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ERRATUM

Please mark your copy or copies of Smithsonian Contributions to

Astrophysics, vol. 5, no. 5 ("On the Motion of Satellites with Critical

Inclination"), to effect the following correction:

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July 25, 1961

Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 5

ON THE MOTION OF SATELLITES

WITH CRITICAL INCLINATION

SMITHSONIAN INSTITUTION

Washington, D.C.

1961
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Contents

Fast

Libration of an Earth Satellite With Critical Inclination: Yusuke Hagihara 39

Motion oj a Particle With Critical Inclination in the Gravitational Field of

a Spheroid: Yoshihide Kozai 53

m

Libration of an Earth Satellite

With Critical Inclination

By YUSUKE HAGIHARA *

Theories on the motion of an earth satellite

have been worked out by Brouwer (1959), Garfinkel (1959), Kozai (1959), Vinti (1959), and

others. In all except Vinti's treatment, a

divisor of the form 1—5 cos2 / appears, where /

denotes the inclination of the satellite's orbit

to the earth's equator. Brouwer notes that

this method of solution cannot be applied to

this case, because the terms containing this

divisor become infinitely large for an earth

satellite having an orbital inclination with the

critical value cos"1 (1/^5).

In the present paper I propose to discuss the

behavior of the motion of an earth satellite

whose orbital inclination is approximately equal

to this critical value by applying my general

theory of libration (Hagihara, 1944) to the

motion of asteroids and natural satellites. We

follow Brouwer's notation and begin with his

equations of motion, after carrying out a first

transformation to eliminate all short period

terms in the differential equations. It will be

shown by a canonical transformation to these

transformed equations that the case for the

critical inclination corresponds to the case of

double points in a polar diagram in which the

radius vector represents cos / and the argument

represents its argument of perigee. This diagram represents the integral of the transformed

equations, which are obtained by putting the

Hamiltonian function equal to a constant.

When the angular variable describing a motion varies between a limited interval smaller

than 2T, and does not reach either the value 0

or 2T, the motion is said to be one of libration.

When, on the contrary, it varies from 0 to 2x

and makes a complete rotation around the

circle, the motion is said to be one of revolution.

The motion resembles that of a pendulum

hanging from the end of a string whose other

end is attached to a fixed support. When the

pendulum makes a complete rotation around

the supporting point, its motion is one of

revolution. When the pendulum returns to

the stable equilibrium point in the vertical

position below the supporting point, without

ever reaching the unstable equilibrium point

vertically above its supporting point, and oscillates around the stable equilibrium point, its

motion is one of libration.

In the orbit of an earth satellite there are two

kinds of double points, a center and a nodal

point. Around the double point corresponding

to the center, the pericenter of the orbit makes

a libration about either of its nodes, ascending

or descending. Values of the inclination at

either of the nodes, for such a libration, are

limited to a narrow interval. An earth satellite

having an inclination whose value at the nodes

lies outside these narrow limits has a motion of

revolution. The boundary of these two kinds

of motion consists of those asymptotic motions

that approach either of the double points of the

second kind corresponding to nodal points, as

shown in figure 3.

Equations of the problem

Following Brouwer (1959) we take Delaunay's

variables,

i Smithsonian Astrophyslcal Observatory.

39
40 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

H=G cos I,

Z=mean anomaly,

g== argument of pericenter,

h=longitude of ascending node.

so that we are left with the discussion of the

canonical equations

dg ZF dG=J>F

dt~ dG' dt bg'

After eliminating all short-period terms by

Delaunay's transformation, Brouwer considered

the following terms in the Hamiltonian functions:

F =F0+F1+F2+AiF2,

Fo==2L2

'

*

l

~UG*\ 2~r2G2

1

~5

t

£

l 0

16 \GS

G

7

 G5

'8I J

1

where we have dropped the primes used in

Brouwer's notation for each of the variables;

k% and kA represent the coefficients of expansion

of the earth's potential, and n contains the

mass of the earth.

