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Perturbations of near-circular orbits by the earth's gravitational potential
Planet.
Space Sci. 1966, Vol. 14, pp. 433 to 444.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
PERTURBATIONS OF NEAR-CIRCULAR ORBITS BY THE EARTH’S GRAVITATIONAL POTENTIAL G. E. COOK Royal Aircraft Establishment, Farnborough,
Hants
(Received 13 January 1966) Abstract-The behaviour of near-circular orbits in the Earth’s gravitational potential is discussed in detail. The major axis rotates at a non-uniform rate or oscillates about a position of equilibrium, depending on the initial conditions. 1. INTRODUCTION
The Earth’s gravitational potential has two important perturbing effects on a satellite orbit. First, although the orbital plane remains inclined at an almost constant angle to the equator, it precesses about the Earth’s axis in the direction opposite to the satellite’s motion. Second, provided the eccentricity is not too small, the major axis rotates at a nearly constant rate due to the even zonal harmonics; while the eccentricity, and hence the perigee radius, undergoes a sinusoidal oscillation due to the effect of the odd zonal harmonics. For near-circular orbits, however, the rate of rotation of the major axis can be markedly non-linear, or even oscillatory, while the variation in eccentricity is no longer sinusoidal. The total change in eccentricity is normally small, but near the critical inclination (63.4”) it can be extremely large. The effect of odd zonal harmonics on orbits of small eccentricity has been treated previously by KozaW and Izsak.@) Here the analysis is extended to include the general odd harmonic and the nature of the motion is discussed in detail. 2. EQUATIONS
OF MOTION
The orbit of a satellite is normally specified by six osculating elliptic elements. These are the semi major axis a, the eccentricity e, the argument of perigee w, the inclination i, the right ascension of the ascending node Q and the modified mean anomaly at the epoch x*. These elements are not suitable for defining an orbit of small eccentricity since their time variations contain singularities at e = 0. It is well known that these singularities can be removed by introducing the transformation 5 = e cos w
(1)
7 = e sin wI
and by computing the perturbation in cu + M, M being the mean anomaly, rather than the perturbations in IXand M separately. Since only the long-period development of the orbit is being considered here, the variation of o + M will not be evaluated. The rates of change of e and u) due to a disturbing function U’ are given by Lagrange’s planetary equations :(3)
P=f(l~~2)l’2[(1 _e~)l~2?$-~] b=-
1
e
1-
( 1 w
cot i au' -. @~)~/~(l - e2)lj2 ai
e2 li2 aU’
-ae
I
433
(2)
434
G. E. COOK
From equations (1) and (2) we obtain
(=
(p~)-~/~[~ ((1- e2) z -
(
-
(1 - e2)l12 g
cos u)
- e( 1 - e2)-l12 cot i ‘g)
+ = (pQ)-r/a [’e (( 1 - e2) g +
(1 - e2)l/” g)
- (1 - e2)l12$)
(1 _ e2)1/2‘g 1
sin ~01
(3)
sin (0
- e(1 - e2)-l12 cot i -au!cos co . ai 1
1
It is convenient to retain e and w in the analysis until the final equations for $ and rj are obtained. 3. DEVELOPMENT
OF DISTURBING FUNCTION
On neglecting longitude-dependent terms, the Earth’s gravitational potential U at an exterior point distant r from the centre, and having geocentric latitude 4, may be expressed in terms of spherical harmonics:
where p is the product of the gravitational constant and the mass of the Earth, and R is the Earth’s equatorial radius. The J,, are constants and P,(sin 4) are Legendre polynomials of degree n and argument sin c$. Terms in (5) can be developed using the addition theorem for zonal harmonies:(4) P,(sin 5) = P,(cos i)P, cos 5 ( ) +2
5 @-s)! 8=l(n
+s)!
