348
78494
84733
90972
12
11
10
09
08
07
90972
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91595
97834
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98457
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93467
99705
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07190
95338
01576
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02200
08437
96586
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97210
03448
09685
93
94
95
96
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15922
22158
28394
10309
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17169
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06
05
04
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97
98
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52709
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53332
02
01
00
10
9
8
7
6
5
4
3
2
1
COS t23--
COS tOl-
1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
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.99 55620
55026
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53
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53405
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52124
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53223
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46
45
44
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57
58
59
60
61
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42
41
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38
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36
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67
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70
71
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32
31
30
29
28
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42941
42269
41593
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26
25
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22
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20
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18
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31934
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16
15
14
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87
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89
90
91
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12
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08
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93
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06
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02
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6
5
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SIN t23-
SIN tOl--
1
2
3
4
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6
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10
99
98
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00
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.06 27905
34176
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96
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43
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COT t23--
TAN tOl--
•
1
2
3
4
5
6
7
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49
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COT t23-
TAN t02-
1
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7
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5
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COT t22--
TAN t02--
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1
2
3
4
5
6
7
8
9
10
49
48
47
50
51
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GOT t22--
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1
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5
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COT 1
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TAN tl2--
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COT tl2--
APPENDIX
COSINES AND SINES map directions in right-handed rec-
tangular coordinates. It is impossible to be right-handed or
left-handed in two dimensions. The Z axis around which direc-
tions are mapped on the back cover is normal to the plane of the
paper at the intersection of the X and Y axes. By definition,
positive rotation carries points on the positive X axis around
toward the positive Y axis. In right-handed coordinates, the
positive direction along the Z axis is the direction in which the
thumb of the right hand would point if the fingers of the right
hand were curled around the Z axis in the sense of positive rota-
tion. The positive Z axis must therefore project upward from the
back cover toward the viewer.
WHOLE TURNS added to or subtracted from rotation around
the Z axis have no effect on directions in the X-Y plane. The
direction determined by any finite rotation around Z may be
defined by an angle in the range ± Vi turn from the positive X
axis.
THE FIGURE on the back cover maps all possible directions
around the Z axis as the collection of points at unit distance from
Z in the plane of the paper. These points are defined both in
terms of the angle, </), and in terms of the corresponding x, y
right-handed rectangular coordinates. By definition:
cos (f) = X sin 4> — y
cot (f) = xly tan </> = ylx
THESE TABLES map cosines, sines, cotangents, and tangents
in the range of positive values of <f> up to V4 turn. The following
rules for determining the sense and magnitude of cosine and
sine for any value of may be verified from the figure on the
back cover:
cos (f) is negative if, and only if \(f)\> t25; (1)
sin (f> is negative if and only if (f> is negative; (2)
\cos\ = \sin\ (f> ± t25; and \sin\ = \cos\ (f> ± t25; and, (3)
\cot\ (f) = \tan\ (f> ± t25, and \tan\ = |cof 1 0* f25 (4)
NOTE: When a synihol is represented between vertical brackets, thus: ]Ai, only the magnitude
of the object. A, is specified.
!A| is always taken as positive, and \X\ = V A^
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