At first we have two integrals,

L=constant,

H= constant,

in which F is of the form:

F-F0=A1(G)+Aa(G)+B2(G).COB29. (1)

As F does not contain time t explicitly, we have

a further integral,

or,

i^= constant, (2)

^(GO +^(G[) +^a(^) • cos 20= (7,

and the equations reduce to:

dG

dt

dg

dt

=-2B2(G)sir\2g, (3)

cos 2g. (4)

motion occurs on the curve representing

(2),

2g=C,

or, when rewritten,

cos zg- (5)

from which we conclude that we must have

By combining integrals (3) and (5) we obdG

2^/[B2(G)]2-[C-A1(G)-A2(G)]3

As the values of G are restricted by the above

inequality (6), the ratio G/H=l/cos I is restricted; hence cos I is bounded on both sides,

and | cos 7| can not reach the value of +1 , as

we shall see later. Thus I oscillates about a

certain value 70. We must find out whether

there is a libration for g.
Poincare transformation

LIBRATION OF A SATELLITE—HAGIHARA

Then

41

We make the following canonical transformation according to Poincare:

x=4G COS 2<7,

y=ifG sin 2g.

The transformed equations are

dt dx'

where

9fl»

2EP

H

5F / 15 /L\2

IP _9 \

^L\l6W r*~16/

cos 2fir,

with

=T6 irf U; ' ^ =

s z^ \H) ' ^=

If ^— =—-=0. thenx=constant, y=constant, 5x dy '

and we have an equilibrium solution which corresponds to the double points of the curveF(x, y)

=C representing one of the integrals. Take

the first order term Fi with regard to k2. We

suppose that kt is of the same order as k\.

P__Atr SIP "I IP

1

 O I ~/~ 2 I ..2\2 I /~2 I ..2\a

Fx=0 at x*+y2=G= ±V3 H, or cos / = ±l/VB,

or 7=54?7 and *—54?7. The variations of

G and ^ are very slow, being of the second order.

Now

bF1_jAJP_ r 5H2 1

dx-tf+yrl tf+y2)

2

]'*'

ZAJP rL

-g-1 and -r?-± both vanish at (i) x=y=0, (ii)

V , (iii) x

2

+y2=». We shall discuss each of these three cases.

(i) x=y=Q has no physical meaning, because

G—0 and cos /=» . cos / should be bounded

by |cos 7|^1 , or Gr=z

2

+y2^H. |cos 7| = 1

corresponds to an equatorial orbit. It does not

correspond to a double point.

(ii) G—±^5 H corresponds to Brouwer's

critical inclination cos /=±1/V5 ; that is,

/=63?4 and -K—63?4. Henceforth we suppose that G is always positive and that H is

negative when cos /< 0 and positive when

cos i>0 . Then Au A2, At, B2j £ 4 > or < 0

depending on whether cos / > or < 0 and

H> or <0 . For a retrograde motion of the

mean anomaly we suppose cos /<0 , so that

we always take the mean motion

(iii) x

2

+t/2=a> corresponds to a polar orbit

with /=7r/2. Since

dF .

dG

dG2~

2 \

3A,

(

15H2

G

2

5H2

G

2

+ 0**

) G*

we have

^=0 , at G=±JTEJ2 H and ».
42 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Hence F has a minimum at G=+-y[5H and a The motion is given by the equations

maximum at G=—f5H, and points of inflection , ZA IP T 5H2

at G=±V 15/277 and #=«>. " —=. . .' ^ A 1 —

For G=l/cosI=x2Jry

2

, F1=Cl has three

roots if

one root if

25V5

• n;

and no root if

If we take cos I=H/(x2+y2) as the ordinate,

the situation is more clearly seen; cos 7=0 or

£?=co corresponds to a polar orbit I=±ir/2.

The values cos 7= -f-1, G— +77, or cos 7= — 1,

G=— 77 both correspond to an equatorial orbit.

Figure 1 shows the curve for Fi/(Ai/2) with

G/H as the ordinate; figure 2 shows the same

curve with cos I=H/G as the ordinate. The

arrows show the sense of rotation of the

pericenter.

Now we have

dg

dt'

<iF\

dG~~

3AtIP

'2(^+y2)

4

5772

and dg/dt is positive or negative depending

on whether G2<5H2 or G2>5H2. Hence

the motion of the pericenter is direct for

|cos 7|>1/V5, that is, for |TT/2-7|>26?6 ; and

is retrograde for |cos 7|<1/V5, that is, for

|TT/2—7|<26?6; and stationary for | cos 7|=

1/V5, that is, for |TT/2—7|=26?6. Any point on

the critical circle x

2

-\-y2=G=-/E77 is an equilibrium point; that is, the pericenter is at rest

with the fixed value of 7 but with an arbitrary

value of g, the argument of the pericenter.

