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3. When a star of mass M is devoid of a nuclear energy source it will keep the same luminosity L for quite a while by consuming its gravitational energy through contraction. Assume the star can remain in a state of hydrostatic equilibrium and its gravitational energy is given by: GM2 Ω = -α- R where G is the gravitational constant, a is a constant and R is the radius of the star. (a) Prove that the rate of contraction of its radius R can be expressed by: RO 22 dR Ř= dt (1+) where R, is the star's initial radius when it starts to contract and: aGM2 T = 2R.L . [19 marks] (b) Calculate how long a star with a luminosity of 1000 Lo, a mass of 8 M., an initial radius of 3 Ro, that has exhausted its reserve of hydrogen, would be able to continue to shine assuming its energy output remains the same. You can assume that a = 0.5. [6 marks]