In nature we have 4 fundamental forces. One phenomenological framework widely used to search for deviations from GR is the fifth force formalism (e.g., Fischbach et al. 1986), which considers deviations from Newtonian gravity in which the gravitational potential takes the form of a Yukawa potential. We will apply this potential to the center of the galaxy, near the supermassive black hole, for an arbitrary star with a mass m << M, where it takes the following form: U = GMm r 1 + er/ . (3) We then consider an arbitrary star with a mass m For the rest of the question we are going to ignore contribution from general relativity. Hint: You already encountered the Yukawa potential in problem set 6. Here it appears in a slightly different form. i. (15pt) Show that the equation of motions for the r component can be written as: r = l 2 m2r 3 GM r 2 1 + er/ GM r e r