PS(cosi)P
..(cos~)coss(u-;),
n
where the P,’ are associated Legendre functions, i is the inclination of the orbit to the equator and U,the argument of latitude, is equal to w + 8,&Jbeing the true anomaly. For the general term in the disturbing function we have
. Terms in (6) containing the true anomaly can be writtenc5) as functions anomaly M using Hansen’s coefficients X;(n+l)*m: a r (-1
n+1
exp jmf3 =
2 X;(n+l),m exp jvA4, v=-cc
(6)
of the mean
LOW ECCENTRICITY
wherej = 2/z. u
SATELLITE ORBITS
435
The result is
n
v=--00
X$+1)*0 9 exp jvM
x aexpj(s
(cu -3
+ VM)]
where W denotes the real part. On neglecting short-period terms, we obtain u
=
-J
?I
I”
“R
0! nf1[
P,(cos
a
i)P,(O)Xp+1)*0
2 (n-d!
P2(cos s=l(I2 +s)!
+2
i) P,s(O)Xp+l),s
cos s co -
(
;
)I .
(7)
The Hansen coefficients in (7) are given in terms of hypergeometric functions by the relation Xr(n+l),m =
“) F(m + 1 - IZ,1 - n; Iy1+ 1; j39,
(1 + p2)“(1 - a8,1W2”(-8,-(-‘~
(8)
where p=
e
1 + (1 - e2)lj2 ’
Since X;(n+l),s = 0 ifs = n, terms containing sectorial harmonics are eliminated from the summation. Also Pns(0) vanishes for (n + s) odd. Hence the only terms remaining have (n + s) even and IZ# s. 4. EFFECT OF EVEN HARMONICS
The second harmonic is the only even harmonic which need be retained in the analysis, since J, is of order 1O-3while all higher J,, are of order 10-6. From (7) we have U, = -J2 5
0
f
“p,(cos
i)P2(0)X,-3,0.
The explicit form for U, is u2
=
J2 5
(!j3(l-
From equations (3) and (4) the contributions
e2)-3/2 (2 COS2i -&).
of J2 to & and 4 are given by
t2 = -3J2(5)1’2(;)2(I
-if)
q2 = 3J2(91’2(;)2(
1 - if)‘,,
where f = sin2 i, if terms of order e2 are neglected.
7
(9) (10)
G. E. COOK
436
5. EFFECT OF ODD HARMONICS
For IZodd we have, from equation (7), P aR n+l i: (n-s)! s=1(n + s)! 0
U,=-2J,+
Since the Hansen coefficients appearing in this equation are of order es and we are confining our attention to orbits of small eccentricity, only the first term in the summation need be retained. On using equation (S), we have XO-(n+l)*l = g(n - I)e + O(e3), so that -
P,l(cos i)P,l(O)e sin w + O(e3) .
From (3) and (4) the contributions of the odd harmonics to t and 4 are given by
60~= Z:J, %dd
=
($)1’2 (5)nin=) P,'(cos
i)P,l(O)
+O(e2)),
(11) (12)
o(e2)9
where the summation is over all odd )2for n 2 3. Explicit forms for the associated Legendre functions appearing in the above equation can be obtained from their definition : . . (2n - 2t)! (-1)” (13) P,Ycos9 = y 7 (cos9n-1-2t, (n
_
1
_
2t),
(n
_
t)
!
t!
where the summation over t is from 0 to (n - 1)/2. 6. MOTION
of the even and odd harmonics given by equations
On combining the contributions (9) to (12), we have
i=-krj+C (14)
rj = k5,
I
where k = 3 ($)““J2(f)“(1 C = &l/2
2 Jn@
nHj
-;f) Pnl(0)P,l(cos i).
The solution of equations is given by 5 = A cos (kt + a)
q = A sin (kt + a) + C/k, A and u being constants of integration depending on the initial conditions. elements can be recovered using the relations e = (E2 + q2)l12 co = tan-l (r/5). It is evident that e2 varies sinusoidally with time.
(1% The conventional
LOW ECCENTRICITY
SATELLITE ORBITS
437
In the (E, 7) plane the motion is represented by a circle of radius A, centre (0, C/k), as shown in Fig. 1. The length of a radius vector from the origin to a point on the circle represents the eccentricity, while the angular distance of the radius vector from the positive &axis gives the argument of perigee. If A > IC/kl, the major axis rotates with period 2rr/lkI and the eccentricity varies between A - C/k and A + C/k. Since the circle is traced out at a uniform rate, the motion of perigee can be extremely rapid if A exceeds IC/kl by only a small amount. Although there is a large variation in w as e -+ 0, there is a correspondingly large variation in the mean anomaly. It can readily be shown that the variation of the quantity o + M remains finite. When the magnitude of A greatly exceeds that of C/k, the variation of e with cc)is approximately sinusoidal.
