LITERAL
EXPRESSIONS OF
FOR THE
THE
CO-ORDINATES
MOON
I. The First Degree Terms
S. R. B O U R N E
Computer Laboratory, Cambridge, England
(Received 14 December, 1971) Abstract. This paper describes a method for finding literal expressions for the first order terms in the M o o n ' s co-ordinates. The method is based on the use of rectangular co-ordinates and was originally proposed by Euler. The variation curve and the terms dependent on the first power of the Lunar eccentricity have been obtained. These results are compared with those of Hill and a number of errors in Hill's results have been found.
1. I n t r o d u c t i o n
This paper describes a new method for determining the co-ordinates of the Moon. These will be obtained in the form of algebraic expressions where the values of the arbitrary constants are retained symbolically. The only information that is assumed about these constants is their order of magnitude so that the theory developed can be applied to other similar problems in Celestial Mechanics. The method used was originally proposed by Euler and is based on the use of a set of rectangular axes, with the Earth at the origin, rotating about the z axis so that the mean position of the Sun lies on the x axis. Referred to these axes and using the notation given in Appendix A, the equations of motion of the Moon may be written:
D2u q- 2rnDu + ~ m 2 (u -t- s)
KU
af2~
r3
as
(1.1)
D2s - 2roDs + ~zm2 (u + s)
D2 z
_ m2 Z
1.3
~U
Kz - - • 0(21 /.3 ~ 2 {~Z
(1.2)
where f2t is the potential that has been described in Appendix B. The second of these equations is the complex conjugate of the first so that either equation is sufficient. In subsequent analysis only the first of these two equations will be used. The initial step in the solution of Equation (1.1) is to assume that f 2 t - 0 and this Celestial Mechanics 6 (1972) 167-186. All Rights Reserved Copyright 9 1972 by D. Reidel Publishing Company, Dordrecht-Holland
168
s.R. BOURNE
corresponds to the following constraints: (i) the Sun's orbit is circular and has infinite radius; (ii) the Moon's orbit lies in the ecliptic. A particular solution of the resulting equation can then be found using a method first devised by Hill and later modified by Poincar6 (1892). The solution so constructed is known as the variation curve since its principal periodic term is called the variation. It contains two arbitrary constants, namely the parameter m and the phase constant to. In the first part of Section 2 the variation curve is constructed using Poincar6's method. This method is one of successive approximation and is well suited to automatic computation. The variation curve does not contain any of the small parameters e, e', ~ or k, and the next step in the solution of Equation (1.1) is to introduce the Moon's eccentricity e. This corresponds to finding the general solution of Equation (1.1) subject to the restriction that f2~ = 0 and was first carried out by Hill (1878, 1886). The method he used was to transform Equation (1.1) into the form
(1.3) where 0 is a function of m and is periodic in z. Equation (1.3) is known as Hill's equation and he solved it by means of an infinite determinant. This method is not particularly well suited to computers and in Section 2.2 a method has been devised to solve the equation without resorting to an infinite determinant. The solution of Equation (1.3) is determined except for a scale factor and it is this scale factor that is later associated with the Moon's eccentricity. Also involved in the solution of this equation is the determination of a characteristic value known as c. This constant is determined in such a way that no secular terms appear in the final solution and corresponds to a motion of the perigee. The solution of Equation (1.1) constructed so far contains all the terms that depend on m and the first power of e. To obtain the remaining first degree terms the restriction that O1 = 0 must be removed. Instead, that part of f21 that depends linearly on e' or ct must be included and this is discussed in Section 2.4. The solution of Equation (1.2) follows similar lines to the solution of the Hill Equation (1.3). There are no zero order terms in the latitude that would correspond with the variation curve; the arbitrary constant that arises as a scale factor in Equation (1.2) is identified with k, the tangent of the angle of the inclination of the Moon's orbit to the ecliptic. Thus the latitude, z, is a quantity of the first order of smallness. In Section 2.3 the initial solution of Equation (1.2) is described and as with the longitude equations it is necessary to calculate a characteristic value, known as g, that corresponds to a motion of the node. A number of expressions for the co-ordinates have been found and checked where possible by substitution into the equations of motion. These results are presented in Section 3 together with a comparison with the results of Hill (1878, 1886, 1895) and Cowell (1896). The notation used is summarised in Appendix A and in Appendix B the disturbing function, used throughout this theory, is derived.
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON, I
] 69
2. The First Degree Terms 2.1.
T H E VARIATION CURVE
The first step in the solution of Equations (1.1) and (1.2) is to ignore those terms that depend on e', c~ or z. This is equivalent to imposing the following restrictions on the Main Problem: (i) the Sun's orbit is circular and has infinite radius; (ii) the Moon's orbit lies in the ecliptic. Under these conditions Equation (1.2) is redundant and Equation (1.1) becomes:
D2u "Jr-2rnDu + -} m 2 (u + S)"
KU
?' 3
-o
(2.1.1)
since ~21 is of order e' or e and may therefore be ignored. This equation will now yield those parts of the motion of the Moon's perigee that depend on m and e. The variation curve or intermediate orbit is a particular solution of Equation (2.1.1) that is periodic in the independent variable 9 and that does not depend on the Moon's eccentricity e. Expressed in terms of the real co-ordinates x and y the initial conditions for this particular solution are: x=a,
y=0
?=b
at
t = 0.
