GLG130 S
Assignment #1

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.

Quantitative questions

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 $\rho$ a distance r from the center of the sun is

{GM_\odot\rho\over r^2}
\end{displaymath} (1)

The mass of the sun is $M_\odot=2\times10^{33}$ grams, while the gravitational constant $G=6.67\times10^{-8}$. As you will show, the gas density $\rho$ will drop out of the problem; you don't need to know it. In fact, you could actually calculate it.

The inward force of gravity is balanced by the outward force of the gradient of gas pressure dP/dr,

{dP\over dr}={GM_\odot\rho\over r^2}
\end{displaymath} (2)

As discussed in class, this is the is the equation describing hydrostatic equilibrium.

The gas pressure is described by the ideal gas law

P={\rho k T\over \mu}.
\end{displaymath} (3)

P is the gas pressure and T is the temperature of the gas. These, together with the density, vary throughout the sun. $k=1.38\times10^{-16}$ ergs/$^\circ K$ is Boltzman's constant, and $\mu$ can be taken to be equal to the mass of a proton, about 10-24 grams.

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 $P/R_\odot$, where $R_\odot=7\times10^{10}$ cm is the radius of the sun. In the same spirit, replace r by $R_\odot$. 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 $T=G\ldots$.

c) Plug in the numerical values for G, $R_\odot$ 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, $M_\oplus=6\times10^{27}$ grams, and the radius of the Earth, $R_\oplus=6\times10^8$ cm. You may assume that the temperature of the atmosphere is the same at all heights, say $T=300^\circ$ 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, $\rho$, $M_\oplus$ and $R_\oplus$.

b) Now use the ideal gas law (3) to eliminate P. You should find something like $H=kT\ldots$.

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?

Norm Murray