Course:ASTR402/Assignments

Problem Set 1 (Due September 18)

Brightness Temperature

From the equation of radiative transfer derive an equation describing how the brightness temperature changes as radiation passes through a thermally emitting gas. You may neglect scattering and assume that the emission is in the Rayleigh-Jeans limit. Solve this equation to give the brightness temperature as a function of optical depth, assuming that the gas has a constant temperature.

Neutrino Blackbody

1. Not only is the universe bathed in a sea of blackbody photons from the early universe, it is also full of thermal neutrinos. Calculate the blackbody spectrum for neutrinos. Unlike photons there can only be zero or one neutrinos in a particular spin and momentum state. What is the ratio of the Stefan-Boltzmann constant for neutrinos to that of photons?
2. Rederive the Einstein coefficients for an atom in equlibrium with a neutrino field.

Toy Atmosphere (Rybicki & Lightman 1.9)

A spherical, opaque object emits as a blackbody at temperature $T_{c}$. Surrounding this central object is a spherical shell of material, thermally emitting at a temperature $T_{s}$. The material absorbs only in a narrow line around a frequency $\nu _{0}$. The width of the line $h\Delta \nu \ll \ kT$.

Sketch answers to the following questions for $T_{c}<T_{s}$ and $T_{c}>T_{s}$.

1. What is the spectrum along a line of sight that passes through the outer shell but doesn't hit the central object (path B)?
2. What is the spectrum along a line of sight that passes through the outer shell and does hit the central object (path A)?

Power-law Atmosphere (ASTR 500 only)

Assume the following

• The Rosseland mean opacity is related to the density and temperature of the gas through a power-law relationship,

$\kappa _{R}=\kappa _{0}\rho ^{\alpha }T^{\beta };\,$

• The pressure of the gas is given by the ideal gas law;
• The gas is in hydrostatic equilibrium so $p=g\Sigma$ where $g$ is the surface gravity; and
• The gas is in radiative equilibrium with the radiation field so the flux is constant with respect to $z$ or $\Sigma$.

Calculate the temperature of the gas as a function of $\Sigma$.

Problem Set 2 (Due September 25)

Propagation through a Metal

Suppose we have a conducting medium, so that the current density ${\vec {J}}$ is related to the electric field ${\vec {E}}$ by Ohm's law:

${\vec {J}}=\sigma {\vec {E}}$

where $\sigma$ is the conductivity (units of s^-1). Investigate the propagation of electromagnetic waves in such a medium and show that:

1. The wave vector ${\vec {k}}$ is complex. What is it?
2. The waves are attentuated as they propagate. What is the corresponding absorption coefficient? N.B. the absorption follows the energy of the wave.

Maxwell's equations before Maxwell

Show that Maxwell's equations before Maxwell, that is, without the "displacement current" term, $c^{-1}{\frac {\partial {\vec {D}}}{\partial t}}$, unacceptably constrained the sources of the field and also did not permit the existence of waves.

Coulomb gauge (ASTR 500 only)

Derive the equations describing the dynamics of the electric and vector potentials in the Coulomb gauge

$\nabla \cdot {\vec {A}}=0$

Look at the equation for the electric potential. What is the solution to the electric potential given the charge density $\rho$? Why is this called the Coulomb gauge?

How does the expression for the scalar potential in the Coulomb gauge differ from that in the Lorentz gauge? What is strange about it? Is it physical?

Now look at the equation for the vector potential. Show that the LHS can be arranged to be the same as in the Lorentz gauge but the RHS is not just the current but the current plus something else.

Show that the RHS can be expressed as

${\frac {4\pi }{c}}\left({\vec {J}}-{\vec {J}}_{\rm {long}}\right)$

where

${\vec {J}}_{\rm {long}}=-{\frac {1}{4\pi }}\nabla \int {\frac {\nabla '\cdot {\vec {J}}}{|{\vec {x}}-{\vec {x}}'|}}d^{3}x$

Problem Set 3 (Due October 2)

Dipoles

Two oscillating dipole moments (radio antennas) ${\vec {d}}_{1}$ and ${\vec {d}}_{2}$ are oriented in the vertical direction and are a horizontal distance $L$ apart. They oscillate in phase at the same frequency $\omega$. Consider radiation at an angle $\theta$ with repect to the vertical and in the vertical plane containing the two dipoles.

1. Show that ${\frac {dP}{d\Omega }}={\frac {\omega ^{4}\sin ^{2}\theta }{8\pi c^{3}}}\left(d_{1}^{2}+2d_{1}d_{2}\cos \delta +d_{2}^{2}\right),$ where $\delta \equiv {\frac {\omega L\sin \theta }{c}}.$
2. Thus show directly that when $L\ll \ \lambda$, the radiation is the same as from a single oscillating dipole of amplitude $d_{1}+d_{2}$.

Cloud

An optically thin cloud surrounding a luminous object is estimate to be 1 pc in radius and to consist of ionized plasma. Assume that electron scattering is the only important extinction mechanism and that the luminous object emits unpolarized radiation.

1. If the cloud is unresolved (angular size smaller than the angular resolution of the detector), what is the net polarization observed?
2. If the cloud is resolved, what is the polarization direction of the observed radiation as a function of position on the sky? Assume only a single scattering occurs.
3. If the central object is clearly seen, what is an upper bound for the electron density of the cloud, assuming that the cloud is homogeneous?

Synchrotron Cooling

A particle of mass $m$, charge $q$, moves in a plane perpendicular to a uniform, static, magnetic field $B$.

