Some procedures are presented for computer development of Hansen coefficients. The method of Von Zeipel and Andoyer was found to be most efficient. A table extends the method from 7th to 12th order

Cherniack, J. R.

English

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1972021973
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::. Research in Space Science
.,_.;;i_,, SAO Special Report No. 346
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,._.•* COMPUTATION OF HANSEN COEFFICIENTS
_':_
:,, J. I-_. Cherniack
";i" :.L:
:Fa::
'_.. July 27, 1972 .
/
:"; Smtthsontan Institution
_. Astrophysical Observatory
ii Cambridge, Massachusetts 02138
I09-020
1972021973-TSB03
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I
TABLE OF CONTENTS
Section Pa_
ABSTRACT .................... , .................. ill
1 INTRODUCTION.................................... 1
NOTATION 2
3 SOME METHODS OF COMPUTING HANSEN COEFFICIENTS ...... 4
4 OUTLINE OF THE VON ZEIPEL- ANDOYER METHOD ......... 5
5 DETAILS .......
5.1 Computation;fJ_'fo;;'>-';_'O:::: : ::::::::::::: 68 ii
5.2 Computation of X_ k ............................. 7
p_ O"
5 3 Computation of X_ k 7O.
6 THE PROGRAMING ................................. 8
7 REFERENCES 9 '._
APPENDIX A-I r'_
i
1972021973-TSB04
!ABSTRACT
This paper describes some procedures for computer development of Hansen
coefficients. The method of Von Zeipel and Andoyer is found most efficient. A table
extends the method from 7th to 12th order.
RESUME
On d_crit ici certains procedes pour obtenir le developpe-
ment dee coefficients d'Haneen _ l'aide ,l'un ordinateur. La
m_thode de Yon Zeipel et Andoye_ e gt_ trouv_e la plus effi-
7_me _mecace. Une table prolonge le mffthode du au 12 ordre.
°
B BTO_ CTaT_e OHHCMBaeTc2 npo_e_ypa paSBHTH_ EOS_HMEeHTOB
PaHoeHa c HOMO_ BBM. MeT0_ _OH 3znen_ E AH_oepa _Mn Ha_eH
HaH6o_ee B_$eETHBHMM. Ta6gH_a pac_p_eT MeTO_ OT 7 r_ _O _2r2 _ _
HOp2_Na.
b
J
III "
1972021973-TSB05
!COMPUTATION OF HANSEN COEFFICIENTS
J. R. Cherniack
1. INTRODUCTION
CayleyVs famous paper "Tables of the Development of Functions in the Theory of
Elliptic Motion" (1861) contains tables of Hansen coefficients to Seventh order in the
eccentricity. We recently required H_sen coefficients to higher order in ex_ml_ter-
accessible form and experimented with techniques for their computation. The method
of Andoyer (1903) and Von Zeipel (1912) is an order of magnitude faster than any other
technique we found. It is little lmow_ as far _s we can determine, appearing in
English only in Izsak_ Gerard, Efimba_ and Barnett (1964) and there almost paren- _:
thetically.
g
In this note, we briefly describe some of the slower procedures and theN.on Zeipel-
Andoyer (VZA) method and extend a table o_ Izsak to 12th order.
.'!
q
Tlds research was supported in part by grant NGR 09-015-002 from the National
Ae_nautics and SpaCe Administration.
1972021973-TSB06
2. NOTATION .._
Our notation will be the following _'!'i
r, the radius vector; i.
a, the semtmaJor axis;
v, the true anomaly; ._
M, the mean anomaly;
e, the eccentricity;
x = exp(lv); ,,:
z = exp(tM); .2.
the Hansen coefficients X_j k which are power series in e, and their generating function :'2
X_ k are defined by ' _:_ •
(r/a)n xk = X_ k= k z j . (1) _i.
If the real and imaginary parts of (I) are separated, we can write :,)_.
(r/a) n cos (kv) = C k cos (JM) , ...
it:J=O .... .,
(2)
(r/a) n sin (kv) ffi k sin (JM) ,
J=l -
0
2
where
c_,k = X_Ok
,k =..+j _ J=1,2,3... .
1972021973-TSB08
3. SOME METHODS OF COMPUTING HANSEN COEFFICIENTS
Since Cayley's original method was designed for hand computation_ it is not _
particularly adaptable for development on a computer. Tisserand (1888, vol. 1, p. 249)
expresses Hansen coefficients as a series involving Bessel functions and hypergeometric
serieS. Ttsserand's method was programed with the symbol-malflpulatlng system
SPASM and was so slow that it stimutated the literature search that culminated in this
note. One coefficient that required 180 sec to com oute was eventually found in 30 msec
by use of VZA.
The next method tried was direct application of Lagrange's Inversion Theorem (for
a description, see Brown and Shook (1933)). Although attractive in some special cases,
it is much too slow for general use.
The next method is the simplest; we compute the base set X 1' 0, X" 1, o, X0, 1
and X0' -1 by any method. Sin_e 0
can be computed from the base set by multiplication. This method is especially .,o
attractive if tables of Xn_ k ape desired. Twenty seconds of computer time is required
for each multiplication if 12th-order terms in e are retained. A special-puz_pse pro-
gra_., could be Written that is 3 to 5 l_Imes faster for the symbolic multiplication. ,,
All the previous methods have the following defects:
a) The methods require manipulation (multiplication or differentiation) of poly- ._
nomials, a rather slow class of operations. _
• : _
b) The computations must be done with rational-fraction coefficients. This is
about 10 times slower than is possible with integer coefficients.
The VZA method, which is next described_ avoids both these defects at the cost of
some added storage and complexity. _
4
...................... ^
1972021973-TSB09
i:''
;ii i 4. OUTLINE OF THE VON ZEIPEL-ANDOYER METHOD
In Section 5_ we show how to define Jp, a(n_ k}_ which are polynomials in n and k
•_ with integer coefficients. For compactness, we write J rb k for J k). These
.. p, O' p_o'(n'
polynomials are computed only once and are saved for later use.
If a particular I_tansen coefficient is desired to _r'(eM)_ M _ 12, we need only
,q
i!) iii!iiii' evaluate at most (M + 1)/2 of the J's for particular values of n, k, p_ a. Denominators
• must now be affixed to the Jvs, and the resulting fractions reduced to lowest terms.
: _,_' Since polynomial evaluation is about an order of magnitude faster for integers than for
-".•'.:.'_' ra_onai fractions, our ability to do most of the computation in terms of integers
•' *" _ affords significant advantages when exact re_mlts are required.: ' ".7
• ;_ '
"_.*_._, ,
...!_...:_
; _,_,_'.
r ),,,,
,_,./_.,

