7 Case II: Tangential, Normal, and Orthogonal Components
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11.7 Case II: Tangential, Normal, and Orthogonal Components
277
We are concerned with transforming from R0 ; S0 to T 0 ; N 0 . The radial direction
component of G is
R0 Á G uO r D T 0 .uO T uO r / C N 0 .uO N uO r /
(11.133)
and in the transverse direction it is
S0 Á G uO Â D T 0 .uO T uO Â / C N 0 .uO N uO Â /
(11.134)
The scalar products are evaluated knowing that uO T and uO N are along and
perpendicular, respectively, to the velocity vector. So we may write
v D rP uO r C rfP uO Â D v uO r
(11.135)
rP
D uO r uO T D uO Â uO N
v
(11.136a)
Then the scalar products
rfP
D uO Â uO r D
v
uO r uO N
(11.136b)
The first parts of Eq. (11.136) can be seen from dot products of Eq. (11.135) by uO r
and uO Â , respectively. The second parts are from the unit vectors geometry. From the
elliptic motion we have
a.1 e2 /
1 C e cos f
p
r2 fP D h D k2 Ma.1
rD
(11.137a)
e2 /
(11.137b)
Evaluation of rP from the first and elimination of fP by means of the second yields
p
k Me sin f
rP D p
a.1 e2 /
p
k M.1 C e cos f /
rfP D
p
a.1 e2
p
k M.1 C e2 C 2e cos f /1=2
vD
p
a.1 e2 /
(11.138)
(11.139)
(11.140)
278
11 General Perturbations Theory
Then Eqs. (11.136) become
uO r uO T D uO Â uO N D
e sin f
rP
D
2
v
.1 C e C 2e cos f /1=2
(11.141a)
1 C e cos f
.1 C e2 C 2e cos f /1=2
(11.141b)
uO Â uO T D uO r uO N D
Substituting Eqs. (11.141) into Eqs. (11.133) and (11.134), we have
1 C e cos f
p
N0
1 C e2 C 2e cos f
(11.142)
1 C e cos f
e sin f
S0 D p
T0 C p
N0
2
1 C e C 2e cos f
1 C e2 C 2e cos f
(11.143)
e sin f
R0 D p
T0
1 C e2 C 2e cos f
With these equations and T 0 , N 0 given, one can substitute for R0 and S0 in
Eqs. (11.131) and obtain the perturbations in the elements due to a force that
has been resolved into tangential, normal, and orthogonal components (McCuskey
1963, pp. 147–149).
We must emphasize that some authors use a different convention for the NTW
frame (Vallado 2001, p. 163). In this convention, uO T is parallel to the direction of
the velocity vector, the normal axis uO N points away from the attraction center, and
uO W D uO N uO T . In this case, the transformation in Eq. (11.142) is written as (note
the sign differences in the N 0 terms)
e sin f
1 C e cos f
R0 D p
T0 C p
N0
1 C 2e cos f C e2
1 C 2e cos f C e2
(11.144a)
1 C e cos f
S0 D p
T0
1 C 2e cos f C e2
(11.144b)
e sin f
p
N0
1 C 2e cos f C e2
11.8 Expansion of the Third-Body Potential
To obtain the time rate of change of the orbital elements, the disturbing function
R must be expanded as an infinite series, where the orbital elements appear in the
coefficients or arguments of trigonometric series. It is also possible to expand the
disturbing function as a power series, for artificial satellites, or as a Chebysehev
series, as per Carpenter (1966).
Let m and m0 be the perturbed and perturbing masses, let r and r0 be their
distances from the central mass m1 , and let r0 > r at all times (recall Fig. 5.7).