(2.1.2)
Writing Equation (2.1.1) in terms of the real co-ordinates x and y we have: 5d- 2 m ~ -
3mZx +
rex ?,3
= 0
and
toy y + 2rn2 + - 3 = 0
(2.1.3)
r
and it may be seen that these equations are unaltered when y is replaced by - y and t by - t . The same is true for x and so it follows that, with the given initial conditions, the variation curve is symmetrical with respect to both the x and y axes. The expressions for x and y are therefore of the form: x = a ~ A, cos (2r + 1) T y=aZB,
s i n ( 2 r + 1) z
r
and so the expression for u may be written oO
u = a Z ak exp (2k + 1) iz
(2.1.4)
--o0
where the coefficients ak are real. The quantity a is a scale factor that will be discussed later. The problem now is to determine the coefficients ak, but before proceeding with this it is convenient to eliminate the term tcu/r a from Equation (2.1.1). This is achieved
170
s.R. BOURNE
by multiplying (2.1.1) by Ds and its conjugate by Du and adding to give 3m 2
(D2u Ds + D2s Du) +
2
(Du + Ds) (u + s)
K
r3 (uDs + sDu) = O,
which integrates directly to give 2~c
DuDs + 88m 2 (u + s) 2 -+- 2C =
r
(2.1.5)
where C is the constant of integration. Substituting this expression for tc/r into (2.1.1) yields after simplification
sfi + 2i ms~ -~
~
9
2
4
m2
m2us =
(3u 2 + 15s 2) + C
(2.1.6)
8
where dots denote differentiation with respect to z. To reduce the left hand side of this equation to a linear form we substitute u = (U
and
s= -lS.
Equation (2.1.6) then becomes:
S U + {~ + 2p} iS(] + 89iSU + 89,~0 - {1 + 2p + -]-p2} US 2 ?Yt m
{3U2~2 + 15S 2 ~ - z } + C .
8
(2.1.7)
(In the left hand side of this equation the parameter m has been replaced by p following Poincar6.) Representing U, S and C by the series U -- U 0 + t n 2 u l + "'" + rrt2quq + "'" S = SO + m2sl
+'"+
rrt2qSq + ' "
C--
+...+
m2qcq + . . .
c O + m2cl
the method of solution is to find uq, sq and cq, assuming that the values of uq_ 1 ---Uo, ... are known. The coefficients u~, s~ and ci are determined by repeated approximation and we may suppose that Uo = So--1 since, for any solution /2, S and C, there exists a corresponding solution 20, 2S and 22C. By a suitable choice of C, therefore, we may remove the scale factor from U and S. The scale factor, a, introduced in the expression (2.1.4) will then have to be found by reference to Equation (2.1.1) that contains •. The terms of order m ~ in Equation (2.1.7) give Co
=
-
+ 2p + 88
so that Uo, So and c o are now known. We may therefore substitute k
U
1 + ~ Uqm 2q + e 1
171
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON~ I
k
S = 1 + ~ Sqm 2q Jr- YI 1
k
C = cO + ~
Cqm 2q-Jr- 6C
into Equation (2.1.7) and assuming that Uo,..., Ua, So,...,Sa, Co,...,Ca are known we obtain the following equation determining e, r/and 6c: + {~ + 2p} i~ + 89i0 - {1 + 2p + { pZ} (e + ~/)
=
Ek+l,
(2.1.8)
where Ek+ 1 is of the form {~, cos (2nz) + ifl, sin (2nz)} n
and is obtained by collecting together those terms of order (2.1.7). Since e and r/are complex conjugates let them be given by:
m 2k+2
from Equation
e = Z {2, cos (2nz) + i/t, sin (2nz)} n
~/= Z {2, cos (2nz) - i/t, sin (2nz)},
(2.1.9)
n
where 2, a n d / t , have to be determined so that U and S are correct to order m 2k+ 2. Substituting the expressions (2.1.9) into Equation (2.1.8) and equating real and imaginary parts leads to the simultaneous equations { - 4n 2 - 2 (89 + 2p + -} p2)} 2, + {n - 2n (3 + 2p)}/t, = % { - n - 2n (3 + 2p)} 2,
+ { - 4n 2 } / t .
= fin
where n takes those values corresponding to all the terms of Ek+l. Since % and ft, are known coefficients the solution of these squations for 2, a n d / t , is A2. = - 4n2c~. + 2n (1 + 2p) ft,
A/t, = 4n (1 + p) c~, - (4n 2 + 1 + 4p +
(2.1.10)
where A is given by A = 2n2 {(8n 2 - 2 ) -
( 4 p - p2)}.
(2.1.11)
This solution is valid except when n --0 and this term, since it is not periodic, may be removed by setting ~C
--
-
-
O~0 .
