General Astronomy, 29:61
Fall, 2004
Sixth Homework Set...October 9, 2004.
Due October 15, 2004

Show calculations and give reasons for your answers. Be sure to put the answers in the right physical units. Note: Two of the problems (# 1 and # 6) are ``big persons'', and have twice the points of the others .

(1)...10pts A satellite is in a highly elliptical orbit around the Earth, and its orbital plane is the Earth's equatorial plane. The semi-major axis is $3.5 R_{\oplus}$ ($1 R_{\oplus}=$ 1 Earth radius) and the eccentricity is 0.57.
(a) How high above the surface of the Earth is the satellite at perigee (closest approach to Earth)?
(b) You are observing the satellite at perigee. In 6 minutes, it goes through an angle of $7^{\circ}$. What angle will it move through at apogee? Hint: Make the assumption (unjustified) that the angles you measure are the same that would be observed at the center of the Earth.
It will help if you draw a diagram of the orbit before you start. That diagram will also help you with problems 2,3, and 4 below.

(2) Read this whole question through before trying to come up with the answer. Think about your satellite TV dish. The satellite TV system receives signals transmitted from a spacecraft in outer space. To receive the signal, the antenna has to point directly at the spacecraft. Small satellite TV antennas have no capability of pointing at different directions in the sky, like astronomical telescopes; they are bolted to the side of a building. The spacecraft that sends the transmissions, on the other hand, is moving on an orbit in space. Given all the above, what can you conclude about the orbital period of the spacecraft as it orbits the Earth? Be sure and explain your answer.

(3) Following on question #2, why must TV satellite orbits be highly circular, i.e. possessing a very low eccentricity?

(4) What is the radius of the orbit of a TV satellite? Hint: You need the answer to # 2.

(5) In class we discussed the exponential decay law for the number of radioisotopes as a function of time, and showed that at $t=T_{\frac{1}{2}}$ ( $T_{\frac{1}{2}}$ is the half life of the radioisotope) there are half as many radioactive nuclei as at $t=0$. Use this exponential decay law formula to prove that for any time $t$, there are twice as many nuclei as there will be at time $t+T_{\frac{1}{2}}$.

(6) ...10 pts A rock is analysed for its age of formation, using radioisotope dating. The radioisotope used is $A$ which decays to isotope $B1$ of element $B$, $A \rightarrow B1$ with a halflife $T_{\frac{1}{2}} = 5.0$ Gyr (billions of years). $B2$ is an isotope of element $B$ which is not the daughter product of radioactive decay, and does not decay itself. The rock contains zones with two minerals having different affinity for elements $A$ and $B$. When the zones are analysed, they give the number of nuclei of $A$, $B1$, and $B2$ shown in the table below. How long ago did the rock form? Hint: Begin by identifying which equation you need to give you the answer, then use the data table to provide the numbers you need. Be sure and show your work.

Zone	$N_A$	$N_{B1}$	$N_{B2}$
1	200	183	400
2	300	235	400

(7) What is the highest altitude angle that will ever be seen for the Moon here in Iowa City? What is the lowest altitude angle? Hint: Think about the orbital elements of the Moon's orbit, and the 18.6 year precession period of its orbit.