Calculating the distance to a star

[Physics Dad]

I have made an effort to answer this question, and would like to know if my thinking is correct. I would appreciate any feedback.

Thank you!

Homework Statement

Two stars in the sky have similar effective temperatures, masses and apparent brightness. However, star 1 has a surface gravity which is 10 times higher than star 2. If star 1 is 1pc away, what is the distance to star 2?

Homework Equations

i) g_*=GM_*/R_*²
ii) f_*=L_*/4πd_*²
iii) L_*=4πR_*²σT_*eff⁴
iv) m_*1-m_*2=-2.5log(f_*1/f_*2)

The Attempt at a Solution

Based upon the question, I can assume that the mass (M_*), the effective temperature (T_*eff) and the apparent magnitude (m_*) of both stars is the same.

First of all, I will start with the gravity which tells me that:

g_*1=10g_*2

so using equation i) I can say that:

GM/R_*1²=10GM/R_*2²

I can then rearrange to get a ratio of radii which will tell me that...

R_*12/R_*22=10^-1

I can then calculate the ratio of fluxes by substituting equation iii) into equation ii),= and doing some cancelling to find that...

f_*1/f_*2=d_*2²(R_*1²/R_*2²)

I know that the value of R_*1²/R_*2²=10^-1 so I can sub this value in

as I know that both apparent magnitudes are the same, I can say that m_*1-m_*2=0 and then sub in my ratio of fluxes to equation 4 to get...

0=-2.5log(d_*2²/10)

I can then solve this equation for d_*2 and I get...

d_*2=3.16pc

This seems feasible given that star two would have a radius 10 times bigger than star 1, but I would appreciate any help or advice anyone can offer.

Thank you!

 

[trurle]

d_*2=3.16pc

This seems feasible given that star two would have a radius 10 times bigger than star 1, but I would appreciate any help or advice anyone can offer.

Thank you!

Answer is correct, but it imply radius ratio of 3.16, not 10. 10 is the surface areas ratio.