In the determination of the variation curve, therefore, the secular terms are removed by finding a correction to the constant C. This is discussed in more detail by Brouwer and Clemence (1961, p. 344). The variation solution, denoted subsequently by Uo and So, may be determined by the above method to any desired order in the parameter m. This solution contains two arbitrary constants, namely m and to, and so, for a general solution of Equation
172
S.R. BOURNE
(2.1.1), a further two arbitrary constants are required. These will be introduced in Section 2.2. It was noted earlier that the solution to Equation (2.1.1) has been determined to within a scaling factor. In subsequent work, this scale factor, a, may be removed by cancellation from the equations except where it occurs in the expression for tc/r 3. The value of a is defined to be the coefficient of ~o in Uo(-1 and does not change later in the theory. The constant tc may therefore be evaluated once the variation curve has been obtained. Using Equation (2.1.1) we find that ~c =
ro
{DZuo+ 2m Duo + 12 rn 2 (Uo + So)}.
(2.1.12)
U0
Having found the value of tc we may now dispense with the scale factor provided that it is cancelled from the remaining terms of the equations of motion. The introduction of the symbol p to replace m in Equation (2.1.7) is an artifice to reduce the labour involved in its solution. The expressions for 2, a n d / , , may then be expressed in terms of c~, and ft, multiplied by rational functions of p. The factor m 2 only appears as a multiple of the non-linear term in this equation thus allowing the solution to proceed along powers of m 2. 2.2. THe HILL EQUATION It is shown in Brouwer and Clemence (1961, pp. 349-360) that the part of the motion of the Lunar perigee that depends on the Moon's eccentricity e may be determined when the solution of the equation (2.2.1)
0+0~=0 is known. The function 0 is given by 0=-
}
75+m 2 +1(/) u
DZs Ds t- 2m
}2 (2.2.2)
-x4 [--~u ~ D s J
89 (Du
D
and so, from the form given for u and s in Equation (2.1.4), 0 may be written O0
0=1+
j=l
0j cos(2j )
m2J.
where 0j is of order Equation (2.2.1) is Hill's equation that he solved using an infinite determinant. The method of solution described here is one of successive approximation that is better suited to automatic computation than Hill's. To solve Equation (2.2.1) we assume that 0 is represented by the expression O0
E Z 0, cos (2jz + cv + e). --O0
(2.2.3)
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON, I
173
The arbitrary constants in this solution are E and e and to ensure that ff is of the f o r m described it will be necessary to determine a value for c in terms of the coefficients 0i. This corresponds to a m o t i o n of the perigee a b o u t the m e a n and is used to prevent secular terms f r o m appearing in the solution (Brown, 1896, p. 53). By considering only the constant part of 0, E q u a t i o n (2.2.1) becomes
0+
=o
so that the first approximation to ~ is (2.2.4)
= EQo cos ('c + e).
We may absorb ~0 into the arbitrary constant E and in view of the solution (2.2.4) the first a p p r o x i m a t i o n to e is 1. Therefore c - 1 + 0(m) and = E cos
+
+ o (.,).
By writing u for r and v for cz, Equation (2.2.1) becomes G,,, + 2cG,, + c2~o,~,~+ 0~ = 0
(2.2.5)
where 02
N o w let m
and c = g + r/r+1, where 0 and ~ are the solutions of (2.2.5) correct to order rn r and er+ 1 and qr+ 1 are the corrections to 0 and g of order rn r+ 1. By substituting Q and c into E q u a t i o n (2.2.5) and retaining only those terms of order mr+ 1 we find that e,, + 2e,~ + e~ + e = 2r/cos v - Kr+ 1, where Kr+ 1 consists of the terms of order m r+ 1 in 0.. + 2g0.v + ?2~v~ + 00. The f o r m of Kr+ 1 is therefore
K;+l
cos
(2nu + v)
and so a particular integral of (2.2.6) is
e,+ 1 =
-~ K;"+ 1 cos (2nu 4n 2 + 4n n
+ v)
(2.2.6)
174
s . R . BOURNE
the summation excluding n = 0 and n = - 1. The term K~ 1 may be removed by setting -- -1 gK ~
r/r+1
1
thus determining that part of c of order m '+ 1. To remove the other resonant term, K7+~1, we resort to the freedom available in 0. In particular we may add to 0 any complementary function whose coefficient is of order m', since its contribution to K,+ 1 will be at least of order m '§ 1. The expression g, = A, cos ( 2 u - v), where A, is of order m', satisfies Equation (2.2.5) to order m" and its contribution to K,+I of order m "+ 1 is (2.2.7)
2mg, v + 2mgv~ + 2rng since, to the first order in m, c=l+m and 0=1
+2m.
Substituting for g into the expression (2.2.7) gives 4 m A , cos (2u - v) as the term to be added to K.+ 1 when g. is added to O. The t e r m Kr+11 may therefore be removed by adding 1 - - K.+ 11 COS (2u 4m
v)
to
Er+ 1 "
The presence of the divisor 4m in these terms reduces their order by one and it is therefore necessary to carry these approximations to order m k+ 1 in order to obtain a solution correct to order mk. To obtain the corrections to u and s once Equation (2.2.1) has been solved, certain transformations are required. They may be found in Brouwer and Clemence (1961) and are the inverse of the transformations used to obtain Equation (2.2.1). The corrections 6u and as are given by flu = (p' - q) Du as = (p' + q) Ds
(2.2.8)
where q = 0 r and r is defined by 2Dr
D (Du Ds)
r
Du Ds
(2.2.9)
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON, I
175
The equation giving p' in terms of q is
2q Dp'= { - m DuDs + W1Du - WzDs}, DuDs where
w, =
(2.2.10)
Ks
(u + s)
r3
and W2 is the complex conjugate of W1. From Equation (2.2.9)
K (DuDs) 112
where K is a constant ot integration. The value of q is therefore given by
q
0
(DuDs)l/e
(2.2.11)
where the constant K has been absorbed into the arbitrary constant E that is a factor of 0. Having regard to the form of u and s, Equation (2.2.10) may be written oO
Dp' = Z Pj (m) cos (2ju + v) ~
o0
and on integration this gives O0
~
P~ (m) sin (2ju + v)
2j + c
+ K'.