The circumstance is quite different when we

consider the second order terms of F. The

orbit is determined by the Hamiltonian

1P_A1[ , , 1

 2 L +

3772

(x2+2/2)

2

1 7T3

or

x

2

+y2= constant.

dt (x2+

dy_ SArIP Fx 5H*^ "I x

We have

Put

x=r cos ^, y= r sin

then r*= constant and

5i72

Hence

= —27C, a constant.

g=-K(t-tQ).

K=0 for the critical inclination. We have

G=r2=Hjcos 7=constant, 7=constant, g=constant. Thus, if we take only the first order

terms, the orbit is an ellipse with rotating

pericenter and rotating node. Further,

dh

dt1 • ' -v XX ' **M

dt l=7lo(t — to),

For the critical inclination the pericenter does

not move, whatever g is, but the node rotates

with the mean motion,

in the retrograde sense for

and

in the direct sense for

cos I— —=•

V5
LIBRATION OF A SATELLITE—HAGIHARA 43

CL-H

6 2

FIQUBB 1.—Curves for Fi/(Ai/2) against OfB, where F\ represents the first order term with coefficient At in the

Hamiltonian function and GfH=l/coa I.

First order effects

So far the discussion has dealt with the first

order effects in the case in which the earth is

supposed to be spheroidal. The center of

mass of the earth is taken to be the coordinate

origin, so that Brouwer's term AiF2 does not

enter. We next consider the effect of k% and

kt, that is, the terms F2 and A4F2. The position

of the pericenter was arbitrary in the first order

discussion for the satellite with critical inclination, but when we consider the second order

effects the pericenter will be seen to be fixed,

as it should be from physical considerations.

We see after some computation that

at(i)

dz~~dy '

= « , or cos 7=0,

567711—Cl-
44 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

- E C CO5 I

EQUATORIAL

ORBIT

POLAR

ORBIT

EQUATORIAL

ORBIT

FIGURE 2.—Curves for Ftl(Ai/2) against cos /, where Fi represents the first order term with coefficient

Hamiltonian function.

in the
LIBRATION OF A SATELLITE—HAGEHARA 45

a polar orbit; and (ii)

The polar orbit

For a polar orbit, for which

= 0 .

cos 7=0,

H=Gcos 7=0,

we have

therefore

h=constant.

The node is stationary with an arbitrary position of the node. For H=0 we have, by

omitting the second order terms,

3 n*k2 .. . v

The motion of the pericenter is retrograde for a

polar orbit. The eccentricity given by

and

9+a2L3

6*

oscillates between two finite limits with the

period

As h is a constant, the node is stationary and the

inclination remains constant, and a polar orbit

always remains a polar orbit.

Second order effects

In order to examine the case in which

1 —n

and

we put

and solve for p. Denote the solution for p

by P=PI and p=pn, respectively, for x

2

=

H and for x

2

= — -JE H. We then obtain the

equations

pie/z

Thus each of the critical points is displaced

by this amount of the first order from the

corresponding point in the first order approximation. The critical point should be situated

on the x-axis in the second order approximation;

that is, the value of 6 or 2g for the critical

point should be 0 or T. Hence the critical

points, which were considered to be the equilibrium points in the first order approximation,

are unstable except for the point on the x-axis

on the critical circle; that is, if we consider the

second order terms, they should tend to approach the point on the x-axis; in other words,

the pericenter should tend to be on the equator

and to coincide with the node.

We consider the equations,

C+A1(r)+A2(r)+Ba(r) cos 0=0,

dO A,

eo*

Here

Hence r, that is, O=H/cos I, is bounded on

both sides and I oscillates between finite

limits.

A[ (r) is small near the critical point and is

zero at the critical point, and changes sign in

passing through it. As dg/dt=Q at the critical

point, the pericenter is stationary and g is in

direct motion for |cos 7|>1/V5, that is, for

|T/2—7|>26?6; and is in retrograde motion for

|cos 7|<1/V5, that is, for |ir/2-7|<26?6.
46 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The nearer the pericenter to the critical point,

the slower is the motion.

As

the motion of the node is in the retrograde

sense near the critical point, cos / = + 1/V5, and

the direct sense near cos / = — l/V^. The node

is stationary for a polar orbit; \dh/dt\ increases

as / decreases from T/2 to 0 and is a maximum

for an equatorial orbit.

