Download The Hill Sphere

The Hill Sphere
(Source: http://www.jgiesen.de/astro/stars/roche.htm)
An astronomical body's Hill sphere is the region in which it dominates the attraction of
satellites. It is named after John William Hill (1812–1879).
Determining the Hill Sphere radius
The satellite or moon (mass μ) is orbiting the star (mass M) with the same angular velocity ω
at the distance R+r as the planet (mass m) at the distance R (permanent full moon position).
The equilibrium condition for the planet is:
m ω2 R = G m M/R2
ω2 = GM/R3
The satellite is dragged by the combined gravitational forces exerted by the star and the
planet:
μ ω2 (R+r) = G μ M/(R+r)2 + G μ m/r2
Inserting ω2:
G μ M (R+r)/R3 = G μ M/(R+r)2 + G μ m/r2
M (R+r)/R3 = M/(R+r)2 + G m/r2
M (R+r)3 r2 = M R3 r2 + m R3 (R+r)2
m R3 (R+r)2 = M r2 (R3+3R2r+3Rr2+r3) - M R3 r2
m R3 (R+r)2 = M r3 (3R2+3Rr+r2)
For r<<R: (R+r)2 ≈ R2, and 3Rr+r2 ≈ 0. The equation simplifies:
m R5 = 3 M r3 R2
m R3 = 3 M r3
r = R [m/(3M)]1/3
Sun
Earth
Hill_sphere.doc
mass
orbit radius
Hill sphere
r
orbit radius
moon
1.99×1030 kg
149,600,000
km
1,496,000
km
384,400 km
= 1/4 r
5.97×10
24
kg
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Can moons have moons?
Source:
http://curious.astro.cornell.edu/our-solar-system/the-moon/44-our-solar-system/the-moon/general-questions/104-can-moons-hav
e-moons-intermediate
Yes, in theory, moons can have moons. The region of space around a satellite where a
sub-satellite can exist is called the Hill sphere. Outside the Hill sphere, a sub-satellite would be
lost from its orbit about the satellite.
An easy example is the Sun-Earth-Moon system. Earth is a satellite of the Sun and the Moon is a
sub-satellite orbiting Earth. The Moon orbits the Earth because the Moon is 370,000 km from
Earth, well within Earth's Hill sphere, which has a radius of 1.5 million km (0.01 AU or 235
Earth radii). Loosely speaking, the Hill sphere defines the space where the Earth's gravity is
more important than the Sun's gravity on another object. If the Moon somehow ended up outside
Earth's Hill sphere, the Moon would orbit the Sun instead of the Earth just like all the other
planets, asteroids, and comets. For comparison, Jupiter's Hill sphere has a radius of 0.35 AU
which is much much larger than the Earth's Hill sphere. This is because Jupiter is more massive
than the Earth and has a stronger gravitational pull, but more importantly because it is further
from the Sun than Earth so the Sun's gravity is weaker at Jupiter than at Earth. This gives Jupiter
a lot of gravitational influence on the space around it. Having such a big Hill radius could
explain why Jupiter has a lot of moons and can affect the orbits of passing comets so strongly.
為什麼木星有很多 (已知 60 餘個) 木衛 ?
Can the Moon have a moon?
Yes, the Moon could have a sub-satellite. If we look at a system of the Earth, Moon, and a
sub-satellite, the same idea as above applies. The Moon has its own Hill sphere with a radius of
60,000 km (1/6th of the distance between the Earth and Moon) where a sub-satellite could exist.
If an object lies outside the Moon's Hill sphere, it will orbit Earth instead of the Moon. The only
problem is that the sub-satellite cannot stay in orbit around the Moon indefinitely because of
tides.
The Moon, like almost all other moons in the solar system, is in synchronous rotation about the
Earth meaning it shows the same face to Earth at all times (its rotation period about its own axis
is the same as its orbital period about the Earth), which is a result of tidal forces between the
Earth and Moon. These are the same tidal forces that cause the high and low tides on Earth. In
this configuration, any object within the Hill sphere of the Moon will have its orbit decay due to
tides! That means the orbit of any sub-satellite of the Moon will shrink over time. In other words,
the distance between the sub-satellite and the Moon will get smaller and smaller until the
sub-satellite crashes into the Moon or the lunar tides rip the sub-satellite apart!
How does the Moon exist if it is a sub-satellite itself?
The reason this argument does not apply to the Sun-Earth-Moon system is that the Earth itself
does not synchronously rotate (nor do any of the planets) about the Sun like the Moon and other
satellites do around the planets. This allows the Moon to have a stable orbit around the Earth.
What about man-made lunar orbiters? How do they survive?
Lunar orbiters only orbit the Moon for a few years, a very short time by astronomical standards.
Man-made satellites can stay in orbit around the Moon or any moon for the duration of a mission
because tidal effects require thousands or millions or more years, depending on the system, to
cause the loss of a sub-satellite. Because of this we can leave a man-made satellite in stable orbit
around a moon for a few years using the spacecraft's rocket thrusters to correct for any changes
in its orbit.
Hill_sphere.doc
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