Remarks on hill lunar theory - part ii, direct lunar orbits as solutions to differential equatio

Stumpff, K.

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NASA TN D-1541
TECHNICAL
D-1541
NOTE
REMARKSON HILL'SLUNARTHEORY.
PARTI!
Karl Stumpff
Goddard Space Flight Center
Greenbelt, Maryland
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON March 1963
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REMARKSON HILL'SLUNARTHEORY.
PARTII
Karl Stumpff
Goddard Space Flight Center
SUMMARY /3 vr, 
In 1879 G. W. Hill, using his differential equation system, found a
series of direct lunar orbits, which represent a simplification of the
restricted three-body problem (neglecting solar parallax and eccen-
tricity). If T is the period of revolution time, and T = 27rm is fixed, the
initial conditions, _0(m), _0(m), and (to = V0 = 0), are dependent upon m
and give periodic solutions. Hill obtained these functions in the form
of power series, with respect to m, which converges rapidly for small
distances of the satellite; and he gave their first coefficients.
In the present discussion, Hill's coefficients are successively de-
termined from linear equations and are in the form of closed trigono-
metric expressions. Although this method--which does not extend
beyond the third order--is only an interesting attempt at achieving the
same goal via a different mathematical route, it does not extend as far
as Hill's method (which may be carried out to the fifth or sixth order
without difficulty).

CONTENTS
Summary ....................................... i
INTRODUCTION .................................. 1
HILL'S DIFFERENTIAL EQUATIONS ................... 1
CONCLUDING REMARKS ............................ I0
RECOMMENDATIONS .............................. II
iii

REMARKS ON HILL'S LUNAR THEORY.
PART II
by
Karl Stumpff*
Goddard Space Flight Center
INTRODUCTION
In his famous writing of 1878, G. W. Hill¢ found a series of direct lunar orbits as solutions of
his differential equation system. This series consitutes a simplification of the probl_me restreint
(by discounting the solar parallax). Hill found that if W is the (synodic) circuit time of the satellite
and T = 2_ is fixed, the initial conditions are G0(m), _0(m), (_o = Vo = 0), and are dependent upon m
as a parameter and give such periodic solutions. He obtained these functions in a power series with
respect to m, which converge rapidly for small distances of the satellite (short circuit times, small
m), and gave their first coefficients. In the following examination, another method will be presented
which leads to the same goal, but which permits these coefficients to appear as finite trigonometric
expressions.
HILL'S DIFFERENTIAL EQUATIONS
Hill's differential equations
: 7)
_+2_: - v)
r 3 ,
(1)
where r 2 = 52 + 72, are based upon the rectangular coordinates
of a massless satellite in reference to its central body (planet)
and in a coordinate system which revolves at the same angular
velocity, n = 1, as the planet which orbits the sun (Figure 1). The
location of the sun S is therefore assumed to be in the direction of
the negative _-axis, at any desired distance, so that the parallax
*NAS-NASA Research Associate; Professor Emeritus, G_ttingen University.
S
17
,," -"%,$
I
I o
I
Figure 1-- Geometric representation
of a massless satellite s of a planet
P.
tHill, G. W., "Researches in the Lunar Theory," in The Collected Worhs o/George William Hill, Vol. I, Washington, D. C.: Carnegie
Institution, 1905.
of the sun can be discounted. Under these circumstances, Equations 1 apply for the satellite S.
Periodic solutions can be expected if we let the satellite start from a point (go, 0) on the g-axis with
a velocity (0, _0), whereby the initial (positive) velocity must occur perpendicular to the g-axis and
vo will depend to a certain extent upon go" Every periodic solution of this sort is symmetric to both
axes, since Equations 1 do not change when g is exchanged with - g and v with -v. If T = 2_m iS the
circuit time in such a periodic orbit, then the total of these orbits also can be characterized by de-
termining go = go (m), _o = _o (m) as functions of the parameter m.