Motion near the critical points

In order to study the nature of the motion near

the critical points, we set

and suppose that p is of the order one-half with

regard to k2. By retaining the terms up to the

second order in our integral (2) we obtain

50V& 625 V5

JI/A 1875V5~Ll6VflV 16.

8B2 f

125V5I

— 1 COS0

de 11

25W

\2SH

B2

125H

Bi

we determine the value of C. For the

equilibrium point P(y=0, cos 0=-fl):

625V5

["15/ZV 9

U6W

8B2

For the equilibrium point P'(y=0, cos 0=

1):

25V5 625VH

pi5^z y 9"| 8^2

16J 125V5 5VH

We have

at p=0, and

and

respectively, for the points P and P'. Hence

F is a minimum at P and a maximum at P \

We set C= CQ-\-72, C=Co+ci, respectively,

for the critical points P and P''. Substituting

in equation (2), F=C, we get forP;

Q A op r*i /r\ 2 ~l

C2 ^ p"* ^ -I YT I —1 ' (1 — COS 0)

10V5 125V5L5W/ J v

3125^

-li . (l-cos<?)=0;

forP'

To study the behavior of the curve passing

through the critical point, we put c2=0 and

Cj=0, respectively, in these two equations.

We get no root for p in the case 0=0+e for P,

and two roots for p in the case 0=*-+e for P'.

Hence P is a center and P' is a saddle point in

Poincare"'s terminology.

The distance between the two intersections

of the x-axis with the two branches passing

through P' of the curve represented by the

integral (2), that is, F=C, gives the width of
LIBRATION OF A SATELLITE—HAGIHABA 47

the domain of libration. By putting c'2=0,

cos 0= + l for this curve, we get:

4y2

5*JbAx

or

e r

 4V2

Put

6 A .

and note that

/ „ , Bt /I

/ R _|_5<

1

KIT-1**

and is of the order of the square of eccentricity,

as

and provided that

1+P >

We assume further that

1

and write

Then from the equation F=C we obtain for P:

P

2=c2—b2 (1—cos 0);

p

2=<h'-\-b2 (1+cos 0).

forP':

A case of libration.—For P we put b2—c2

= b2 cos 0O and the equation we have to solve is

-7-=—Oi V^2 V c o s 0—cos doWe have

Hence the pericenter g makes a libration.

The amplitude of the libration increases as

{b2—c2)lb2 decreases. The half-amplitude tends

to 0O=T/2, extending to the unstable double

points in the limit (b2—c2)/b2-*O and the motion

is asymptotic to these double points. The period

of the libration is

2 J-e

dd

s 0—cos 0O

and becomes infinite in the limit (ba—t

(which is a case of asymptotic motion to the

critical point), while it tends to zero in the limit

(b2—c2)/b2—>l or c2/62-K), that is, at the equilibrium point.

We have

2dz

Vcos0— cos 0O

with

m

dt,

1 0

- = ta n -> a= cos 0O= (fi2—c2)/b2,

m \ 4 2o , 1+a

1—a 1—a

According to the formulas in the author's work

on the general theory of libration (Hagihara,

1944) we put

4a 2-3 (9—8a2

) a

9i~21(\-aY'

then we have

As

dz \C2 j .

^(s) has two real roots, Zi>0>22i and two

conjugate complex roots, and z and 6?(z) are

both real. We take u as the real variable, and

dz

with
48 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

and 2iuf as the two periods of the elliptic

functions. While u varies from 0 to 2a>, t varies

from 0 to V^/^2 • «• The period T of the

libration with regard to t is ^c2/b2 •«. Then

from the theory of elliptic functions we have:

t 2 ^J^

^+JL cot «•+*. ± s i n

with

- j^f n=?o>, u=j Vc2 • it - *0) •

case o/ revolution.—For P' we have

and we have to integrate:

and, by inverting, we get

. 6 . y=sin »=sin g

ii.it n^O

where

Thus the period T with regard to t is

ATT

or roughly

as can be seen by expanding the square root in

the numerator in powers of k2, and the mean

motion of the pericenter is of the second order.

For 6=0 we have L—O= constant, and

L/H=l/cos /«±V5. In the equation (2),

F=C, the coefficient b2 of cos 2g vanishes and

we get p2=constant and dO/dt=constant. The

motion of the pericenter is one of revolution.

