Gravity Of Extended Bodies

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Gravity Of Extended Bodies

Discussion

tidal forces

The tides, tidal forces, prolate spheroid, Roche limit

Let…

r =separation between planet and moon
ab =radius of planet and moon, respectively
mamb =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 
r3b2 
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
 =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

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