3. Below is a plot of the measured rotation curve as a function of radius for th

Question

3. Below is a plot of the measured rotation curve as a function of radius for the Large Magellanic Cloud, a satellite galaxy of the Milky Way. The data points show the measured rotational velocities of the stars and hydrogen gas (HI), while the inferred rotation curves for dark matter and stars are shown with dashed and dotted lines. The rotation curve due to all mass is the solid line 120 100 80 60 40 Stellar Mass -DM mass - Total HI data Stellar data 4 10 r (kpc) (a) At what radius r is the enclosed mass (M(

Explanation / Answer

a. for a mass m enclosed in radius r

the rotational velocity v is related by

v^2 = Gm/r

m = v^2*r/G

so for enclosed mass of dark matter = enclosed mass of stars

both v and r form the grpah should be same for the dashed line( dark matter) and stars (dotted line)

hence this intersection is at r = 2kpc

b. at 6 kpc, v = 80 km/s

now r = 6 kpc = 1.851*10^20 m

hence

m = v^2*r/G = (80,000)^2*1.851*10^20/6.67*10^-11 = 1.776*10^40 kg

c. v for stars at r = 6 kpc = 1.851*10^20 m is

v = 60,000 m/s

so m = v^2*r/G = 9.9904*10^39 kg

so fraction of LMC made up of stars = 9.9904/10*1.776 = 0.56252279265772519145832489160825

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