Chapter 4.4, Problem 71E

Magnitude of Stars The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by

$M=-2.5\log \left(\frac{B}{B_0}\right)$

where B is the actual brightness of the star and B₀ is a constant.

1. (a) Expand the right-hand side of the equation.
2. (b) Use part (a) to show that the brighter a star, the less its magnitude.
3. (c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero.

To determine

To expand: The equation $M=-2.5\log \left(\frac{B}{B_0}\right)$.

Answer to Problem 71E

The expansion of $-2.5\log \left(\frac{B}{B_0}\right)$ is $\left(-2.5\right)\left(\log B-\log B_0\right)$.

Explanation of Solution

Law used:

Let $a$ be positive no, with $a\ne 1$. Let $A,B$ and C are any real no with $A>0$ and $B>0$.

${\log }_a\left(\frac{A}{B}\right)={\log }_{a}A-\log B$

Calculation:

Here, the given expression is $-2.5\log \left(\frac{B}{B_0}\right)$.

Use the law of logarithm, ${\log }_a\left(\frac{A}{B}\right)={\log }_{a}A-\log B$ and $\log \left(A^C\right)=C\log A$, express $-2.5\log \left(\frac{B}{B_0}\right)$ as,

$-2.5\log \left(\frac{B}{B_0}\right)=\left(-2.5\right)\left(\log B-\log B_0\right)$

Therefore, the expansion of $-2.5\log \left(\frac{B}{B_0}\right)$ is $\left(-2.5\right)\left(\log B-\log B_0\right)$.

To determine

To show: The magnitude of a star decreases as when the brightness of a star increases.

Explanation of Solution

Use the equation $\left(-2.5\right)\left(\log B-\log B_0\right)$ in part (a), which gives the equation of magnitude of a star.

Here, B is brightness of the star. In the equation $\left(-2.5\right)\left(\log B-\log B_0\right)$ the value of brightness increases means $\log B$ also increases (logarithmic function is an increasing function).

Therefore, the value of $\left(-2.5\right)\left(\log B-\log B_0\right)$ decreases.

That is, the magnitude of a star decreases as when the brightness of a star increases.

Example

Take $B=1$  then, the magnitude of a star is $\left(-2.5\right)\left(\log 1-\log B_0\right)$.

Compute the value of $\left(-2.5\right)\left(\log 1-\log B_0\right)$.

$\begin{array}{c}\left(-2.5\right)\left(\log 1-\log B_0\right)=\left(-2.5\right)\left(0-\log B_0\right)\\ =2.5\log B_0\end{array}$

When, the value of $B=1$ then, the magnitude of a star is $2.5\log B_0$.

Take $B=2$ then, the magnitude of a star is $\left(-2.5\right)\left(\log 2-\log B_0\right)$.

Compute the value of $\left(-2.5\right)\left(\log 2-\log B_0\right)$.

$\begin{array}{c}\left(-2.5\right)\left(\log 2-\log B_0\right)=\left(-2.5\right)\left(\log 2-\log B_0\right)\\ =2.5\left(.30-\log B_0\right)\end{array}$

When, the value of $B=2$ then, the magnitude of a star is $2.5\left(.30-\log B_0\right)$.

$2.5\log B_0>2.5\left(.30-\log B_0\right)$

That is, the magnitude of a star decreases when the brightness of a star increases.

To determine

To show that: The magnitude of Betelgeuse is 5 magnitudeless bright than Albiero.

Explanation of Solution

Given:

The brightness of Betelgeuse is 100 times brighter than Albiero.

Compute the magnitude of Albiero.

Assume that B is the brightness of Albiero and take $B_0=1$.

Use the equation of part (a) $\left(-2.5\right)\left(\log B-\log B_0\right)$, which gives the equation of magnitude of a star.

Therefore, the magnitude of Albiero is $M_1=\left(-2.5\right)\left(\log B-\log 1\right)$.

$M_1=\left(-2.5\right)\log B$ (1)

Compute the magnitude of Betelgeuse.

The brightness of Betelgeuse is 100B (because B is the brightness of Albiero ) and $B_0=1$

Therefore, the magnitude of Betelgeuse is $M_2=\left(-2.5\right)\left(\log 100B-\log 1\right)$.

$\begin{array}{c}{M}_2=\left(-2.5\right)\log 100B\\ =\left(-2.5\right)\left(1og100+\log B\right)\\ =\left(-2.5\right)1og100+\left(-2.5\right)\log B\\ =\left(-2.5\right)1og100+M_1\end{array}$

On further simplification the value of $M_2$ becomes,

$\begin{array}{c}{M}_2=-5+M_1\\ =M_1-5\end{array}$

$M_2=M_1-5$ (2)

That is, the magnitude of Betelgeuse is 5 magnitude less bright than Albiero.