The effect of the earth’s oblateness on the orbit of a near satellite

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Abstract
The equations of motion of a satellite in an orbit over an oblate earth in vacuo are solved analytically, by a perturbation method. The solution applies primarily to orbits of eccentricity 0⋅05 or less. The accuracy of the solution for radial distance should then be about 0⋅001%, and the error in angular travel about 0⋅001% per revolution. The earth’s oblateness has four main effects on the motion: (1) The orbital plane, instead of remaining fixed, rotates about the earth’s axis in the opposite direction to the satellite, at a rate of 10⋅00(R/r̄)3.5 cos α deg./day, where α is the inclination of the orbital plane to the equator, R the earth’s equatorial radius and r̄ the satellite’s mean distance from the earth’s centre. (2) The period of revolution of the satellite, from one northward crossing of the equator to the next, is 14⋅5 √(R/r̄) sin2α sec greater for an inclined orbit than for an equatorial orbit. (3) The radial distance r from the earth’s centre changes. For a given angular momentum the mean r is 14⋅5 R/r̄ nautical miles greater for a polar orbit than an equatorial one. Also, during each revolution r oscillates twice, the amplitude of the oscillation being 0⋅94 (R/r̄) sin2αn. miles. (4) The major axis of the orbit rotates in the orbital plane at a rate of 5⋅00(R/r̄)3.5 (5 cos2α —1) deg./day. Thus it rotates in the same direction as the satellite if α < 63⋅4°, or in the opposite direction if α > 63⋅4°. A brief comparison is made between theory and observation for Sputniks 1 and 2.