Astronomical observations of our Milky Way galaxy indicate that it has a mass of
ID: 1776597 • Letter: A
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 5.9 104 light years from its center. (a) What should the orbital period (in y) of that star be? 254300000 Correct: Your answer is correct. y
(b) If its period is 6.4 107 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.
Explanation / Answer
(A) for circular orbit,
G M m / r^2 = m w^2 r
w^2 = G M / r^3
w^2 = (6.67 x 10^-11)(8 x 10^11 x 2 x 10^30) / (5.9 x 10^4 x 365 x 24 x 3600 x 3 x 10^8)^3
w = 7.833 x 10^-16 rad/s
T = 2pi / w = 8.02 x 10^15 sec
T (in years) = 2.54 x 10^8 years
(B) w =2 pi / (6.4 x 10^7 x 365 x 24 x 3600 s)
w = 3.113 x 10^-15 rad/s
w^2 = G M / r^3
putitng values,
M = 2.527 x 10^43 kg
M = 1.27 x 10^13 solar years
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