## 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$\cdot $05 or less. The accuracy of the solution for radial distance should then be about 0$\cdot $001%, and the error in angular travel about 0$\cdot $001% per revolution. The earth's oblateness has four main effects on the motion: (1) The orbital plane, instead of remaining fixed, rotates about the earth's axis in the opposite direction to the satellite, at a rate of 10$\cdot $00(R/$\overline{r}$)$^{3\cdot 5}$ cos $\alpha $ deg./day, where $\alpha $ is the inclination of the orbital plane to the equator, R the earth's equatorial radius and $\overline{r}$ 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$\cdot $5 $\surd $(R/$\overline{r}$) sin$^{2}\alpha $ 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$\cdot $1 R/$\overline{r}$ 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$\cdot $94(R/$\overline{r}$)sin$^{2}\alpha $ n. miles. (4) The major axis of the orbit rotates in the orbital plane at a rate of 5$\cdot $00(R/$\overline{r}$)$^{3\cdot 5}$ (5 cos$^{2}\alpha $ - 1) deg./day. Thus it rotates in the same direction as the satellite if $\alpha $ < 63$\cdot $4 degrees, or in the opposite direction if $\alpha $ > 63$\cdot $4 degrees. A brief comparison is made between theory and observation for Sputniks 1 and 2.