(2.2.12)
~0O
By choosing K' to be zero the initial conditions referred to in (2.1.2) remain true. Substituting the values given by (2.2.11) and (2.2.12) into the expressions (2.2.8) for 5u and 6s we find that 6u = E~ Z {Aj cos (2jz + cz) + iBj sin (2jz + cz)).
(2.2.13)
J
The constant E may be identified with the Moon's eccentricity, e, the latter being defined by Brown as the coefficient of i sin(cz) in the expansion of (-* 5u. From Equation (2.2.13) this coefficient is EBo and since B0 is a polynomial in m, its inverse can be constructed to any prescribed order in m. By substituting E - e B o ~ into 5u and 5s we may assign to e the same meaning as other Lunar theorists. 2.3. THE MOTIONOF THE NODE The latitude, z, is a quantity of the first order of smallness and so, neglecting terms of order 2 2 in Equations (1.1) and (1.2), the principal part of the motion of the node is described by the equation OZz
-
+m
.z=0.
(2.3.1)
176
S.R.BOURNE
The form of this equation indicates that the solution, z, contains a factor that is an arbitrary constant. This constant, k, may be identified with y, the inclination of the Moon's orbit to the ecliptic, and in the case of the Moon is a first order quantity. To find that part of the solution of Equation (2.3.1) that depends on the first power of k we take the value of r as defined by the variation curve. The expression {tc/r 3 + m 2} is then a known series of the form 1 + Z ~j cos (2j'c) and it will be noticed that this series is of the same form as the function 0 that occurs in Hill's Equation (2.2.1). We may therefore use the same algorithm to solve Equation (2.3.1) as was used in Section 2.2, the only difference being the interpretation of the arbitrary constants that arise. The solution of (2.3.1) may be written in the form z = ak ~ Zj sin (2jz + g'c + ~7),
(2.3.2)
J
the constants of integration being k and ~/. The value of g has to be determined in the same way that r was determined in Section 2.2 to prevent secular terms from appearing in the solution. From Equation (2.3.1) using the variation value for r it is possible to determine the part of g that depends on m. For the remaining parts of g and z new terms must be added to r and this is dealt with in Section 3. The arbitrary constant, k, is fixed in a similar way to the constant E defined earlier. It is defined to be the coefficient of the term 2 sin (g z) in the expansion of z and this corresponds to choosing the value of Z o as 2. The motion of the node is discussed in greater detail by Adams (1877) and by Cowell (1896). 2.4. THE
REMAINING FIRST DEGREE TERMS
It has been shown in Sections 2.2 and 2.3 how the terms in u that contain the first power of e and the terms in z that contain the first power of k may be determined. The expression for the latitude is now correct to the first order of small quantities. In the expression for u, however, the terms of first degree in e' and c~ have yet to be found. To find the terms in u dependent on e' it is necessary to include further terms in the expansion of 01 than have been included so far. In the above work f21 was truncated by ignoring those terms dependent on e' and ~. Retaining only those terms dependent on e', the expression for f2~ may be written
f at3 =
3m 2
} -
88 ( u +
_
m2r 2
{r } 73
1
(2.4.1)
all the subsequent terms in Equation (B12) of Appendix B being at least of order a. Representing the expression (2.4.1) by the symbol 092 the equation determining the
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON, I
177
part of u dependent on e' may be written, from Equation (1.1), as
D2u + 2mDu +-}m 2 (u + s)
~u
aa~2
r3
as
(2.4.2)
Before this equation can be solved, it is necessary to establish the form of the expression for - @o~2/~s). In Appendix B it is shown that a
!
!
(2.4.3)
= 1 + O(e')
r
and (2.4.4)
S~ = 89(u + s) + 0 (e')
and that the angle w' appears as the argument of a periodic function. Using the expressions (2.4.3) and (2.4.4) and the variation values for u and s we may, from Equation (2.4.1), write -(&o2/c~s) in the form
ae'~ Z {Aj cos (2jz + w') + iBj sin (2j~ + w')}.
(2.4.5)
J
Since w' is the mean anomaly of the Sun we have (2.4.6)
w' = n' t + ew,
where ew, is a suitable arbitrary phase constant. In terms of the quantities m and z the expression (2.4.6) may be written (2.4.7)
W' = m'~ + 8w,.
