# White dwarf v. sun magnitudes

1. May 7, 2011

### hfitzgerald

1. The problem statement, all variables and given/known data
This is a past exam question that I am trying to get my head around but keep getting stuck
Show that if a white dwarf were at the centre of the solar system instead of the sun, it would have an apparent visual brightness of -18.6mag.
Assume that the white dwarf has surface temperature 3 times that of the sun and a radius 0.01 times that of the sun.
The sun has absolute bolometric magnitude 4.75mag and the visual magnitude of the white dwarf will be 3 times smaller than it's bolometric magnitude.

2. Relevant equations
L=4πR2σT4
Mbol=-2.5log(Lbol)+5+constant (absolute bolometric magnitude)
m1-m2=-2.5log(L1/L2)
m-M=-2.5log(d) – 5 (M=absolute magnitude, m=apparent magnitude, d=distance in parsecs)

3. The attempt at a solution
L(sun)=4πR2σT4
L(dwarf)= 4π(0.01R)2σ(3T)4=34 *0.012 *Lsun
M(dwarf)-M(sun)=- 2.5log(L(dwarf)/L(sun))=-2.5log(34*0.012)=5.23
M(dwarf)=5.23 +M(sun)=-5.23+ 4.75= 9.98
m=5log(d) – 5 +M
d=1AU =4.84813681 × 10-6 Parsecs
m=5log(4.84813681 × 10-6) – 5 + 9.98=-21.6