If r = 0 is the moment at which the satellite crosses the positive g-axis at right angles, then the
periodic motion has the form
g(r) -- A I coszrr + A 3cos3z_r + A s cosSz_-r + -.- ,
W(r) = B lsinzrr + B 3 sin3vr + B s sinSv'r + "-.
(2)
where v = i/m = 2_/T. For circuit times as brief as desired (m -0)--hence for very close orbits--the
perturbations bythe sun become as small as desired. The above equations (Equation 2) now change
into
g(r) = a cos z_, _(r) = a sin_-_r , (3)
or into the form of uniform circular motion, for circles are the only undisturbed orbits which are
symmetric to both axes. If we differentiate Equation 3 any number of times, we find, for this bound-
ary case and for r = 0,
m2-g0(=.) _- {-l).g°'
m2n-t_o2--1 = (-1)--I_o
The above equations mean that, for m _ 0, the quantities
= -- - 1 , fll = ÷m'_"-- 1 ,
go so
g_4)
G- 4 = + m 4 _ _ 1 ,
77o
f13 = -m3_-o _ 1 ,
g0_')
a6 -- -m6 --_0 _ 1 ,
vo<s)
/3S -- +m5 --_0 _ 1 ,
(4)
all approach unity.
If for the sake of abbreviation we set 1//r 3 = /_, then the differential equations (Equation 1) can
be written:
= 2_ - _(_-3) ; and _ = -2_ - v# .
If we differentiate these equations any number of times and if we consider that uneven derivatives of
and _ and even derivatives of v disappear for _- = 0, we then obtain the system:
- m2
+"_' _o'": _'_o-[;o(.o-")+Co;o]
_o
-_' Co_,>= _.:,>-[c_',(_,o-3)+_o_o +CoCo''>]
- m 3
_o
% : -2_o-_o,,
+m $
_o _o_' ---_:o'"-(_o_+_o_)
- m 7 / _
:--_e_'> - (,_'>,+1o'Z'o_.+ s%,('>y,VTo(7)
If these equations are multiplied by the factors to their left and if Equation 4 is substituted they
become:
% = -2m_, - 3m_ +¢ . Zl = ¢ '
=4 = -2_ - _n 2% +¢_2 ' Z3 = -2m=_ + ¢_1 '
a 6 = -2m_5 - 31112 a 4 + _V 4 , fl$ = - 2llla 4 ÷ ¢Y3 ,
where
_0
¢ = mU; o and ¢ = 3a = rn (--_; (6)
and, with
#2 = - m2 -- = + m4 -- = ....
' _0 ' ' (7)
v3 = 33 + 3#231 , v4 = a4 + 6;2a2 + ;4
vs = 3s ÷ 10;2 33 + 5;4 31 ' v6 = % + 15_2_4 ÷ 15;4% + ;6 "
(8)
If the quantities ;2, ;4,
substituting their values for r = 0 into Equation 7, the following system of recurrence formulas is
obtained for v i :
N1 = _1 '
• .. are eliminated from Equation 8 by forming the even derivatives of _ and
W5 = fi5 -- 5_4V 1 ÷ lOft 3 _2 - 10% v 3 + 5fl,v4 ,
v 2 = -(1 +1) a2 + (1 +2)fllvl ,
v 4 = -(1 + 1} % + (3 + 4} 33 _1 - (3 + 6) % v 2 + (1 + 4) fll v3 '
we = -(l+l)a 6 + {5+6)flS'_' 1 - (lO+lS)ct4_" 2 + (lO+20)f13v 3 - (S+lS)a2z_ 4 + (1+6)31z.' 5
_(9)
In these formulas we again recognize the binomial coefficients of uneven order in the coefficients
of vl with uneven indices, while the coefficients of "i with even indices are made up of the binomial
coefficients of uneven and subsequent even orders. In all cases, the signs are alternating and the
sum of the coefficients is unity. However, this means that all vi, just as the % and 3_, approach
