<blockquote>
  <p>Is the SOI a spherical region or a oblate-spheroid-shaped region?</p>
</blockquote>

<p>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</p>

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

<p>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</p>

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

<p>Tada! The equation of a sphere!</p>

<p>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.</p>

<p>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 $Q_A = Q_B$.</p>

CC BY-SA 3.0
2015-02-24T12:59:27.517Z
2015-02-24T12:59:27.517Z
David Hammen
<p>The Wikipedia article on <a href="http://en.wikipedia.org/wiki/Sphere_of_influence_(astrodynamics)" rel="noreferrer">Sphere of influence</a> states that:</p>

<blockquote>
  <p>"A <strong>sphere</strong> of influence (SOI) in astrodynamics and astronomy is the
  <strong>oblate-spheroid-shaped</strong> region around a celestial body where the
  primary gravitational influence on an orbiting object is that body."
  (Emphasis added.)</p>
</blockquote>

<p>It then gives an expression for the radius of the sphere of influence. </p>

<p>Is the SOI a spherical region or a oblate-spheroid-shaped region? 
If it is an oblate-spheroid-shaped region, then why?</p>

<gravity>
Is the SOI a spherical region or a oblate-spheroid-shaped region?
Christo
# Is the SOI a spherical region or a oblate-spheroid-shaped region?

(Asked by: username_0 on 2015-02-23)

The Wikipedia article on [Sphere of influence](http://en.wikipedia.org/wiki/Sphere_of_influence_(astrodynamics)) states that:

> "A **sphere** of influence (SOI) in astrodynamics and astronomy is the
> **oblate-spheroid-shaped** region around a celestial body where the
> primary gravitational influence on an orbiting object is that body."
> (Emphasis added.)

It then gives an expression for the radius of the sphere of influence.

Is the SOI a spherical region or a oblate-spheroid-shaped region?
If it is an oblate-spheroid-shaped region, then why?

**Tags:** gravity

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## 1 answers

### Answer by username_1 on 2015-02-24. Score: 3 | Accepted answer

> 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 $Q\_A = Q\_B$.


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