Thus the motion changes from libration to

revolution for a nearly circular orbit with

vanishingly small eccentricity in the scope of

the present approximation.

We have assumed that Ba^-zBi. If Ba=

5

-i?4, which, as noted by Kozai (1960), is the

o

case J\=Ji corresponding to Vinti's (1959)

assumption; then the term with cos 2g in the

present approximation vanishes, irrespective of

the value of the eccentricity. This is a case

of revolution, and the question of libration

does not arise. If B2<C.-zBif then libration

o

changes into revolution, and revolution into

libration.

Risumi.—To sum up the situation: The

pericenter makes a libration about the ascending or the descending node when the orbital

inclination is near the critical inclination

/~63?4 or /~116?6. The position of the

pericenter at either of the nodes is stable. The

position of the pericenter at the middle point

between the two nodes is unstable. The

motion of the pericenter is retrograde for

and is direct for

and is stationary for

=26?6.

The situation is illustrated in figure 3, where the

Roman numerals along the curves correspond

to the value of C illustrated in figure 2.
LIBRATION OF A SATELLITE—HAGIHARA 49

o

a.
50 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

As G is restricted between two limits, -jl—e2

is restricted between two limits. Thus the

eccentricity oscillates between two limits and

the oscillation has the same period as G or /

and g, that is, the period of the libration.

The motion of the node,

is retrograde for cos /> 0 and direct for cos

/<0 . The absolute value of the mean motion

of the node is equal to

25 V5 LW*

for the critical inclination. It has a maximum

* for an equatorial orbit.

Effect of the neglected terms

If we take further terms A3F2 and A5F2 of

Brouwer into our discussion, the situation is

somewhat different. With the transformation

x—^G cos g, y=^G sin g, I have worked out

the computations to a certain stage. The

double points at <7=0 and g=ir are modified

to be not exactly on the x-axis and we have, as

before, the double points g=ir/2 and g=Sr/2, but

libration may still occur about the double

points near g—0 and g=v on the critical circle,

if the coefficients of the terms containing

Brouwer's ^30 and Aw, together with those of

the terms considered in the present paper,

satisfy certain relations. The term containing

Am is the cause of the modification of the

double points for g=0 and g=ir, because the

term vanishes at the critical points. But the

question of libration is discussed by the term

Let

«a= l

2W 1875

V5

Then the motion of the pericenter is determined

by the equation

^=-ai[(c 2 -2&2 )±e 2T/ 2 )-(e 2+3/ 2 ) sin g

+262 sin2 0+4/2 sin3 g]1'*,

depending on whether the double point corresponds to g=ir/2 or <7=3ir/2. The behavior

of the motion near g=0 or g=ir is discussed by

the equations

-j-=—ai [(c2—b2)—e2 sin g

+262 sin2 y-/ 2 sin Sg]1/2,

dg

dt = — ai [(c2—362)—e2 sin g

+262 sin2 g-j2 sin 3g]1/2,

depending on whether g «0 or g ~v. A detailed

discussion with explicit expressions appears

in the succeeding article by Kozai (p. 53).

The author wishes to thank Dr. Kozai for

information on the present state of research

on the orbits of earth satellites, and for the

discussions relating to the present problem.

References

BBOUWER, D.

1959. Solution of the problem of artificial satellite

theory without drag. Astron. Journ., vol.

64, pp. 378-397.

GARFINKEL, B.

1959. The orbit of a satellite of an oblate planet.

Astron. Journ., vol. 64, pp. 353-367.

HAGIHARA, Y.

1944. On the general theory of libration. Japanese Journ. Astron. Geophys., vol. 20,

pp. 29-43.

KOZAI, Y.

1959. The motion of a close earth satellite.

Astron. Journ., vol. 64, pp. 367-377.

1961. Motion of a particle with critical inclination in the gravitational field of a spheroid. Smithsonian Contr. Astrophys.,

vol. 5, no. 5, pp. 53-58.

VINTI, J. P.

1959. New method of solution for unretarded

satellite orbits. Journ. Res. Nat. Bur.

Standards, vol. 63B, pp. 105-116.
NO. s LIBRATION OF A SATELLITE—HAGIHARA 5 1

Abstract

Brouwer (1959) has noted that for the orbit of an earth satellite with inclination 63?4 with reference to the earth's

equator, his solution based on Delaunay's method fails because of the appearance of a factor G*-5H* in the denominator.