unity for m40. The same also applies for ¢ and ¢, as can be seen from Equation 5. With the help
of Equations 5 and 9, a i and /3, can be determined one after another, in the order of increasing indi-
ces (3,, %, 33, %, ""), as functions of m, ¢, %
4
The orbit of a satellite which originates from (_0, 0) at time _ = 0 with the velocity (0, ;70) will
be periodic if it reaches the point (-_0, 0) at _ = ( 1/2)W = vrm. Therefore the periodicity conditions can
be written:
_(v,n) = - _o ' v(_rm) = 0 , (10)
or, if we set up the Taylor series,
(_m) 2 (ran)4
_(_) : -_o : _:o+_;o +_o (4) ÷ "'" '
• _ (v'n)Sv(_m) = 0 = r'mVo + . "Vo + _ Vo(s) ÷ "'" "
If we divide by {o and substitute the quantites of Equation 4, we can then write
772 _4 _6
f(m) = 0 = 2 - a2 _ + a4"-4T. - a6-_T'+ "'"
,rr3 "/75
g(m) = 0 = fll _ - '83_ + fls-_. .... "
(11)
On the other hand,
f(m) = fo
g(rn) = go
, m 2 m 3
+ fo m + fo_ + fo_ + "'"
, . m2 ., m"+
+ gom + go_ + go_ + ""
+o 1
0 ,
(12)
if the index 0 is based upon m = 0 and the derivatives according to m indicated by primes. The Equa-
tions 12 are identities, since they are fulfilled for any given value of m. Thus the requirement of
periodic orbits is given by
fo -- fo fo = fo
i - 0 _ mgO = gO g gO
..... o , t (13)
= "'' = 0 ,
n) n)In general, then, if n is any positive whole number or zero and if %( , _( are understood to be the
n th derivatives of the quantities a i , Zt with respect to m for m = 0, then
_2 _4 _6 "1
fo = 2 -2T +4T. -"6T ÷ .... 1 + cos_ = 0 ,
W3 _$ _7
go = _ --_-.+5-_.=-_. + .... sin_ = 0 ,
(14)
since all a i, fll assume the value 1 for m- 0, and furthermore
The Equations 14 are identically fulfilled and only show that the arrangements were correct.
For additional computations we shall make an analysis of Equations 15 for n = 1 explicitly; i.e.,
(15)
f 0'
g0'
i Tr2 , 7T4 7"/6
= 0 = -% _+ % -4T- a'6 --_-" + "'"
, 773 W5
= 0 = /_1' 77- _3 -_-t + /_$' "_- .... .
(16)
We have shown above that it is possible to represent the quantities %, fli as functions of m, ¢ and _.
Since ¢ and ¢ assume the value 1 for m = O, and if we set
m2 m3
¢(m} = 1 + aim + a2-2"T + aa-_-. + ...
m 2 m a
_b(m) = 1 + blm + b 2 _ + b3-_- ' * .,.
(17)
it follows, for m = 0, that
¢0 (n) = a , ¢0 (a) : b, (18)
From Equations 5 and 9 we obtain, one after another, the equations:
ill' = ¢',
a2' = -2/31 - 2m,8l' - 6m + ¢' ,
s i
_3 = -2% - _%' +¢'B1 +_1 '
, = 2 , ¢, (-2%' + ' ,0. 4 -2_ 3 - 2rr¥3 3' - 6ma 2 - 3m 0- 2 + v 2 + ¢ 6_1_ l ) etc.
If, in these and the following formulas, we set m = 0 and consider that the %, ill, _, ¢, and ¢ go
towards 1 and the derivations ¢', ¢' towards al, b,, then, after some brief calculations, we get
f
a2 _ al - 2 , fll _ bl '
a 4' --.6b 1- (al-2) , /33' _ b I + (al-2) ,
a s' _ 48b I - 21(a 1-2) , 19b, 7(a, 2).
as' _ 234b1 - 113(ai-2)' fiT' _ 109bl - 51(al-2) .
