How Bright Would a White Dwarf Appear if It Replaced the Sun?

[hfitzgerald]

Homework Statement

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.

Homework 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)

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

 

[mark726]

mag

Hello there! Your attempt at solving this problem is on the right track, but there are a few minor mistakes that need to be corrected.

Firstly, the equation L=4πR2σT4 is the luminosity of a star, not its absolute bolometric magnitude. The absolute bolometric magnitude equation is Mbol=-2.5log(Lbol)+constant. This constant is usually taken to be 4.75 for the sun, but it can vary for different stars.

Secondly, the equation m1-m2=-2.5log(L1/L2) is used to compare the apparent magnitudes of two objects, not their absolute magnitudes. For this problem, we want to find the absolute magnitude of the white dwarf, so we need to use the equation M1-M2=-2.5log(L1/L2).

With these corrections in mind, we can solve the problem as follows:

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= -0.48

Now, we can use the equation m-M=-2.5log(d) – 5 to find the apparent magnitude of the white dwarf:

m-M=-2.5log(d) – 5
-18.6 - (-0.48) = -2.5log(d) – 5
-18.12 = -2.5log(d)
d = 10^-7.25 Parsecs

Therefore, if a white dwarf were at the center of the solar system, its apparent visual brightness would be -18.6mag.