Consider a binary system of gravitating spheres. Each sphere hasmass M and radiu

Question

Consider a binary system of gravitating spheres. Each sphere hasmass M and radius R. The distance between the center of spheres isD; the spheres revolve on a circular orbit with radius D/2.

a.) Determine the period of revolution.
b.) Calculate the minimum period of revolution if the spheres aremade of Osmium
c.) Calculate the minimum period of revolution if the spheres aremade of neutron-star material

I figured out the first part pretty easily. But cant figure out whyit would matter if the spheres are made of Osmium and neutron-star.Their density would increase but how would that affect theperiod?

Explanation / Answer

Yeah, you should have got   period = 2 (D3 / 2 G M )1/2 . You could replace M with   (4/3) R3      but that doesnthelp you to determine the minimum period of revolution unless youmake a weird assumption: .        Which is...? Well,the fastest period (i.e. minimum period) would relate to thelargest M, right? So this would be the case where the radius ofeach star is D/2. . In that case...    M = (4/3) (D/2)3   = (1/6) D3 . If you sub this into your period equation, you get .    period = 2 ( D3 /2G (1/6) D3 )1/2= 2 ( 3 / G )1/2   . So you can actually calc a value for the minimum period usingjust the densities of osmium and a neutron star. . If you sub this into your period equation, you get .    period = 2 ( D3 /2G (1/6) D3 )1/2= 2 ( 3 / G )1/2   . So you can actually calc a value for the minimum period usingjust the densities of osmium and a neutron star. .

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