Gravity of Extended Bodies

Discussion

tidal forces

The tides, tidal forces, prolate spheroid, Roche limit

[magnify]

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. Worlk it!

gtidal = Gmb  ⎛
⎝		 (r + a)2 − (r − a)2 ⎞
⎠		  = Gmb  ⎛
⎝		 (r2 + 2ra + a2) − (r2 − 2ra + a2) ⎞
⎠
(r − a)2(r + a)2 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

Polar radius a, equatorial radius c. The flattening factor (also called oblateness) is …

ℰ =  a − c
a

gravity inside & outside

Two ways to solve problems …

  1. in general …
g(r) = − G  ⌠⌠⌠
⌡⌡⌡		 r̂ dm & Vg = − G  ⌠⌠⌠
⌡⌡⌡		 dm
r2 r2
all space all space
  1. for spherically symmetric mass distributions (something like Gauss's Law)
r r
g(r) = −  G ⌠
⌡		  ρ(r) 4πr2 dr r̂ & Vg(r) = −  ⌠
⌡		  g(r) · dr
r2
0 ∞

Prove that the gravitational field inside a uniform mass distribution is 0. This is the key.

The Physics Hypertextbook

  • No condition is permanent.