Gravity of Extended Bodies
Discussion
tidal forces
The tides, tidal forces, prolate spheroid, Roche limit
Derive the tidal force formula. Despite being called a force, it's more common to care about the field — that is, the gravitational field strength, also known as the acceleration due to gravity.
| g = | Gm |
| r2 |
Consider the tidal force of the Moon acting on the Earth. Let…
| r = | separation between the center of the Earth and the Moon |
| Δr = | radius of the Earth, the object experiencing the tide |
| m = | mass of the Moon, the object causing the tide |
The tidal force on the Earth is the difference between the gravitational field on its front (the point nearest the Moon) and the gravitational field at its center (the place we normally think of as the location of the Earth).
| Δg = | gfront | − | gcenter | |
| Δg = | Gm | − | Gm | |
| (r − Δr)2 | r2 |
Work that algebra. Work it!
| Δg = |
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| Δg = |
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| Δg = |
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Now the part that feels like a cheat. Eliminate the "small" terms — those with the most Δr in them. That gives us an approximate solution that's good enough for most purposes.
| Δg ≈ Gm | ⎛ ⎜ ⎝ |
2rΔr | ⎞ ⎟ ⎠ |
| r4 |
Simplify.
| Δg ≈ | 2GmΔr |
| r3 |
Let's derive the tidal force equation again, but this time let's do it the easy way — using calculus.
| Δg ≈ | dg | Δr |
| dr |
| Δg ≈ | d | ⎛ ⎜ ⎝ |
Gm | ⎞ ⎟ ⎠ |
Δr |
| dr | r2 |
| Δg ≈ | 2GmΔr |
| r3 |
Wow.
Roche limit
Good, now derive the Roche limit equation.
| gtidal | = | gsurface | |||||
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= |
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= |
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| rroche = rmoon | ∛ | 2mplanet |
| mmoon |
For the Earth and the Moon…
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flattening
Oblate spheroid
Equatorial radius a, polar radius c. The flattening factor (f), also called oblateness (Υ uppercase upsilon) or ellipticity (ε lowercase epsilon), is…
| f = | a − c |
| a |
Similar to eccentricity, which is used for elliptical orbits and prolate spheroids.
| e = √ | ⎛ ⎜ ⎝ |
1 − | b2 | ⎞ ⎟ ⎠ |
| a2 |
| e = | √(a2 − b2) |
| a |
gravity inside & outside
Two ways to solve problems. In general…
| g(r) = − G | ⌠⌠⌠ ⌡⌡⌡ |
r̂ dm |
| r2 |
where…
| g(r) = | gravitational field vector at any location in space |
| G = | gravitational constant |
| dm = | infinitesimal mass |
| r = | vector pointing out from infinitesimal mass to any location in space |
| r̂ = | direction of r |
| r = | magnitude of r |
Since
| r | ||
| Vg(r) = − | ⌠ ⌡ |
g(r) · dr |
| ∞ |
We get
| Vg(r) = − G | ⌠⌠⌠ ⌡⌡⌡ |
dm |
| r |
For systems with spherical, cylindrical, or planar symmetry…
∯ g · dA = −4πGm
For spherically symmetric mass distributions…
| r | |||
| g(r) = − | G | ⌠ ⎮ ⌡ |
ρ(r) 4πr2 dr r̂ |
| r2 | |||
| 0 |