Chapter 23, Problem 7Q

(a)

To determine

The distance to M51 in megaparsecs (Mpc), if the distance modulus of galaxy M51 is $m-M=29.2$, as may be judged from the figure which is presented below.

Answer to Problem 7Q

Solution:

$\text{6}\text{.9 Mpc}$

Explanation of Solution

Given data:

The difference between the apparent magnitude and absolute magnitude is given by the folloing expression.

$m-M$=$29.2$

Formula used:

Apparent magnitude (m) is related to absolute magnitude (M) and the distance to the galaxy (d) in parsecs, by the following equation.

$m-M=5\log d-5$

Explanation:

Now recall the formula of apparent magnitude and absolute magnitude.

$m-M=5\log d-5$ …… (1)

Use the following equation.

$m-M\text{ = 29}\text{.2}$ …… (2)

Compairing equation (1) and (2),

$\begin{array}{c}5\log d-5=29.2\\ 5\log d=34.2\\ \log d=\text{}\frac{34.2}{5}\\ =6.84\end{array}$

Solve further.

$\begin{array}{c}d={10}^{6.84}\\ =6.9\times {10}^6\text{ pc}\left(\frac{1\text{ Mpc}}{{10}^6\text{ pc}}\right)\\ =6.9\text{}\text{Mpc}\end{array}$

Conclusion:

The distance to M51 is $\text{6}\text{.9 Mpc}$.

(b)

To determine

The the distance to M51 in megaparsecs (Mpc), if the distance modulus of galaxy M51 is $m-M=29.6$, as may be judged from the figure which is given below.

Answer to Problem 7Q

Solution:

$\text{8}\text{.3}\text{Mpc}$

Explanation of Solution

Given data:

The difference between the apparent magnitude and absolute magnitude is 29.6.

Formula used:

Apparent magnitude (m) is related to absolute magnitude (M) and the distance to the galaxy (d) in parsecs by the following relation.

$m-M=5\log d-5$

Explanation:

Recall the formula of apparent magnitude and absolute magnitude.

$m-M=5\log d-5$ …… (3)

It is given that,

$m-M=29.6$ …… (4)

Comparing equation (3) and (4),

$\begin{array}{c}5\log d-5=29.6\\ 5\log d=34.6\\ \log d=\text{}\frac{34.6}{5}\\ =6.92\end{array}$

Solve further.

$\begin{array}{c}d={10}^{6.92}\\ =8.3\times {10}^6\text{ pc}\left(\frac{1\text{ Mpc}}{{10}^6\text{ pc}}\right)\\ =8.3\text{}\text{Mpc}\end{array}$

Conclusion:

The distance to M51 is $8.3\text{Mpc}$.

(c)

To determine

The difference between the distances to M51, when the different values of $m-M$ are taken $\left(29.2\text{and}\text{29}\text{.6}\right)$ and compare this difference with the 750 kpc distance from Earth to M31, the Andromeda galaxy.

Answer to Problem 7Q

Solution:

The difference is $1.4\text{Mpc}$, which is greater than the distance to M31.

Explanation of Solution

Introduction:

Andromeda Galaxy: This is the spiral galaxy of the distance 2.5 million ly from the Earth and it is a major galaxy, nearest to the milky way. It is also known as Messier 32 or M31.

Explanation:

Obtain the values of distance to galaxies from the above parts with respect to the two different values of $m-M\left(29.2\text{ and}\text{}\text{29}\text{.6}\right)$.

Therefore, the distance value for $29.2$ is $6.9\text{Mpc}$ and for $29.6$ is $8.3\text{Mpc}$.

The difference is given blow.

$\begin{array}{c}\text{Difference}\text{=}\text{8}\text{.3 Mpc}-\text{6}\text{.9 Mpc}\\ \text{=}\text{1}\text{.4 Mpc}\end{array}$

Conclusion:

From the calculation, it is observed that the difference is $1.4\text{}\text{ Mpc}$, which is larger than the distance to M31 $\left(\text{750 kpc}\right)$.