(a)
A;
c/k
. (b)A
f
FIG.la and b. MOTIONOFORBIT IN (5, q) PLANE.
If A -=cjC/kl, the major axis oscillates about w = 7~12if C/k > 0 and about w = -7~12 if C/k < 0. The motion has a period 2?r/lkl and an amplitude sin-l (IAk/CI), while the eccentricity varies between lC/kl - A and jC/kl + A. For an orbit starting at the point (0, C/k) in the (E, q) plane there is no perturbation in either the eccentricity or the argument of perigee. 7. DISCUSSION To illustrate the behaviour of near-circular orbits we retain only the first four odd harmonics since these are, at present, the most accurately determined. The appropriate
438
G. E. COOK
I ,
/
I
I
I
J3TO J9
TERMS
INCLUDED
14
I.0
08
‘56
13';L 0'
__
5.2
//
/
I
/’
0
-32
t
-0.4
I C
1
c
15 INCLlNATlON
FIG. 2.
VARIATION
OF
C/k
WJTH
E -
7
DEqREEs
ORBITAL INCLINATION FOR
R/a = 0.9.
80
9i
LOW ECCENTRICITY
SATELLITE ORBITS
439
values of J, are 106J, = 1082.7 -2.56 -0.15 106J, = -0.44 106J, = 0.12. 106J3 = 106J, =
The explicit form for C is C=
-~($)1’z(~)3sini(J3(l +;
--if) J,(;)4(1
--~J5(~)“(1--~f+~f2)
-;f+!?fL
$f")
R6
-gJg-
a (I(
Since both k and the J3-term in C possess the factor (1 - if),
C/k:is of the form
sin i + terms in J5, J,, etc.
C/k = -g(J,/J&R/a) .
-._
3
i3
3
t--
e
2
,
0
ARGUMENT
FIG.
3.
VARIATION
OF
OF ECCENTRICITY
PERIGEE
WITH
ORBITS HAVING
w
ARGUMENT
R/a = 0.9.
OF PERIGEE FOR FOUR
440
G. E. COOK
The variation of C/k with inclination for R/a = 0.9 is shown in Fig. 2. For most values of inclination C/k is positive, so that rapid motion of perigee will normally occur when o is between rr and 27r if A > C/k. For inclinations near 66” the odd harmonics have very little effect on the eccentricity; but the total variation in e can become large as the critical inclination is approached. In practice the theory developed here becomes invalid as i + sin-l 2/d/5, since the effects of the higher even harmonics must be taken into account when the factor (1 - $j’) becomes small, of order 103.
Fm.4.
VARLWION
OF ECC~TRI~ITY WITH TIME FOR AN ORBIT WITH AN INCLINATION OF 60" AND R/a = 0.9.
Figure 3 shows the eccentricity plotted against argument of perigee for four orbits with R/a = O-9 and 5 = lo’, 7 = 0 initially. These orbits have A > C/k and the form of the variation of e differs markedly from a simple sine curve, which is the appropriate form for an orbit with moderate or large eccentricity (see, for example, Ref. 6). As noted by Izsak,@) a satellite having an orbit with a very small eccentricity can still be used in the determination of odd harmonics. Equations of the form (15) should be fitted to the observed orbital elements after correcting for the effects of luni-solar perturbations, radiation pressure and air drag. The resulting value of C/k then gives one linear equation between the odd J,,.
LOW ECCENTRICITY
441
SATELLITE ORBITS
An example of the variation of eccentricity with time is shown in Fig. 4, while typical variations of co with time are illustrated in Figs. 5 and 6 for orbits having the same initial conditions as Fig. 3. As noted earlier, the rate of rotation of the major axis can be extremely rapid when w is near 37~12. 40
35
50 w DEGREES
0
IO
20
30
40
50
60
TIME-DAYS
FIG.~. VARIATION OF ARGUMENT OF PERIGEE WITH TIME FOR TWO INITIAL ECCENTRICITIESOF 0.001 (R/U = 0.9).
