Smithsonian
Contributions to Astrophysics

VOLUME 5, NUMBER 5

ON THE MOTION OF SATELLITES
WITH CRITICAL INCLINATION

SMITHSONIAN INSTITUTION

Washington, D. C.

1961

Publications of the Astrophysical Observatory

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Contents

Publications of the Astrophysical Observatory

Libration of an Earth Satellite With Critical Inclination: Yusuke Hagihara

Equations of the problem

Poincaré transformation

First order effects

The polar orbit

Second order effects

Motion near the critical points

Effect of the neglected terms

References

Abstract

Motion of a Particle With Critical Inclination in the Gravitational Field of a Spheroid: Yoshihide Kozai

The equations of motion

Equilibrium points

Trajectories in the (ξ, η)-plane

Trajectories near the equilibrium points

Small eccentricity

Node and mean anomaly

References

Abstract

Libration of an Earth Satellite With Critical Inclination

By YUSUKE HAGIHARA [1]

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 cos^2 I appears, where I 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

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 I 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 2π, and does not reach either the value 0 or 2π, the motion is said to be one of libration. When, on the contrary, it varies from 0 to 2π 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.

[1] Smithsonian Astrophysical Observatory.

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,

H=G cos I,

l=mean anomaly,

g=argument of pericenter,

h=longitude of ascending node.

After eliminating all short-period terms by Delaunay’s transformation, Brouwer considered the following terms in the Hamiltonian functions:

where we have dropped the primes used in Brouwer’s notation for each of the variables; k_2 and k_4 represent the coefficients of expansion of the earth’s potential, and µ contains the mass of the earth.

At first we have two integrals,

L=constant,

H= constant,

so that we are left with the discussion of the canonical equations

in which F is of the form:

As F does not contain time t explicitly, we have a further integral,

or,

and the equations reduce to:

The motion occurs on the curve representing the integral (2),

or, when rewritten,

from which we conclude that we must have

By combining integrals (3) and (5) we obtain the time t,

As the values of G are restricted by the above inequality (6), the ratio G/H=1/cos I is restricted; hence cos I is bounded on both sides, and |cos I| can not reach the value of +1, as we shall see later. Thus I oscillates about a certain value I_0. We must find out whether there is a libration for g.

Poincaré transformation

We make the following canonical transformation according to Poincaré:

The transformed equations are

where

and we have an equilibrium solution which corresponds to the double points of the curve F(x, y) =C representing one of the integrals. Take the first order term F_1 with regard to k_2. We suppose that k_4 is of the same order as

Then

or and . The variations of G and g are very slow, being of the second order.

Now

both vanish at (i) x=y=0, (ii) , (iii) x^2+y^2=∞. We shall discuss each of these three cases.

(i) x=y=0 has no physical meaning, because G=0 and cos I=∞. cos I should be bounded by |cos I|≦1, or G=x^2+y^2≧ H. |cos I|= 1 corresponds to an equatorial orbit. It does not correspond to a double point.

(ii) G=± H corresponds to Brouwer’s critical inclination cos I=±1; that is, and . Henceforth we suppose that G is always positive and that H is negative when cos I<0 and positive when cos I>0. Then A_1, A_2, A_4, B_2, B_4> or <0 depending on whether cos I> or <0 and H> or <0. For a retrograde motion of the mean anomaly we suppose cos I<0, so that we always take the mean motion n_0>0.

(iii) x^2+y^2= ∞ corresponds to a polar orbit with I=π/2. Since

we have

Hence F has a minimum at G=+H and a maximum at G=–H, and points of inflection at and G=∞.

For G=l/cosI=x^2+y^2, F_1 =C_1 has three roots if

one root if

and no root if

If we take cos I=H/(x^2+y^2) as the ordinate, the situation is more clearly seen; cos I=0 or G=∞ corresponds to a polar orbit I=±π/2. The values cos I=+1, G=+H, or cos I= – 1, G=–H both correspond to an equatorial orbit.

