The "Sphere of Influence" of the Moon is not really spherical

A tool often employed in the early stages of space mission design is the patched conics approximation. In this formalism, a trajectory is approximated by a sequence of two-body orbits, defined with respect to an appropriately changing primary body, according to the updated position of the spacecraft. The validity of this approach rests on the assumption that in each portion of the trajectory the acceleration of the spacecraft mostly arises from the gravitational attraction by a single body, which dominates over all the other perturbations. For instance, to an excellent degree of approximation, the motion of an interplanetary probe is mostly governed by solar gravity, except when in the immediate vicinity of a planet.

The "Sphere of Influence" (SOI) of a celestial body with respect to a distant, major primary is formally defined below. At the lowest order of approximation, the SOI is a sphere centered on the secondary body, and of radius $R_S = D \, \left(\frac{m}{M}\right)^{2/5}$, where the primary and the secondary have mass $m$ and $M$, respectively, and $D$ is their mean distance. The spherical approximation is increasingly accurate as the mass ratio $m/M$ decreases. 

In the case of the major planets of the solar system with respect to the Sun, $m/M$ is $\lesssim 1\%$, and thus their SOI's are very nearly spherical, with a radius that is much smaller than the semimajor axis of their heliocentric orbits. Conversely, with a mass ratio of $\approx 0.012$, the boundary of the SOI of the Moon with respect to the Earth is significantly non-spherical. In this post I discuss in detail the derivation of the accurate shape of the lunar SOI, and put it in context with other solar system examples.

 

Definition of the SOI

The equations of motion of a test particle under the gravitational attraction of $m$ and $M$ can be written equivalently with either $m$ or $M$ as the origin of the coordinates, as (refer to the figure below):

$\dfrac{d^2{\bf r}}{dt^2} = {\bf a}_m^p + {\bf a}_m^d = -\dfrac{G m}{r^3}{\bf r} - G M \left( \dfrac{1}{d^3}{\bf d} + \dfrac{1}{\rho^3}{\boldsymbol{\rho}} \right)$ 

or 

$\dfrac{d^2{\bf d}}{dt^2} = {\bf a}_M^p + {\bf a}_M^d = - \dfrac{G M}{d^3}{\bf d} - G m \left( \dfrac{1}{r^3}{\bf r} - \dfrac{1}{\rho^3}{\boldsymbol{\rho}} \right)$

In other words, the first form emphasizes the gravitational attraction to $m$ as the primary acceleration, treating the term due to $M$ as the perturbation, and vice versa. In both cases note that, having chosen the origin of the coordinates coincident with the primary, the perturbation contain a direct and an indirect term.

 

Clearly, the relative advantage of either form depends on the ratio of the disturbing over primary acceleration. In the course of the trajectory propagation, it can be convenient to switch from on the description to another, according to which form results in the smallest ratio. This observation leads naturally to the definition of the SOI as the region bounded by the surface over which the ratio of the perturbing over primary acceleration is the same in the two descriptions.

 

The accurate shape of the SOI boundary

Intuitively, the SOI boundary is a closed surface which encloses $m$ (for $m < M$), and is axially symmetric with respect to the line connecting $m$ and $M$. The radius of the bounding surface as a function of the angle $\alpha$ can be calculated from the defining equation:

$\dfrac{||{\bf a}_m^d||}{||{\bf a}_m^p||} = \dfrac{||{\bf a}_M^d||}{||{\bf a}_M^p||}$

Substituting the expressions for the accelerations given above, and introducing the definitions $x \equiv r/\rho$, $\nu \equiv \cos \alpha$, the radius of the SOI is implicitly defined by the equation:

$x^4 = \left(\dfrac{m}{M}\right)^2 \left(\dfrac{d}{\rho}\right)^4 \left[ \dfrac{1 + x^4 - 2\nu x^2}{1 + (d/\rho)^4 + 2 (d/\rho) (\nu x - 1)} \right]^{1/2}$     with     $(d/\rho) = \sqrt{1 + x^2 - 2\nu x}$. 

