Orbital data derived from photoreduced Baker- Nunn observation

Gaposchkin, E. M.

English

NASA Technical Reports Server

. I  
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b 
S M I T H S O N I A N  I N S T I T U T I O N  
A S T R O P H Y S I C A L  O B S E R V A T O R Y  
Research i n  Space Science 
S P E C I A L  R E P O R T  
Number 232 
PUBLICATION O F  ORBITS DERIVED 
FROM PHOTOREDUCED BAKER-NUNN OBSERVATIONS 
E. M. Gaposchkin 
December 12, 1966 
CAMBRIDGE, MASSACHUSETTS, 021 38 
https://ntrs.nasa.gov/search.jsp?R=19670010072 2020-03-16T18:09:50+00:00Z
S A 0  Special Repor t  No. ' 232 
PUBLICATION OF ORBITS DERIVED 
FROM PHOTOREDUCED BAKER-NU" OBSERVATIONS 
E. M. Gaposchkin 
Smithsonian Institution 
A s trophy sic a1 Ob s e rvato r y 
Cambridge, Massachusetts,  021 38 
6 -66- 25a 
Section 
TABLEOFCONTENTS 
Page 
BIOGRAPHICAL NOTE . . . . . . . . . . . . . . . . . . . . . . . . . . .  i v  
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v 
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 3  
ii 
LIST OF TABLES 
. 
Table 
1 
2 
3 
4 
5 
Page 
Summary  of final orbital  information published by the 
Smithsonian Astrophysical Observatory . . . . . . . . . .  6 
Values of the constants GM and J used in orbit  
calculation . . . . . . . . . . . . . . . . . . . . . . . . . .  8 
Zonal harmonics  used in orbi t  calculation . . . . . . . . .  9 
T e s s e r a l  harmonics  used in orbit  calculation . . . . . . .  10  
Coordinates of the Baker-Nunn stations used in orbi t  
calculation . . . . . . . . . . . . . . . . . . . . . . . . . .  11 
iii 
BIOGRAPHICAL NOTE 
Mr.  Gaposchkin graduated in e lec t r ica l  engineering f r o m  Tufts University, 
Medford, Massachuset ts ,  in 1957. He received a Degree of Numerical  Anal- 
y s i s  i n  1959 f r o m  Cambridge University, England, and is now working to- 
ward  a doctorate in geophysics at Harvard  University. 
Since joining the staff of the Smithsonian Astrophysical Observatory in  
1959, Gaposchkin h a s  held positions as p rogrammer  and Division Chief of the 
Computations Division, and i s  now a mathematician in the Resea rch  and 
Analysis Department. He has  helped to develop the basic  computer program 
used in all analyses of satell i te motion. 
His main  in te res t s  include satellite geodesy and geophysics, and applied 
mathematics .  
i v  
ABSTRACT 
The p rec i se  orbi ta l  information published by the Smithsonian Astrophysical 
Observa tory  has  been computed using var ious numerical  values for the geo- 
physical  constants. 
used. 
information. 
This  repor t  details the numerical  values of the constants 
The fur ther  r epor t s  of orbital data will include the same relevant 
V 
PUBLICATION OF ORBITS DERIVED FROM 
PHOTOREDUCED BAKER-NUNN OBSERVATIONS 
E. M. Gaposchkin 
The Smithsonian Astrophysical Observatory has  published final orbital  
information (Staff of SAO, 1962, 1963, 1964a, and 1964b; r e fe r r ed  to  a s  E-1 , 
E-2, E-3, E-4, respect ively)  that has  been the by-product of cer ta in  inves- 
t igations at the Observatory.  Our purpose h e r e  i s  to detail  information that 
will  define these resu l t s  m o r e  precisely ( see  Tables  1 to 5) .  
we plan to  supply such supporting information with each publication. 
In the future,  
The tables  w e r e  compiled by re fer r ing  direct ly  t o  the orbit f i les.  In 
some  cases ,  the information could not be re t r ieved  (e. g. , the station coor-  
dinates).  Where the information is uncertain,  it is appropriately designated. 
We now confine ourselves  to a few r e m a r k s  about these  data. 
A. The data published are the d i r ec t  resu l t  of the DO1 program. The 
distinction between mean and smoothed elements  is based on the method of 
running the program. 
pract ical ,  a r e  available elsewhere (Gaposchkin, i 964). W-e mention, with- 
out discussion, only aspec ts  necessary  for  this  summary.  