If we substitute these expressions into Equation 16, we get two linear equations:
Abl - B(al-2 ) : 0 ,
Cb I - D(a t-2) : 0 ,
(19)
the coefficients of which are represented by the n-series:
_4 7T6 n8
A = 6_-. - 48 _-. + 234_ .... ,
77 2 n 4 n 6 -r/-8
B = _T+-_T/- 21-_. + 113-_. .... ,
77 3 /7 5 77.7
C = _r --_-. + 19-gT- 109-_"T + ''' ,
n 3 -/75 n 7
D = -_. + 7"_"! - SI"_'T+ "'"
This series can be easily summed, for from
T/2 7?4 776 7-/8
cos_ = 1 --_. +'_'. --_. +'_-. .... ,
1 1 .2 .4 _6 .s
-_cos2_ = -_- - 2-_. + 8-_-.I - 32-_-. + 128_-. ....
7T2 W4 776 _8
2_sinrr = 4-_-_-8-_-t + 12_-Ft- 16-_-+ ...
we obtain by summation:
1 3 _/._. Tr4 _6 _8 )cos_ +-_-cos2_ +2_sin_ = _-+ +W_-. - 21-_-. + 113-_-. ....
3
= _+B = -2.
Similarly we representA, C, D by closed expressions: specifically,
A = -3 + 2cosvz + cos27r + 3_sinn = -4 ,
3 1
B = -_-÷ cos_ +_'cos27z + 2_sin_ - 2 ,
C = 2sinTz + sin2_ - 3vzcosrr = 37z ,
1
D = sin_z +-_-sin2_ - 2_cosTz = 27z .
J (20)
Since the determinant of the homogeneous system (Equation 19), BC-AD = 2n, differs from zero, it
follows that
a z = 2, b t = 0 • (21)
After al, b I are known, we can determine a2, b 2 in a corresponding manner from the pair of
equations f0 = 0, go = 0. From these a pair of linear equations results:
Ab 2 - Ba 2 = X2 ,
Cb2 - Da2 = Y2 '
(22)
in which A, B, C, D are defined as in Equation 20, and in which X2, Y_ represent new _r-series. The
coefficients of this 77-series can be determined numerically by using Equation 21. With the values
(Equation 20) forA,...D, we find the general expressions
Ab n - Ba. = Xn ' an = - _" Xn - _ Yn '
>
1
Cbn - Dan = Yn ' bn = -X. --_'Yn '
(23)
where x, v are known, if all a i, b i up to i = n - 1 are present numerically.
By carrying out this relatively easy analysis to the order n = 3, we obtain:
X I = 3 - 2cosvz - cos2_ - 4_sinTz = -2B = +4 ,
19 3 3
X 2 = -3 + -_-cosTz + cos2_7 -_-cos37r - 7_sinTr = -4 ,
15 5
X a = -_-- 2cosTr -_-cos2w - 3 cos3cr - 367zsin_r = +10 ,
Yl = -2sinw - sin2n + 4_cosn : -2D = -47z ,
19 3 3 3
Y2 = - T sinw + sin2w ---_sin3w +)-cosw = "_ ,
5
Y3 = -22sin=---_--sin 2w - 3sin3_ + 36wcosw = -367r .
(24)
The first coefficients of the development (Equation 17) are therefore known, and we obtain
and
m s m3
_b(m) = 1 + 2m + 9 _ + 57 _ + "'"
II m 2 rn a
¢(m) = i +-'_- -_-. + 26-_-. + "'" •
But according to Equation 6, we have
m 2 m 2 _)0
¢ - 1"03 e:' ¢ m _,
so that we can also calculate _ and 9 from Hill's power series for _0 and v0" This calculation yields:
m2 m3 1649 m* 1235(5511 ms
¢(m) = 1 + 2m + 9 _ + 57 _ + "-4--- _ + 324 _ + "'" , (25a)
i1 m2 m3 3857 m4 23153 ms
¢(m) = I +-2- -_-. + 2(5"3T/+ 2-'_U" "_'.-. + 18 -ST'. + "'" ' (25b)
and completely verifies our analysis.
