• .,.. .

,_. .,._:. _:'!,.. ..._,._...,_ .;.., _., ._ ';- " . o ':_,_. ' o ',.... "....... _,).?,' .. : ...... .,_: ;.... o-.Y"-...,"_..

1972021973

k 4

:2' "

:::, ,:!

::. Research in Space Science

.,_.;;i_,, SAO Special Report No. 346• fr ,;,

,._.•* COMPUTATION OF HANSEN COEFFICIENTS

_':_:,, J. I-_. Cherniack

";i" :.L:

:Fa::

'_.. July 27, 1972 .

/

:"; Smtthsontan Institution

_. Astrophysical Observatoryii Cambridge, Massachusetts 02138

I09-020

1972021973-TSB03

; , ; . •

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 ii5.2 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

JIII "

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 NationalAe_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 :'2X_ 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 .,oattractive 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+kp,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_ whichk

became the heart of a Fortran program that computes integer values of jX_,ofr_:.m

O"

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,0 ' ,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

A-f1

1972021973-TSD01

.

I

4" ,n

x o

IN !0 _ e,l ,0 $ N 0 _O

4

:_: • .. = . , . ° = . . _,

p,. e- • 0 O

¢.,=

._,;_

.° ,_

_'_, A-13 •

] 97202] 973-TSD03

• ,41,41 0 e •

-+"+ _ _ +-++i

,, : _ [ . o _ +: .-.?

°i!-)---:i-!;, .= - ., , +_

,,+_ ¢ °r .:+

;, + _ _ e ':-L

o +';" 7 7 f " iII

A-14 _

.l

"1972021973-TSD04

o %. _ :" .," _ . ° : -.= "=' .°

., . ® Y ._

• i !n #

•.n -n _

'!A-15 :,_

1972021973-TSD05

BIOGRAPHICAL NOTE

JEROME R. CHERNIACK grm_ated from City College of New YorJ_J_nJ958 and hasbeen with the Observatory since 1989.

i'

NOTICE

%'7...

I_.;'; ' This series of Special Reports was instituted under the supervision

'_ of Dr. F. L. Whipple, Director of the Astrophysical Observatory of the_:., Smithsonian Institution, shortly after thelaUnching of the first artificial

earth satellite on October 4, 1957. Contributions come f_o m theStaff," of the Observatory.

,,b' :

.:_. First issued to ensUre the immediate dissemination of data for satel-

lite tracking, the. reports have contin_edto provide a rapi_i distributionof catalogs of satellite observations, o_bital information, _d prelimi-nsry results of •data analyses prior to formal publication in the appro-priate .journals. The Reports are also used extensively for the rapid

' publication of preliminary or special results in other :fields of astro-phy s ic s.

The Reports are regularly distributed to all institutions partici-pating in t_e U. _. space• research program and to individual scientists .....

_/:" Lwho reqUest them from the PUblications Division, Distribution Section,'Smithsonian AstrophySical Observatory, Cambridge, MassaChusetts

02138.

0 .

,_,,_i_ _.•

1972021973-TSO09