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White dwarf v. sun magnitudes

by hfitzgerald
Tags: luminosity, magnitude, sun, white dwarf
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May7-11, 05:01 PM
P: 2
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
Mbol=-2.5log(Lbol)+5+constant (absolute bolometric magnitude)
m-M=-2.5log(d) 5 (M=absolute magnitude, m=apparent magnitude, d=distance in parsecs)

3. The attempt at a solution
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
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