1. We believe that the Universe is 10 to 14 billion years old. What are the three strongest pieces of evidence supporting this belief?
2. How old is the solar system? What evidence is there to support your answer?
3. Name and describe the four major mechanisms by which planet-size bodies gain heat.
4. List characteristics of the solar system that are major clues to the formation process.
5. Describe the evidence from observations of other stars that planets form in disks around young stars.
1. In class we used the equations describing hydrostatic equilibrium for a self-gravitating ideal gas to find the temperature in the core of the sun. In this problem you will go through the same problem yourself, using order of magnitude estimates. Recall the physical setting: the sun is a ball of gas held together by self-gravity. If gravity were the only force acting, the gas would collapse down to a much smaller size. However, the gas is held up by gas pressure, or more precisely by the gradient of gas pressure.
We need two equations, one to
describe the hydrostatic equilibrium between the self-gravity and the
gas pressure gradient, and one relating the gas pressure and density
to the gas temperature T. We'll start with the gravitational force experienced by a
layer of gas with density
a distance r from the center of the
The inward force of gravity is balanced by the outward force of the gradient of
gas pressure dP/dr,
The gas pressure is described by the ideal gas law
After all these preliminaries, it is now time for you to do some work:
a) Following my injunction to do an order of magnitude calculation, replace the derivative dP/dr in equation (2) by , where cm is the radius of the sun. In the same spirit, replace r by . Solve the resulting equation for P.
b) Now use the expression given by the ideal gas law (3) to eliminate P for the result you obtained in a). Solve this new equation for T; the result should look like .
c) Plug in the numerical values for G, and so on to find the temperature at the center of the sun.
2. In this problem you will use the equation of hydrostatic equilibrium to find the thickness of the Earth's atmosphere. You will need to use the mass of the Earth, grams, and the radius of the Earth, cm. You may assume that the temperature of the atmosphere is the same at all heights, say Kelven.
a) The Earth differs from the sun in that it is a solid body surrounded by a gas atmosphere, whose height H you are going to find. Hence the estimate for dP/dr also differs, namely dP/dr=P/H. Using equation (2) from the first problem, solve for H in terms of P, , and .
b) Now use the ideal gas law (3) to eliminate P. You should find something like .
c) Once again, plug in numerical values to find the height of Earth's atmosphere.
d) In 1980 Reinhold Messner climbed Mt. Everest, height 8,900 meters or so, without using oxygen tanks. Do you think he suffered any ill effects simply because of the height?