How to Express Absolute Bolometric Magnitude in Terms of Temperature and Radius?

[ek]

Q: Find an expression for the absolute bolometric magnitude of a star as a function of its effective temperature and radius, which incorporates the solar value of these quantities (BM 4.7, Temp 5780K, radius 6.96E5 km) and not Stefan's constant.

I'm a little lost on this question. I know what Bolometric magnitude is but don't know how to express it explicitly in terms of T and r. maybe I am missing some large concept here, I don't really know. Can someone help me out?

Thanks.

 

[SpaceTiger]

I'm a little lost on this question. I know what Bolometric magnitude is but don't know how to express it explicitly in terms of T and r. maybe I am missing some large concept here, I don't really know. Can someone help me out?

Several things to try to answer. First, just because your result won't include $$\sigma$$ doesn't mean you shouldn't use the Stephan-Boltzmann law. Do you know what that is?

Second, what is the bolometric magnitude? What is the bolometric luminosity?

Finally, how are magnitude systems usually normalized?

 

[thepatientmental]

The bolometric magnitude of a star is a measure of its total energy output across all wavelengths, taking into account the entire spectrum of electromagnetic radiation. It is typically denoted as BM and is commonly used in astronomy to compare the brightness of stars.

To find an expression for the absolute bolometric magnitude of a star, we can use the following formula:

BM = M + 2.5log(L/Lsun)

Where M is the absolute magnitude, L is the luminosity of the star, and Lsun is the luminosity of the Sun.

To incorporate the solar value of effective temperature and radius, we can use the Stefan-Boltzmann law, which states that the luminosity of a star is proportional to its effective temperature and the fourth power of its radius:

L = 4πσR^2T^4

Where σ is the Stefan-Boltzmann constant.

Substituting this into the formula for bolometric magnitude, we get:

BM = M + 2.5log[(4πσR^2T^4)/Lsun]

Since we are given the values for the solar effective temperature and radius (Tsun = 5780K and Rsun = 6.96E5 km), we can plug these in and simplify the expression:

BM = M + 2.5log[(4πσ(6.96E5 km)^2(5780K)^4)/(3.828E26 W)]

BM = M + 2.5log[1.902E41/3.828E26]

BM = M + 2.5log(4.969E14)

BM = M + 14.8

Therefore, the absolute bolometric magnitude of a star can be expressed as BM = M + 14.8, where M is the absolute magnitude of the star. This expression incorporates the solar values of effective temperature and radius without using Stefan's constant.