Figure 1 shows the curve for F_1/(A_1/2) 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

and dg/dt is positive or negative depending on whether G^2<5H^2 or G^2>5H^2. Hence the motion of the pericenter is direct for |cos I|>1/, that is, for ; and is retrograde for |cos I|<1/, that is, for ; and stationary for | cos I|= 1/, that is, for . Any point on the critical circle x^2+y^2=G=H is an equilibrium point; that is, the pericenter is at rest with the fixed value of I 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

or

x^2+y^2= constant.

The motion is given by the equations

We have

Put

x=r cos θ, y=r sin θ;

then r^2=constant and

Hence

K=0 for the critical inclination. We have G=r^2=H/cos I=constant, I=constant, e=constant. Thus, if we take only the first order terms, the orbit is an ellipse with rotating pericenter and rotating node. Further,

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

FIGURE 1.—Curves for F_1/(A_1/2) against G/H, where F_1 represents the first order term with coefficient A_1 in the Hamiltonian function and G/H=1/cos 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 Δ_1F_2 does not enter. We next consider the effect of and k_4 , that is, the terms F_2 and Δ_4F_2. 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)

FIGURE 2.—Curves for F_1/(A_1/2) against cos I, where F_1 represents the first order term with coefficient A_1 in the Hamiltonian function.

a polar orbit; and (ii)

The polar orbit

For a polar orbit, for which

we have

therefore

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

The motion of the pericenter is retrograde for a polar orbit. The eccentricity given by

and

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

y=0

and

we put

and solve for ρ. Denote the solution for ρ by ρ=ρI and ρ=ρII, respectively, for x^2=+ H and for x^2=– H. We then obtain the equations

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 θ or 2g for the critical point should be 0 or π. 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,

Here

Hence r, that is, G=H/cos I, is bounded on both sides and I oscillates between finite limits.

is small near the critical point and is zero at the critical point, and changes sign in passing through it. As dg/dt=0 at the critical point, the pericenter is stationary and g is in direct motion for |cos I|>1/, that is, for ; and is in retrograde motion for |cos I|<1/, that is, for .

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 I=+1/, and the direct sense near cos I=–1/. The node is stationary for a polar orbit; | dh/dt| increases as I decreases from π/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 ρ is of the order one-half with regard to k_2. By retaining the terms up to the second order in our integral (2) we obtain

First we determine the value of C. For the equilibrium point P(y=0, cos θ=+1):

For the equilibrium point P'(y=0, cos θ= –1):

We have

at ρ=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=C_0+c_2, , respectively, for the critical points P and P'. Substituting in equation (2), F=C, we get for P:

To study the behavior of the curve passing through the critical point, we put =0 and =0, respectively, in these two equations. We get no root for ρ in the case θ=0+ є for P, and two roots for ρ in the case θ=π+є for P'. Hence P is a center and P' is a saddle point in Poincaré’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 the domain of libration. By putting =0, cos θ= + 1 for this curve, we get:

or

Put

and note that

and is of the order of the square of eccentricity, as

and provided that

We assume further that

and write

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

A case of libration.—For P we put b_2–c_2=b_2 cos θ_0 and the equation we have to solve is

We have

Hence the pericenter g makes a libration. The amplitude of the libration increases as (b_2–c_2)/b_2 decreases. The half-amplitude tends to θ_0=π/2, extending to the unstable double points in the limit (b_2–c_2)/b_2→0 and the motion is asymptotic to these double points. The period of the libration is

and becomes infinite in the limit (b_2–c_2)/b_2→0 (which is a case of asymptotic motion to the critical point), while it tends to zero in the limit (b_2–c_2)/b_2→1 or c_2/b_2→0, that is, at the equilibrium point.

We have

According to the formulas in the author’s work on the general theory of libration (Hagihara, 1944) we put

G(z) has two real roots, Z_1>0>Z_2, and two conjugate complex roots, and z and G(z) are both real. We take u as the real variable, and

and 2iω' as the two periods of the elliptic functions. While u varies from 0 to 2ω, t varies from 0 to The period T of the libration with regard to t is Then from the theory of elliptic functions we have:

or roughly

as can be seen by expanding the square root in the numerator in powers of k^2, and the mean motion of the pericenter is of the second order.