CONCLUDINGREMARKS
The foregoing method of obtaining Hill's coefficients is not intended to be more than an interest-
ing attempt to achieve the same goal via a completely different mathematical route. This attempt is
interesting for two reasons: (1) Because the coefficients of the series (Equations 25a and 25b) are
determined successively from linear equations (Equation 23); and (2) Because they are in the form of
closed trigonometric expressions. It must not be overlooked, however, that this method-- elegant
though it may be -- does not lead as far as Hill' s, the developments of which can be carried out to the
fifth or sixth order without difficulty. Here we could hardly pass beyond the third order-- at least
not with the method used, which is, however, probably not the only possible one. The reason for this
is as follows:
1. The method of determining the coefficients of the trigonometric formulas (Equation 24) may
be demonstrated by using the example of x2, for which we obtain the n-series
7T2 T/.4 7T6 _8
X2 = -(5_- 6_ + 198_- 2190"_. + .... (26)
Then the arrangement
X 2 = a + bcos_r + ccos 2_ + dcos37T + e_rsinTr
= a+b
+ c
I _.2 72.4 )1 --_-., + "_-. ....
1 - 4"_- + 16-47-. ....
+ d 1 - 9-_T + 81-_-. ....
.n2 7-/4 )+ e 2_T. - 4_-_-" ....
(27)
10
leadsto theequations:
a +b + c +d = 0 ,
-b - 4c - 9d+ 2e -- -6 ,
b + 16c+81d- 4e = -6 ,
-b - 64c- 729d ÷ 6e = 198 ,
b + 256c + 6561d - 8e = -2190 ,
(28)
the number of which can be expanded as far as the coefficients of the n-series are known. Only if
the first five of these equations yield the coefficients of the trigonometric arrangement (Equation 27),
will this formula solve the problem exactly; however, when the coefficients-- a, ."e -- calculated by
means of it also fulfill the other equations (Equation 28), a demand which could only be satisfied if it
were possible to find a law (and then to prove it) by which the terms on the right side of Equation 28
-- the coefficients of Equation 26-- would increase.
In the case of the developments (Equation 24), it was possible to check the validity of the form-
ula with at least one of the supernumerary equations and to thereby prove it. Furthermore, in this
case, it is not difficult to find the law of the progression of the coefficients. The agreement with the
results of Hill also vouch for their correctness.
2. The difficulties of extending this method begin with the determination of X4, Y,, for the nu-
merical values of the coefficients of the n-series (Equation 26) increase extraordinarily, and be-
cause until now it has remained uncertain which formula (Equation 27) would achieve the right goal.
A trial of the formula
X4 = a + bcos_ + cos2_ + dcos3n + ecos4n + f_sin
and a corresponding one for ¥4 has not succeeded, probably because the formula requires still more
terms -- at least one with cos 5n.
RECOMMENDATIONS
Further experiments with this mathematical problem could be useful and interesting, and we
shall conclude this report with some suggestions:
1. The expressions of Equation 24 can be given in a simpler form. For instance, we could
write
XI = 2(1-cos_) + 2sin2_ _ 4"rrsin_,
II
and by the substitution of the "c-functions" known from the two-body motion,*
c o = cosX, c 1 = (sin)_)/_ , c 2 = (1-cos)_)/_ 2 , c 3 = (X - sin ;_) /X _ , ,.o
we then obtain, for X = _r:
E- 3 -1)],x2= 2
and
Y_ = _ (%-%) +_ _?
2. Noticeable and displeasing is the fact that the expressions of Equation 24 contain a rather
large number of disappearing terms, which are necessarily created by the algorithm which leads to
them. It should be possible to modify the process in such a way that these superfluous and compli-
cating terms can be avoided at the beginning. Here is a mathematical problem that is not only chal-
lenging, but its solution would be very useful in celestial mechanics wherever the problem of seeking
initial conditions for periodic orbits exists.
*Stumpff, K., "Calculation of Ephemerides from Initial Values," NASA Technical Note D-1415, 1962.
12 sx_-,__, ,.s G-315


REMARKS ON HILL'S LUNAR THEORY. PART II

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