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...

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.42 The difference between apparent and absolute magnitude should yield information about the distance to the star. The table on the following page gives magnitude difference m and distance d, measured in light-years, for several stars.

a. Plot ln d against m and determine whether it is reasonable to model distance as an exponential function of magnitude difference.

 b. Give an exponential model for the data using the logarithm as a link.

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