The present paper shows that this critical inclination corresponds to a case of libration in the ordinary perturbation

theory, as an application of the author's general theory of libration in the motion of asteroids and natural satellites.

The pericenter of the orbit of an earth satellite having an orbital inclination nearly 63?4 can librate about either of its

nodes, ascending or descending, which correspond to the double points of the Hamiltonian function. The motion

near the other double points situated at the middle points between the two nodes is one of revolution, and the motion

is asymptotic at the double point. For an orbit with this critical inclination, the pericenter is stationary but the node

rotates in the retrograde sense when cos /> 0 and in the direct sense when cos J<0. However, the libration may change

to revolution for an orbit with vanishingly small eccentricity. A polar orbit always remains polar, and the node is

stationary but the pericenter moves in a retrograde sense.

Motion of a Particle With Critical

Inclination in the Gravitational Field

of a Spheroid

By YOSHIHIDE KOZAI *

The long-periodic terms in the motion of a

particle moving around a spheroid cannot be

derived by the usual method of successive

approximations if the inclination of the orbit

to the equator of the spheroid has the critical

value. In this paper the problem is studied

by the use of elliptic functions.

The equations of motion

Consider a small particle moving in the gravitational field of a spheroid whose gravitational

potential U at a point (r,5) is expanded into the

series of spherical harmonics,

3

 5 . 2

~

S i n

35 . 4

where /x is the gravitational constant multiplied

by the mass of the spheroid and R is the equatorial radius of the spheroid. In this potential

J2R

2

 is assumed to be a small quantity of the

second order, and J3R? and JtR* are of the

fourth order.

With Delaunay's canonical variables,

G=LTJI=?,

H= G cos i,

Z=mean anomaly,

> Smithsonian Astrophyslcal Observatory.

g=argument of pericenter,

h=longitude of ascending node,

the equations of motion of the particle are

expressed in canonical form. In this case the

Hamiltonian F depends neither on time explicitly nor on h.

By von Zeipel's (1916) transformation from

L, G, H, I, g, and h to L', G', Hf, V, g', and

h\ V can be eliminated from the new Hamiltonian F* (Brouwer, 1959). Brouwer's expressions of F* up to terms of the fourth order are

the following:

* M2 , M4^^ 8 / 1. 3 Hl

~2U~* L3G

3

 \ 4"l"4 G2

_15_

+

1 21288 L10

6

V A _18 & , H

G

5

\ 5 G2 + G*

9

128

L

8

 U

64

G

2

 I, H2

W1 IP

G>

53
54 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

where primes to be attached to the variables

are omitted for simplicity.

As F* depends neither on h nor on /, there

exist the following integrals, in addition to an

energy integral F*=constant;

H— constant,

L=constant.

(2)

Then the Hamiltonian F* is a function of

only two variables, Q and g, which satisfy the

following equations;

dG bF*

17=-?^==2Q

 si n 29+S co s

 9,

dg bF* bP bQ _ bS -^g cos 2g—^ sin g,

(3)

where P, Q, and S are functions depending

only on Q.

By the usual method we integrate these

equations approximately by regarding Q as

a constant and gr as a known linear function of

time t in the right-hand members.

However, if £P/G2=l/5, the mean motion of

g becomes a small quantity of the fourth

order, and the amplitudes of periodic terms in

solutions of 0 and g are of the zero-th order.

This means that g and O in the right-hand

members cannot be regarded as known functions.

The condition H2/G2=l/5 is satisfied if the

inclination is about 63 ?4, which is called a

critical inclination.

Equilibrium points

To study the motion of a particle with an inclination very near to the critical one, we may

write

(4)

thus P, Q, and S are expanded into power

series of x, which is regarded as a first order

quantity. Expressions of P, Q, and S up to

terms of the fifth order are:

S(x)=S1x,

where P2 and P3 are of the second order and the

other coefficients except for Po are of the

fourth order. A constant coefficient Po will

not appear in the following discussion. Pt is of

the second order unless P(x) is expanded

around x=0. It is remarkable that So vanishes.

Full expressions of Pu Qu and Si are as follows:

where

a=—

at?

Here, a and e correspond respectively to the

semimajor axis and the eccentricity, but are not

equal to the osculating values.

