Chapter 17, Problem 33Q

(a)

To determine

The reason so that the equation $m_{\text{B}}-m_{\text{V}}=2.5\log \left(\frac{b_{\text{V}}}{b_{\text{B}}}\right)$ is correct.

Answer to Problem 33Q

Solution: The equation $m_{\text{B}}-m_{\text{V}}=2.5\log \left(\frac{b_{\text{V}}}{b_{\text{B}}}\right)$ for the B-V color index is correct because the relation $m_2-m_1=2.5\log \left(\frac{b_1}{b_2}\right)$, which is used for different stars can be used for indexing the same star if it is seen from other sides, then the provided result is obtained.

Explanation of Solution

Given data:

The B-V color index is related to the color ratio $\left(b_{\text{V}}/b_{\text{B}}\right)$ by the equation:

$m_{\text{B}}-m_{\text{V}}=2.5\log \left(\frac{b_{\text{V}}}{b_{\text{B}}}\right)$

Formula used:

Magnitude difference related to the brightness ratio:

$m_2-m_1=2.5\log \left(\frac{b_1}{b_2}\right)$

Here, $m_1$ is the apparent magnitude of the star 1, $m_2$ is the apparent magnitude of the star 2, $b_1$ is the apparent brightness of the star 1 and $b_2$ is the apparent brightness of the star 2.

Explanation:

Recall the expression that relates the magnitude difference to the brightness ratio.

$m_2-m_1=2.5\log \left(\frac{b_1}{b_2}\right)$

Now, as the above expression is for two different of star but the same expression can be applied if the same star is seen from different sides. So, to obtain the expression for the same, replace the subscript 1 and 2 by B and V, respectively. So, the B-V color ratio $\left(b_{\text{V}}/b_{\text{B}}\right)$ can be explained by the equation shown below:

$m_{\text{B}}-m_{\text{V}}=2.5\log \left(\frac{b_{\text{V}}}{b_{\text{B}}}\right)$

Therefore, the provided equation is correct.

Conclusion:

The expression for the B-V color index is obtained by using the relation, $m_2-m_1=2.5\log \left(\frac{b_1}{b_2}\right)$.

(b)

To determine

The B-V color indices for the Bellatrix, the Sun, and the Betelgeuse by using the data in table 17-1.

Answer to Problem 33Q

Solution:

$-0.23$, $0.68$ and $1.86$ as the B-V color index is smaller, the star will be hotter.

Explanation of Solution

Given data:

Refer the table 17-1, to obtain the value.

The B-V color index is related to the color ratio $\left(b_{\text{V}}/b_{\text{B}}\right)$ by the equation:

$m_{\text{B}}-m_{\text{V}}=2.5\log \left(\frac{b_{\text{V}}}{b_{\text{B}}}\right)$

Formula used:

Magnitude difference related to the brightness ratio:

$m_2-m_1=2.5\log \left(\frac{b_1}{b_2}\right)$

Here, $m_1$ is the apparent magnitude of the star 1, $m_2$ is the apparent magnitude of the star 2, $b_1$ is the apparent brightness of the star 1 and $b_2$ is the apparent brightness of the star 2.

Explanation:

Refer the table 17-1:

The B-V color index is related to the color ratio $\left(b_{\text{V}}/b_{\text{B}}\right)$.

$\begin{array}{rr}\hfill \text{Star}& \hfill {\text{b}}_{\text{V}}{\text{/b}}_{\text{B}}\\ \hfill \text{Bellatrix}& \hfill 0.81\\ \hfill \text{Sun}& \hfill 1.87\\ \hfill \text{Betelgeuse}& \hfill 5.55\end{array}$

Refer the result of part (a):

The B-V color index is related to the color ratio $\left(b_{\text{V}}/b_{\text{B}}\right)$ by the equation:

$m_{\text{B}}-m_{\text{V}}=2.5\log \left(\frac{b_{\text{V}}}{b_{\text{B}}}\right)$

For Bellatrix:

Substitute $0.81$ for $b_{\text{V}}/b_{\text{B}}$.

$\begin{array}{c}{m}_{\text{B}}-m_{\text{V}}=2.5\log \left(0.81\right)\\ =-0.23\end{array}$

For Sun:

Substitute $1.87$ for $b_{\text{V}}/b_{\text{B}}$.

$\begin{array}{c}{m}_{\text{B}}-m_{\text{V}}=2.5\log \left(1.87\right)\\ =0.68\end{array}$

For Betelgeuse:

Substitute $5.55$ for $b_{\text{V}}/b_{\text{B}}$.

$\begin{array}{c}{m}_{\text{B}}-m_{\text{V}}=2.5\log \left(5.55\right)\\ =1.86\end{array}$

The B-V color index is positive for a cold star and negative for a hot star because the hot stars are assigned lower number on the apparent magnitude scale while the cold stars are assigned a higher number.

Conclusion:

The B-V color indices for the Bellatrix is $-0.23$, the Sun is $0.68$, and the Betelgeuse is $1.86$. For the smaller B-V color index, the star will be hotter.