ORBITS HAVING
Figure 7 shows how eccentricity varies with argument of perigee when A < C/k. The three orbits have an inclination of 80’ and R/a = 0.9. The motion is represented by a closed curve since the major axis oscillates about w = 7r/2. 8. EXTRANEOUS PERTURBATIONS The motion of near-circular orbits, as described in section 6, will not normally be much affected by extraneous perturbations. Air drag should have a negligible effect over a
G. E. COOK
442
time interval 277/lkl if perigee is above about 400 km. The second harmonics in the solar and lunar disturbing functions contribute terms proportional to eccentricity in equation (14). These terms can be neglected, however, since their magnitudes are small compared with the .7,-terms. The contribution from the lunar parallactic term does not contain a factor e and can be large if the semi major axis is large. This limits the applicability of the theory to orbits with values of a less than about 3 Earth radii. o-
I
L
f
3-
3-
0
100
200
300
TIME-DAYS
FIG. 6. VAFUTION
OF ARGUMENT OF PERIGEE WITH TIME FOR TWO INITIALECCENTRICITIESOF o-0()1(R/U = 0.9).
ORBITS HAVING
The main extraneous perturbing force is solar radiation pressure, which introduces into equation (14) a term with a maximum value of @(a/,~)~/~, where F is the force per unit massactingon the satellite. For a satellite with a typical area-to-mass ratio, e.g. 0.025 m2/kg, in a near-Earth orbit the magnitude of $F(u/,u)~‘~is small compared with C. In fact the magnitude will be less than 10 per cent of C provided that the inclination is not close to zero or 66”. Except at these inclinations the integrated effect on 5 and 7 will normally be quite
LOW ECCENTRICITY
SATELLITE ORBITS
2 3F PERIGEE
FIG.
7.
443
J
w
EXAMPLES OF THE VARIATION OF ECCENTRICITY WITH FOR ORBITS WITH A < C/k AND AN INCLINATION OF
ARGUMENT
OF PERIGEE
80” (R/u = 0.9).
444
G. E. COOK
small, since the magnitude of the solar radiation pressure terms varies as the orientation of the orbit with respect to the Sun changes. In general the motion will exhibit small oscillations about the solution discussed in section 6. Even if a resonant term occurs, the odd harmonics will dominate the motion; the resonant term will slightly displace the centre of the circle in Fig. 1, while the remaining terms will produce small departures from the circle. Although solar radiation pressure will not greatly affect the value of e, it could have a substantial effect on w if the orbit passes near the origin of the (5, q) plane. Acknowledgement-The author thanks Miss Diana W. Scott for performing the computations. Copyright reserved. Reproduced by permission of the Controller, H.M. Stationery Office.
Crown
REFERENCES 1. 2. 3. 4.
Y. KOZAI, Astronom. J. 64,367 (1959). I.G. IZSAK, The Use of Artificial SuteIZitesfor Geodesy, p. 329. North-Holland, Amsterdam (1963). W. M. SMART, Celestial Mechanics. Longmans Green, London (1953). H. JEFFREYS and B. S. JEFFREYS,Methods of Mathematical Physics (Third edition) p. 646. Cambridge University Press (1956). 5. H. C. PLUMMER, An Introductory Treatise on Dynamical Astronomy p. 44. Cambridge University Press (1918), republished by Dover, New York (1960). 6. D. G. KING-HELE, G. E. COOK and D. W. SCOTT,Planet. Space Sci. 14,49 (1966). Pe3KtMt+--&?TaJIbHO 06cyxc~aeTcR IIOBeAeHIleIlOqTH KPYrOBblX OP611T B FpaBHTalViOHHOM IlOTeHqSiia~e 3eMJlH. rJlaBH3X OCb Bp3LQiWTCE C Hep3BHOMepHOi CKOPOCTbH),HJIIl ?Ke Kone6neTcFI BOKPyr IIOJlOH(eHIIHPaBHOBeCHH, B 33BHCHMOCTI4 OT HZlSUE.HbIX yCJlOBH2t.