The details  of the program,  both theoret ical  and 
a. Each  orbi ta l  element a s  input to  the DO1 program i s  r e p r e -  
sented a s  a polynomial with t r igonometr ic  terms, such a s  
2 E(t)  = E t E l t  t E2t t * .  - 0 
tc Ai s i n  (Bi t Cit) ,  
i= 1 
This  work was  supported in pa r t  by grant  number NsG 87-60  f r o m  
the National Aeronautics and Space Administration. 
1 
where t = T - To; i. e . ,  t is the t ime measu red  f r o m  the epoch 
of the orbit. 
J 
program, and the Ai, Bi, and Ci  a r e  determined theoretically. 
In th i s  instance, the Ai, Bi, and C. r ep resen t  the long-period 
perturbation due to  the ea r th ' s  oblateness. 
B. + C.t  E i w ( t ) .  
theoret ical  development of Kozai ( 1  964). 
a r e  given with the smoothed elements.  
In general ,  the E .  a r e  determined by the DO1 
1 
Therefore ,  
The amplitudes A .  a r e  evaluated f r o m  the 
1 1 1 
Where used, they 
b. The smoothed elements  a re  s imply the values E 
f r o m  orbi ta l  a r c s  of 1 t o  2 weeks. The mean  elements  a r e  the 
values of E(t)  a t  the  epoch (t = 0), given in  the fo rm of a table. 
They differ f r o m  the constant pa r t  of the polynomial only by 
the contribution of the long-period t r igonometr ic  terms. These 
mean  elements  a r e  derived from shor t e r  a r c s  (typically, 4 
days), computed f o r  epochs at 2-day intervals.  Therefore ,  
consecutive a r c s  a r e  determined f r o m  overlapping intervals. 
Ai, derived 
j' 
c. These elements,  a s  contrasted with osculating elements,  a r e  
"mean" elements  in  the sense that they a r e  averages.  In 
other words,  they a r e  the elements  with the short-per iod 
per turbat ions subtracted.  
d. The t i m e s  of the epoch a r e  given in Modified Jul ian Days (MJD), 
defined in  terms of Jul ian Days (JD): 
MJD = J D  - 2 400 000. 5 . 
e. The units of the orbi ta l  elements a r e  degrees  for  angular 
quantities, megamete r s  for l inear  quantities, and revolutions 
for  the mean  anomaly. 
f. The values  of the  SA0 elements  a r e  given for  the argument  of 
per igee a, the right ascension of the ascending node 5 2 ,  and 
inclination I, the eccentricity e, and the m e a n  anomaly M and 
its  der ivat ives  n, n/ 2. They a r e  given a s  functions of t 
2 
measured  in days. The number at the right of each constant 
r ep resen t s  the s tandard e r r o r  for  that element and r e f e r s  t o  
the last digits g iven  
g. The tabulation of "mean" elements  gives the mean (anomalist ic) 
motion n, the orbital  acceleration i ~ /  2,  and the semimajor  
axis a or the geocentric distance of perigee q. 
t h ree  columns, the one headed N indicates the number of ob- 
servat ions used for  the computation of a s e t  of elements;  the 
one headed D, the number of days used; and the one headed u, 
the standard e r r o r  relative to  their  assumed accuracy. 
In the last 
h. The semimajor  axis  a of the orbit  i s  computed f r o m  the m e a n  
motion n by means  of the following formula: 
2 ( s ee  Kozai, 1959, equation (14)), where  we put A2 = ( 3 /  2)J2ae . 
The value of J given in Table 2 is this  A 
M ; n i s  the  m e a n  motion defined as the t ime  derivative of 
the polynomial pa r t  of the mean anomaly; I i s  the inclination; 
and e i s  the eccentricity.  The units for  GM given in Table 2 
are Mm rev2 days-2.  In r epor t s  E-1 t o  E - 4  the mean motion 
was the total  t ime derivative of the mean anomaly, and therefore  
included unwanted long -period effe c t s . 
expressed in 2 2 
m 
3 
i. In  the DOI, the inclination and the argument  of perigee a r e  
r e fe r r ed  t o  the t r u e  equator of date, and the right ascension 
of the ascending node i s  measured  f r o m  the mean equinox of 
1950. 0. 
should s e t  the  node equal to: 
To r e fe r  the orbit to the m e a n  equinox of date,  we 
t 3P508 X 10-5(MJD - 33281) '"date = " ~ O I  
3 
j. Two t ime sys tems a r e  implicit in  the DO1 calculations. The 
first i s  the t ime used fo r  computing the ephemer is ;  it should 
be a uniform s y s t e m  The second, an empir ical ly  determined 
quantity, i s  used for  computing the position of the ea r th  (i. e. , 
the station position). It should be UTI because of the way we 
have defined the s idereal  angle. 
