Question 4. (4 points for (a)-(c), graded for correctness; 2 points for (d)-(f), graded for completion) In astronomy, apparent magnitude is a logarithmic measure of the brightness of a star as seen by an observer on Earth. The apparent magnitude M of a star is related to the brightness B of the observed light by M = -1.09 In (B) where Bo = 2.13 x 10-6 lux. Note that brighter stars have smaller apparent magnitudes. (You will need to use a calculator or computer for this problem to compute the logarithms and exponentials.) (a) As seen from Earth, the star Polaris (the North Star) has a brightness of 3.5 x 10-7 lux. What is the apparent magnitude of Polaris? (b) The brightest star in the night sky is Sirius, with an apparent magnitude of -1.47. However, the daytime sun is approximately 12 billion times as bright as Sirius. What is the apparent magnitude of the sun? (c) Suppose that the brightness of a star is changing over time, so both M and B are functions of t. Differentiate the equation above with respect to t to find an equation relating + to " and B. The apparent magnitude of the variable star Delta Cephei oscillates regularly with a period of several days. The following graph shows some observed data for the brightness of Delta Cephei over the course of its cycle. (Data taken from Engle et al., The Astrophysical Journal 794.1, 2014.) 9 x 10-8 . . 8 x 10-8 . . 7 x 10-8 Brightness (lux) . 0. 6 x 10-8 5 x 10 8 .. 4 x 10 20 40 60 80 100 120 Time (hours) (d) Use this graph to estimate the apparent magnitude of Delta Cephei when it is 50 hours into its cycle. (e) Use this graph to estimate the rate at which the brightness of Delta Cephei is increasing whenever it is 50 hours into its cycle. Make sure to include units in your