Answer:
[tex]\frac{d_{1}}{d_{2}}=0.36[/tex]
Explanation:
1. We can find the temperature of each star using the Wien's Law. This law is given by:
[tex]\lambda_{max}=\frac{b}{T}=\frac{2.9x10^{-3}[mK]}{T[K]}[/tex] (1)
So, the temperature of the first and the second star will be:
[tex]T_{1}=3866.7 K[/tex]
[tex]T_{2}=6444.4 K[/tex]
Now the relation between the absolute luminosity and apparent brightness is given:
[tex]L=l\cdot 4\pi r^{2}[/tex] (2)
Where:
Now, we know that two stars have the same apparent brightness, in other words l₁ = l₂
If we use the equation (2) we have:
[tex]\frac{L_{1}}{4\pi r_{1}^2}=\frac{L_{2}}{4\pi r_{2}^2}[/tex]
So the relative distance between both stars will be:
[tex]\left(\frac{d_{1}}{d_{2}}\right)^{2}=\frac{L_{1}}{L_{2}}[/tex] (3)
The Boltzmann Law says, [tex]L=A\sigma T^{4}[/tex] (4)
Let's put (4) in (3) for each star.
[tex]\left(\frac{d_{1}}{d_{2}}\right)^{2}=\frac{A_{1}\sigma T_{1}^{4}}{A_{2}\sigma T_{2}^{4}}[/tex]
As we know both stars have the same size we can canceled out the areas.
[tex]\left(\frac{d_{1}}{d_{2}}\right)^{2}=\frac{T_{1}^{4}}{T_{2}^{4}}[/tex]
[tex]\frac{d_{1}}{d_{2}}=\sqrt{\frac{T_{1}^{4}}{T_{2}^{4}}}[/tex]
[tex]\frac{d_{1}}{d_{2}}=\sqrt{\frac{T_{1}^{4}}{T_{2}^{4}}}[/tex]
[tex]\frac{d_{1}}{d_{2}}=0.36[/tex]
I hope it helps!
