Question

In a double-star system, two stars of mass 4.6 x 1030 kg each rotate about the system's center of mass at radius 1.9 x 1011 m.

(a) What is their common angular speed?

(b) If a meteoroid passes through the system's center of mass perpendicular to their orbital plane, what minimum speed must it have at the center of mass if it is to escape to "infinity" from the two-star system?

Answer 1

To carry out this exercise, it is necessary to use the equations made to Centripetal Force and Gravitational Energy Conservation.

By definition we know that the Centripetal Force is estimated as

`F_c = M\omega^2R`

Where,

M = mass

`\omega =` Angular velocity

R = Radius

From the 'linear' point of view the centripetal force can also be defined as

`F_c = (GM^2)/(R^2)`

PART A ) Equating both equations we have,

`(M)/(\omega^2R)=(GM^2)/(R^2)`

Re-arrange to find \omega

`\omega = \sqrt{(Gm)/(r^3)}`

Replacing with our values

`\omega = \sqrt{((6.67*10^(-11))(4.6*10^(30)))/((1.9*10^(11))^3)}`

`\omega = 2.115*!0^(-7)rad/s`

Therefore the angular speed is
`\omega = 2.115*!0^(-7)rad/s`

PART B) For energy conservation we have to

`KE_(min) = PE_(cm)`

Where,

`KE_(min) =` Minimus Kinetic Energy

`PE_(cm) =` Gravitational potential energy at the center of mass

Then,

`(1)/(2) mv^2_(min) = (2GMm)/(R)`

Re-arrange to find v,

`v_min = \sqrt{(4GM)/(R)}`

`v_min = \sqrt{(4(6.67*10^(-11))(2.2*10^(30)))/((1.9*10^(11)))}`

`v_min = 5.55*10^4m/s`

Therefore the minimum speed must it have at the center of mass if it is to escape to "infinity" from the two-star system is
`v_min = 5.55*10^4m/s`