Please show legible work, will rate. Astronomical observations of our Milky Way

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Please show legible work, will rate.

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8·x 1011 solar masses. A star orbiting near the galaxy's periphery is 5.8 x 104 light years from its center. (For your calculations, assume that the galaxy's mass is concentrated near its center.) (a) What should the orbital period of that star be? yr (b) If its period is 5.3 x 107 years instead, what is the mass of the galaxy? Such calculations are used to imply the existence of "dark matter" in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies solar masses

Explanation / Answer

(a)

Use Kepler’s third law:

P (in seconds) = 2pi[a sqrt(a/GM) ]

= 2pi[(5.8*10^4*9.46*10^15m) sqrt((5.8*10^4*9.46*10^15m)/(6.67*10^-11*8*10^11*2*10^30 kg) ]

= 7.8169 x 10^15 s

= 2.4787 x 108years

(b)

Similarly, rearranging in terms of mass:

M = 4pi2a3/(GP2)

= 4pi2*(5.8*10^4*9.46*10^15m)3/(6.67*10^-11*(5.3*10^7*365*24*60*60s)2)

= 3.49966 x 10^43 kg

= (3.49966 x 10^43)/(2*10^30) solar masses

= 1.75 x 1013 solar masses

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