reduced observation card  i s  atomic t ime ( A l )  a s  published 
by the U. S .  Naval Observatory; it i s  assumed to be a uniform 
t ime system. In reports  E-1 to E-4  the DO1 computed UT1 
t ime f r o m  
The t ime  given on a photo- 
UTI = A1 t (-12. 154 x l o m 6 )  t (-0. 015379 x X T days, 
where T = T - 37000. This t ime was used in both the s iderea l -  
t ime and the ephemeris  calculations. Therefore ,  the orbi ts  a r e  
r e fe r r ed  t o  the UT1 t ime sys t em as defined above. 
and all subsequent reports  the atomic t ime (as taken f r o m  the 
observation) i s  used in the ephemeris  calculation, and the 
empir ica l  UTI , given as  the difference A1 -UTI on the obse r -  
vation card,  is used in calculating the s iderea l  time. There-  
fore,  these  orbi ts  a r e  r e fe r r ed  t o  the A1 t ime system. 
In E-5 
B. The following comments apply t o  the tables. 
a. In  Table 1, under the columns headed "determination of i3 and 
S2 , 
mined empir ical ly  by the DOI, and a c means  that the t e r m  
was held constant and determined by some other means.  In  
the la t te r  case,  the l inear  t e r m s  were  usually obtained by a 
separa te  l ea s t - squa res  program, which used the values  of the 
element  i n  question at successive epochs. 
a v indicates that the l inear  t e r m  in question was de t e r -  
b. The constants re fer red  to in Table 1 a r e  given in Tables 2 to  5. 
The number given in Table 1 r e fe r s  to the se t  of coefficients 
designated in the subsequent tables. Under t ime reference,  1 
designates UT1 a s  defined above, and 2 designates A l .  
4 
c. In repor t s  E-1 to E - 4  only the short-per iod oblateness 
perturbations w e r e  included, while E-5 contains lunar pe r tu r -  
bat ions and t e  s s er al-  harmonic s perturbations.  
p resence  or  absence of the perturbations and the values of the 
constants used. Unfortunately, a prel iminary se t  of t e s s e r a l  
harmonics  was  used in E-5. 
reference,  but the reader  mus t  be cautioned that they a r e  not 
the se t  distributed by SAO. 
We detail  the 
The values are  given h e r e  for  
5 
N N N N N  
z x z z z  
I 
Z Z Z Z Z  
Z Z Z  z z z z z z  Z Z Z Z Z  
r e e r r  c c c  e >  z z  r * *  > z > c c >  
> 
> > ; > >  
> > ; u >  
> 
u > >  > > > > u  u > > > v u  
u > >  > > > > v  v > > > v v  
z z z  z a  r m a  I E Z  z z Z I z z  
0 
- * s 
- 
N n 
- N - N -  
0 0 0 0 0  
- - c 0 . *  
0 0 0 0 0  
0 0 0 0 0  
0 . 0 . 0 0 -  
" l Y I v l Q Q  
N - N  - N  N - -  
0 0 0  0 0  0 0 0  . - -  c o  0. " l m  
0 0 0  0 0  0 - -  
0 0 0  0 0  0 e o  
m e . -  P O  0 0 0  
* * V I  " l a  Q Q Q  
6 
h 
ci 
G 
0 u 
Y 
'ga- 1 Y .3 
L Z Z Z Z Z Z Z Z  
z z z z z z z z z  
z z z z z z z z z  
r r r r r r c r c  
N 
1 
* c r r r r r r r  
* r r r r r r r r  
c r r r r r z r r  
> > > > > > > > U  
> > > > u u u > u  
7 
Table 2. Values of the constants GM and J used  in orbi t  calculation. 
a 
274. 54269 
274. 53910 
J Set R ef e r ence  
0. 0660705456 1 
0. 0660546000 2 Gaposchkin, 1964; 
Kozai, 1962 
8 
Table 3. Zonal harmonics  used in  orbit  calculation. 
9 
Table 4. T e s s e r a l  harmonics used in orbit  calculation. 