Now the equations to be solved are written as:

sin g+Six}cos g,

^

9

-=2±(Pi+2P2x+3P3x*-Q1cos2g+S1 sin y).

(5)

As the Hamiltonian F* does not depend on

time t explicitly, an equation,

F*=-C, (6)

with a constant C, represents a trajectory in the

foi7)-plane, where £= (H/G)2cos g, i=(Hf6)*

sin g. If derivatives of both G and g with

respect to time vanish at a certain point, this
A PARTICLE IN THE GRAVITATIONAL FIELD OP A SPHEROID—KOZAI 55

is an equilibrium point where the motion in

the (£,i7)-plane is stationary.

There are four equilibrium points of equation

(6) in the (f,7j)-plane, and their coordinates

correspond to;

i) 1=JC Xl-

 Pl

~

Ql

' 2P2

ii) 02=18O°—*

iii) 03=90°,

2P2

iv) 04=27OO, x4= —

where K and Xi are of the second order and

sin K=SI

Trajectories in the (£, ij)-plane

Assume that at a certain epoch t = t0 a trajectory

F*=—C passes through a point corresponding

to x=8 and 0=0O; then the energy integral is

written as:

P(x)-Q(x) cos 2g+S(x) sin g

=P(8) — Q(8) cos 20O+S(5) sin g0. (7)

Here 8 is assumed to be of the second order.

From this integral g is solved as a function of

x as follows:

sin 0=—'"* 4O()—~^^—~^~^' ®

where

A(x,8)=P(x)-P(8) + Q(x) cos 20O

~S(8) sin go-Q(x).

Then from the right-hand member of the differential equation,

-77=—T~(4Q(X)

 si n

 0+5'(x)}cOS 0,

g can be eliminated so that the equation contains x as the only variable, and the equations

become of the following form;

(9)

If terms of higher order are omitted, /i(x)

and/2(x) are written as:

/i(x)=P2x

2

-2Q0sin2fir0, (10)

Here/i(x)=0 corresponds to g=K or 180°—K,

and/2(x)=0 corresponds to 20=180°.

If J2 is positive, P2 is also positive. Therefore in this case/2(x) or/^x) is positive definite,

depending on whether Qo is positive or negative.

Then there are two kinds of solutions, both

expressed by Jacobi's elliptic functions. If Qo

is positive, one has

l2Q0 .

x== V~P~ s m

 ^o cn{/3(<—10), sin 0O},

sin 0=sin gQ sn{p(t—t0), sin 0o},

and if Qo is negative,

(11)

x=-

!

cos 0o cn{/3'(<—10), cos 0O},

(12)

cos y=cos g0 sn{j8'(/—10), cos g0},

where ff and /S' are of the third order and

is of the first order, and

where the signs correspond to those of cos g.

(13)

If Qo is positive, maxima and minima of x

occur at g=K and 0=180°—K, and x vanishes at

0=0o- On the other hand, if Qo is negative, the

maxima and minima occur at 0=90° and —90°.

Trajectories F*= — C are given in figure 1

for two cases, Q<K>0 and Q0<C0. If QO is positive

and 8 is of the second order, 0 cannot make

a complete revolution but has a motion of

libration around the equilibrium point near the

£-axis, unless 0O is very near to 90° or —90°.

However, even though 8 is of the first order,

0 has a motion of libration, if 0O is very near

to K or 180°—K. A trajectory with 0O=±9O°

corresponds to an asymptotic solution.
56 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

FIGURE 1.—Trajectories in the (£,ij)-plane. Top: when Qa is positive; bottom, when Qo is negative.
A PARTICLE IN THE GRAVITATIONAL FIELD OF A SPHEROID—KOZAI 57

Trajectories near the equilibrium points

The solutions discussed in the previous section

will fail if (i) |cos 2<7O±1| is a very small quantity of the second order or if (ii)

is of the sixth order. In these cases x must be

regarded as a quantity of the second order. The

condition (ii) means that the trajectory passes

very near to one of the four equilibrium points.

Let us assume that |cos 2g0—1| is very small.

In this case, expressions of j\(x) and/ 2 0*0 are:

fi(.x)=2Q0

Mx)=P2X

2

-P2f?-2Q0 sin2 g0, (14)

where

2P,

6=e — 2P2

If Qo is positive,/i(x) cannot vanish, that is,

g cannot reach either to +90° or to —90°.

