Question

In our solar system, when the ratio of the orbital periods of two objects has some special values, this corresponds to an orbital resonance. Particularly interesting are the resonance between Jupiter and objects on the asteroid belt.

a. Calculate the semi-major axes that correspond to the following resonances with Jupiter P_J /P_Ast =3 , P_J /P_Ast = 5 2 , P_J /P_Ast = 7 3 , and P_J /P_Ast =2 , where P_J is the orbital period of Jupiter and P_ast the period of that specific point in the asteroid belt. It is given that for Jupiter, the semimajor axis is aJ = 5.2 A.U. Remember: Kepler’s third law: P ² = (4 ? ² ) / (GM) * a ³ when P is measured in years and a in AU.

c. Now, consider an asteroid with a semi-major axis a = 2.2 AU. It is known that some processes gradually change the semi-major axes of asteroids. One of them is he Yarkovsky effect related to solar radiation. Because of this, the semi-major axis of the above asteroid increases with a rate 10-9 A .U . year . How long will it take for the semimajor axis to become 2.5 A.U. ? Explain how this process can “create” meteorites that visit the inner solar system.

Answer 1

a) We are given

P_J / P_ast = 3

We know from kepler's law

P² = a³

so,

we have

( a_J / a_ast )³ = 3

(a_J)³ / (a_ast)³ = 3

(5.2)³ / (a_ast)³ = 3

(a_ast)³ = 46.869

a_ast = 3.605 AU

Similarly for

P_J / P_ast = 5.2

using the same method as above,

(5.2)³ / (a_ast)³ = 5.2

a_ast = 3.0014 AU

use the same method to find for other ratios.

b) The semi - major axis increases by 2.5 AU - 2.2 AU = 0.3 AU

As it increases at 10^-9 AU/year

so, Number of years it will take = 0.3/10^-9 = 3e8 years