General Astronomy, 29:61
Fall, 2004
Eighth Homework Set...October 30, 2004.
Due November 5, 2004

Show calculations and give reasons for your answers. Note: Two of the problems (# 1 and #3 ) are ``big person'' questions, and have twice the points of the others. Problems #7 and #8 require material which will be presented in class on Monday, November 1.

(1) ...10pts: A charged particle with charge $q$ and mass $m$ moves in the $(x,y)$ plane. The magnetic field is given by $\vec{B} = -B_0 \hat{e}_z$. Show that the following trajectory for the particle is a solution to Newton's 2nd Law with the force given by the Lorentz force,

$$\vec{r}(t) = r_0 \left( \cos (\Omega t) \hat{e}_x + \sin (\Omega t) \hat{e}_y \right)$$ (1)

with $\Omega =\frac{q B_0}{m}$

(2) Use the results from the previous problem. If the speed of the particle is $v$, what is the radius of the trajectory? This radius is called the Larmor radius.

(3) ...10 pts Assume that some process modifies the distribution of molecular speeds in a planetary atmosphere, so that the distribution function is

$$N(v) = \frac{2N}{\sqrt 2 \pi} \left( \frac{m}{k_B T}\right)^2 v^3 e^{-\frac{mv^2}{2k_BT}}$$ (2)

instead of a Maxwellian. In the case of ionized gases called plasmas, modifications of this sort actually occur. Assume that $V_{esc}= 6 V_{rms}$, with $V_{esc}$ and $V_{rms}$ defined as they were in class. Would the rate of loss of molecules from this atmosphere be greater or lesser than that of an atmosphere with a Maxwellian at the same temperature? Give reasons for your answers, using plots or arguments based on the properties of the two functions. Be sure and review the discussion in class of how the two speeds $V_{esc}$ and $V_{rms}$ determine the rate of atmospheric loss.

(4) Given data presented in the book and in lecture on the pressure and temperature at the surface of Venus and at the surface of the Earth, (a) calculate the number of molecules/m$^3$ at the surface of Venus. (b) Do the same calculation for Earth. Look at and think about these numbers.

(5) A proton is moving perpendicular to the Earth's magnetic field at a speed of $v=4.4 \times 10^6$ m/sec. Assume that the strength of the Earth's magnetic field is that which we measure at ground level. What is the ratio of the Lorentz force on the proton to the gravitational force between the proton and the Earth? The electric charge of the proton is $q = 1.6 \times 10^{-19}$ Coulombs.

(6) A charged particle has a velocity

$$\vec{v} = 3.0 \times 10^4 \hat{e}_x + 1.2 \times 10^5 \hat{e}_y \mbox{ m/sec }$$ (3)

and it moves in a magnetic field given by

$$\vec{B} = 2.0 \times 10^{-3} \hat{e}_x + 8.0 \times 10^{-3} \hat{e}_y \mbox{ Tesla }$$ (4)

and the charge of the particle is $q = 1.6 \times 10^{-19}$ Coulombs. What is the force acting on the particle?

(7) A sphere with a radius of 2 meters has a surface temperature of 1000 K. What is the power radiated from the sphere in the form of blackbody radiation?

(8) A spherical asteroid is orbit around the Sun with a semimajor axis of 2.5 a.u. It has an albedo of 5 %. What is its equilibrium temperature? Now assume that the albedo is 75 %. What is the equilibrium temperature now? Hint: The solar constant (defined as the flux of energy in the form of sunlight) is inversely proportional to the square of the distance of the object from the Sun.