N2M.8 Suppose you are tracking a star that is orbiting the black hole at the cen
ID: 1774921 • Letter: N
Question
N2M.8 Suppose you are tracking a star that is orbiting the black hole at the center of our galaxy. You have taken three careful measurements of the star's position, and find its positions (separated by 10-day intervals) to have the coordinates 2.10 r! = | 19.65 | Trn, r2-20.11 | Trn, 2.10 19.52 | Tm (N2.11) in some suitable coordinate system (1 Tm 1012 m) where the black hole is at the origin. Using these measurements alone, estimate the mass of the black hole. (Hint: Imag- ine constructing a motion diagram for this situation, but instead of graphically constructing the vector difference you will need, do the equivalent calculation using com ponents. You should not assume that the star's orbit is circular.)Explanation / Answer
Radius of orbit = r = 2.10Tm = 2.10*10^12 m
Period of orbit = T = 40 days = 20*24*60*60 s
Use Kepler’s third law,
T^2= [(4^2)/(GM)]*r^3
Where T = period, M = mass of black hole, t = radius of orbit
Rearranging,
M= [(4^2)/(GT^2)]*r^3
M = [(4^2)/(6.67*10^-11*(40*24*60*60)^2]*(2.10*10^12)^3
M= 4.6*10^35 k
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