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I have the following problem:

"The Sun has an orbital speed of about 220 km s−1 around the center of the Galaxy, whose distance is 28 000 light years. Estimate the total mass of the Galaxy in solar masses."

I know how to solve it in two different ways, considering the galaxy mass as the total mass inside the sphere that the sun is orbiting:

  • using that the gravitational force is equal to the centripetal force;

  • by Kepler's third law

But, in both cases, what I get is the total mass of the Galaxy, in kilograms. To estimate this mass in solar masses, I need to know the solar mass - which is easily found, but it's not given in this problem - and then I divide the value that I find by it.

The question is: is there a way I can obtain this ratio without knowing somehow the solar mass? Is there a way that I can get directly an expression with the ratio M(galaxy)/M(sun) and not depending on the value of the last one?

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The answer is no. The orbital speed of the Sun doesn't depend on its mass, so the mass of the galaxy which you obtain from the orbital speed doesn't depend on the mass of the Sun. So there's no hope that in their ratio the mass of the Sun cancels out.

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    $\begingroup$ Makes sense... I can notice that equaling both gravitational and centripetal forces, Msun cancels in both sides (and if I try to construct the ratio Mgalaxy/Msun, the last one appears in the other side). I was trying to get an expression for the Sun Mass in terms of the distance to the galaxy center and its orbital speed, but now I understood that they are independent, thank you! $\endgroup$
    – Jorge Defaia
    Commented Sep 6, 2019 at 16:51

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