Space Sci. 1966, Vol. 14, pp. 433 to 444.
Pergamon
Press Ltd.
Printed
in Northern
Ireland
PERTURBATIONS OF NEAR-CIRCULAR ORBITS BY THE EARTH’S GRAVITATIONAL POTENTIAL G. E. COOK Royal Aircraft Establishment, Farnborough,
Hants
(Received 13 January 1966) Abstract-The behaviour of near-circular orbits in the Earth’s gravitational potential is discussed in detail. The major axis rotates at a non-uniform rate or oscillates about a position of equilibrium, depending on the initial conditions. 1. INTRODUCTION
The Earth’s gravitational potential has two important perturbing effects on a satellite orbit. First, although the orbital plane remains inclined at an almost constant angle to the equator, it precesses about the Earth’s axis in the direction opposite to the satellite’s motion. Second, provided the eccentricity is not too small, the major axis rotates at a nearly constant rate due to the even zonal harmonics; while the eccentricity, and hence the perigee radius, undergoes a sinusoidal oscillation due to the effect of the odd zonal harmonics. For near-circular orbits, however, the rate of rotation of the major axis can be markedly non-linear, or even oscillatory, while the variation in eccentricity is no longer sinusoidal. The total change in eccentricity is normally small, but near the critical inclination (63.4”) it can be extremely large. The effect of odd zonal harmonics on orbits of small eccentricity has been treated previously by KozaW and Izsak.@) Here the analysis is extended to include the general odd harmonic and the nature of the motion is discussed in detail. 2. EQUATIONS
OF MOTION
The orbit of a satellite is normally specified by six osculating elliptic elements. These are the semi major axis a, the eccentricity e, the argument of perigee w, the inclination i, the right ascension of the ascending node Q and the modified mean anomaly at the epoch x*. These elements are not suitable for defining an orbit of small eccentricity since their time variations contain singularities at e = 0. It is well known that these singularities can be removed by introducing the transformation 5 = e cos w
(1)
7 = e sin wI
and by computing the perturbation in cu + M, M being the mean anomaly, rather than the perturbations in IXand M separately. Since only the long-period development of the orbit is being considered here, the variation of o + M will not be evaluated. The rates of change of e and u) due to a disturbing function U’ are given by Lagrange’s planetary equations :(3)
P=f(l~~2)l’2[(1 _e~)l~2?$-~] b=-
1
e
1-
( 1 w
cot i au' -. @~)~/~(l - e2)lj2 ai
e2 li2 aU’
-ae
I
433
(2)
434
G. E. COOK
From equations (1) and (2) we obtain
(=
(p~)-~/~[~ ((1- e2) z -
(
-
(1 - e2)l12 g
cos u)
- e( 1 - e2)-l12 cot i ‘g)
+ = (pQ)-r/a [’e (( 1 - e2) g +
(1 - e2)l/” g)
- (1 - e2)l12$)
(1 _ e2)1/2‘g 1
sin ~01
(3)
sin (0
- e(1 - e2)-l12 cot i -au!cos co . ai 1
1
It is convenient to retain e and w in the analysis until the final equations for $ and rj are obtained. 3. DEVELOPMENT
OF DISTURBING FUNCTION
On neglecting longitude-dependent terms, the Earth’s gravitational potential U at an exterior point distant r from the centre, and having geocentric latitude 4, may be expressed in terms of spherical harmonics:
where p is the product of the gravitational constant and the mass of the Earth, and R is the Earth’s equatorial radius. The J,, are constants and P,(sin 4) are Legendre polynomials of degree n and argument sin c$. Terms in (5) can be developed using the addition theorem for zonal harmonies:(4) P,(sin 5) = P,(cos i)P, cos 5 ( ) +2
5 @-s)! 8=l(n
+s)!
