Help with relative star flux and luminosity

[DunWorry]

Homework Statement

A star as an apparent visual magnitude of 14 and an absolute visual magnitude of 14.7. I have worked out that its distance is 7 parsecs. The sun has an absolute visual magnitude of 4.8 and an effective temperature of 5800k. If the star has the same effective temperature of the sun, what is its relative radius to that of the sun?

Homework Equations

The Attempt at a Solution

I am really confused here because the stefan Boltzmann law F = $\sigma$T$^{4}$ you can see that if these two stars have the same temperature, they must have the same flux. The only formula I know to link flux to radius is L = $\sigma$T$^{4}$4$\pi$r$^{2}$

We are aiming for something like $\frac{R_{sun}}{R_{star}}$ which = $\frac{L_{sun}}{L_{star}}$ but I can't think of a way to work out luminosity without knowing the radius. Also I have not used any of the information regarding magnitude.

However if I use m = $-\frac{5}{2}$log$_{10}$(Flux) I work out the flux to be 0.012 for the sun and 1.318x10$^{-6}$ for the star, how can this be? as I said before the stefan Boltzmann law F = $\sigma$T$^{4}$ if two stars have the same temperature surely they must have the same flux?


Thanks

 

[mfb]

The magnitude gives the total radiation, this is intensity*surface. Intensity is given by the temperature, but total radiation is not.

As temperature is the same, total radiation is proportional to the squared radius. If you know the relation between the total radiation, you can calculate the relative radius of the star.

 

[gneill]

There's a magnitude\flux relationship you should have in your text or notes:

$m_A - m_B = -2.5 log\left(\frac{f_A}{f_B}\right)$

If you know the two absolute magnitudes, you should be able to find the ratio of the fluxes.

 

[DunWorry]

I don't see how finding the ratio of the fluxes will help me find the relative radius? Also I know if I use m - M = -2.5 log(f / F) I will find probrably find the fluxes are different, but I thought from the stefan Boltzmann law that F = $\sigma$T$_{eff}$$^{4}$

Sigma is a constant, and if T is the same, why are the fluxes different?

 

[cepheid]

At first glance this question seems simple enough: if you know the difference between the two absolute magnitudes of the stars, then you know the ratio of their luminosities.

If you know the ratio of their luminosities, you know the ratio of the squares of their radii (since they have the same surface flux, as you've pointed out).
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Since astronomical terminology seems to be causing some confusion:

luminosity (L) is the thing measured in watts (total power output of star)

surface flux is the thing measured in watts/metre² and given by the Stefan-Boltzmann law (power radiated per unit surface area of the star).

It's confusing, because I think astronomers also use the term flux to describe the power arriving per unit area at a distance r from the source (which I think physicists call irradiance), which is given by L/4πr² (assuming the source is isotropic). This flux is definitely different for the two stars, but you don't really need to consider it here. It is the flux that goes into the magnitude equation, since the magnitude is expressing how bright the thing appears, which is precisely what this quantity measures.

 

[DunWorry]

Ah right so basically its 14.7-4.8 = -2.5 log$_{10}$$\frac{f}{F}$
and you get the ratio of $\frac{f}{F}$ as 1x10$^{-4}$ and to get the ratio of the radius you just square root this to get 0.01 solar radii which is correct. I got confused on the fact that in the formula m - M = -2.5 log$_{10}$$\frac{f}{F}$ the flux does not correspond to the surface flux which is given by the stefan Boltzmann law so although the surface flux is the same as the radius is different for each star the total luminosity is different