The analysis is slightly different for r0 < r, and fails for the Neptune-Pluto case
where the values change with respect to each other. The perturbation function can
11.8 Expansion of the Third-Body Potential
279
be written as
2
RDk m
0
D k 2 m0
Ä
Ä
1
xx0 C yy0 C zz0
r03
r2 C r02
rr0 cos
r03
1=2
2rr0 cos
"
Ä
1
r Á2
rÁ
cos
1
C
2
Dk m
r0
r0
r0
"Ä
r Á2
rÁ
k 2 m0
1C 0
2 0 cos
D 0
r
r
r
2
1=2
0
1=2
rr0 cos
r03
rÁ
cos
r0
#
#
(11.145)
where is the angle between the radius vectors. Expanding the first term in the
bracket by the binomial theorem, we have
Ä
1C
1=2
r Á2
rÁ
cos
2
r0
r0
Â
Ã
r Á2
3
1
rÁ
2
C cos
D 1 C 0 cos C 0
r
r
2
2
Â
Â
Ã
5
r Á3
3
r Á4 3
cos C cos3
C 0
C 0
r
2
2
r
8
15
cos2
4
C
35
cos4
8
Ã
C :::
(11.146)
The groups of trigonometric functions in Eq. (11.146) that appear due to
expanding part of the disturbing function are Legendre polynomials of the first kind,
Pn .cos /. The Legendre polynomials, generally denoted for some argument x by
Pn .x/, are useful when expanding functions such as
1
kr
D p
r0 k
r2 C r02
1
2rr0 cos
D
1
X
.r0 /n
nD0
rnC1
Pn .cos /
(11.147)
Each Legendre polynomial, Pn .x/, is an nth degree polynomial. It may be expressed
using the Rodrigues formula:
Pn .x/ D
1 dn
.x2
2n nŠ dxn
1/n
(11.148)
280
11 General Perturbations Theory
For example, P0 .x/ D 1, P1 .x/ D x, P2 .x/ D 0:5.3x2 1/ and P3 .x/ D 0:5.5x3 3x/.
In our case, the polynomials are then
P0 .cos / D 1
(11.149)
P1 .cos / D cos
(11.150)
P2 .cos / D
P3 .cos / D
P4 .cos / D
D
P5 .cos / D
1
1
Œ3 cos2
1 D Œ3 cos 2 C 1
2
4
1
1
Œ5 cos3
3 cos D Œ5 cos 3 C 3 cos
2
8
1
Œ35 cos4
30 cos2 C 3
8
1
Œ35 cos 4 C 20 cos 2 C 9
64
1
Œ63 cos5
70 cos3 C 15 cos
8
(11.151)
(11.152)
(11.153)
(11.154)
The Legendre polynomials are uniformly bounded, that is
jPn .cos /j Ä 1; n D 0; 1; 2; 3; : : :
(11.155)
1
X
r Án
Pn .cos /
r0
nD0
(11.156)
so the series
is convergent, since r=r0 < 1 by our choice. Combining Eqs. (11.145) and (11.146),
where the r cos =r0 terms cancel, we have
Ä
r Á2
k 2 m0
r Á3
r Á4
R D 0 1 C 0 P2 C 0 P3 C 0 P4 C : : :
(11.157)
r
r
r
r
We must now look at r=r0 and the polynomials Pn .cos / to determine R in terms of
2
the orbital elements. Let us consider the factors in rr0 P2 .cos / separately.
11.8.1 The Factor .r=r0 /2
In Sect. 5.9, we saw Kepler’s equation and how we can expand E in trigonometric
functions of the mean anomaly M. Likewise, we can expand to the e2 term
1
1 2
r
D 1 C e2 e cos M
e cos 2M C : : :
a
2
2
a
D 1 C e cos M C e2 cos 2M C : : :
r
(11.158)
(11.159)
11.8 Expansion of the Third-Body Potential
Â
1
eC 1
2
cos E D
281
Ã
3 2
1
e cos M C e cos 2M
8
2
3 2
e cos 3M C : : :
8
Ã
Â
1
1 2
e sin M C e sin 2M
sin E D 1
8
2
C
C
(11.160)
3 2
e sin 3M C : : :
8
(11.161)
From r=a and a=r,
Â Ã
r
aÁ
r Á a0
D
r0
a
r0
a0
Ä
1
a
D 0 1 C e2 e cos M
a
2
1 2
e cos 2M
2
1 C e0 cos M 0 C e02 cos 2M 0
(11.162)
Squaring Eq. (11.162) yields in terms to order e2
r Á2
a Á2
1 C e2 C e2 cos2 M
D 0
0
r
a
2e cos M
e2 cos 2M
C e0 cos2 M 0 C 2e0 cos M 0 C 4ee0 cos M cos M 0
C 2e02 cos 2M 0 C : : :
(11.163)
Terms such as cos2 M and cos M cos M 0 can be transformed into functions of
multiple angles or sums and differences of angles,
1
.1 C cos 2M/
2
1
cos M cos M 0 D Œcos.M C M 0 / C cos.M
2
cos2 M D
(11.164)
M 0 /
(11.165)
So Eq. (11.162) becomes
Ä
r Á2
3
a Á2
1
1 C e2 C e02
D 0
0
r
a
2
2
2e cos M C 2e0 cos M 0
1 2
5
e cos 2M C e02 cos 2M 0
2
2
2ee0 cos.M
M0 /
2ee0 cos.M C M 0 /
(11.166)
Thus, .r=r0 /2 is the sum of terms of the form Apq cos. pM C qM 0 /, where p and q are
integers, either positive, negative, or zero, and Apq are functions of a; a0 ; e; e0 .
