Below is an image of Fomalhaut b, the first extrasolar planet to be observed directly by visible light, obtained by the Hubble Space Telescope.
Astronomy Picture of the Day 2008 November 14. http://antwrp.gsfc.nasa.gov/apod/ap081114.html. Image Credit: NASA, ESA, P. Kalas, J. Graham, E. Chiang, E. Kite (Univ. California, Berkeley), M. Clampin (NASA/Goddard), M. Fitzgerald (Lawrence Livermore NL), K. Stapelfeldt, J. Krist (NASA/JPL)
The scale of the larger diagram is shown on the lower left; the 13 " refers to the angle, in arc-second, subtended by 100 \mathrm{AU} at the distance of Fomalhaut. The scale of the inset can be determined by size of the small box.
a. A planet is in a circular orbit of radius R and period T. Derive an expression for the mass of a star in terms of R, T, and the gravitational constant G. You may assume that the mass of the planet is much, much less than the mass of the star.
b. From the image, estimate the mass of the star Fomalhaut in solar masses. You may assume that the orbit of Fomalhaut b is circular and ignore any errors associated with the fact that the plane of the image is not coincident with the plane of the orbit. Recall that the radius of the Earth’s orbit around the Sun is 1 AU. You do not need to do an error analysis, but the number of significant digits in your answer ought reflect the accuracy of your answer.