For e=0 we have L=G=constant, and L/H/=1/cos I≈±. In the equation (2), F=C, the coefficient b_2 of cos 2g vanishes and we get ρ^2=constant and dθ/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 If B_2= which, as noted by Kozai (1960), is the case 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 then libration changes into revolution, and revolution into libration.

Résumé.—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 or . 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

The situation is illustrated in figure 3, where the Roman numerals along the curves correspond to the value of C illustrated in figure 2.

FIGURE 3.—Curves for the Hamiltonian function F=C drawn with the constant C as the varying parameter.

As G is restricted between two limits, is restricted between two limits. Thus the eccentricity oscillates between two limits and the oscillation has the same period as G or I and g, that is, the period of the libration.

The motion of the node,

is retrograde for cos I >0 and direct for cos I<0. The absolute value of the mean motion of the node is equal to

for the critical inclination. It has a maximum 3µ^4k_2/L^3H^4 for an equatorial orbit.

Effect of the neglected terms

If we take further terms Δ_3F_2 and Δ_5F_2 of Brouwer into our discussion, the situation is somewhat different. With the transformation , I have worked out the computations to a certain stage. The double points at g=0 and g=π are modified to be not exactly on the x-axis and we have, as before, the double points g=π/2 and g=3π/2, but libration may still occur about the double points near g=0 and g=π on the critical circle, if the coefficients of the terms containing Brouwer’s A_30 and A_50, together with those of the terms considered in the present paper, satisfy certain relations. The term containing A_30 is the cause of the modification of the double points for g=0 and g=π, because the term vanishes at the critical points. But the question of libration is discussed by the term A_50. Let

Then the motion of the pericenter is determined by the equation

depending on whether the double point corresponds to g=π/2 or g=3π/2. The behavior of the motion near g=0 or g=π is discussed by the equations

depending on whether g≈0 or g≈π. 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

BROUWER, 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.

Abstract

Brouwer (1959) has noted that for the orbit of an earth satellite with inclination with reference to the earth’s equator, his solution based on Delaunay’s method fails because of the appearance of a factor G^2-5H^2 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 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 I >0 and in the direct sense when cos I<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 [1]

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,δ) is expanded into the series of spherical harmonics,

where μ is the gravitational constant multiplied by the mass of the spheroid and R is the equatorial radius of the spheroid. In this potential J_2R^2 is assumed to be a small quantity of the second order, and J_3R^3 and J_4R^4 are of the fourth order.

With Delaunay’s canonical variables,

[1] Smithsonian Astrophysical 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, l, g, and h to L', G', H', l', and h', l' 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:

where primes to be attached to the variables are omitted for simplicity.

As F* depends neither on h nor on l, there exist the following integrals, in addition to an energy integral F*=constant;

Then the Hamiltonian F* is a function of only two variables, G and g, which satisfy the following equations;

where P, Q, and S are functions depending only on G.

By the usual method we integrate these equations approximately by regarding G as a constant and g as a known linear function of time t in the right-hand members.

However, if H^2/G^2=l/5, the mean motion of g becomes a small quantity of the fourth order, and the amplitudes of periodic terms in solutions of G and g are of the zero-th order. This means that g and G in the right-hand members cannot be regarded as known functions.

The condition H^2/G^2=1/5 is satisfied if the inclination is about , 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

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:

where P_2 and P_3 are of the second order and the other coefficients except for P_0 are of the fourth order. A constant coefficient P_0 will not appear in the following discussion. P_1 is of the second order unless P(x) is expanded around x= 0. It is remarkable that S_0 vanishes.

