Chapter 4.4, Problem 18E

Special Rounding Instructions For this exercise set, round all regression parameters to three decimal places, but round all other answers to two decimal places unless otherwise indicated.

Magnitude and Distance Astronomers measure brightness of stars using both the absolute magnitude, a measure of the true brightness of the star, and the apparent magnitude, a measure of the brightness of a star as it appears from Earth. The difference between apparent and absolute magnitude should yield information about the distance to the star. The following table gives magnitude difference $m$ and distance $d$, measured in light years, for several stars.

Star	Magnitude difference $m$	Distance $d$
Algol	$2.56$	$105$
Aldebaran	$1.56$	$68$
Capella	$0.66$	$45$
Canopus	$2.38$	$98$
Pollux	$0.13$	$35$
Regulus	$2.05$	$84$

a. Plot $d$ versus $m$.

b. Give an exponential model for the data.

c. If one star shows a magnitude difference $1$ greater than the magnitude difference that a second star shows, how do their distances from the Earth compare?

d. Alphecca shows a magnitude difference of $1.83$. How far is Alphecca from Earth?

e. Alderamin is $52$ light-years from Earth and has an apparent magnitude of $2.47$. Find the absolute magnitude of Alderamin.