General Astronomy, 29:61
Fall, 2004
First Homework Set...August 27, 2004. Due September 3, 2004

Show calculations and give reasons for your answers. Don't go around confused and despondent; if you do not know how to get started, ask us for help. The purpose of problem sets is to promote thinking and lead to understanding, not produce a confiteor of revealed truth.

(1) Show that the function

$$p(z) = p_0 e^{-\alpha z}$$ (1)

satisfies the relation

$$\Delta p = -\alpha \Delta z p$$ (2)

Don't use calculus, in which case it is just a one-liner. Instead, use algebra. The key is to use the series expansion for the exponential function, in which

$${\mbox if } x \ll 1, e^{-x} \simeq 1-x$$ (3)

where $\simeq$ means ``approximately equal to''.

(2) Baseball fans claim that home runs are harder to hit during the World Series (in October) than in the middle of the season in July. Using physical arguments presented in class, discuss why this might be true. For this problem, I don't expect a calculated number, but would like a discussion in which equations and physical concepts are used.

(3) The surface gravity on Mars is 0.381 times that on Earth, and its atmosphere is nearly entirely CO$_2$. Assuming a surface temperature of 260K, calculate the atmospheric pressure scale height on Mars.

(4) Estimate the summer-winter difference in the atmospheric pressure scale height on Earth.

(5) A small cylinder has a radius of 8 millimeters, and a length of 20 millimeters. It has an open end which is closed by an air-tight, but movable piston. It is placed upright, filled with air at room pressure and temperature, and then closed with the cylinder. A 10 kilogram mass is then placed on top of the piston. How far down does the piston move before equilibrium is reached? Assume the gas inside stays at the same temperature (this could be done by making the walls of the cylinder from metal).

(6) Using Figure 11-13, calculate the number density of air molecules at altitudes of 20 kilometers and 80 kilometers. Compare those densities with the number densities at sea level.

(7) Use the JPL solar system simulator (on the course home page) to find the current distance to the planet Saturn. If you transmit a command to the Cassini spacecraft, how long must you wait to receive confirmation that it carried out the command?

(8) You are on a commercial airliner and the altimeter stops functioning. The crew and all the passengers are in a panic. However, some of the aircraft instruments are still functioning and their data is visible on the entertainment center. You see that the pressure outside is $3.1 \times 10^4$ Newtons/m$^2$. You do a quick calculation on an air-sickness bag and figure out the altitude of the aircraft. What is it?