Astronomy 23/223 Homework:

How the sun works.


This homework assignment is going to investigate the internal workings of the sun using the tools of physics and calculus. We will begin with a few comprehension questions and then progress to some mathematical ones.

1.

Name and describe the different forms of energy transfer in the sun. Where in the sun do the different transfers take place?

In the interior of the sun, radiation of gamma ray energy takes place. The zone extends to about 71% of the sun's radius. The process itself is somewhat similar to a random walk. The next zone is one of covection through which currents raise hot material to the surface. In this process, pockets of hot gas expand, become less dense, and rise to the surface. Granulation allows us to see evidence of the process.

2.

How is the energy in the sun "manufactured"? Using Einstein's E=mc2,show how much energy is given off when 4He is formed.

To do this, take thedifference of the mass of two protons and two neutrons and the mass of 4He. Then multiply that mass by the square of the speed of light.
3.

Conservation of Energy.

a) The potential energy in the sun is the energy that would be liberated if it collasped. For a sphere of uniform density this is given by:

Where G is the universal constant, M is the mass of the sun, and R is the radius of the sun. Solve this set of integrals, find values for the constant and properties, and calculate the potential energy of the sun (astronomers usually do this calculation in ergs). This is actually an approximation. To see a more accurate derivation check out this page.

 

b) The thermal energy of the sun can be described by:

Where N is the number of particles, k is the Boltzmann factor, and T is the average temperature in the sun. This equation comes from assuming that hydrogen is an ideal gas and calculating its kinetic energy. Using 8E6 K for the average internal temperature of the sun and 2E57 for the number of particles, calculate the energy. Compare your answer with part A. Because of the approximation in the previous part, your answer may be off by a factor of 2.

4.

Pressure. This problem will find the pressure profile of the sun. To begin, we will assume that the sun is a spherically symmetrical object. It's mass then, is simply the sum of its density at each radius times the surface area at that radius. In calculus, it is writen as follows:

or in differential form:

We are assumping hydrostatic equilibrium which means that the pressure in terms of radius can be writen as follows:

Solve the differential form of the mass equation for the density. Then substitute it in for the density in this pressure equation. Multiply each side of the pressure equation by dr and integrate from the center of the sun to the surface (from the r=0 to R). Use known quantities to then calculate the pressure at the center of the sun.

 

A bit of help: When you're doing the last integral. The pressure at the surface is zero. How much mass is there at the center? How much mass is enclosed at the surface?

 

 

 

 

 

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Last updated on 28 May, 2004 .