Homeworks 6

Homework 6 due on 10/12/2009

Problem 1

Assume that you need on average 1 kilowatt of power continuously, day and night, to support your life style. Some of the energy is used as electricity, but much is used to manufacture and run your automobile and other possessions. One Watt is $10^{7}$ ergs/sec.

1. Suppose you live for 70 years, how many ergs of energy will you use up?

          1 W = 10E7 erg/sec

          70 years = 2207520000 seconds

          2207520000 sec * 10E7 erg/sec = 2.208E17 ergs

2. Using $E=mc^{2}$, how much mass would have to be converted to energy to create this much energy?

          2.208E17 ergs = 2.208E10 J

          m = E/c^2 -->  2.208E10 J / 3E8^2 m/s = 2.45E-7 kg

3. Suppose the energy is created by fusing hydrogen into helium. How many grams of hydrogen would be needed? Remember that only 0.71% of the mass is converted to energy during this reaction.

          E = (0.71m)* c^2

          m = E / 0.71c^2 = 2.208E10 / (0.71* 3E8^2) = 3.455E-7kg --> 3.455E-4 g          (Love aug)

Problem 2

What, specifically, is a theoretical model for a star? How are theoretical models produced? What are the most important physical processes that need to be included to give accurate models?

Problem 3

Describe hydrostatic (pressure) equilibrium in stars. Hydrostatic equilibrium means that stars must have higher pressure at their centers than at their surfaces. Why? What kinds of behavior can a star have if it is not in hydrostatic equilibrium? Give examples of stars not in hydrostatic equilibrium.

Problem 4

How many kinds of neutrinos are there? Why is this important for measuring the properties of the sun? There are three kinds of neutrinos, not counting the anti-particles. This is important for measuring the properties of the sun because for so long we were only observing roughly 1/3 of the neutrinos that we should have. At some point in the nineties it was discovered that there are are actually three varieties, and that two thirds of the neutrinos were decaying into the exotic varieties. Once this was rectified, the neutrino problem was solved!

Problem 5

Estimate the central density and temperature of a star with one solar mass and radius. (Hint: Use the equations for hydrostatic equilibrium, assume an ideal gas, and a 'one zone' model with reasonable boundary conditions). Compare the results with values from literature.

The central pressure is given by

$P_{c}\;=\;{\frac {GM_{o}\rho }{R_{o}}}$

From the Ideal Gas Law,

$P\;=\;{\frac {{\rho }kT}{{\mu }m_{H}}}$

Assuming an ideal gas,

${\frac {{\rho }kT}{{\mu }m_{H}}}\;=\;{\frac {GM_{o}\rho }{R_{o}}}$

$T\;=\;{\frac {{\mu }m_{H}GM_{o}}{kR_{o}}}$

For one solar mass and one solar radius,

$T\;=\;{\frac {(1.750)(1.673*10^{-27}\;kg)(6.67*10^{-11}\;N{\frac {m^{2}}{kg^{2}}})(1.989*10^{30}\;kg)}{(1.38*10^{-23}\;{\frac {J}{K}})(6.955*10^{8}\;m)}}$

$T\;=\;4.045*10^{7}\;K$

Compared to literature, this is off by a factor of 2.57.

Using the equation,

$kT_{c}\;=\;({\frac {\pi }{6}})^{1/3}G{\mu }m_{H}{\rho }_{c}^{1/3}M^{2/3}$

Solving for ${\rho }_{c}$,

${\rho }_{c}\;=\;(kT_{c})^{3}[({\frac {\pi }{6}})^{1/3}G{\mu }m_{H}M^{2/3}]^{-3}$

${\rho }_{c}\;=\;[(1.38*10^{-23}\;{\frac {J}{K}})(40.45*10^{6}\;K)]^{3}[({\frac {\pi }{6}})(6.67*10^{-11}\;N{\frac {m^{2}}{kg^{2}}})(1.75)(1.673*10^{-27}\;kg)(1.989*10^{30}\;kg)^{2/3}]^{-3}$

${\rho }_{c}\;=\;1.129*10^{4}\;{\frac {kg}{m^{3}}}$

This is off by a factor of about 14.5 with the value from literature.