We may now express (2.4.5) in the form
ae'~ ~ {Aj cos (2j~ + re'c)+ iBj sin (2j~ + m~)},
(2.4.8)
J
the phase constant ew, being omitted since it will always be associated with an argument of the form jm'c for integer j. Suppose then that ue, is the part of u that depends on the first power of e'. We may express Ue, as the power series in m Ue,
=
ae'~ (UOe, + mU*e, + m2u e'+'") 2
(2.4.9)
where the coefficients UJe, may be written
u~, = E {C~ cos (2kv + m r ) +
iD~ sin (2kz + m,)}.
(2.4.10)
k
Suppose then that the values of #e' have been determined up to and including the term UeP,- 1. Denoting the part of Ue, correct to order m p- 1 by v_ lue,, we may substitute into Equation (2.4.2) the expression
auo + v-lUe, + ae'~u~,
178
S.R. BOURNE
for u and, by neglecting those terms whose order exceeds m p, we obtain
D:u~, + 2Du~, + -~ ~Ue, + g,} = = E {Aj cos (2j
+
iBj sin (2j
+
J
- ~coefficient of ae'mP~ from D2u + 2mDu + ~rn 2 (u + s)
(
~-5
9
(2.4.11) The value of u and s used inside the last brackets of this equation is auo +p-llle, and is therefore a known series of the same form as the expression (2.4.8). Let the entire right hand side of Equation (2.4.11) be written as Z {Aj. cos (2jz + re'c)+ iB} sin (2jz + re'c)}. J
Then on substituting the expression (2.4.10) for u~,, we find that E {(3 + #2) C{ + 2/~D~} cos (#z) + E {2#C{ +/I2D~} i sin (#z) = k
k
= ~ {A" cos (/~z) + iB'~ sin (#z)},
(2.4.12)
u
where # = 2k + m is the argument of a general periodic term. The range of summation k is now defined to be the same as that for r on the right hand side, the latter range being known. We now have two simultaneous equations that determine C~ and D~ in terms of the known quantities A~ and B~. These may be solved provided that the determinant, A, of the coefficients is non zero. Since A =/z4-/z z these equations are soluble provided that/z 4=0 or ]~2~: 1. These conditions on # correspond to the conditions (2k + m) 4:0
or
~ 1
and since m is irrational these conditions are always satisfied. By a process of successive approximation, therefore, we may determine Ue,. The determination of u~ follows similar lines by including those parts of f21 that depend on the first power of a. In this case the relevant parts of f21 are of the form
ae~ Z {Ek cos (2kz) + iFk sin (2kz)) k
and since there are no terms for which k = 0 the corresponding linear equations may be solved.
3. Results The results presented here have been obtained using the C A M A L system (Bourne and Horton, 1971a, b) that provides facilities for manipulating Poisson series on the computer. In this system both rational and floating point arithmetic are available and for these calculations the rational arithmetic has been used throughout. All solutions
L I T E R A L E X P R E S S I O N S F O R T H E C O - O R D I N A T E S OF T H E M O O N , I
179
obtained have been verified by substituting them back into the equations of motion. The coefficients for the variation curve have also been checked by substitution in Hill's linear equations (Hill, 1878, p. 135) and these are satisfied to order m 12. The variation curve differs from the value given by Hill (1878, p. 142) and these differences are given in Figure 1. An interesting error in Hill's value is in the term of order m 9 in a4. The numerator given by Hill is divisible by 3. The discrepancies found do not affect Hill's numerical results since these were calculated independently. In the present development the variation curve has been obtained to order m 12. Coefficient
Hill
Corrected
1576553 a-4
a-3
28901376
71
7477
m 7
1920
a-2
65239
m 8
403200
2674679587
m 9
63504000
238200053
m 8
368640000 8085846833
m 9
58060800000 37269889 96337920 Fig. 1.
m 8
1105920000
--44461407673
a4
m 9
14224896000
29030400000
a2
m 8
627200
79400351 a-2
m7
215040
14086643 a-3
m 9
57802752
46951 a-8
795829
m 9
m9
146886277
m 9
537600000
m 9
- - 11098919887
m 9
14515200000 18638507
m9
48168960
C o r r e c t i o n s to Hill's value for u0.
As a further check on Hill's variation curve it has been substituted into both the equations of motion and the linear equations. In both cases there is a residue of order m 7. The errors in Hill's variation curve propagate into the values given for r cos(v) and r sin(v) (Hill, 1878, p. 143). The next step in the solution of the equations of motion is to find the first order terms and this has been carried out for that part of u dependent on the Lunar eccentricity. These terms are denoted by Ue and are found by solving Hill's Equation (2.2.1). The value of Ue correct to order 10 is given in Figure 4. In order to solve Hill's equation the function 0 is required. Hill's value for 0 (Hill, 1895, p. 35) contains three errors and these are shown in Figure 2. The solution of Hill's equation involves the determination of co, the motion of the perigee. The corrections to the value given by Hill (1895, p. 39) are presented in Figure 3 and the coefficients given here agree with those of Deprit (1971).
180
s.R. BOURNE Coefficient
00
Corrected
Hill 49359583
49539583
m 1~
124416
12A416
01
-- 2641291011773 11943936000
0~
252382507 3456000 Fig. 2.
Term
ml0
ml0
m8
-- 2813929549973 1194393600 126191191 1728000
ml0
m8
Corrections to Hill's value for 0.
