### How to calculate magnitude of a star in a triple star system?

• dogtag

7 years ago

I want to find a star's magnitude in a triple star. I already knew the the magnitude of the triple star and the other two star, but I don't quit know how to solve it. Is there a way to find it?

• 7 years ago

Basically you need to convert between luminosities (which you can add) and magnitudes using
$$M-M_\odot=-2.5\log_{10}(L/L_\odot)$$
Let's call the total luminosity $L_0$ and magnitude $M_0$ and the individual luminosities and magnitudes $L_1$, $L_2$ and $L_3$ and $M_1$, $M_2$ and $M_3$.
Then, you have the total luminosity of the system, directly
$$L_0/L_\odot=10^{-0.4(M_0-M_\odot)}$$
and as the sum of the components
$$L_0/L_\odot=(L_1+L_2+L_3)/L_\odot=10^{-0.4(M_1-M_\odot)}+10^{-0.4(M_2-M_\odot)}+10^{-0.4(M_3-M_\odot)}$$
Solving these equations for $M_3$ gives
$$M_3-M_\odot=-2.5\log_{10}\left(10^{-0.4(M_0-M_\odot)}-10^{-0.4(M_1-M_\odot)}-10^{-0.4(M_2-M_\odot)}\right)$$
I'm assuming you have absolute magnitudes, but you can rewrite the formulae in terms of apparent magnitudes using
$$M=m+5(1-\log_{10}d)$$
but I think the result then also depends on the distance. Surely the absolute magnitude of the Sun needs to be subtracted from the LHS of eqn. 4? @RobJeffries: You're right! Fixed.