According to the third Kepler's law (see https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law):
T2a3≈4π2GM where a=10kpc≈3.086×1020m is the radius of the orbit, T=2×108years≈6.307×1015s is the period, G=6.674×10−11N⋅m2/kg2 is the gravitational constant, and M is the mass of the Galaxy. The equality is approximate because we do not take into account the mass of the Sun. But it is clear that it is much much less then the mass of the Galaxy, thus, we can neglect it.
For M obtain:
M=GT24π2a3M=6.674×10−11⋅(6.307×1015)24π2⋅(3.086×1020)3≈4.294×1043kg Answer. 4.294×1043kg
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