Hi Im having trouble with this problem Two stars of mass m

Hi, I\'m having trouble with this problem ,

Two stars of mass m = 0.41 and M = 77.00 (in units of solar masses) separated by a distance d = 6.00 (in units of AU, the earth-sun distance) revolve in circular orbits about their (common) center of mass as shown in the figure below.

What is the orbital period of the larger star in years?

Thanks for any help!

Solution

The angular velocity of the objects are relative to each other while the angular momentum of the objects of mass M and m are conserved.

The momemtum of the mass M is given by,

L = I x omega or M r^2 x omega where r is (d - r1)

The momentum of mass m is given by,

L = m r1^2 x omega1

Equating the above relation we obtain the relation between the angular velocities of the two masses,

M(d-r1)^2 x omega = mr1^2 x omega1

solve for the omega in terms of omega1,

omega = m * r1^2*omega1 / M (d-r1^2)

or replaced with values and simplified to get,

omega = [0.005*r1^2 / (r1 - 6)^2] * omega1

The period of the mass M is the T = 1 / f

where f = 2*pi / omega ------- revolutions per year

Now T = omega / 2*pi

T = {[0.005*r1^2 / (r1 - 6)^2] * omega1} / 2*pi

T = 0.0008 * omega1 * r1^2 / (r1-6)^2 where omega1 = V1 / r1

T = 0.0008 * [r1 / (r1-6)^2] * V1

Also, the center of mass remains is proportionally placed i.e M / m = r1 / r2

r1 = M / m * (d - r1) = [ 77 / 0.41] * (6 - r1)

solve for r1 = 5.94 AU

The time period T for mass M is ,

T = 0.0008 * [5.96 / (5.96 - 6)^2] * V1

T = 2.98 * V1