Jenab said:
That's what I used to think the Hill radius was. But I did some simulations on the three-body problem, Earth-Sun-test particle, and it looks like the outer stability radius is
Rmax = (D/3) { (Me+m) / Ms }^(1/3)
Where
Me is the Earth's mass.
Ms is the sun's mass.
m is the mass of the test particle.
D is the distance between Earth and Sun.
If D=1.4959787E+8 km, Me=5.976E+24 kg, Ms=1.989E+30 kg, and m=0, then
Rmax = 719553 km
Which is about half of your estimate.
Jerry Abbott
From what I can tell, the original usage "The Hill Sphere" is correct - the term "the Hill radius" is similar but omits some important proportionality factors.
The results I get are based on exact theory rather than simulations (but with some numerical approximations due to a series expansion so that the formula is only good for large mass ratios. The mass ratio of the earth/sun is very large, however).
The formula can be found in the
Wikipedia
The relevant formula is
r = a (m/3M)^(1/3)
as per the Wikipedia article. This is different by a factor of roughly 2:1 from your result.
Here a is the orbital semimajor axis, m is the mass of the smaller body, and M is the mass of the primary.
The exact theory is only good when there are only three bodies, though - with more than three bodies, objects undoubtedly need to be inside the Hill sphere for the orbit to be stable.
I'll go a bit into the theory behind this result.
There is a constant of motion of the 3 body problem, called the Jacobi integral. You can see the mathematical formula for it
here
It's basically the Hamiltonian of the problem re-written in terms of position and it's derivatives (like the Lagrangian). This form of the Hamiltonian is often known as "the energy function". Because it's a constant of motion, for any orbit this quantity is the same everywhere.
There are some sample plots of the behavior of this function, plotted as an iinequality for some specific mass ratios. The function is plotted as an inequality because it involves both position and it's derivative, and the two dimension plot only shows the position terms. These plots are located
here
As long as the Jacobi integral is above a critical value, there is a bounded region that the third body cannot leave, as it does not have sufficient energy.
The critical value of the Jacobi integral occurs at the Lagrange points L1 and L2. This is where the numerical approximations come in - there is no exact formula for the location of the Lagrange points, but they are approximately located at
a (m/3M)^(1/3)
from the secondary, which gives the radius of the Hill sphere.
Note that for the mass ratios of .01 plotted, the so-called "Hill sphere" is really rather egg-shaped. For the Earth-sun case, the value of the mass ratio should be much larger, and the region should look much more spherical.