This equation can be solved numerically using a standard root-finding algorithm (see also [1] for a dedicated discussion of the numerical challenges connected with this specific equation). A suitable starting value, based on the spherical approximation, is $x_0 = (m/M)^{2/5}$.

To develop some intuition on the shape of the function $r(\alpha)$, we can also construct approximate solutions, by expanding the accelerations in terms of Legendre polynomials, and then truncating the series at the first or second term. Following this approach, sketched in Section 8.5 and Problem 8-16 of [2], we can write:

$\dfrac{a_m^d}{a_m^p} = \dfrac{M}{m} x^3 \sqrt{1+3\nu^2} \left[1 + \dfrac{6\nu^3}{1+3\nu^2} x + O(x^2) \right]$ 

$\dfrac{a_M^d}{a_M^p} = \dfrac{m}{M} \dfrac{1}{x^2} [1 - 2\nu x + O(x^2)]$ 

Dropping the linear terms in $x$ in the brackets from both expressions, we arrive at the already familiar spherical approximation:

$\dfrac{M}{m} x^3 \sqrt{1+3\nu^2} = \dfrac{m}{M} \dfrac{1}{x^2} \ \ \Rightarrow \ \  \dfrac{r}{\rho} = \left(\dfrac{m}{M}\right)^{2/5} (1+3\nu^2)^{-1/10} \approx \left(\dfrac{m}{M}\right)^{2/5}$

where the term $(1+3\nu^2)^{-1/10}$ is always very close to one, and can therefore be omitted at this level of approximation.

Retaining the linear terms, we obtain the lowest order non-spherical correction:

$\dfrac{r}{\rho} = \left[ \left(\dfrac{m}{M}\right)^{-1/5} (1 + 3\cos^2 \alpha)^{-1/10} + \dfrac{2}{5} \cos \alpha \left( \dfrac{1+6\cos^2\alpha}{1+3\cos^2\alpha} \right) \right]^{-1}$ 

 

The lunar SOI 

The bounding surface of the lunar SOI is plotted in the next figure, where the radius of the surface was calculated numerically from the complete equation (blue line), as well as from the spherical and first order approximation (purple and orange lines, respectively). All objects are drawn to scale.

 

 

For the lunar case the spherical approximation gives $R_{\rm S, Moon} \approx 66200$ km (purple). We can see from the figure that the surface is approximately centered on the Moon, and it is compressed along the Moon-Earth direction, more significantly toward the Earth ($\alpha=0^\circ$) than away from it ($\alpha=180^\circ$). We also note that the first order approximation gives a very accurate representation of the exact solution found numerically.  To appreciate their differences the radius as a function of the angle from the Earth-Moon axis is shown in the next figure:

We note that the radius of the actual surface is slightly larger than $R_S$ in the vicinity of $\alpha=\pm 90^\circ$; the first order approximation already captures qualitatively this behavior. 

To gain more insight into the shape of the SOI, we can consider the effect of varying the mass parameter. In the next figure, I have plotted the radius of the SOI scaled to the distance between $m$ and $M$ for the three values $m/M = 0.122$, $0.0123$, and $2.4\cdot 10^{-4}$ (roughly corresponding to the pairs Charon-Pluto, Moon-Earth, and Titan-Saturn, respectively). Thanks to the normalization, the SOI's and the location of $m$ and $M$ for these different systems can be plotted to scale in the same figure.

 

At the relatively large mass ratio of the Charon-Pluto pair, the asymmetry in the near vs. far side of the more massive body is more pronounced, and the center of the surface does not coincide with the smaller body. The opposite is observed for the Titan-Saturn mass ratio, with the lunar case displaying an intermediate behavior. 

Physically, the elongated shape in the direction perpendicular to the axis is characteristic of "tidal" perturbative accelerations, such as $a_m^d$ and $a_M^d$. The offset of the SOI center observed at large $m/M$ is mostly a geometric effect, due to the non-negligible size of the SOI in comparison with the distance between the two attracting bodies.

 

References

[1] Burrows, R.R., 1966. The classical "sphere-of-influence" (No. NASA-TM-X-53485).

[2] Battin, R.H., 1999. An introduction to the mathematics and methods of astrodynamics. AIAA.