Is the SOI a spherical region or a oblate-spheroid-shaped region?
Is the SOI a spherical region or a oblate-spheroid-shaped region?


The sphere of influence is neither a sphere nor an oblate spheroid. It is a surface with no name. An approximation of this surface is

$$\left(\frac r R\right)^{10}(1+3\cos^2\theta) = \left( \frac m M \right)^4$$

This is neither a sphere nor an oblate spheroid, and this is but an approximation. Thefull expression is an absolute mess. Dropping the factor of $(1+3\cos^2\theta)^{1/10}$ (which is close to one) yields

$$r = \left( \frac m M \right)^{2/5} R$$

Tada! The equation of a sphere!

The true surface is defined in terms of two ratios. Consider two gravitating bodies, call them body A and body B. From the perspective of an inertial frame, the acceleration of a tiny test mass toward these two bodies is given by Newton's law of gravitation. These two bodies accelerate toward one another as well, so a frame based at the origin of either body is non-inertial.

From the perspective of a frame at the center of body A, the acceleration of the test mass is the inertial frame acceleration of the test mass toward body A plus the inertial frame acceleration of the test mass toward body B less the acceleration of body A toward body B. Denote the acceleration of the test mass toward body A as the primary acceleration and the difference between the inertial frame accelerations of the test mass and body A toward body B as the disturbing acceleration. Finally, define $Q_A$ as the ratio of these two. Now do the same for a frame with origin at the center of body B. The sphere of influence is that surface where $Q_A = Q_B$.