Question

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8 ✕ 10¹¹ solar masses. A star orbiting near the galaxy's periphery is 5.8 ✕ 10⁴ lightyears from its center.

(a) What should the orbital period (in y) of that star be?

(b) If its period is 6.9 ✕ 10⁷ y instead, what is the mass (in solar masses) 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.

Answer 1

Using Kepler's 3rd law:

T^2 = (4*pi^2/(G*M))*r^3

M = Mass of Milky way galaxy = 8*10^11 Solar masses = 8*10^11*2.0*10^30

r = radius of galaxy's periphery = 5.8*10^4 lightyears

1 lightyear = 9.461*10^15 m, So

r = 5.8*10^4*9.461*10^15 m

Using these values:

T = sqrt[4*pi^2*r^3/(G*M)]

T = sqrt [(4*pi^2*(5.8*10^4*9.461*10^15)^3)/(6.67*10^-11*8*10^11*2*10^30)]

T = 7.818*10^15 sec

Since 1 yr = 3.154*10^7 sec, So

T = (7.818*10^15)/(3.154*10^7)

T = 2.47876*10^8 years = 2.5*10^8 yrs

Orbital period = 2.5*10^8 yrs

Part B.

Now if Period = 6.9*10^7 yr

T = 6.9*10^7*3.154*10^7 sec = 2.17626*10^15 sec

from 3rd law of kepler

M = 4*pi^2*r^3/(G*T^2)

M = 4*pi^2*(5.8*10^4*9.461*10^15)^3/(6.67*10^-11*(2.17626*10^15)^2)

M = 2.06494*10^43 kg

Since 1 solar mass = 2*10^30 kg

M = 2.06494*10^43/(2*10^30)

M = 1.03*10^13 solar masses

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