Color Index and Intrinsic Magnitude of a Star

[Kelli Van Brunt]

Homework Statement

A UBV photometric (UBV Johnson’s) observation of a star gives U = 8.15, B = 8.50, and V = 8.14. Based on the spectral class, one gets the intrinsic color (U – B)o = -0.45. Determine the intrinsic magnitudes M_U, M_B, and M_V of the star (take, for the typical interstellar matters, the ratio of total to selective extinction R = 3.2). The star is known to have radius of 2.3 solar radii, absolute bolometric magnitude of -0.25, and bolometric correction (BC) of -0.15.

Relevant Equations

B - V = (B - V)o + E(B - V)
U - B = (U - B)o + E(U - B)
R = Av / E(B - V) = 3.2
m - M = 5log(r/10) + A
E(U - B) = 0.72*E(B - V)

I first determined $E_{U - B}$ by using the second equation listed above: $$U - B = (U - B)_0 + E_{U - B}$$ $$8.15 - 8.50 = -0.45 + E_{U - B}$$ Then since the ratio to the B-V color excess is known, I solved for that and obtained $E_{B - V} = 0.14$. Using the ratio of total to selective extinction and the fact that $E_{B-V} = A_B - A_V$, we get $A_V = 0.44$ and $A_B = 0.58$. (So far this is in line with the given solution.)

From here, the solution given for this problem takes the following step: it uses $V = M_V + A_V$, giving $8.14 = M_V + 0.44$ and solving for $M_V$, $M_B$, and $M_U$ from there, obtaining $M_V = 7.69$, $M_B = 7.91$, and $M_U = 7.46$.

I don't understand why this is a valid method. I've looked in my textbook and it corroborates what I assumed to be correct - that the equation for V is $V = M_V + 5log\frac{r}{10} + A_V$ to account for the "spreading out" of the light in addition to the extinction. So is the $5log\frac{r}{10}$ automatically accounted for in $A_V$ in this case? If not, what am I missing? Thanks to anyone who can offer help.

There is some extra information in the homework statement that was used for further parts of this question; I don't believe it's relevant to this part, but I left it there in case it has some implication here. One other possibility is that I am misunderstanding the term "intrinsic magnitude" in the question statement, which I took to mean "absolute magnitude." Is there a difference here?

 

[Adrmay]

The equation for V that you have given is only valid for objects that are at a distance of 10 parsecs. In this case, the star is much farther away, so the equation needs to be modified. The extra term in the equation (the one with $5log\frac{r}{10}$) accounts for the fact that the star is not at a distance of 10 parsecs, and thus the light from the star has been spread out over a larger area. Therefore, it is less bright than it would be if it were at a distance of 10 parsecs, and this term accounts for that.The "intrinsic magnitude" in this case refers to the magnitude of the star if it were at a distance of 10 parsecs, without any extinction or other effects. In other words, it is the absolute magnitude of the star.