Astronomy 1 / Section 3 (S. Myers)

Problem Set #7 (due Mon 8 April 1996 5pm)

1. As you hopefully found on the midterm, the faint companion to the bright star Sirius A, called Sirius B, has a mass of 0.98 solar masses, but a radius of only 0.025 solar radii! Using the values for Msun and Rsun in the back of the book, calculate the mean density of Sirius B in kg/m^3 (the volume of a sphere is 4pi R^3/3). This is an example of a white dwarf, where the electrons and nuclei are packed close enough together so that it is degenerate.

   What I also intended you to calculate was the surface gravity of Sirius B relative to the Earth's gravity. The gravitational force on a mass m at the surface of a spherical body of mass M and radius R is given by F = m g, where the surface gravity g = GM/R^2. The weight w of a mass m is then just given by w = m g/g_E, where g_E = GM_E/R_E^2 at the surface of the Earth (radius R_E, mass M_E). Calculate the ratio of surface gravities g/g_E for Sirius B. This is how many times heavier you would be on the surface of Sirius B!

2. A neutron star is a star of mass M = 1.5 solar masses compressed to a radius of only 10 km! Calculate the mean density of a neutron star and the surface gravity relative to that of the Earth. This is about as dense as you can get and still have more or less normal matter.

3. In the collapse of a star to a white dwarf or neutron star, the angular momentum L = MvR, where v is the rotation velocity, is conserved. The rotation frequency f (in revolutions per second) is given by f = v/2pi R, so L =2pi Mf R^2 is constant in the collapse. The rotation period P of the Sun is about 25 days, so calculate the rotation frequency of the Sun in rotations per second f = 1/P (remember to convert days to seconds!).

   Assuming that the angular momentum L = 2pi Mf R^2 remains constant if the Sun were to collapse to a neutron star of radius 10 km from its current radius of 7 x 10^5 km, what would the rotation frequency of the resulting neutron star be? What is the rotation period in seconds?

   Neutron stars rotating this fast have actually been observed! Pulsars are neutron stars that emit beams of light that we see flash by our view once per revolution. The pulsar in the center of the Crab Nebula rotates once every 0.033 seconds. The fastest pulsars know rotate once every 0.001 seconds (1000 times a second)! Its a strange universe out there.

4. If you were to compress the mass of the Sun even more than in a neutron star, down to a radius of 3 km, something very weird would happen. At this radius, called the Schwarzschild Radius, the surface gravity is so strong that light itself is unable to escape! If you were to compress the Sun to this radius or smaller, then no light could shine out of it, and you would be left with what is called a black hole. Calculate the surface gravity of a 1 solar mass black hole with radius 3 km relative to the Earth's surface gravity.

   Using the centripetal acceleration formula v_orb^2 = GM/R, calculate the orbital velocity v_orb at the black hole ``surface''. If you calculate the escape velocity, the velocity needed to escape entirely from the gravity of the object, you actually get the formula v_esc^2 = 2GM/R. What is the significance of the number that you get from this? (Hint: compare it to another important velocity.) Feel free to speculate on this, and read the part in the book on black holes.