282
11 General Perturbations Theory
11.8.2 The Factor P2 .cos /
We have the polynomial
P2 .cos / D
First we want the form of cos
3
1
C cos2
2
2
(11.167)
in terms of the orbital elements. From Eq. (11.66),
O
r D xOi C yOj C zkO D PO C ÁQ
(11.168)
D r cos f D a cos E ae
p
Á D r sin f D a sin E 1 e2
(11.169)
and
(11.170)
we have
O D aŒ.cos E
r D PO C ÁQ
p
e/PO C . 1
O
e2 sin E/Q
(11.171)
O are unit vectors, functions of ; !; i, as given in Eqs. (11.62). An
where PO and Q
equation similar to Eq. (11.171) can be written expressing r0 to appropriate unit
O 0 , where the prime denotes the disturbing planet. So we can write
vectors PO 0 and Q
cos
Á
Á
0 O0
O
O0
PO C ÁQ
P C Á0 Q
r rO 0
D
D
rr0
rr0
0 O O0
O PO 0 C Á0 PO Q
O 0 C ÁÁ0 Q
O Q
O0
P P C Á 0Q
D
rr0
Examining a term such as 0 PO PO 0 =rr0 , since D a.cos E e/ and
e0 /, using the expansion for cos E in Eq. (11.158) we have
0
(11.172)
D a0 .cos E0
Â ÃÄ
3 2
3
1
a Á a0
e C .1
e / cos M C e cos 2M
D
rr0
r
r0
2
8
2
Ä
3
3 0
3 02
1
C e2 cos 3M
e C .1
e / cos M 0 C e0 cos 2M 0
8
2
8
2
0
C
3 02
e cos 3M 0
8
(11.173)
A typical product from the two brackets is of the form cos pM cos qM 0 , which can
be transformed into the form Œcos. pM C qM 0 / C cos. pM qM 0 /=2.
11.8 Expansion of the Third-Body Potential
283
Products of these terms by series expressions for .a=r/ and .a0 =r/ can be reduced
in a similar manner. So the product 0 =rr0 takes the form Bpq cos. pM C qM 0 /,
where p and q are integers, and Bpq are functions of a; a0 ; e; e0 . The product PO PO0 D
P1 P01 C P2 P02 C P3 P03 , where the components are
P1 D cos
cos !
sin
sin ! cos i
(11.174)
P2 D sin
cos ! C cos
sin ! cos i
(11.175)
P3 D sin ! sin i
(11.176)
and similarly for P01 ; P02 ; P03 . We can write cos i D 1 2 sin2 .i=2/ and, since i is a
small quantity, sin.i=2/ i=2 D . So cos i D 1 2 2 . Then P1 D cos. C !/ C
2 2 sin ! sin , and, using 2 sin ! sin D Œcos. C !/ cos.
!/,
P1 D .1
2
P01 D .1
02
/ cos.
/ cos.
C !/ C
0
2
!/
cos.
C !0/ C
02
0
cos.
(11.177a)
!0/
(11.177b)
O can be expressed similarly.
Other components of PO and Q
We are interested in the form of PO PO0 . From Eqs. (11.177), the product of P1 P01
will consist of terms of the form cos. C!/ cos. 0 C! 0 /, which reduce to sums such
0
as Œcos. C!C 0 C! 0 /Ccos. C!
