An accretion disc may form around a black hole. This is a thin disc of orbiting matter spanning radii r = Rin to Rout around the black hole. We assume that Rout » Rin and so we make the simplifying approximation that Rout→+00. The disc radiates according to the following equation 3 GM M D(t,0)= ' (¹-[-]"). 4 T3 Here, r and are the usual polar coordinates with the origin at the centre of the disc. G is the gravitational constant, M is the mass of the black hole, Rin is the disc inner radius, M is the accretion rate - all these are constants. (a) Integrate D(r, 0) over the surface of the disc to find the total radiation output of the disc. (b) Find the total radiation in the case of Rin = 6GM/c².
An accretion disc may form around a black hole. This is a thin disc of orbiting matter spanning radii r = Rin to Rout around the black hole. We assume that Rout » Rin and so we make the simplifying approximation that Rout→+00. The disc radiates according to the following equation 3 GM M D(t,0)= ' (¹-[-]"). 4 T3 Here, r and are the usual polar coordinates with the origin at the centre of the disc. G is the gravitational constant, M is the mass of the black hole, Rin is the disc inner radius, M is the accretion rate - all these are constants. (a) Integrate D(r, 0) over the surface of the disc to find the total radiation output of the disc. (b) Find the total radiation in the case of Rin = 6GM/c².
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![An accretion disc may form around a black hole. This is a thin disc of orbiting matter
spanning radii r = Rin to Rout around the black hole. We assume that Rout » Rin and so
we make the simplifying approximation that Rout → +∞o. The disc radiates according to
the following equation
3 GM
D(r, 0) =
1 CM (1-[B]").
4 3
Here, r and are the usual polar coordinates with the origin at the centre of the disc. G
is the gravitational constant, M is the mass of the black hole, Rin is the disc inner radius,
M is the accretion rate - all these are constants.
(a) Integrate D(r, 0) over the surface of the disc to find the total radiation output of the
disc.
(b) Find the total radiation in the case of Rin = 6GM/c².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03ff2dc9-8e4f-4b4d-86ce-92f77e5400ec%2Fd5939635-9bde-411e-b8ba-dddf062b0d74%2Fclkksqq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:An accretion disc may form around a black hole. This is a thin disc of orbiting matter
spanning radii r = Rin to Rout around the black hole. We assume that Rout » Rin and so
we make the simplifying approximation that Rout → +∞o. The disc radiates according to
the following equation
3 GM
D(r, 0) =
1 CM (1-[B]").
4 3
Here, r and are the usual polar coordinates with the origin at the centre of the disc. G
is the gravitational constant, M is the mass of the black hole, Rin is the disc inner radius,
M is the accretion rate - all these are constants.
(a) Integrate D(r, 0) over the surface of the disc to find the total radiation output of the
disc.
(b) Find the total radiation in the case of Rin = 6GM/c².
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