# Which moon is best? (for gravity assists)

Inspired by my answer to Is a ballistic Jovian capture using the Galilean moons possible from interplanetary entry? in which I discovered that Callisto offers a stronger gravity assist than Ganymede despite being less massive.

This is because Callisto is further from Jupiter than Ganymede and thus, in a trajectory gravitationally dominated by Jupiter, an object will encounter Callisto with significantly less relative velocity ($$V_{\infty}$$). The strength of gravity assist can be realized as the amount it deflects a trajectory:

$$\delta = 2 \cdot sin^{-1} \biggr ( \frac{1}{1 + \frac{r_p \cdot V_{\infty}^2}{\mu}} \biggr)$$

The strength can also be quantified as a $$\Delta V$$:

$$\Delta V = V_{\infty} \cdot \sqrt{2-2 \cdot \cos{\delta}}$$

Question: Which moon in the solar system can provide the strongest gravity assist considering an object on an interplanetary arrival trajectory (initially hyperbolic at the host planet) deflection and $$\Delta V$$ wise?

• That's a very complex question, because obviously you would not only be affected by the moon's gravity, but also the planet the moon orbits. In the case of Jupiter, that is some massive (pun intended) influence Nov 20, 2021 at 12:02
• @CuteKItty_pleaseStopBArking the complexity lowers significantly in the case where the parent planet /body accounts for the high high majority of system mass (like with Jupiter). Consider a gravity assist around Jupiter itself; the Sun's gravity doesn't play a big role outside of determining $V_{\infty}$ for the hyperbolic flyby of Jupiter Nov 20, 2021 at 16:47
• Gee, this is interesting!
– uhoh
Nov 20, 2021 at 22:18

Using a selection from Wikipedia's List of natural satellites (moons with listed masses) I, with some simplifying assumptions, calculated both the deflection, $$\delta$$, and $$\Delta V$$ for each moon.

In all cases the moons were assumed to be in a circular orbit around the parent planet. To find the $$V_{\infty}$$ for the encounter, I calculated the orbital speed, $$V$$, about the parent planet at the moon's orbital radius for a $$V_{\infty}=0$$, technically parabolic but more or less hyperbolic (and fair between planets), arrival. I then subtracted and added the moon's orbital speed (circular orbit speed at moon's semi-major axis) to get the $$V_{\infty}$$.

Differing encounter geometry between the trajectory and the moon means that $$V_{\infty}$$ is bounded by $$V-V_{moon}$$ as a minimum (prograde) and $$V+V_{moon}$$ as a maximum (retrograde).

The equations given in the question can then be used where the flyby close approach, $$r_p$$, is the moon's radius (+500 km for Titan).

Here is the retrograde case (logarithmic scale): And here is the prograde case (logarithmic scale): Zooming in on the top right (logarithmic scale): In $$\Delta V$$ terms; a prograde Titan gravity assist is strongest at $$1.63$$ $$km/s$$. In deflection terms; a prograde gravity assist by the Moon is strongest at $$140°$$.

### P.S.

The Moon exhibits a unique variation on the plot from retrograde to prograde, different from any other moon (at least of those shown in the zoomed in view). All of the other moons have a positive 'slope', if you will, while the Moon has a negative 'slope' from retrograde to prograde. I believe this is because, as Wikipedia states:

[The Moon] is the largest natural satellite in the Solar System relative to the size of its planet

But I don't understand how this mechanism works. I did make two cool Desmos plots though showing the variation from prograde to retrograde:  • Some moons smaller than Pluto's moon Charon are included. Pluto isn't a "planet", but can you/your reference tell wherea its large moon would fall on these plots? Jul 23, 2022 at 1:27