AST 307 · Introductory Astronomy Fall 2003

AST 307
Homework #10
Due Monday Nov 24

1. Fortunately for astronomy students, Newton's laws can be used even around black holes, as long as you are far enough from the black hole that your orbital speed is small compared to the speed of light.
One of the more interesting effects on an object orbiting around a black hole is caused by the tidal force, which results from the difference between the gravitational force on two sides of an object if one side is closer to the hole, so it feels a greater gravitational force than the other side. Imagine you are orbiting 750 km away from a 5 solar mass black hole.
a) What is your orbital speed? This should be less than the speed of light (although not tremendously less) so we can use Newton's laws.
b) To make a very simplified model of your body, assume that 1/2 of your mass is in your feet and 1/2 is in your head, and that your head is 2m from your feet. Assume that your feet are pointing toward the black hole, so they are 749,999m from the hole and your head is 750,001m from the hole. Calculate the force on your feet and the force on your head. (Keep as many digits as you can in your calculator, so you can tell that these two numbers are different.)
c) The tidal force is the difference between these two forces. If you are orbiting with an acceleration that is right for the force on your head, there will be too much force on your feet, and they will be pulled away from your head. What is the value of the tidal force? Give your answer both in Newtons and in pounds. Are you willing to volunteer to be the first astronaut to orbit this close to a black hole?

2. I would like to travel to the center of the Milky Way to see if there really is a black hole there. I've invented a new rocket that has enough thrust to make my spaceship accelerate at 1g (9.8 m/s^2), so I won't feel weightless.
a) Using Newton's laws, calculate how long it would take me to get up to the speed of light with that acceleration. (Newton's laws don't really work once I get close to the speed of light, but I could get up to about 7/10 the speed of light in that time, and about .99c in 10 times that time.)
b) Assuming I did get up to .99c, how long would it take me to get to the Galactic center?
c) That's a long time, but it's really not so bad because when I'm going that fast my clocks slow down, and so does my heart, and my aging. The formula for the slowing of time is that clocks slow down by a factor of

How long would the trip seem to me? Should I try it?

14 November 2003
Astronomy Program · The University of Texas at Austin · Austin, Texas 78712
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