I 
2 
3 
3 
3 
4 
4 
4 
4 
5 
5 
5 
5 
5 
6 
6 
6 
6 
6 
8 
8 
8 
9 
10  
11 
12 
1 3  
1 3  
15 
15  
15  
m 
2 
1 
2 
3 
1 
2 
3 
4 
1 
2 
3 
4 
5 
1 
2 
3 
4 
5 
1 
2 
4 
1 
1 
1 
1 
12 
13 
12 
13 
14 
c1 m(X l o - 6 )  
2. 46 
1. 91 
0. 77 
0. 39 
-0.54 
0. 37 
0. 8 4  
-0. 04 
-0.10 
0. 6 1  
-0.55 
-0.22 
0. 09 
-0. 04 
0. 0 9  
-0.03 
-0.01 
-0.28 
0. 00 
0. 10  
-0.15 
0.11 
0. 07 
-0. 09 
-0.10 
0. 01 
-0. 05 
-0. 03 
-0. 06 
0. 01 
SI m(X 
-1. 30 
0. 24 
-0.59 
1. 54 
-0.43 
0. 65 
-0.20. 
0. 33 
-0.10 
-0.25 
-0. 09 
0. 06 
-0. 57 
-0. 06 
-0. 33 
0. 02 
-0. 46 
-0.45 
0. 10 
0. 06 
0. 04 
0. 05 
-0.12 
0. 02 
-0. 01 
-0.01 
0. 08 
0. 03 
-0. 06 
-0. 02 
Set 
1 
Reference 
Gaposchkin and 
Izsak, 1965 
10  
- 
9 o.
3 
m 
> 
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W 
N t- 
o' 
P 
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m 
0. 
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w 
m 
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m 
9 
t- 
9 
m 
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t- 
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N 
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- 
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m 
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t- m 
m 
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N 
9 
rc 
N 
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0 
9 
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t- 
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oc 
9 
N d 
0 
-r 
o r d  
- ~ m d m a r - m o ' o - ~  
0 0 0 0 0 0 0 0 0 - - 3  
0 0 0 0 0 0 0 0 0 0 0 0  
o ' o . o . o . o . o ' m o . o ' m o ' c r  
N 
v 
0 o. m 
9 L n  
r 
N 
W 
0 m 
m 
P 
- d t- 
N 
N 
P m 
m 
0 
b 0
P 
9 m 
9 
P o. m
5 N
N 
9 
0 m 
3 
w N 
m 
- 
N 
N 
9 
t- 
W 
N 
m 
ln 
t- 
N t-
ln 
- 
m 
N m 
0 -- 
N 
o' 
m 
m 
IC m
m 
m 
m m s 
9 W 
o d d d o d d o d o d d  
I 4  0 I 
m 
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2 
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r- 
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r- 
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01 
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0 
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o l d  
- ~ m ~ m a r - m o ' o - ~  
3 0 0 0 0 0 0 0 0 - 3 -  
3 0 0 0 0 0 0 0 0 0 0 0  
> o ' o ' m m o ' o ' m o ' o ' o ' o .  
1 1  
* 
01 
VI 
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0. 
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d 0 
N 
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0 m
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0 
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9 
N 
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9 
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m 
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u; 
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t- 
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m 
m 
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9 0
N 
m 
m 
9 
9 
d 
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o o o o o o o o o - - ~  
0 0 0 0 0 0 0 0 0 0 0 0  
0 . 0 . 0 . 0 . 6 0 . 0 . 0 . c n 0 . 6 6  
d 
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n 
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6 
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2 
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3 
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m 
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m 
m 
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0 -
6 0.
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9 
d 0. 
3 
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0 
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r- 
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W 
9 r-
m 
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m 
N 
0 
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9 r- 
0. 
m 
9 m 
0 
N 
m 
'$ 
9 0
9 
9 
d 
- ~ m a m . ~ r - m ~ o - ~  
0 0 0 0 0 0 3 0 0 - - -  
0 0 0 0 0 0 0 0 0 0 0 0  m U . U . m c n 6 0 . c 6 0 . 6 6  
12 
REFERENCES 
GAPOSCHKIN, E. M. 
1964. Differential  o rb i t  improvement (DOI- 3). Smithsonian Ast ro-  
phys. Obs. Spec. Rep. No. 161, p. 47. 
GAPOSCHKIN, E. M . ,  AND IZSAK, I. G. 
1965. K set of t e s s e r a l  harmonics  and C-4 s e t  of station coordinates  
(pre l iminary  resu l t s  which are  distributed internally). 
KOZAI, Y. 
1959. The motion of a close e a r t h  satellite. Astron. Jou rn . ,  vol. 64, 
pp. 367-377. 
1962. Numerical  resu l t s  f rom orbi ts .  Smithsonian Astrophys.  Obs. 
Spec. Rep. No. 101, p. 5. 
1964. New determinat ion of zonal harmonics  coefficients of the e a r t h ' s  
gravitational potential. Smithsonian Astrophys. Obs. Spec. 
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Publication of orbits derived from photoreduced Baker-Nunn observations

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