The Physics
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Opus in profectus

# Gravity of Extended Bodies

## Discussion

### tidal forces

The tides, tidal forces, prolate spheroid, Roche limit

Let…

 r = separation between planet and moon a, b = radius of planet and moon, respectively ma, mb = mass of planet and moon, respectively

Derive the tidal force formula.

 gtidal = gfront − gback gtidal = Gmb − Gmb (r − a)2 (r + a)2

Work that algebra. Work it!

 Gmb − Gmb (r − a)2 (r + a)2
 Gmb ⎛⎝ (r + a)2 − (r − a)2 ⎞⎠ (r − a)2(r + a)2
 Gmb ⎛⎝ (r2 + 2ra + a2) − (r2 − 2ra + a2) ⎞⎠ r4 − a4

Simplify.

 gtidal = Gmb ⎛⎝ 4ra ⎞⎠ r4 − a4

Super-simplify.

 gtidal ≈ 4Gmba r3

Good, now derive the Roche limit.

 gtidal ≈ gsurface 4Gmab ≈ Gmb r3 b2
 r ≈ b ∛ 4ma mb

### flattening

Oblate spheroid

Equatorial radius a, polar radius b. The flattening factor (also called oblateness) is…

 ε = a − b 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