Hill
Corrected
-- 1547804933375567
-- 1547775442175567 40768634880
40768634880
mll
m 1~
-- 818293211836767367 4892236185600 Fig. 3.
-- 818429336556024967 4892236185600
Corrections to Hill's value for co.
.< 1/4 + 5/32 mr2 + 2417/3072 mt3 + 25633/36864 mt4 + 2685221/1769472 mt5 + 28093511/2654208 mr6 + 166188135929/2548039680 mt7 + 15163524127/477757A,4 mt8 + 100112167608846277/73383542784000 mt9 8 > c o s ( t + ct) +<
1/4 + 7/32 mt2 + 1489/3072 mt3 + 25073/36864 mt4 + 4847077/1769472 m t5 + 34250947/2654208 mt6 + 188037400441/2548039680 mt7 + 136176204979/382205952 mt8 + 112407147672547973]73383542784000 mt9 > i sin(t + ct)
+<
3/4 -- 45/32 m -- 531/128 mt2 -- 21139/2048 m t 3 -- 245881/12288 m t4 -- 10742797/294912 m t5 --88579349/884736 mt6 -- 336481254193/679477248 mt7 -- 28083421706623/10192158720 mt8 --34833304967464231/2AA,6118092800 mt9 > COS(t--et)
+<
--3/4 +45/32 m + 579/128 mt2 + 23743/2048 mt3 + 262783/12288 mt4 + 2999545/73728 mt5 + 109360835/884736 mt6 + 430941397225/679477248 mt7 + 34949672319823/10192158720 mt8 + 20958641071755353/1223059046400 mt9 > i sin(t--ct)
+<
3/16 mt2 + 27/64 mt3 + 89/192 mt4 + 94849/184320 mt5 + 123553/368640 mt6 -- 3154559239/2654208000 mt7 -- 3275523992933/1114767360000 m t 8 + 4329773145961607]312134860800000 mt9 > cos(3t + ct)
+<
3/16 mt2 + 27/64 mt3 + 341/768 mt4 + 71641/184320 mt5 + 3493/40960 mt6 -- 3613917811/2654208000 mt7 - - 3119874346817/1114767360000 mt8 + 4621739182826093/312134860800000 mt9 > i sin(3t + ct)
--
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON~ I
+<
15/32 m + 277/128 mt2 + 35969/61AA, mt3 + 84125/9216 mt4 + 810089/8847360 m t 5 -- 1375348843/33177600 m t 6 -- 18791200551047/254803968000 mt7 + 1545796647796651/3822059520000 mt8 + 52248524946532332983/12842119987200000 mt9 > cos(3t -- ct)
+ < 15/32 m + 277/128 mr2 + 36689/6144 mt3 + 91073/9216 mt4 + 25586801/8847360 mt5 --617514791/16588800 mt6 -- 23314165379303/254803968000 mt7 + 942085360961898/3822059520000 m t 8 + 43601091632183649167/12842119987200000 mt9 > i sin(3t -- ct) +<
177/1024 mr4 + 3447/5120 mr5 + 780991/614400 mt6 + 216549551/137625600 mt7 + 787299688727/867041280000 mr8 1520732268990073/485543116800000 mt9 > cos(5t + ct) -
+<
-
177/1024 mr4 + 3A,A7/5120 mt5 + 767941/61A,A,00mt6 + 199267151/137625600 mt7 + 500879793527/867041280000 mt8 1765808127489673/485543116800000 mr9 > i sin(5t + ct) -
-
+ < 45/128 mt3 + 137/64 mt4 +911059/122880 mt5 +131100257/7372800 mt6 + 179383253089/6193152000 mt7 + 22280358416363/867041280000 mt8 + 28826891147063693/971086233599999 mt9 > cos(5t -- ct) + < 45/128 mt3 + 137/64 mt4 +906559/122880 mt5 + 128859557/7372800 mr6 + 171370665139/6193152000 mt7 + 18789668130263/867041280000 m t 8 + 54571542878226079/2913258700800000 mt9 > i sin(5t -- ct) + < 529/2072 mt6 + 266307/286720 mt7 + 177151103/72253440 mt8 + 7592305384199/1820786688000 mt9 > cos(7t + ct) + < 529/3072 mt6 + 266307/286720 mt7 + 351498181/144506880 mt8 + 7348904096699/1820786688000 mt9 > i sin(7t + ct) + < 2655/8192 mt5 + 79593/32768 mt6 + 36646765/3670016 mt7 + 18462240037/642252800 mt8 + 396638989642013/6473908224000 mt9 > cos(7t- ct) + < 2655/8192 mt5 + 79593/32768 mt6 + 36500605/3670016 mt7 + 18231153937/642252800 mt8 + 385640301890213/6473908224000 mt9 > i sin(7t--ct) + < 46825/262144 mt8 + 3920785/3211264 mt9 > cos(9t + ct) + < 46825/262144 mt8 + 3920785/3211264 mt9 > i sin(9t + ct) + < 2645/8192 mt7 + 3945485/1376256 mt8 + 5252550941/385351680 mt9 > cos(9t--ct) + < 2645/8192 mt7 + 3945485/1376256 mt8 + 20463011/1505280 mt9 > i sin(9t--ct) + 702375/2097152 mt9 c o s ( l l t - - c t ) + 702375/2097152 mr9 i s i n ( l l t - - c t ) Fig. 4.
