Chapter 7.2, Problem 58PPS

(a)

To determine

To find: how many times as bright is a magnitude $3$ star as a magnitude $10$ star?

Answer to Problem 58PPS

  $627.64$ times as bright as a magnitude $10$ star.

Explanation of Solution

Given information:

Given, the magnitude of $9$ star is $2.51$ times as bright as a magnitude of $10$ star and a magnitude of $8$ star is $2.51*2.51={2.51}^2$ times as bright as a magnitude of $10$ star.

Calculation:

Calculate the expression

So the magnitude of $9$ star is

  $\frac{{2.51}^{10}}{{2.51}^9}={2.51}^{10-9}={2.51}^1=2.51$

Hence the magnitude of $9$ star is $2.51$ times as bright as a magnitude of $0$ star.

A magnitude $8$ star is

  $\frac{{2.51}^{10}}{{2.51}^8}={2.51}^{10-8}={2.51}^2$

Hence the magnitude of $8$ star is ${2.51}^2$ times as bright as a magnitude of $10$ star.

It follows that a magnitude $3$ star is

  $\frac{{2.51}^{10}}{{2.51}^3}={2.51}^{10-3}={2.51}^7=627.64$ times as bright as a magnitude $10$ star.

Therefore, $627.64$ times as bright as a magnitude $10$ star.

(b)

To determine

To find: write an expression to compare a magnitude $m$ star to a magnitude

Answer to Problem 58PPS

The magnitude $m$ star would be $\frac{{2.51}^{10}}{{2.51}^m}={2.51}^{10-m}$ times as bright as a magnitude $10$ star.

Explanation of Solution

Given information:

Given, the magnitude of $9$ star is $2.51$ times as bright as a magnitude of $10$ star and a magnitude of $8$ star is $2.51*2.51={2.51}^2$ times as bright as a magnitude of $10$ star.

Calculation:

Calculate the expression

We can say that the magnitude $m$ star would be $\frac{{2.51}^{10}}{{2.51}^m}={2.51}^{10-m}$ times as bright as a magnitude $10$ star.

Therefore, the magnitude $m$ star would be $\frac{{2.51}^{10}}{{2.51}^m}={2.51}^{10-m}$ times as bright as a magnitude $10$ star.

(c)

To determine

To find: does your expression hold is true? Give an example.

Answer to Problem 58PPS

A full moon would be $1,557,742,231$ times as bright as a magnitude $10$ star.

Explanation of Solution

Given information:

Given, the magnitude of $9$ star is $2.51$ times as bright as a magnitude of $10$ star and a magnitude of $8$ star is $2.51*2.51={2.51}^2$ times as bright as a magnitude of $10$ star.

Calculation:

Calculate the expression

According to the expression, a full moon would be

  ${2.51}^{10-m}={2.51}^{10-(-13)}={2.51}^{23}=1,557,742,231$ times as bright as a magnitude $10$ star.

Therefore, A full moon would be $1,557,742,231$ times as bright as a magnitude $10$ star.