4 m
B _
I
...... :" The computation begins with eep_ations (3) and continues in the order
,?_.L. ,
' .'i?:-,
•,.,':.i jr b k jn, k
." 2_ 0 P I_ 1
• ',' '. ,t.'
9 1 9 9 ''' '
Equation (4) is used to compute the polynomials in the first column, and equation (,5)
: ";a is used in all other cases.
.' _ * 5.2 Compu_tton of Xn' k
::" " Pt",.r
:, !,': 2"
.,,, Xn,k xn,-k
_,_...; P, ff= ff_p p < O"
'_":', jrb k
.. ,..... _, k p, a
._...:, p >_ a {6)
..::.. p, O" 2P+apla!..,.,.: ( )
•2";?.;
5.3 Computation of _ k
a • (7)
p-c=j+k
p,a>_O
1972021973-TSB13
6. THE PROGRAMING
Izsak published jn, k for p, _ m 0_ p + a __ 7. We continued the calculation to
p + (r _ 12, using the general-purpose symbol-manipulating system SPASM (Hall and
Chernlack_ 1969). The computation to order 8 took 1.5 min_ and the extension of the
computation to order 12 took all additional 13.5 rain of CDC 6400 time.
The output of our program was in the form of Fortran DATA statem_nls_ which
k
became the heart of a Fortran program that computes integer values of jX_,o fr_:.mO"
specific values of n_ !% p, and 0".
• Y
?
1972021973-TSC01
7. REFERENCES
t
A NDOYER_ H.
1903. Contribution/I la th6orie des petites planets dont le moyen mouvement est
sensiblement double de celui de Jupiter. Bull. Astron., vol. 20,
pp. 321-356.
BROWN, E.W., and SHOOK, C. A.
1933. Planetary Theory. Cambridge University Press, Cambridge.
CAYLEY, A.
1861. Tables of the developments of functions in the theory of elliptic motion.
Mem. Roy. Astron. Soc., vol. 29, pp. 191-306.
.. . HALI_ N.M., and CHERNIACK, J. R.
1969. Smithsonian Package for Algebra and Symbolic Mathematics. Smithsontan
Astrophys. Obs. Spec. Rep. No. 291, 49 pp. ....
IZSAK, I. G., GERARD, J. M., EFIMBA_ R., and BARNETT, M. P.
1964. Construction of Newcomb operators on a digital computer. Smtthsonian _,
Astrophys. Obs. Spec. Rep. No. 140, 103 pp.
TISSERAND, F. i_"
1888. Trait_ de C_leste M_canique. Gauthter-Villars, Paris ....
VON ZEIPEI_ H. _
1912. _r le calcul des operatuers de Newcomb. Ark. Mat. Astro. Fys., .
vol. 8, no. 19, 9pp.
_a
i
r, "
1972021973-TSC02
APPENDIX
This Appendix contains a brief example, as well as a table of jn,. 1,_forg¥
0_< p+ ___ 12.
The example is the computation of r/a to _(e). From equation-(1), we have
r/a = X1' 0 ,
oO
= _._ Zj _ X I'0 e_G (byequation(7))p,O" 'j=-- p-o'=j
p, o'_
= XI'0,00+ \IX1'0,01z-1 + x1'1,0 z 1) e + 0'(e 2) • (A-l)
From Section 5.2, it fonows that ,,;,4
4,,0 0 10 ....O, ' ,i
,)
=1 •
i
A-I .,_
1972021973-TSC03
' " ': ,' _,i_ " _ ,,: _¢:_/_2_ ''__ : ,., • :a _,' " ' : •
!, , ,_ ,_., ,,
_:. Slmllarlyp
._!','.;;:, X 1, = X 1, 0
• '( O, 1, 0 '
... = j19 _2
n=l
= (2k - n)/21 k=_0. 9
= -1/2 .
9
i"_'_. Substituting in equation (A-l) gives
"L ,:,
'_':_" r/a = i - e(z-I + z)/2+ _(e2) ,I.
.:. e 2
.. .' ;' =l-ecosM+ _/'()
:l
: y..
,-,;_
2:-"
!- ,, , _' .... J,.+., ,,
4, N
' "z "'= o" .
• "i +' +" " .'+
-, ,;, .<,
+Pil1 f_Z m 4 +
_++, . +, ,,.-.. +
° + +t ++ "+,+,-
++++ +++::.+.++! .- .+.++. +: . ++ ,.,; ..,,. , ;
i .;, ,,,,' ..' _ " ,_,,. +- - ) +,++• .,.,. - .'; . +,': +'+
- " ,+ ,, " _, +' ._ T 7 T . _'
I$ II II n II It II hi I1 II It It H
,.+,
o o o - : - o = _" _ = ,* *;
-% ..% .,% .% _ -% _ '; ,..% _ "; ,_
,,,5 ++
....... , - , ..... ..+.._+ -:+j .... ,?- +
-- +..... " " + _ +, t_j.,+ ,,,+,eke,+.._+ __++aa..
•,k;, --+ + , + •
+, ou . . ;o
1972021973-TSC05



