PHYS 183 Lecture Notes - Lecture 22: Electron Degeneracy Pressure, Pauli Exclusion Principle, Degenerate Matter

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greywildebeest28

21 Apr 2018

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The most massive stars form black holes (m>20 solar masses) Intermediate mass stars for neutron stars (8

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Astronomers have observed a small, massive object at the centerof our Milky Way galaxy. A ring of material orbits this massiveobject; the ring has a diameter of about 15 light years and anorbital speed of about 200 km/s.
A) Determine the mass of the massive object at the center of theMilky Way galaxy.
Take the distance of one light year to be 9.461Ã10¹⁵m. Iwas able to get this it is 4.26Ã10³⁷kg.
B) Express your answer in solar masses instead of kilograms, whereone solar mass is equal to the mass of the sun, which is1.99Ã10³⁰. I got this to it is 2.14Ã10⁷ solarmasses
C)Observations of stars, as well as theories of the structure ofstars, suggest that it is impossible for a single star to have amass of more than about 50 solar masses. Can this massive object bea single, ordinary star? I got this it is no.
D) Many astronomers believe that the massive object at the centerof the Milky Way galaxy is a black hole. If so, what is itsSchwarzschild radius? Express your answer in meters.
E)Would a black hole of this size fit inside the earth's orbitaround the sun? The mean distance from the sun to the earth is1.5Ã10^11m.
Please solve part D and E. For part D please show equations andsolve all the way through. I worked it twice and I am getting myanswer wrong. I am afraid I am making a silly mistake.

The radius (Rh) and mass (Mh) of a black hole are relatedby Rh = 2GMh/c squared, where (c) is the speed of light. Assumethat the gravitational acceleration (ag) of an object at a distance(ro)= 1.001(Rh) from the center of a black hole is given by (ag) =G M / r squared (it is, for large black holes).
A. What is (ag) at (ro) for a very large black hole whosemass is 1.25 × 1014 times the solar mass of 1.99× 1030 kg?
B.  If an astronaut with a height of 1.73 m is atrowith her feet toward this black hole, what is the difference ingravitational acceleration between her head and her feet?

Black holes are objects whose gravitational field is so strong thatnot even light can escape. One way of thinking about this is toconsider a spherical object whose density is so large that theescape speed at its surface is greater than the speed of light, c.If a star's radius is smaller than a value called the Schwarzschildradius Rs, then the star will be a black hole, that is, lightoriginating from its surface cannot escape.
(a) For a nonrotating black hole, the Schwarzschild radius dependsonly upon the mass of the black hole. Show that it is related tothat mass M by Rs = (2GM)/c2.
(b) Calculate the value of the Schwarzschild radius for a blackhole whose mass is ten solar masses.
-----(ANSWER in km)

P.S. The problem gives the hint "Use conservation of energy, and anescape speed equal to the speed of light to derive the expressionfor the Schwarzchild radius of a non-rotating black hole."

Thank you so much for any help!

PHYS 183 Lecture Notes - Lecture 22: Electron Degeneracy Pressure, Pauli Exclusion Principle, Degenerate Matter

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