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Homework Statement
Two equal-mass stars maintain a constant distance apart of ## 8.0 *(10^11) ## m it is eight times ten to the eleventh, not eight times ten to the first power times one. I don't know why Latex forbids exponents of two digits or more and rotate about a point midway between them at a rate of one revolution every ##12.6## years. What is the mass of each star?
Homework Equations
(1) ## a_R=\frac{v^2}{R} ##
(2) Law of universal gravitation ##\frac{GM}{R^2}##
3. The attempt at a solution
My solutions guide manual says "set the gravity force on one of the stars equal to the centripetal force..." ##F_G = G\frac{M^2}{d^2} = F_R = M\frac{v^2}{d/2}## then appears in the guide. Here ## d= 2R = ##diameter. I am confused because I don't see how ##d## can be used in place of ##R## in equation for universal gravity. Additionally, where did ##M^2## come from? Is it the combined mass of the two stars? Finally, in the last part ## = M\frac{v^2}{d/2}## what is M? The two stars are orbiting each other, so there is nothing in the center. Why isn't ##M = 0##kg?
PS: The question states "two equal mass stars..." Correct me if I'm wrong but I am under the assumption that if two objects orbit each other (or a central point between them) their masses MUST be equal or nearly equal... Is this always true?
Thank you.