Question

Astronomers classify stars according to their brightness by assigning them a stellar “magnitude.” The higher the magnitude the dimmer the star. The dimmest stars visible to the naked eye have stellar magnitudes of 6. The table below shows the relative brightness of different stellar magnitudes.

Answer 1

(a)
`b=100(0.4)^(m-1)`

(b) The relative brightness for a stellar magnitude of 9 is 0.07.

Explanation:

Let the relative brightness be given by 'b' and stellar magnitude by 'm'.

(a)

Given:

The magnitude of relative brightness decreases as the magnitude of stellar increases.

From the table, we can conclude that the ratio for any two consecutive values of 'b' is the same and is equal to 0.4.

`(40)/(100)=(16)/(40)=(6.3)/(16)=(2.5)/(6.3)=(1)/(2.5)=0.4`

Now, we know that, for a common ratio 'r' of a given series, the series is called a geometric series.

The
`n^(th)` term of a geometric series is given:

`a_n=a_1r^(n-1)`

Now, for the given table,
`m=n, b=a_n , r=0.4, a_1=100`

Therefore, the equation that gives the relative brightness in terms of stellar magnitude is given as:

`b=100(0.4)^(m-1)`

(b)

Given:

The stellar magnitude is,
`m=9`

The equation to find relative brightness is:

`b=100(0.4)^(m-1)`

Plug in 9 for 'm' and solve for 'b'. This gives,

`b=100(0.4)^(9-1)\\b=0.07`

Therefore, the relative brightness for a stellar magnitude of 9 is 0.07.