THE ROCHE LIMIT

DEFINITION

The Roche limit is the distance from a primary body (eg a planet) at which tidal forces will disrupt a small body (eg an asteroid) that is only held together by gravitation force (ie no other chemical forces holding it together). If the secondary body approaches the primary closer than the Roche limit it will pulled apart by tidal forces from the primary body. It is thought that this is how Saturn's rings were formed.

The Roche limit is found in many areas of astrophysics including the physics of black holes. Closer to home it may be used to decide whether loose surface material on an asteroid may be pulled off the surface of the asteroid during a close encounter with the Earth or other planetary body.

DERIVATION - 1

The simplest derivation may be done with reference to the diagram below:

The gravitational attraction between the two small bodies is:

F_a = G m² / ( 2 r )²

F_t = G M m / ( d - r )² - G M m ( d + r )²
= G M m [ ( d + r )² - ( d - r )² ] / [ ( d - r )² ( d + r )² ]
= G M m [ 4 d r ] / d⁴
= 4 G M m r / d³

d_R = ( 2 M / m )^1/3 s

DERIVATION - 2

A more useful formulation considers the problem of a boulder or a small grain of sand (mass μ) on the surface of a small asteroid (mass m) as per the diagram below:

The gravitational force which holds the Boulder on the surface of the asteroid is:

F_a = G m μ / r²

F_t = G M μ [ 1 / d² - 1 / (d - r)² ]
= G M μ [ (d - r)² - d² ] / [ d² (d - r)² ] = G M μ [ d² - 2 d r + r² - d² ] / [ d⁴ - 2 d³ r + d² r² ]

F_t = 2 G M μ r / d³

d_Roche = r ( 2 M / m )^1/3

M = 4 π ρ_M R³ / 3

m = 4 π ρ_m r³ / 3

d_Roche = R ( 2 ρ_M / ρ_m )^1/3
≈ 1.26 R ( ρ_M / ρ_m )^1/3

APPLICATION

The above formula might be used to determine if an asteroid in a close encounter with the Earth is likely to 'shed' loose material from its surface. The radius and density of the Earth is known, but you may have to estimate the asteroidal density.

If the asteroid is a "Rubble pile" it is likely to be totally disrupted if it comes closer than the Roche limit.

The Roche limit is important in many areas of astrophysics, including situations involving high energy processes.

Edouard Roche who, in the 1850's, developed the concept of the limit named after him.

Australian Space Academy