PS(cosi)P
..(cos~)coss(u-;),
n
where the P,’ are associated Legendre functions, i is the inclination of the orbit to the equator and U,the argument of latitude, is equal to w + 8,&Jbeing the true anomaly. For the general term in the disturbing function we have
. Terms in (6) containing the true anomaly can be writtenc5) as functions anomaly M using Hansen’s coefficients X;(n+l)*m: a r (-1
n+1
exp jmf3 =
2 X;(n+l),m exp jvA4, v=-cc
(6)
of the mean
LOW ECCENTRICITY
wherej = 2/z. u
SATELLITE ORBITS
435
The result is
n
v=--00
X$+1)*0 9 exp jvM
x aexpj(s
(cu -3
+ VM)]
where W denotes the real part. On neglecting short-period terms, we obtain u
=
-J
?I
I”
“R
0! nf1[
P,(cos
a
i)P,(O)Xp+1)*0
2 (n-d!
P2(cos s=l(I2 +s)!
+2
i) P,s(O)Xp+l),s
cos s co -
(
;
)I .
(7)
The Hansen coefficients in (7) are given in terms of hypergeometric functions by the relation Xr(n+l),m =
“) F(m + 1 - IZ,1 - n; Iy1+ 1; j39,
(1 + p2)“(1 - a8,1W2”(-8,-(-‘~
(8)
where p=
e
1 + (1 - e2)lj2 ’
Since X;(n+l),s = 0 ifs = n, terms containing sectorial harmonics are eliminated from the summation. Also Pns(0) vanishes for (n + s) odd. Hence the only terms remaining have (n + s) even and IZ# s. 4. EFFECT OF EVEN HARMONICS
The second harmonic is the only even harmonic which need be retained in the analysis, since J, is of order 1O-3while all higher J,, are of order 10-6. From (7) we have U, = -J2 5
0
f
“p,(cos
i)P2(0)X,-3,0.
The explicit form for U, is u2
=
J2 5
(!j3(l-
From equations (3) and (4) the contributions
e2)-3/2 (2 COS2i -&).
of J2 to & and 4 are given by
t2 = -3J2(5)1’2(;)2(I
-if)
q2 = 3J2(91’2(;)2(
1 - if)‘,,
where f = sin2 i, if terms of order e2 are neglected.
7
(9) (10)
G. E. COOK
436
5. EFFECT OF ODD HARMONICS
For IZodd we have, from equation (7), P aR n+l i: (n-s)! s=1(n + s)! 0
U,=-2J,+
Since the Hansen coefficients appearing in this equation are of order es and we are confining our attention to orbits of small eccentricity, only the first term in the summation need be retained. On using equation (S), we have XO-(n+l)*l = g(n - I)e + O(e3), so that -
P,l(cos i)P,l(O)e sin w + O(e3) .
From (3) and (4) the contributions of the odd harmonics to t and 4 are given by
60~= Z:J, %dd
=
($)1’2 (5)nin=) P,'(cos
i)P,l(O)
+O(e2)),
(11) (12)
o(e2)9
where the summation is over all odd )2for n 2 3. Explicit forms for the associated Legendre functions appearing in the above equation can be obtained from their definition : . . (2n - 2t)! (-1)” (13) P,Ycos9 = y 7 (cos9n-1-2t, (n
_
1
_
2t),
(n
_
t)
!
t!
where the summation over t is from 0 to (n - 1)/2. 6. MOTION
of the even and odd harmonics given by equations
On combining the contributions (9) to (12), we have
i=-krj+C (14)
rj = k5,
I
where k = 3 ($)““J2(f)“(1 C = &l/2
2 Jn@
nHj
-;f) Pnl(0)P,l(cos i).
The solution of equations is given by 5 = A cos (kt + a)
q = A sin (kt + a) + C/k, A and u being constants of integration depending on the initial conditions. elements can be recovered using the relations e = (E2 + q2)l12 co = tan-l (r/5). It is evident that e2 varies sinusoidally with time.
(1% The conventional
LOW ECCENTRICITY
SATELLITE ORBITS
437
In the (E, 7) plane the motion is represented by a circle of radius A, centre (0, C/k), as shown in Fig. 1. The length of a radius vector from the origin to a point on the circle represents the eccentricity, while the angular distance of the radius vector from the positive &axis gives the argument of perigee. If A > IC/kl, the major axis rotates with period 2rr/lkI and the eccentricity varies between A - C/k and A + C/k. Since the circle is traced out at a uniform rate, the motion of perigee can be extremely rapid if A exceeds IC/kl by only a small amount. Although there is a large variation in w as e -+ 0, there is a correspondingly large variation in the mean anomaly. It can readily be shown that the variation of the quantity o + M remains finite. When the magnitude of A greatly exceeds that of C/k, the variation of e with cc)is approximately sinusoidal.
