Understanding the Hill Sphere: Exploring the Formula & Its Implications

In summary, the Hill sphere is the area in which a celestial object is the dominant force. The formula for calculating the radius of the Hill sphere is R_{H} = a \left(\frac{M_{\mathrm{minor}}}{M_{\mathrm{major}}}\right)^{2/5}. This is different from the formula for the Lagrange points L1 and L2, which are found at R_L = a \left(\frac{M_{\mathrm{minor}}}{3M_{\mathrm{major}}}\right)^{1/3}. The derivation for the Hill sphere can be found in Astrodynamics lecture notes, not the Wikipedia article which discusses the Laplace Sphere.
  • #1
Ryan H
15
0
The Hill sphere of a celestial object is the area in which it is the dominating force in the area.

I was wondering how the formula was determined.

If the mass of the smaller body is m, and it orbits a heavier body of mass M with a semi-major axis a and an eccentricity of e, then the radius r of the Hill sphere for the smaller body is, approximately:

Equation: [tex]r \approx a(1-e)\sqrt[3]{m/3M}[/tex]

I believe I understand how to determine eccentricity and the semi-major axis, I'm just searching for how the whole thing comes together.

http://en.wikipedia.org/wiki/Hill_sphere"

The derivation on the wikipedia article didn't help much.
 
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  • #2
Ryan H said:
The Hill sphere of a celestial object is the area in which it is the dominating force in the area.

I was wondering how the formula was determined.

If the mass of the smaller body is m, and it orbits a heavier body of mass M with a semi-major axis a and an eccentricity of e, then the radius r of the Hill sphere for the smaller body is, approximately:

Equation: [tex]r \approx a(1-e)\sqrt[3]{m/3M}[/tex]

I believe I understand how to determine eccentricity and the semi-major axis, I'm just searching for how the whole thing comes together.

http://en.wikipedia.org/wiki/Hill_sphere"

The derivation on the wikipedia article didn't help much.

The reason why it didn't help is that it's wrong. The derivation is for the Laplace Sphere, which is a related concept but distinct mathematically...

as discussed here on Physforums before

...there's Astrodynamics lecture notes available online covering it, but I'll have to track them down.
 
Last edited by a moderator:
  • #3
qraal said:
The reason why it didn't help is that it's wrong. The derivation is for the Laplace Sphere, which is a related concept but distinct mathematically...

as discussed here on Physforums before

...there's Astrodynamics lecture notes available online covering it, but I'll have to track them down.

I tracked down my old copies of the University of Wisconsin EMA 550 lectures and the Wikipedia entries. The equation for the Sphere of Influence aka the Hill sphere has been confused with the Lagrange Points L1 & L2. They're similar concepts in some features, so it's an understandable confusion. So it goes like so...

[itex]R_{H} = a \left(\frac{M_{\mathrm{minor}}}{M_{\mathrm{major}}}\right)^{2/5}[/itex]

...and the L1 & L2 points are found (+/-) in line with the the minor body's semimajor axis around the major body via...

[itex]R_L = a \left(\frac{M_{\mathrm{minor}}}{3M_{\mathrm{major}}}\right)^{1/3}[/itex]
 

What is the Hill Sphere?

The Hill Sphere, also known as the Roche Sphere, is a region of space around a celestial body (such as a planet or star) where its gravitational influence is dominant over the gravitational influence of other nearby objects.

How is the Hill Sphere calculated?

The Hill Sphere is calculated using the formula: r = a(1-e)(m/3M)^1/3, where r is the radius of the sphere, a is the semi-major axis of the celestial body's orbit, e is the eccentricity of the orbit, m is the mass of the celestial body, and M is the mass of the larger object it is orbiting.

What is the significance of the Hill Sphere?

The Hill Sphere is important because it determines the boundaries of a celestial body's gravitational influence. Objects within the Hill Sphere are more likely to be captured by the celestial body's gravitational pull and become its satellites.

How does the Hill Sphere affect satellites?

The Hill Sphere affects satellites by determining their stability. Satellites within the Hill Sphere will likely remain in orbit, while those outside of it may experience orbital perturbations and eventually become unbound from the celestial body.

Can the Hill Sphere change over time?

Yes, the Hill Sphere can change over time due to changes in the mass or orbital parameters of the celestial body or the larger object it is orbiting, such as a planet or star. For example, if a planet gains or loses mass, its Hill Sphere will also change accordingly.

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