! 0 /=2. The products from PO PO0 and other
scalar products in Eq. (11.172) are of the form Cj cos .j1 C j2 0 C j3 ! C j4 ! 0 /,
where ji ; i D 1; 2; 3; 4 are integers and Cj are functions of and 0 . From this
analysis of the form of 0 P1 P01 =rr0 , we can see that the form of the perturbation
function will be
X
R D k 2 m0
Cp .a; a0 ; e; e0 ; ; 0 /
p
cos p1 M C p
2M 0 C p3
0
C p4
C p5 ! C p6 ! 0
(11.178)
where pi ; i D 1; : : : ; 6 are integers. The expression for R can be used in Eq. (11.108)
to obtain the perturbations in the orbital elements. Let M D nt C ; M 0 D n0 t C 0 ,
so
p1 M C p2 M 0 D . p1 n C p2 n0 /t C p1 C p2
0
(11.179)
Denote the angular argument in R by
Â D . p1 n C p2 n0 /t C p1 C p2
0
C p3
C p4
0
C p5 ! C p6 ! 0
(11.180)
If we consider the orbital elements of the perturbing body m0 as constants,
Eq. (11.180) becomes
Â D . p1 n C p2 n0 /t C p1 C p3
C p 5 ! C Â0
(11.181)
284
11 General Perturbations Theory
0
where Â0 contains the contributions from p2 ; p4 ; p6 ;
R D k 2 m0
X
Cp cosŒ. p1 n C p2 n0 /t C p1 C p3
0
;
; ! 0 . Then
C p 5 ! C Â0
(11.182)
p
where the summation refers to all pi ; i D 1; : : : ; 6. Then
@R
D
@
(
@R
D
@
k 2 m0
0;
k 2 m0
P
p
X
Cp p1 sin ; p1 Ô 0
p1 D 0
(11.183a)
Cp p3 sin Â
(11.183b)
X
@R
D k 2 m0
Cp p5 sin Â
@!
p
(11.183c)
p
X @Cp
@R
D k 2 m0
cos Â
@e
@e
p
Â ÃX
@Cp
1 2 0
@R
i
D k m cos
cos Â
@i
2
2 p @
X @Cp
@R
D k 2 m0
cos Â
@a
@a
p
k 2 m0
X
p
(11.183d)
(11.183e)
Ã
Â
@n
sin Â
Cp p1 t
@a
(11.183f)
where @n=@a D 3n=2a. Substituting Eq. (11.183e) into Eq. (11.108d) we have
P D
k2 m0 cos.i=2/ X @Cp
@R
D
cos Â
p
e2 @i
2na2 sin i 1 e2 p @
1
p
2
na sin i 1
(11.184)
Usually, m0 is small compared with the central mass; hence, as a first approximation,
the elements on the right of Eq. (11.184) are assumed constant. The derivative of P
becomes a periodic function of t alone, if p1 and p2 are not both zero. If p1 and p2
are both zero, P is a constant, say A. We can consider P separated, so
P DAC
X
Bp cosŒ. p1 n C p2 n0 /t C
1
(11.185)
p
where 1 D p1 0 C p3 0 C p5 !0 C 0 , and the zero subscript denotes fixed
elements. Integrating Eq. (11.185) we have
D
0
C At C
X
p
Bp
sinŒ. p1 n C p2 n0 /t C
. p1 n C p2 n0 /
1
(11.186)
11.8 Expansion of the Third-Body Potential
285
where p1 and p2 are not both zero. The linear part of the change in , At, is known
as a secular perturbation term. From Eq. (11.184), we see that, if p1 n C p2 n0 D
0, a secular term will arise. This requires commensurability in the periods of the
perturbed and perturbing planets. If P and P0 are these periods, then
n
P0
D 0 D
P
n
p2
p1
(11.187)
and p1 and p2 are integers. Jupiter and Saturn approach the ratio 5:2. The nature of
the periodic perturbations in an element such as depends on the magnitude of Bp ,
and on p1 nCp2 n0 . Bp is not large in the solar system. If p1 nCp2 n0 is large for a given
pair of values . p1 ; p2 /, then for these values of p1 ; p2 will have periodic terms of
small amplitude and short period. These are short-period inequalities. If, for a pair
of values p1 ; p2 , the quantity p1 n C p2 n0 is small, then will have a perturbation
with a large amplitude and a long period. These are long-period inequalities. From
the equations for all the elements, it can be seen that all the elements, except a, will
exhibit secular and periodic changes, from analysis made to the first order in m0 ,
such as that for . For the semimajor axis, however,
aP D
2k2 m0 X
Cp p1 sin ;
na
p
p1 Ô 0
(11.188)
aP 0;
p1 D 0
(11.189)
The semimajor axis oscillates about the mean value a0 , where a D a0 C ı.a/ with a
period
PD
2
p1 n C p2 n0
where
ı.a/ D
Â
Ã
2k2 m0 X
p1
cos Â
Cp
na0
p1 n C p2 n0
(11.190)
It has been shown that to the third power in m0 , there is no secular change in a.