(a)
A;
c/k
. (b)A
f
FIG.la and b. MOTIONOFORBIT IN (5, q) PLANE.
If A -=cjC/kl, the major axis oscillates about w = 7~12if C/k > 0 and about w = -7~12 if C/k < 0. The motion has a period 2?r/lkl and an amplitude sin-l (IAk/CI), while the eccentricity varies between lC/kl - A and jC/kl + A. For an orbit starting at the point (0, C/k) in the (E, q) plane there is no perturbation in either the eccentricity or the argument of perigee. 7. DISCUSSION To illustrate the behaviour of near-circular orbits we retain only the first four odd harmonics since these are, at present, the most accurately determined. The appropriate
438
G. E. COOK
I ,
/
I
I
I
J3TO J9
TERMS
INCLUDED
14
I.0
08
‘56
13';L 0'
__
5.2
//
/
I
/’
0
-32
t
-0.4
I C
1
c
15 INCLlNATlON
FIG. 2.
VARIATION
OF
C/k
WJTH
E -
7
DEqREEs
ORBITAL INCLINATION FOR
R/a = 0.9.
80
9i
LOW ECCENTRICITY
SATELLITE ORBITS
439
values of J, are 106J, = 1082.7 -2.56 -0.15 106J, = -0.44 106J, = 0.12. 106J3 = 106J, =
The explicit form for C is C=
-~($)1’z(~)3sini(J3(l +;
--if) J,(;)4(1
--~J5(~)“(1--~f+~f2)
-;f+!?fL
$f")
R6
-gJg-
a (I(
Since both k and the J3-term in C possess the factor (1 - if),
C/k:is of the form
sin i + terms in J5, J,, etc.
C/k = -g(J,/J&R/a) .
-._
3
i3
3
t--
e
2
,
0
ARGUMENT
FIG.
3.
VARIATION
OF
OF ECCENTRICITY
PERIGEE
WITH
ORBITS HAVING
w
ARGUMENT
R/a = 0.9.
OF PERIGEE FOR FOUR
440
G. E. COOK
The variation of C/k with inclination for R/a = 0.9 is shown in Fig. 2. For most values of inclination C/k is positive, so that rapid motion of perigee will normally occur when o is between rr and 27r if A > C/k. For inclinations near 66” the odd harmonics have very little effect on the eccentricity; but the total variation in e can become large as the critical inclination is approached. In practice the theory developed here becomes invalid as i + sin-l 2/d/5, since the effects of the higher even harmonics must be taken into account when the factor (1 - $j’) becomes small, of order 103.
Fm.4.
VARLWION
OF ECC~TRI~ITY WITH TIME FOR AN ORBIT WITH AN INCLINATION OF 60" AND R/a = 0.9.
Figure 3 shows the eccentricity plotted against argument of perigee for four orbits with R/a = O-9 and 5 = lo’, 7 = 0 initially. These orbits have A > C/k and the form of the variation of e differs markedly from a simple sine curve, which is the appropriate form for an orbit with moderate or large eccentricity (see, for example, Ref. 6). As noted by Izsak,@) a satellite having an orbit with a very small eccentricity can still be used in the determination of odd harmonics. Equations of the form (15) should be fitted to the observed orbital elements after correcting for the effects of luni-solar perturbations, radiation pressure and air drag. The resulting value of C/k then gives one linear equation between the odd J,,.
LOW ECCENTRICITY
441
SATELLITE ORBITS
An example of the variation of eccentricity with time is shown in Fig. 4, while typical variations of co with time are illustrated in Figs. 5 and 6 for orbits having the same initial conditions as Fig. 3. As noted earlier, the rate of rotation of the major axis can be extremely rapid when w is near 37~12. 40
35
50 w DEGREES
0
IO
20
30
40
50
60
TIME-DAYS
FIG.~. VARIATION OF ARGUMENT OF PERIGEE WITH TIME FOR TWO INITIAL ECCENTRICITIESOF 0.001 (R/U = 0.9).