Ue to order 10.
181
182
S.R.
BOURNE
Expressions for the latitude and the motion of the node have been obtained and are listed here in Figures 5 and 6. The latitude zl is given correct to order m 1~ The value for g agrees with that given by Cowell (1896) to order m s and in Figure 5 the terms of order m 9, m a~ m11, not obtained by Cowell, are shown.
1711851619 679477248
m 9
§
300364819183 40768634880
Fig. 5.
m 1~
§
33552548605553 4892236185600
m 11
Terms of order m 9, m 1~ m 11 in #0.
sin(gt) § < 3/16 mt2 § 1/2 mt3 § 197/384 mr4 § 3067/18432 mt5 -- 348A,A,9/1105920 mt6 --257557433/265420800 mt7 --36359330219/15925248000 mt8 - - 8858566190209/1911029760000 mt9 --5342875515568159/802632499200000 m t 10 > sin(2t § § < 3/8 m § 29/32 mt2 § 2029/1536 mr3 f 18875/18432 mt4 +44633/442368 mt5 -- 5569883/5308416 mt6 --6659773567/2548039680 mr7 - - 801031045801/152882380800 mt8 -- 152716349343161/18345885696000 mt9 - - 8601761005446173/1100753141760000 m t 10 > sin(2t--gt) + < 25/256 rnt4 + 803/1920 mt5 § 377701/460800 mr6 § 101154847/110592000 rot7 § 235562237/573A, A,0000 mt8 -- 24285139772478/26011238400000 m t 9 - - 129664600352464537/32774160384000000 mt 10 > sin(4t § gt) § < 9/128 mr3 § 105/512 mt4 § 8553/40960 mt5 -- 123367/819200 mt6 --270482311/294912000 mr7 -- 34491329623/17694720000 mt8 - - 190994021606809/59454259200000 m t 9 -- 108550165172266009/24970788864000000 m t 10 > sin(4t--gt) § < 833/12288 mt6 § 27943/71680 mt7 § 386808239/361267200 mt8 § 3318737997637/1820786688000 mr9 § 498242863668241/254910136320000 m t 10 > sin(6t § gt) § < 75/2048 mr5 § 18797/122880 mr6 § 18593731/68812800 mt7 § 4615828951/28901376000 mr8 -- 29226596856839/72831467520000 mr9 - - 15738621516321323/10196405452800000 m t 10 > sin(6t--gt) §
3537/65536 mt8 § 18638507/48168960 mt9 § 23383473271/17340825600 m t 10 > sin(8t -+-gt)
+<
833/32768 mt7 § 1926881/13762560 mr8 § 8128627261/23121100800 mt9 § 13695396713027/29132587008000 m t 10 > sin(8t--gt) § 732413/15728640 m t 10 sin(10t § gt)
§ < 10611/524288 m'9 § 71283131/513802240 m t l 0 > sin(lOt--gt)
Fig. 6.
zl to order m 1~
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON, I
183
Acknowledgements
My thanks are due to Dr D. Barton for his assistance with the algorithm for solving Hill's equation. This work was supported initially by the Science Research Council to whom I am grateful. I would also like to thank the Master and Fellows of Trinity College, Cambridge, for their support during the completion of this work. Appendix A NOTATION
The notation used is, as far as possible, the same as that used by Brown (1896) and is given here for reference. Where applicable unprimed symbols refer to the M o o n and primed symbols to the Sun. n, n' - the observed mean angular velocities of the M o o n and Sun. x, y, z - the co-ordinates of the M o o n referred to a set of rectangular axes with the Earth at the origin and whose x y plane coincides with the ecliptic. Further the mean position of the Sun is assumed to lie on the x axis so that the frame of reference is rotating about the z axis with angular velocity n'. , y, - t h e co-ordinates of the Sun referred to axes parallel to those used for the X, Moon but whose origin coincides with the centre of gravity of the Earth and the Moon. ,u - the sum of the masses of the Earth and Moon. m p - the mass of the Sun. a, e - the scale factor and eccentricity of the Moon's orbit. a,e - the semi-major axis and eccentricity of the Sun's orbit. f ' , w' - the true and mean anomalies of the Sun. ?, - t h e tangent of the angle of inclination of the Moon's orbit relative to the ecliptic. = x cos ( f ' - w') + y sin ( f ' - w'). 81 2 = X 2 +y2 and r 2 = X 2 Ar- y 2 + Z 2" 0 t
r
t2
U
t
=
X,2 + y , 2 .
= x +iy
S--x~iy.
rn = n ' / v ,
Y
= exp (iz), 0~
and
D-
tc = p l y 2,
"c= v ( t - to).
d/cl .
= a/a'.
