General Astronomy, 29:61
Fall, 2005
Eighth Homework Set...October 28, 2005.
Due November 4, 2005

Show calculations and give reasons for your answers. Be sure to put the answers in the right physical units when that is necessary. Be sure to note that not all problems have the same number of points. Problems # 1, 3, and 5 are worth 10 points rather than 5.

(1) ...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

(2) What is the thermal speed (taken as the root-mean-square or rms speed of the molecules) of carbon dioxide molecules in the atmosphere of Venus? You need to look at Figure 11-14 on p336 for some information.

(3) ...10pts A planet has a mass of $10^{24}$ kg and a radius of 2000 km. The surface temperature is 300K. (a) Calculate the escape speed from the surface. (b) Calculate the thermal speed of a carbon dioxide molecule. (c) Would the planet be able to retain a carbon dioxide atmosphere over a period of time equal to the age of the solar system? Justify your answer.

(4) Look at Figure 9-20g from the textbook, which presents data on craters on four of the moons of Jupiter. Based solely on the data given in this figure, what would you say about the geological histories of Callisto and Europa? Based on this figure, what would you expect for the appearance of these objects? You will need to read the text in the chapter accompanying this figure to understand this.

(5) ...10 pts Suggestion: read this one through completely and think about it before starting to work it out. Planet X has an atmosphere which has two molecular species, A and B. The escape speed from Planet X is 8 speed units. Imagine that the distribution of molecular speeds for neither molecule is Maxwellian, but instead given by the equations below. (Note this is not physically impossible, and in fact something similar happens in the interplanetary medium, but it does take some unusual and weird physics to produce).

For molecule A, the distribution of molecular speeds is

$$N_A(v) = N_0 e^{-(\frac{v-V_0}{a_G})^2}$$ (1)

where $v$ is the molecular speed, $V_0$ is the mean speed, $V_0=5$ speed units, and $a_G$ is the width of the speed distribution, $a_G=2$ speed units.

For molecule B, the distribution of molecular speeds is

$$N_B(v < 4) = 0.70 N_0 v$$ (2)

if $v \leq 4$ speed units, and

$$N_B(v > 4) = 0.70 N_0 \left[1-(v-4)/6\right]$$ (3)

if $v \geq 4$ speed units. $N_B(v) = 0$ if $v > 10$ speed units.

Assume, as is the case for the Maxwellian distribution, that collisions will cause the distribution to be regenerated if some molecules are lost, so that the shape of the distribution of molecular speeds for A and B will remain the same. That is, as molecules are lost, only $N_0$ changes with time.

Now for the question. Which molecule, A or B, will be lost by planet X first? You have to describe your arguments. Hint: this problem can be satisfactorily answered in a graphical fashion, i.e. by use of careful graphs of $N_A(v)$ and $N_B(v)$, together with use of the ideas presented in lecture. Alternatively, if you know integral calculus, you can use it to analyse this problem. Again, you have to understand the physics behind the mathematics. Think about what is going on to cause a planet to lose a gas in an atmosphere.

(6) Calculate the thermal speeds of oxygen (O$_2$) and nitrogen ($N_2$) in the air at room temperature.