ORBITS HAVING
Figure 7 shows how eccentricity varies with argument of perigee when A < C/k. The three orbits have an inclination of 80’ and R/a = 0.9. The motion is represented by a closed curve since the major axis oscillates about w = 7r/2. 8. EXTRANEOUS PERTURBATIONS The motion of near-circular orbits, as described in section 6, will not normally be much affected by extraneous perturbations. Air drag should have a negligible effect over a
G. E. COOK
442
time interval 277/lkl if perigee is above about 400 km. The second harmonics in the solar and lunar disturbing functions contribute terms proportional to eccentricity in equation (14). These terms can be neglected, however, since their magnitudes are small compared with the .7,-terms. The contribution from the lunar parallactic term does not contain a factor e and can be large if the semi major axis is large. This limits the applicability of the theory to orbits with values of a less than about 3 Earth radii. o-
I
L
f
3-
3-
0
100
200
300
TIME-DAYS
FIG. 6. VAFUTION
OF ARGUMENT OF PERIGEE WITH TIME FOR TWO INITIALECCENTRICITIESOF o-0()1(R/U = 0.9).
ORBITS HAVING
The main extraneous perturbing force is solar radiation pressure, which introduces into equation (14) a term with a maximum value of @(a/,~)~/~, where F is the force per unit massactingon the satellite. For a satellite with a typical area-to-mass ratio, e.g. 0.025 m2/kg, in a near-Earth orbit the magnitude of $F(u/,u)~‘~is small compared with C. In fact the magnitude will be less than 10 per cent of C provided that the inclination is not close to zero or 66”. Except at these inclinations the integrated effect on 5 and 7 will normally be quite
LOW ECCENTRICITY
SATELLITE ORBITS
2 3F PERIGEE
FIG.
7.
443
J
w
EXAMPLES OF THE VARIATION OF ECCENTRICITY WITH FOR ORBITS WITH A < C/k AND AN INCLINATION OF
ARGUMENT
OF PERIGEE
80” (R/u = 0.9).
444
G. E. COOK
small, since the magnitude of the solar radiation pressure terms varies as the orientation of the orbit with respect to the Sun changes. In general the motion will exhibit small oscillations about the solution discussed in section 6. Even if a resonant term occurs, the odd harmonics will dominate the motion; the resonant term will slightly displace the centre of the circle in Fig. 1, while the remaining terms will produce small departures from the circle. Although solar radiation pressure will not greatly affect the value of e, it could have a substantial effect on w if the orbit passes near the origin of the (5, q) plane. Acknowledgement-The author thanks Miss Diana W. Scott for performing the computations. Copyright reserved. Reproduced by permission of the Controller, H.M. Stationery Office.
Crown
REFERENCES 1. 2. 3. 4.
Y. KOZAI, Astronom. J. 64,367 (1959). I.G. IZSAK, The Use of Artificial SuteIZitesfor Geodesy, p. 329. North-Holland, Amsterdam (1963). W. M. SMART, Celestial Mechanics. Longmans Green, London (1953). H. JEFFREYS and B. S. JEFFREYS,Methods of Mathematical Physics (Third edition) p. 646. Cambridge University Press (1956). 5. H. C. PLUMMER, An Introductory Treatise on Dynamical Astronomy p. 44. Cambridge University Press (1918), republished by Dover, New York (1960). 6. D. G. KING-HELE, G. E. COOK and D. W. SCOTT,Planet. Space Sci. 14,49 (1966). Pe3KtMt+--&?TaJIbHO 06cyxc~aeTcR IIOBeAeHIleIlOqTH KPYrOBblX OP611T B FpaBHTalViOHHOM IlOTeHqSiia~e 3eMJlH. rJlaBH3X OCb Bp3LQiWTCE C Hep3BHOMepHOi CKOPOCTbH),HJIIl ?Ke Kone6neTcFI BOKPyr IIOJlOH(eHIIHPaBHOBeCHH, B 33BHCHMOCTI4 OT HZlSUE.HbIX yCJlOBH2t.