3-8. Hill's Equations and Hill Radius
Jovian satellite
The Hill sphere is a volume around 2 interior to the Lanrange points L1 and L2. The volume
where satellite orbits are stable and escape to a circumstellar orbit is forbidden. The radius of
the Hill sphere, RH, is important dynamical parameter as shown below.
For example, if we are interested in the motion of a satellite orbiting 2 at distance of r2<<RH,
then we can ignore the solar gravity. This is because the satellite id deep within 2 potential
well and its motion is essentially two-body motion.
However, if the satellite is in a wide orbit around 2 such that r2~RH, then we cannot ignore
the solar gravity, that is, we must treat this as three-body problem.
Similarly, if particle P is in a low-energy orbit about the primary 1, and it approaches 2to
within a distance r2~RH, then large changes in P's orbit are likely (gravitational scattering).
But if the separations stay large, i.e., r2>>RH, then the effects of gravitational scattering is
weak.
Such a system was originally derived by Hill (1878).
Using Eqs. (3-16) and (3.17),
æ x+m
x -m ö
x˙˙ - 2ny˙ - n 2 x = -çm1 3 2 + m2 3 1 ÷
r1
r2 ø
è
æ m1 m2 ö
2
y˙˙ + 2nx˙ - n y = -ç 3 + 3 ÷ y
è r1 r2 ø
(3.16)
(3.17)
we have two equations for small mass ratio 1≈1 (i.e. 2≈0),
æx
x -1ö
x˙˙ - 2 y˙ - x = -ç 3 + m2 3 ÷
r2 ø
è r1
æ1 m ö
˙y˙ + 2 x˙ - y = -ç 3 + 32 ÷ y
è r1 r2 ø
y
(3.200)
(3.201)
(x,y)
r1
(-, 0)
r2
O
(+, 0)
x
1
We now transform the x-axis such that
x1+x leaving the y-axis unchanged and
=r2. Since we are now considering
motion close to the planet, we can assume
that x, y, and  are small quantities.
Neglecting higher powers of 2 we have
r12(1+2x) and Eqs. (3.200) and (3.201)
can be written
r1
(x,y)
r2
x
O
(-1-, 0)
(-, 0)
Hill’s equation
æ 1 m2 ö
y˙˙ + 2 x˙ = -ç 3 + 3 ÷ y + y
r2 ø
è r1
= -{1- 3x + O( x 2 )} y =-
m2
r23
yº
¶U H
¶y
m2
r
3
2
(3.202)
y+y
Hill’s equation
(3.203)
where
and
(3.204)
Inspection of Eq. (3.202)
reveals that the radial force vanishes
when 33=2. This leads to the definition of the Hill’s sphere as the sphere of radius
(3.207)
surrounding the secondary mass.