This, of course, becomes important when considering the stability of the planetary
orbit. The analysis discussed can, of course, be applied to the other elements. The
expressions derived are only to first order, so they only contain the factor m0 .
We started with the assumption that the elements were constants. If we now were
to re-substitute with the secular and periodic terms for the elements, we should
obtain the second approximation with terms in t2 , which would all contain the factor
m02 ; and thus higher approximations are derived.
The semimajor axis, as mentioned, and the eccentricity, are critical for the
stability of the system. If the eccentricity changed sufficiently, we might have a close
286
11 General Perturbations Theory
approach between planets and a radical change in orbits. In 1776, Lagrange showed
that the semimajor axis was stable to the first order. In 1809, Poisson showed the
same result applied when m02 terms are included. The higher order condition is less
certain. Hagihara (1957, 1972) has discussed the stability question. Laskar (2008)
and Laskar et al. (2011) presented a number of important results on the subject.
There are many more possible approaches to perturbation theory using various
coordinates such as rectangular coordinates, Hansen variables, and so on. Likewise,
other series such as Chebyshev series can be used in place of trigonometric series
(McCuskey 1963, pp. 153–158).
11.9 The Earth-Moon System
The motion of the Moon around the Earth influenced by the Sun is a complex
problem. This is due to the large mass ratio of the Moon with respect to the Earth
and the proximity of the Moon to the Earth. The primary mass, m1 , is the Earth, m,
the perturbed mass, is the Moon, and m0 is the Sun. The Moon’s orbit is inclined
about 5ı to the ecliptic; for this discussion we will neglect this inclination. Also
we will neglect the eccentricity of the Earth’s orbit around the Sun. The Earth’s
eccentricity e D 0:016 has only a second order effect on the analysis. The problem
then is reduced to a two-dimensional one.
Let be the celestial longitude of the Moon and 0 the longitude of the Sun, as
shown in Fig. 11.3.
0
From Eq. (11.157), with D
the perturbation function becomes
RD
Ä
r Á2
k 2 m0
1 C 0 P2 cos
0
a
a
0
C :::
(11.191)
Fig. 11.3 Sun-Earth-Moon
geometry
Moon
Sun
′
′
Ψ
Ψ′
11.9 The Earth-Moon System
287
where a0 is the assumed circular radius of the Earth’s orbit. R will be truncated at
second order terms in .r=a0 /. From the Legendre coefficients we have
0
P2 Œcos.
/ D
0
1 C 3 cos 2.
4
/
and since .r=a0 /2 D .a=a0 /2 .r=a/2 , R may be written as
RD
Ä
k 2 m0
1 a Á2 r Á2
3 a Á2 r Á2
1
C
D
cos 2.
a0
4 a0
a
4 a0
a
0
/
(11.192)
R must be expressed in terms of the elements of the Moon’s orbit. Neglecting the
terms in Eq. (11.166) containing e0 , which has been assumed to be zero, we have
Ä
3e2
a Á2
r Á2
1
C
D
a0
a0
2
2e cos M
1 2
e cos 2M
2
(11.193)
in terms of second order in e. Since the Moon, Sun, and Earth have been assumed
to be in the same plane, we take D C ! C f , where f is the true anomaly of the
Moon. Then
r Á2
cos 2
a
0
D
r Á2
cos 2f C 2
a
C!
0
(11.194)
which is expanded to
r Á2
cos2 f cos 2
C!
a
h rÁ
ih rÁ
i
2
sin f
cos f sin 2
a
a
2
0
r Á2
cos 2
a
C!
C!
0
0
(11.195)
p
From r cos f D a cos E ae and r sin f D a sin E 1 e2 , and the expansions of
cos E and sin E in terms of M, we have
Â
Ã
3 2
3
1
r
cos f D 1
e cos M C e cos 2M C e2 cos 3M
a
8
2
8
3
e
2
Â
r
sin f D 1
a
(11.196)
Ã
5 2
3
1
e sin M C e sin 2M C e2 sin 3M
8
2
8
(11.197)