Chapter 8, Problem 3P

To determine

Complete the table of apparent magnitude, absolute visual magnitude, distances, and parallaxes.

Answer to Problem 3P

Complete table is as follows-

$m_v$	$M_v$	$d\left(\text{pc}\right)$	$p\left(\text{arc}\text{second}\right)$
7	7	10	0.1
11	1	1000	0.001
6	-2	400	0.025
4	-2.98	250	0.040

Explanation of Solution

Given:

Incomplete table

$m_v$	$M_v$	$d\left(\text{pc}\right)$	$p\left(\text{arc}\text{second}\right)$
-	7	10	-
11	-	1000	-
-	-2	-	0.025
4	-	-	0.040

Formula used:

Distance of star in parsec is calculated as

$d\left(pc\right)=\frac{1}{p\left(\text{arc}\text{seconds}\right)}$

Absolute magnitude of the star is given as

Calculation:

Parallax of star is calculated as

$\begin{array}{l}10=\frac{1}{p\left(\text{arc}\text{seconds}\right)}\\ p=0.1\end{array}$

Apparent magnitude of the star is calculated as

$\begin{array}{c}{m}_v-M_v=-5+5{\log }_{10}\left(d\right)\\ m_v-7=-5+5{\log }_{10}\left(10\right)\\ m_v=7\end{array}$

Parallax of star is calculated as

$\begin{array}{l}1000=\frac{1}{p\left(\text{arc}\text{seconds}\right)}\\ p=0.001\end{array}$

Absolute magnitude of the star is calculated as

$\begin{array}{c}{m}_v-M_v=-5+5{\log }_{10}\left(d\right)\\ 11-M_v=-5+5{\log }_{10}\left(1000\right)\\ M_v=1\end{array}$

Distance of star is calculated as

$\begin{array}{l}d=\frac{1}{0.025}\\ p=400\text{pc}\end{array}$

Apparent magnitude of the star is calculated as

$\begin{array}{c}{m}_v-M_v=-5+5{\log }_{10}\left(d\right)\\ m_v-\left(-2\right)=-5+5{\log }_{10}\left(400\right)\\ m_v=6.0\end{array}$

Distance of star is calculated as

$\begin{array}{l}d=\frac{1}{0.040}\\ p=250\text{pc}\end{array}$

Apparent magnitude of the star is calculated as

$\begin{array}{c}{m}_v-M_v=-5+5{\log }_{10}\left(d\right)\\ 4-M_v=-5+5{\log }_{10}\left(250\right)\\ M_v=-2.98\end{array}$

Therefore the complete table is as follows

$m_v$	$M_v$	$d\left(\text{pc}\right)$	$p\left(\text{arc}\text{second}\right)$
7	7	10	0.1
11	1	1000	0.001
6	-2	400	0.025
4	-2.98	250	0.040

Conclusion:

Complete table is as follows-

$m_v$	$M_v$	$d\left(\text{pc}\right)$	$p\left(\text{arc}\text{second}\right)$
7	7	10	0.1
11	1	1000	0.001
6	-2	400	0.025
4	-2.98	250	0.040