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Doubly averaged effect of the Moon and Sun on a high altitude Earth satellite orbit

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Abstract

Infinite series expansions are obtained for the doubly averaged effects of the Moon and Sun on a high altitude Earth satellite, and the results used to interpret numerically integrated examples. New in this paper are: (1) both sublunar and translunar satellites are considered; (2) analytic expansions include all powers in the satellite and perturbing body semi-major axes; (3) the fact that retrograde orbits have more benign eccentricity behavior than direct orbits should be exploited for high altitude satellite systems; and (4) near circular orbits can be maintained with small expenditures of fuel in the face of an exponential driving force one forI a<I<Ib, whereI b=180°−I a andI a is somewhat less than 39.2° for sublunar orbits and somewhat greater than 39.2° for translunar orbits.

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Abbreviations

a :

semi-major axis

A lk :

coefficient defined in Equation (11)

B lk :

coefficient defined in Equation (24)

C km :

coefficient defined in Equation (25)

D, E, F :

coefficients in Equations (38), (39)

e :

eccentricity

H k :

expression defined in Equation (34)

\(\bar H_t \) :

expression defined in Equation (35)

I :

inclination of satellite orbit on lunar (or solar) ring plane

J 2 :

coefficient of second harmonic of Earth's gravitational potential (1082.637×10−6 R 2E )

K k, Lk, Mk :

expressions in Section 4

\(\bar K_l , \bar L_l , \bar M_l \) :

expressions in Section 4

p=a(1−e 2):

semi-latus rectum

P l :

Legendre polynomial of degreel

q :

argument of Legendre polynomial

\(r = \frac{p}{{1 + e\cos \psi }}\) :

radial distance of satellite

R E :

Earth equatorial radius (6378.16 km)

R, S, W :

perturbing accelerations in the radial, tangential and orbit normal directions

syn:

synchronous orbit radius (42 164.2 km=6.6107R E)

t :

time

T :

satellite orbital period

T′:

orbital period of perturbing body (Moon)

T e :

period of long periodic oscillations ine for |I|<I a

T s :

synodic period

U :

gravitational potential of lunar (or solar) ring

x, y, z :

Cartesian coordinates of a satellite with (x, y) being the ring plane

γ:

coefficient defined in Equation (20)

Δβ:

average change in orbital element β over one orbit (β=a, e, I, Ω, ω)

ɛ1,ɛ2ɛ3 :

unit vectors in thex, y, z coordinate directions

ɛ r ,ɛ s ,ɛ w :

unit vectors in the radial, tangential and orbit normal directions

η=ω+ψ:

angle along the orbital plane from the ascending node on the ring plane to the true position of the satellite

θ:

angle around the ring

μ:

gravitational constant times mass of Earth (3.986 013×105 km s−2)

μ′:

gravitational constant times mass of Moon (or Sun)

μ m :

gravitational constant times mass of Moon (μ/81.301)

μ s :

gravitational constant time mass of Sun (332 946 μ)

π:

ratio of the circumference of a circle to its diameter

ϱ:

radius of lunar (or solar) ring

ϱ m :

radius of lunar ring (60.2665R E)

ϱ s :

radius of solar ring (23455R E)

ψ:

true anomaly

ω:

argument of perigee

ω0 :

initial value of ω

μ i :

critical value of ω in quadranti(i=1, 2, 3, 4)

Ω:

longitude of ascending node on ring plane

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Author notes

  1. Michael E. Ash

    Present address: Charles Stark Draper Laboratory, Inc., Cambridge, Mass., USA

Authors and Affiliations

  1. Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Mass., USA

    Michael E. Ash

Authors
  1. Michael E. Ash
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Additional information

This work was sponsored by the Department of the Air Force.

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Ash, M.E. Doubly averaged effect of the Moon and Sun on a high altitude Earth satellite orbit. Celestial Mechanics 14, 209–238 (1976). https://doi.org/10.1007/BF01376321

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  • DOI: https://doi.org/10.1007/BF01376321

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