Full expressions of P_1, Q_1, and S_1 are as follows:

Here, α 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:

As the Hamiltonian F* does not depend on time t explicitly, an equation,

with a constant C, represents a trajectory in the (ξ,η)-plane, where ξ=(H/G)2cos g, η=(H/G)2 sin g. If derivatives of both G and g with respect to time vanish at a certain point, this is an equilibrium point where the motion in the (ξ,η)-plane is stationary.

There are four equilibrium points of equation (6) in the (ξ,η)-plane, and their coordinates correspond to;

where k and x_1 are of the second order and

Trajectories in the (ξ,η)-plane

Assume that at a certain epoch t=t_0 a trajectory F*=—C passes through a point corresponding to x=δ and g=g_0; then the energy integral is written as:

Here δ is assumed to be of the second order.

From this integral g is solved as a function of x as follows:

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

g can be eliminated so that the equation contains x as the only variable, and the equations become of the following form;

where the signs correspond to those of cos g.

If terms of higher order are omitted, f_1(x) and f_2(x) are written as:

Here f_1(x)=0 corresponds to g=κ or 180°—κ, and f_2(x)=0 corresponds to 2g=180°.

If J_2 is positive, P_2 is also positive. Therefore in this case f_2(x) or f_1(x) is positive definite, depending on whether Q_0 is positive or negative.

Then there are two kinds of solutions, both expressed by Jacobi’s elliptic functions. If Q_0 is positive, one has

If Q_0 is positive, maxima and minima of x occur at g=k and g=180°—κ, and x vanishes at g=g_0. On the other hand, if Q_0 is negative, the maxima and minima occur at g=90° and —90°.

Trajectories F*=—C are given in figure 1 for two cases, Q_0>0 and Q_0<0. If Q_0 is positive and δ is of the second order, g cannot make a complete revolution but has a motion of libration around the equilibrium point near the ξ-axis, unless g_0 is very near to 90° or —90°. However, even though δ is of the first order, g has a motion of libration, if g_0 is very near to κ or 180°—κ. A trajectory with g_0=±90° corresponds to an asymptotic solution.

FIGURE 1.—Trajectories in the (ξ,η)-plane. Top: when Q_0 is positive; bottom, when Q_0 is negative.

Trajectories near the equilibrium points

The solutions discussed in the previous section will fail if (i) |cos 2g_0±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 2g_0 — 1| is very small.

In this case, expressions of f_1(x) and f_2 (x) are:

If Q_0 is positive, f_1(x) cannot vanish, that is, g cannot reach either to +90° or to –90°. And solutions of the equations (9) are,

where g_0 is κ or 180° — κ depending on whether cos g is positive or negative. The angular velocity β has the same expression as that in (13).

If |cos 2g_0+1| is very small, then

f_2(x) has two real roots, and by assuming that Q_0 is positive, one has

In this case any trajectory will cut the ξ-axis, and tends to make a complete revolution around the origin. On the other hand, if

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 Q_0 is negative should be derived in a similar way.

Small eccentricity

If Q_0 is very small the problem becomes very complicated, because in this case , J_2J_3, and J_2 J_4 terms must be added to the Hamiltonian. However, if Q_0 is very small because of a factor e^2, similar coefficients in the , J_2 J_3, and J_2 J_4 terms are also very small, due to the same factor e^2, and are neglected.

In this case κ is of the first order, and

Therefore, at least one of D_1 and D_2 is positive.

If D_1 is positive and D_2 is negative, g has a motion of libration around the equilibrium point near the ξ-axis, and the amplitude of x is of the second order. If D_1 is negative and D_2 is positive, the libration will occur around the equilibrium point on the η-axis.

However, if both D_1 and D_2 are positive, g has a motion of libration around one of the equilibrium points, if the two roots of one equation f_1(x)=0 are both larger than or both less than the two roots of f_2(x)=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,

If Q_0 is positive and x is expressed by equation (11), then

where hs is the secular part of h.

If Q_0 is negative, sin g_0 in equation (22) must be replaced by cos g_0.

Similar results are obtained for the mean anomaly. If Q_0 is positive, the solution is:

where ls is the secular part of l 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 planètes. 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 arc 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.