Appendix B THE DISTURBING FUNCTION
The form of disturbing function used is given by m
r
f2 = # ~ r (r '2 - 2 r ' $ 1 + r2) 1/2
m
r'
t
m
rS
r '2
1
(B1)
184
S.R.BOURNE
and although f2 is not the true potential, the differences between it and the true value are so small, in the Lunar theory, that it is usual to ignore them. The use of this form of the potential does not remove any of the essential difficulties involved in solving the equations of motion; it does, however, make the analysis more convenient to handle. For a derivation of the expression (B 1) and for a discussion of the differences between f2 and the true potential the reader is referred to Brown (1897-1908). For the Lunar theory described here we require that the potential be expanded in terms of the variables u, s and z and the parameters m, e', ~, w' and this is facilitated by writing (B 1) in the form: O0
=
v2
__
-[- 2 m 2
r
(a' 3
a
\-/1
-a'
n-
2
t
n-
2
rn
a ;':2P"
2
where P , ( x ) is the nth Legendre polynomial. The term m ' / r ' has been omitted from (B2) since it is independent of the co-ordinates of the Moon and will not therefore be present in the equations of motion. Also, m' has been replaced by its elliptic value n ,2 a , 3 .
From the well known properties of an ellipse a'
=
r'
dE'
(B3)
dw'
where E' is the eccentric anomaly given by E ' = w' + e' sin E'.
(B4)
The expansion of E' in powers of e' may be found from Equation (B4) by successive approximations, the first such approximation being Eo = w'. Subsequent values of E', correct to order n + 1 in e', may be calculated using the iterative procedure: !
!
E,+ 1 = e sin (E', + w'). The form of the solution for E' is therefore O0
emsn(Wt)
E' = w' + ~ n=l
where e, is a periodic function. We then have, using Equation (B3),
a'
= 1+
l
r
n = 1
e'" de. dw'
(BS)
*
The function St is defined by the equation S: = x cos ( f ' = 89
w') + y sin ( f ' + lse~(I '-w')
w')
(B6)
and may be found as a function of e' and w' w h e n f ' has been expressed in terms of
LITERAL EXPRESSIONS FOR THE CO-ORDINATES OF THE MOON~ I
185
these parameters. Again, from the properties of an ellipse, it is known that: d f '7= ( a ' ) 2 dw- \77] (1
-
(B7)
e'2) 1/z.
However, the expansion of a'/r' and hence of its square is already known from (B5). We may therefore rewrite (B7) in the form: df'
oo = 1+
dw'
e'"r
Z
(B8)
n= 1
where ~b, is a known function, periodic in w'. Integrating (B8) and setting the arbitrary constant so t h a t f ' - w ' when e'--0 we obtain: f'-
w'= ~
n=1
e'"
f
r
(B9)
and by substituting this result into the expression (B6) for $1 we find that: $1 = 89(u + s) + {terms of order e'},
(B10)
We may now expand the expression for (2 given by Equation (B2) along powers of e' and a with the angle w' only appearing as the argument of a periodic function. Using the expressions (B5) and (B10) for a'/r' and $1, that part of 2(2/v 2 independent of e' and a is given by: 2~
- - -1- m2 (k (U -[- S) 2 - ( u , +
(Bll)
r
since P2 (x) = (3x 2 - 1)/2 and /,,2
--US+Z
2.
Separating this part of the potential from those terms of order e' or ~ we may write (B2) in the form: -2- f 2 = -2x - + 88 2 (u + s) 2 _ V2
m 2 z 2 -- m 2 u s + ~ 1
r
where f21 is given by: (21 __ 3 m 2
a'
+ 2m2 \ r s j
$2
.=3
I(U 4
-[- S) 2 } - - m 2 r 2
-- 1 ~ +
(\r'] \-rs]
a ~- 2 P.
00
--'EWn. 2
(B12)
186
s.R. BOURNE
References' Adams, J. C.: 1877, Monthly Notices Roy. Astron. Soc. 38, 43-49. Barton, D.: 1966, 'A New Technique for the Lunar Theory', P h . D . Thesis, Cambridge University. Bourne, S. R.: 1969, 'Automatic Algebraic Manipulation and its Application to the Lunar Theory', P h . D . Thesis, Cambridge University. Bourne, S. R. and Horton, J. R.: 1971a, The CAMAL System Manual, Computer Laboratory, University of Cambridge. Bourne, S. R. and Horton, J. R. : 1971b, The Design of the Cambridge Algebra System, Proc. SYMSAM II. Brouwer, D. and Clemence, G. M. : 1961, Methods of Celestial Mechanics, Academic Press. Brown, E. W. : 1897-1908, Mem. Roy. Astron. Soc. 53, 39-116; 163-202; 54, 1-63; 57, 51-145; 59, 1-103. Brown, E. W. : 1896, An Introductory Treatise on the Lunar Theory, Cambridge Univ. Press. Brown, E. W.: 1895, Am. J. Math. 17, 318. Cowell, P. H. : 1896, Am. J. Math. 18, 99. Delaunay, C. E. : 1860--1867, Mem. Acad. Sci. Paris 28 and 29. Deprit, A. : 1971, Astron. J. 76, 273-276. Hill, G. W. : 1878, Am. J. Math. 1, 5-26, 129-147, 245-260. Hill, G. W. : 1886, Acta Math. 8, 1-36. 9 Hill, G . W . : 1895, Ann. Math. 9, 31-41. Meusen, P.: 1971, Celest. Mech. 3~ 289-311. Poincar6, H.: 1892-1899, Legons de M~canique Cdleste, 2 part 2, Gauthier-Villars, Paris.
