Where's that moon?

We are bounded in a nutshell of Infinite Space: Week 10: Worksheet #18: Problem #1: Where's that moon?  

1. Hill spheres. One outcome of planet formation is systems of satellites around planets. Now you may ask yourself, why do some planets have moons 10s of millions of kilometers away, while the Earth’s moon is only 400,000 km away. To answer this question we need to think about how big of a region around a planet is dominated by the gravity of a planet, i.e. the region where the gravitational pull of the planet is more important than the gravitational pull of the central star (or another planet).

a) Gravitational forces. Put a test mass somewhere between a star of mass \(M_S\) and a planet of mass \(m_P\) at a distance \(r_P\) from the star. Make a drawing marking clearly these characteristics as well as the distance r between the test particle and the planet. Write separate expressions for the gravitational force on the particle from the star and on the particle from the planet. At what distance r from the planet are the two forces balanced? This distance approximates the radius of the Hill sphere, which in the case of planet formation is the sphere of disk material which a planet can accrete from.

(b) Planetary Hill radii. Calculate the Hill radii for Earth, Jupiter, and Neptune. How do they compare with the separation between the planets and their most distant moons? 

a. With these hill spheres, the defining characteristic is the point of equilibrium at which the point mass would be find between the two masses. This is expressed with the law of universal gravitation: \[ F_{G_{r_P}} = F_{G_{r_P - r}} ,\] and reconfiguring these equations, we find how: \[ \frac{G M_S m_{part} }{(r_P - r)^2} = \frac{G m_P m_{part}}{r^2}\] \[\frac{M_S}{(r_P - r)^2} = \frac{m_P}{r^2} \] \[\frac{r^2}{(r_P - r) ^2} = \frac{m_P}{M_S},\]  which can be further simplified till finding: \[ \frac{r}{r_P - r} = \sqrt{\frac{m_P}{M_S}}\] \[ r = \sqrt{\frac{m_P}{M_S}} (r_P - r)\] \[ r(1+ \sqrt{\frac{m_P}{M_S}}) = \sqrt{\frac{m_P}{M_S}} r_P ,\] and thus we have \[ r = \frac{\sqrt{\frac{m_P}{M_S}} r_P}{ 1+ \sqrt{\frac{m_P}{M_S}}},\] which can be given a final derivation till a fairly simple relationship emerges: \[ r = \frac{\sqrt{m_P} r_P }{ \sqrt{M_S} + \sqrt{m_P}}\]

b. Now, we can take the equation we have found and find the hill radii for several planets in our solar system. But first, let us define a couple of values, such as the mass of the Earth: \[M _\oplus = 5.9 \times 10^{24} kg \] \[ M_\odot = 2 \times 10^{30} kg,\] and therefore: \[ 1 ~M_\odot = 3.3 \times 1- ^5 M_\oplus ,\] similarly: \[ 1 ~AU = 1.5 \times 10^8 km\]

First, for the Earth itself, we can find its hill radius: \[ r_\oplus = \frac{\sqrt{M_\oplus} r_P }{ \sqrt{M_S} + \sqrt{M_\oplus}},\] plugging in values, we find that: \[ r_\oplus = \frac{\sqrt{1 M_\oplus} (1 ~ AU) }{ \sqrt{3.3 \times 10 ^5 M_\oplus } + \sqrt{ 1 M_\oplus}} ,\] \[ r_\oplus = \frac{1}{5.7 \times 10^{2}}\] \[ r_\oplus \approx 2 \times 10^{-3} AU \approx  3 \times 10^5 km, \] and comparing it to the radius at which the Moon is found, we see that the hill radius we have calculated is similar to the actual distance to the Moon to a factor less than 2.

Now for Jupiter. Following the same process, we find that: \[ r_{Jup} = \frac{\sqrt{M_{Jup}} r_P }{ \sqrt{M_S} + \sqrt{M_{Jup}}},\]  \[ r_{Jup} = \frac{\sqrt{320 M_\oplus} \cdot 5.2 ~AU} { \sqrt{3.3 \times 10^5 M_\oplus } + \sqrt{ 320 M_\oplus } },\] which means the hill radius is \[ r_{Jup} \approx 0.15 AU \approx 2.2 \times 10^7 km ,\] which which is practically the same as the actual distance to (one of) the moons of Jupiter, \(2.4 \times 10^7 km\).   

And finally Neptune, the hill radius is: \[ r_{Nep} = \frac{\sqrt{M_{Nep}} r_{Nep} }{ \sqrt{M_S} + \sqrt{M_{Nep} } }\] \[ r_{Nep} = \frac{\sqrt{17 M_\oplus} \cdot 30~ AU}{ \sqrt{3.3 \times 10^5 M_\oplus } + \sqrt{17 M_\oplus}  },\]  and therefore it results in:\[ r_{Nep} \approx 0.2 ~AU \approx 3 \times 10^7 km,\] which is but a factor less than two away from the precise measurement of Neptune’s moon’s distance.