How to find the distance to a cluster using the apparent magnitude

ppy

Homework Statement

The main sequence can be approximated by a relation of the form L$\propto$T^5 .
In Cluster A, a main sequence star called Star α has the same B – V colour as the Sun, and a V-band apparent magnitude of 11.3.
The effective temperature of Star α is 5800K. In Cluster B, a main sequence star called Star β has an effective temperature of 9000K and a V-band apparent magnitude of 17.0.
Derive an approximate value for the distance to Cluster B. {9}

I have already shown that Cluster A lies at an approximate distance of 200pc by using its absolute magnitude as 4.8 which was stated.

Homework Equations

m$_{a}$-M$_{a}$=5log$_{10}$(d$_{a}$)-5
m$_{b}$-M$_{b}$=5log$_{10}$(d$_{b}$)-5

L=4$\pi$σR$^{2}$T$_{eff}$$^{4}$ ( However as in the question earlier on it says L is proportional to T^5 I don't think we can use this equation)

The Attempt at a Solution

I have tried numerous ways but I can't seem to get the distance for cluster B on its own.

Also I do not really see how you can use L$\propto$T$^{5}$ in the equations for the apparent and absolute magnitude.

I am so confused any help would be great, thanks

Mentor
##L\propto T^5## can be expressed as ##\displaystyle \frac{L}{T^5}=\text{constant}## or ##\displaystyle\frac{L_1}{T_1^5} = \frac{L_2}{T_2^5}## for two stars 1 and 2.

This is not ∝T^4 as it takes into account how the radius depends on the temperature.

You know the temperatures of both stars and the luminosity of one star, this allows to calculate the luminosity of the other star.

ppy
Hi,

I am still a little confused, how do I know the luminosity of star a, I know the distance but wouldn't I need to no the flux or am I supposed to use m$_{a}$=-2.5log(L)+5log(d)+ constant. If so how can I use this equation when I don't know what the constant is or am I supposed to ignore it?

Thanks again.

Mentor
Hi,

I am still a little confused, how do I know the luminosity of star a, I know the distance but wouldn't I need to no the flux or am I supposed to use m$_{a}$=-2.5log(L)+5log(d)+ constant. If so how can I use this equation when I don't know what the constant is or am I supposed to ignore it?

Thanks again.
See my post, you just have to insert values in that formula to get the absolut luminosity of star B.
You can (but do not have to) determine the constant if you like.

ppy
ok, so I now think I have the luminosity of star B how I can I use this to find the distance to star B

Mentor
You can compare the absolute luminosity with the apparent luminosity. Be careful in which way the color influences the values you have.

ppy
in which equations are there 2 different types of luminosity. I only know of the absolute magnitude and the apparent magnitude but not an absolute luminosity and an apparent luminosity. I am very confused.