I'm really stuck. I need to figure out the mass/light ratio of a galaxy in solar units. I know its mass is 5.7 x 1010 solar masses. I know its absolute magnitude (-17.3) and distance (29 Mpc). I'm feeling dumb here because I think this ought to be clear to me, but I'm getting hung up at the magnitude - luminosity - light part.
Here's what I think: mass/light = 100.4*(Magnitude galaxy-magnitude Sun) x (mass galaxy/mass Sun)
I get a mass to light ratio of around 90. Am I doing this right?
Yes, the result is about 80 (unit: solar mass per solar intensity).
Taking "light" as meaning intensity (measured from Earth in units of power per unit area) the ratio, $R$, is:
$ R = \frac{m_g}{I_g} = \frac{5.7 \times 10^{10} m_{sun}}{I_g \times \frac{I_{sun}}{I_{sun}}} = \frac{5.7 \times 10^{10} m_{sun}}{\frac{I_g}{I_{sun}} \times I_{sun}} $
where $m_g$ is the mass of the galaxy, $I_g$ is the intensity of the galaxy and $I_{sun}$ is the intensity of the Sun.
$ \frac{I_g}{I_{sun}} $ can be computed from:
$ M_g - M_{sun} = -2.5\log_{10} ( \frac{I_g}{I_{sun}} ) $
where M are magnitudes.
Rearranging and using a value of 4.83 for the absolute magnitude of the Sun:
$ \frac{I_g}{I_{sun}} = 10^{-0.4(M_g - M_{sun})} = 10^{-0.4(-17.3 - 4.83)} = 7.1 \times 10^8$
Inserting $ \frac{I_g}{I_{sun}} $:
$ R = \frac{5.7 \times 10^{10} m_{sun}}{7.1 \times 10^8I_{sun}} = 80\ \frac{m_{sun}}{I_{sun}} $
