Astronomy Homework 7- HJ
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School
University of Michigan *
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Course
102
Subject
Astronomy
Date
Apr 3, 2024
Type
Pages
4
Uploaded by CoachRabbitMaster1086
1) Schwarzschild Radii
Calculate the Schwarzschild radius (in kilometers) for each of the following. A 1×10^8 M
Sun black hole in the center of a quasar. A 6 M Sun black hole that formed in the supernova
of a massive star. A mini-black hole with the mass of the Moon. Estimate the Schwarzschild
radius (in kilometers) for a mini-black hole formed when a super advanced civilization
decides to punish you (unfairly) by squeezing you until you become so small that you
disappear inside your own event horizon. (Assume that your weight is 50 kg)
2) The Crab Pulsar Winds Down
Theoretical models of the slowing of pulsars predict that the age of a pulsar is
approximately equal to p/2r, where p is the pulsar's current period and r is the rate at
which the period is slowing with time. Observations of the pulsar in the Crab nebula show
that it pulses 30 times a second, so that p = 0.0333 second, but the time interval between
pulses is growing longer by 4.2×10^
−
13 second with each passing second, so that
r=4.2×10^
−
13 second per second. Using that information, estimate the age of the Crab
pulsar. How does your estimate compare with the true age of the pulsar, which was born in
the supernova observed in 1054?
My predicted age for the Crab pulsar was approximately 1,258 years. However, the true age
of the pulsar, which was born in the supernova observed in 1054, would be around 958
years old. So, my estimate is a fair amount larger than the actual observed age of the pulsar.
3) A Water Black Hole
A clump of matter does not need to be extraordinarily dense in order to have an escape
velocity greater than the speed of light, as long as its mass is large enough. You can use the
formula for the Schwarzschild radius Rs to calculate the volume (4/3)
π
R^3 inside the event
horizon of a black hole of mass M. What does the mass of a black hole need to be in order
for its mass divided by its volume to be equal to the density of water (1g/cm^3 )?
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4) Energy of a Supernova
In a massive star supernova explosion, a stellar core collapses to form a neutron star
roughly 10 kilometers in radius. The gravitational potential energy released in such a
collapse is approximately equal to GM^2/r where M is the mass of the neutron star, r is its
radius, and G=6.67×10
−
11 m^3 /(kg s^2 ) is the gravitational constant. Using this formula,
estimate the amount of gravitational potential energy released in a massive star supernova
explosion. How does it compare with the amount of energy released by the Sun during its
entire main-sequence lifetime?
Therefore, the energy released by a massive star supernova explosion is compared to the
energy radiated by the Sun during its entire main sequence lifetime at
.
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