Homework 6 - UMass Astronomy

Homework 6
Astronomy 228 – Spring 2012
Due: Class Time on Thursday, March 29
Answer the following questions. Be certain to show all of your work for full
credit. Each question is worth 10 points.
Question 1: Any mass object can become a black hole if it is sufficiently
collapsed so to be smaller than its Schwarzschild radius. What is the
Schwarzschild radius for an object with the mass of the Earth ?
Question 2: Pulsars are rapidly rotating collapsed objects. Consider a 1
solar mass white dwarf star with a radius of 5000 km. To explain the light
variations of pulsars, a white dwarf would need to rotate at a rate of at least
1 revolution per second. For the white dwarf star compute the gravitational
acceleration and the centripetal acceleration at the surface on the equator.
Compare these two accelerations, do you think white dwarf stars can spin
that rapidly ?
Question 3: Hopefully you found in Question 2 that white dwarf stars
cannot be pulsars. Now consider a 2 solar mass neutron star with a radius
of 10 km. Again to explain the light variations of pulsars, this neutron stars
would need to rotate at a rate of at least 1 revolution per second. For the
neutron star compute the gravitational acceleration and the centripetal
acceleration at the surface on the equator. Compare the the two
accelerations. Is it possible for neutron stars to rotate this rapidly ?
Question 4: Consider a 2 solar mass neutron star with a radius of 10 km.
What is the escape speed from its surface ? How does this compare with the
speed of light ?
Question 5: For the neutron star in Question 4, compute the mean
separation between neutrons. How does this compare with the “size” of a
neutron of about 1x10 -13 cm ?
Questions 6: A Type Ia supernova produces about 1x10 51 ergs of energy in
the thermonuclear explosion. Compare this energy with an estimate of the
total amount of energy that the Sun produces over its main sequence
lifetime.

Questions 7: A Type Ia supernova at its peak in brightness has an absolute
visual magnitude of -19.3. With our modern day telescopes and detectors,
we can detect astronomical objects with apparent visual magnitudes of +28.
How far away can we see detect a Type Ia supernova (you might want to
express your answer in Mpc or millions of parsecs) ?
Questions 8: Cyg X-1 is a bright X-ray emitting binary system believed to
be composed of a 25 solar mass O-type supergiant (O Iab) star and a 9 solar
mass black hole. The orbital period is 5.6 days. Assuming circular orbits,
compute the separation between the two stars. The inner Lagrangian point
(L1) lies between the two objects (assume this is the point where the
gravitational force of the A-type supergiant star and the black hole on an
object is equal). How far is the L1 point from the center of the O-type
supergiant star ?
Question 9: The O-type supergiant star in the Cyg X-1 system has a
luminosity of 4x10 5 solar luminosities (1.6x10 39 ergs s -1 ) and a surface
temperature of 28,000 K. Compute the radius of the O-type supergiant star.
Using the result from question 8, is it plausible that matter from the A-type
supergiant is spilling onto the black hole ?
Question 10: Figure 13.11 in the textbook shows a composite H-R diagram
for globular clusters in the Milky Way. Use the information given in
Appendix E and what we know about the main sequence lifetime of stars to
estimate the age of this composite star cluster. Explain your answer.