You see a galaxy which lies at distance D from the earth. You measure a flux from this galaxy of f. If you assume a mass-to-light ratio of r (i.e. r is the number of kilograms needed to generate one watt of radiation), derive an equation for the stellar mass m of the galaxy.

4*pi*f*r*D^2

Stellar mass , m= L * r where L is the luminosity.

Luminosity can be derived by rearranging the usual inverse-square law equation

To derive the equation for the stellar mass of the galaxy, we need to make use of the distance, flux, and mass-to-light ratio. Here's how you can derive it:

1. The flux received from the galaxy at distance D can be related to its luminosity (L) using the inverse square law of radiation:
F = L / (4πD^2)
where F represents the observed flux.

2. The luminosity of the galaxy can be related to the stellar mass (m) and mass-to-light ratio (r) as:
L = r * m
where L represents the luminosity.

3. Substitute the equation for luminosity (L) derived in step 2 into the equation for flux (F) derived in step 1:
F = (r * m) / (4πD^2)

4. Rearrange the equation to solve for the stellar mass (m):
m = (4πD^2 * F) / r

Therefore, the derived equation for the stellar mass (m) of the galaxy is:
m = (4πD^2 * F) / r

To derive an equation for the stellar mass (m) of the galaxy using the given information, we can make use of the flux (f) and the distance (D) to obtain the luminosity (L) of the galaxy. Then, by utilizing the mass-to-light ratio (r), we can relate the stellar mass (m) to the luminosity (L).

The luminosity (L) is the total amount of energy radiated by the galaxy per unit time and can be calculated using the formula:

L = 4πD^2f

where:
- L is the luminosity of the galaxy,
- D is the distance from the galaxy to the Earth, and
- f is the measured flux from the galaxy.

Next, we can relate the luminosity (L) to the stellar mass (m) using the mass-to-light ratio (r). The mass-to-light ratio (r) represents the amount of stellar mass required to generate a unit amount of luminosity. Therefore, we have:

L = mr

Solving this equation for the stellar mass (m), we get:

m = L / r

Substituting the expression for luminosity (L), we obtain:

m = (4πD^2f) / r

So, the equation for the stellar mass (m) of the galaxy is given by:

m = (4πD^2f) / r