A-'7 +'
........... ! ...... + + " _'.++++"_ + ++ . i
++ o .. : ",, i,- o+ +r_:+,.._.... + "-" - _+' . __:--++::+_.___ '+ - .......... m
+ : ++_,,+'_++ _° " +v+=_"+:_+'+"u l_'_'-"_" + ++. ,, o .... .0 ._,. .+ +,+ .... +..,+ ' u+, ,, _ ,+. +, v +..... +_'l___ _+ " '++ . , + +
197202197g-TSC09

I
++
A-9 ',
,_. .-- -_+ +.. +<._* .- ,+++ ....... • , , ..... _+_++_, ........ + .......
; .,.+ _ ,+ . . +,.. ,'_+_- _ . .+ . + .. ,+'o ,, ,_?_.i + _'_+ + - o'_ + - +'
"197202"1973-TSC'11

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1972021973-TSD05

BIOGRAPHICAL NOTE
JEROME R. CHERNIACK grm_ated from City College of New YorJ_J_nJ958 and has
been with the Observatory since 1989.
i'

NOTICE
%'7...
I_.;'; ' This series of Special Reports was instituted under the supervision
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_:., Smithsonian Institution, shortly after thelaUnching of the first artificial
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," of the Observatory.
,,b' :
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_/:" Lwho reqUest them from the PUblications Division, Distribution Section,
'Smithsonian AstrophySical Observatory, Cambridge, MassaChusetts
02138.
0 .
,_,,_i_ _.•
1972021973-TSO09


Computation of Hansen coefficients

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