And solutions of the equations (9) are,

I

P2e

2+2Q0sin~g~0

2Q cos0(i t0), (15)

where g0 is K or 180° —K depending on whether

cos g is positive or negative. The angular

velocity /S has the same expression as that

in (13).

If |cos 2<7o+l I is very small, then

cos2 go, (16)

where

x=Y l- ^ -» + .. . sin 0>O

*=U-tTM^- -...sin<K0

Then, if

P2U

2

-2Q0 cos2

12{x) has two real roots, and by assuming that

<2o is positive, one has

eos^^^ocos^^^, ^ (17 )

In this case any trajectory will cut the {-axis,

and tends to make a complete revolution around

the origin. On the other hand, if

U

2—-^ cos2

 go<O,

the trajectory cannot cut the {-axis, and

cos g= cosh P(t-t0). (18)

The solutions (17) and (18) containing

hypertrigonometric terms are valid only if the

distance to one of the equilibrium points is of

the second order.

Solutions when Qo is negative should be

derived in a similar way.

Small eccentricity

If Qo is very small the problem becomes very

complicated, because in this case J\, J3J3, and

JvJi terms must be added to the Hamiltonian.

However, if Qo is very small because of a factor

e

2

, similar coefficients in the J\, JaJz, and J2J4

terms are also very small, due to the same

factor e2, and are neglected.

In this case K is of the first order, and

-2(Q0+Q1S) sin2 g0,

2

+2(Qo+Ql8) cos2 g0. • (19)

The discriminants Dx and D2 of the quadratic

equations/i(x)=0 a.ndJ2(x)=0 are;

D1=(Pl-Ql+2P28)2-\-8Pi(Q0+QlS) sin2 g0,

D2=(Pl+Ql+2P28)2-8P2(Q0

«) cos2 flro.(2O)

Therefore, at least one of Dx and D2 is positive.
58 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

If A is positive and D2 is negative, g has a

motion of libration around the equilibrium

point near the £-axis, and the amplitude of x

is of the second order. If Dx is negative and

D2 is positive, the libration will occur around

the equilibrium point on the 17-axis.

However, if both Dx and D2 are positive, g

has a motion of libration around one of the

equilibrium points, if the two roots of one

equation/i(z)=0 are both larger than or both

less than the two roots of/2(a;)=0. Otherwise,

g can make one complete revolution, and x is

periodic with an amplitude of the second order.

Node and mean anomaly

The longitude of the node is derived by the

equation,

dh 3 1 /, , 5

If Qo is positive and x is expressed by equation

(11), then

(22)

where h, is the secular part of h.

If Qo is negative, sin g0 in equation (22) must

be replaced by cos g0.

Similar results are obtained for the mean

anomaly. If Qo is positive, the solution is:

(23)

where lt is the secular part of I and E is an

elliptic integral of the second kind.

The solutions in the present paper are similar

to those for the commensurable case for the

characteristic asteroids (Hagihara, 1944), but

are more simple than the asteroidal case.

The author is grateful to Prof. Y. Hagihara

for helpful discussions.

NOTE ADDED IN PROOF: After this manuscript

had been sent to the printer, G. Hori's paper

entitled "The Motion of an Artificial Satellite

in the Vicinity of the Critical Inclination" was

published in the Astronomical Journal (vol. 65,

pp. 291-300, 1960). His theory also is based

on Brouwer's (1959) expression of F*; however,

he follows rather different procedures.

References

BROUWER, D.

1959. Solution of the problem of artificial satellite theory without drag. Astron. Journ.,

vol. 64, pp. 378-397.

HAGIHARA, Y.

1944. On the general theory of libration. Japanese Journ. Astron. Geophys., vol. 20,

pp. 29-43.

VON ZEIPEL, H.

1916. Recherches sur le mouvement des petites

planetes. Ark. Mat. Astron. Fys., vol.

11, No. 1, pp. 1-58.

Abstract

The motion of a particle with a critical inclination is treated on the basis of Brouwer's expression of the transformed Hamiltonian F*. In the potential of the central spheroid, it is assumed that the second harmonic is of the

second order, and that the third and the fourth harmonics are of the fourth order. The solutions are classified into

three types: the case of libration around one of four equilibrium points, the case of a complete revolution, and the

asymptotic case.

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