PROBLEM SET #5

ASTR 3730

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_pc², 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 cm²/gm. 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,  wher 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?