Homework 7 - ASTRO 102

1) Schwarzschild Radii Calculate the Schwarzschild radius (in kilometers) for each of the following. A 1×10 8 MSun black hole in the center of a quasar. A 6 MSun 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) 3 * 10^8 km 17.7 km 1.1 * 10^-7 km 7.4 * 10^-29 km 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 /2 r , 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? 3.97 * 10^10 seconds or about 1259 years which is about 309 years more than the true 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 R s to calculate the volume (4/3) πR 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 )? 1.35 * 10^8 MSun 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? About 10^46 Joules of gravitational potential energy is released. This is about 102 times that of the sun’s energy released.