PROBLEM SET #5
ASTR 3730
Read Chapters 10, 11, 15, 16, 18
1. Assume the Sun was initially 70% hydrogen by mass. Calculate the
number of H nuclei available for nuclear reactions. If each nucleus yields 0.03m_{p}c^{2},
and only 13% is available for consumption (no mixing), what will be the lifetime of the
Sun on the main sequence?
2. The neutrino is an elusive subatomic particle that interacts with
matter through the weak nuclear force. Its opacity is about 10^{20} cm^{2}/g.
Calculate the mean free path of a neutrino through solid lead.
3. Consider that local pressure can be given by P=nvp where n is the
particle density, v the velocity, and p the momentum. We also have dx
= n^{1/3}, where dx is the size of space occupied
by the particle. Use the Heisenberg uncertainty principle ( dx dp ~ h) and the definition of momentum to derive a formula for the
pressure in a White Dwarf.
4. We can estimate the gravitational pressure in the following way.
Assume there is a mass equal to the mass M of the star a distance R/2 from another
mass M. Use Newton's Law to derive a formula for this force. This is roughly the force
that a star uses to hold itself together. Assume the pressure P is force divided by area,
where the area is that of a sphere of radius R/2. Derive a formula for gravitational
pressure.
5. Set the pressures from parts 3 and 4 to be equal. Change density to
mass and radius. Solve for radius. This formula should resemble the white
dwarf radius formula from class, except for a fixed constant.
6. White Dwarf Calculations
a) Calculate the radius of a white dwarf with exactly one solar mass.
b) Calculate the average density of this star.
c)How big would the Earth be if it had the same density?
d)Let's calculate the cooling time of this white dwarf.
i) Assume the star behaves classically (not quantum mechanically).
Write an expression for its thermal energy content as a function of temperature.
ii) Write an expression for its radiative energy loss as a function of
temperature.
iii) Noting that one of these expressions is the derivative of the
other, write an expression for the change of temperature with time as a function of
temperature.
iv) Solve the simple differential equation to get T vs. time.
v) How long does the star take to cool from 100,000K to 10,000K?
From 10,000K to 1000K?
e) If the electrons were replaced with neutrons, how big would the star be?
7. Problem 16.7 of Carroll and Ostlie
8. Imagine that a star only 1pc from Earth goes supernova and
achieves
a luminosity of 10^{11} solar luminosities.
a) What is the flux at Earth in ergs/cm^{2}/s?
b) By what factor is the flux on the Earth's surface increased over that from the Sun
alone?
c) What will be the temperature of the Earth when it reaches a new thermal equilibrium?
d) Would life on Earth survive?
e) How distant would the supernova have to be if the temperature on the Earth is to remain
below 140F?
f) Assuming the density of stars in the galaxy is 0.1 per cubic parsec, there are 10^{11} stars in the galaxy, and there is one supernova every 30
years, how often would a supernova that raises T above 140F occur?
g) Severe high energy radiation would hit the Earth if the supernova were within 20pc. How
often does this happen?
9. Neutron Stars
a) A marshmallow weighing 5g is dropped from a large height onto a 1 solar mass neutron
star. How much energy is released when it impacts the surface? How does this compare with
a nuclear bomb?
b) In Larry Niven's famous story Neutron Star the hero does a close orbit around a
neutron star while hanging on for dear life. How long does a close orbit around a neutron
star last? Is the story accurate from a physics point of view?
c) Tidal force on an object can be computed by taking the difference of gravitational
attraction between the front and the back of an object. Estimate the tidal force on our
hero. Can he survive?
