The Physics
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Opus in profectus

Gravity of Extended Bodies

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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

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. In general…

 
g(r) = − G  ⌠⌠⌠
⌡⌡⌡
 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
 =  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 
r2
0