1. Calculate the total energy radiated per unit time, expressing it in terms of the constants already defined and the ratio $\gamma =1/{\sqrt {1-\beta ^{2}}}$ of the particle's total energy to its rest energy. You can assume that the particle is ultrarelativistic.
2. If at time $t=0$ the particle has a total energy $E_{0}=\gamma _{0}mc^{2}$, show that it will have energy $E=\gamma mc^{2}<E_{0}$ at a time $t$, where $t\approx {\frac {3m^{3}c^{5}}{2q^{4}B^{2}}}\left({\frac {1}{\gamma }}-{\frac {1}{\gamma _{0}}}\right).$

Classical HI

A particle of mass $m$ and charge $q$ moves in a circle due to a force ${\vec {F}}=-{\hat {r}}{\frac {q^{2}}{r^{2}}}$. You may assume that the particle always moves non-relativistically.

1. What is the acceleration of the particle as a function of $r$?
2. What is the total energy of the particle as a function of $r$? The potential energy is given by $-q^{2}/r$.
3. What is the power radiated as a function of $r$?
4. Using the fact the $P=-dE/dt$ and the answer to (2), find $dr/dt$.
5. Assuming that the particle starts with $r=r_{i}$ at $t=0$, find the value of $t$ where $r=0$.
6. Let's assume that $q=e$, the charge of the electron, and $m=m_{e}$, the mass of the electron. Write your answer in (4) in terms of $r_{i}$, $r_{0}$ (the classical electron radius) and $c$.
7. What is the time if $r_{i}=0.5\mathrm {\AA}$ (for an hydrogen)?
8. Compare this to the lifetime of a hydrogen atom.

Problem Set 4 (Due October 11)

Rapidity

1. Show that the rapidity of two successive Lorentz boosts in the same direction are additive.
2. Derive the addition of velocities formula from the fact that the rapidities are additive.
3. Show that the speed of light is therefore constant.

Here are some possbly useful relations for the hyperbolic functions:

$\sinh \left(z_{1}+z_{2}\right)=\sinh z_{1}\cosh z_{2}+\cosh z_{1}\sinh z_{2}$

$\cosh \left(z_{1}+z_{2}\right)=\cosh z_{1}\cosh z_{2}+\sinh z_{1}\sinh z_{2}$

$\tanh \left(z_{1}+z_{2}\right)={\frac {\tanh z_{1}+\tanh z_{2}}{1+\tanh z_{1}\tanh z_{2}}}$

Superluminal

An object emits a blob of material at speed $v$ at an angle $\theta$ to the line of sight of a distant observer.

1. Find what is the apparent transvese velocity inferred by the observer (i.e. the angluar velocity on the sky times the distance to the object).
2. Show that $v_{\rm {app}}$ can exceed $c$; find the angle for which $v_{\rm {app}}$ is maximum, and find the maximal value.

Cooling Particle

A particle of rest mass $m$ moves with velocity $v$ in frame $K$. In its rest frame $K'$ the particle emits some of its internal energy $W'$ in the form of isotropic radiation.

1. Argue that there is no net reaction force on the particle and it remains at rest in $K'$.
2. What is the total momentum of the emitted radiation as seen in frame $K$?
3. Since this momentum is emitted into the forward direction, does the particle slow down as a result? If so, how can this be reconciled with the fact that the paritcle remains at rest in $K'$? If not, how can this be reconciled with the conservation of momentum?

Hyperspace (ASTR 500 only)

Suppose athat an observer at rest with respect to the fixed distant stars sees an isotropic distribution of stars. That is, in any solid angle $d\Omega$ he sees $dN=N(d\Omega /4\pi )$ stars, where $N$ is the total number of stars he can see.

Suppose now that another observer (whose rest frame is $K'$) is moving at a relativistic velocity $\beta c$ in the $z$ direction. What is the distribution of stars seen by this observer? Specifically, what is the distribution function $P(\theta ',\phi ')$ such that the number of stars seen by this observer in his solid angle $d\Omega '$ is $P(\theta ',\phi ')d\Omega '$? Let $\theta$ be the angle between the $z'$ and the line of sight. Check to see that $\int P(\theta ',\phi ')d\Omega '=N$ and that $P(\theta ',\phi ')=N/4\pi$ as $\beta \rightarrow 0$. In what direction will the stars "bunch up," according to the moving observer?

Problem Set 5 (Due October 23)

Bremsstrahlung

Consider a sphere of ionized hydrogen plamsa that is undergoing spherical gravitational collapse. The sphere is held at uniform temperature, $T_{0}$, uniform density and constant mass $M_{0}$ during the collapse and has decreasing radius $R_{0}$. The sphere cools by emission of bremsstrahlung radiation in its interior. At $t=t_{0}$ the sphere is optically thin.

1. What is the total luminosity of the sphere as a function of $M_{0},R(t)$ and $T_{0}$ while the sphere is optically thin?
2. What is the luminosity of the sphere as a function of time after itbecomes optically thick in terms of $M_{0},R(t)$ and $T_{0}$?
3. Give an implicit relation in terms of $R(t)$ for the time $t_{1}$ when

the sphere becomes optically thick.

1. Draw a curve of the luminosity as a function of time.

Synchrotron Radiation

An ultrarelativistic electron emits synchrotron radiation. Show that its energy decreases with time according to

$\gamma =\gamma _{0}\left(1+A\gamma _{0}t\right)^{-1},A={\frac {2e^{4}B_{\perp }^{2}}{3m^{3}c^{5}}}.$

Here $\gamma _{0}$ is the initial value of $\gamma$ and $B_{\perp }=B\sin \alpha$. Show that the time for the electron to lose half its energy is

$t_{1/2}=\left(A\gamma _{0}\right)^{-1}$

How do you reconcile the decrease of $\gamma$ with the result of constant $\gamma$ for motion in a magnetic field?

Problem Set 6 (Due October 30)

Particles in a Box

A reasonable model for the neutrons and protons in a nucleus is that they are confined to a small spherical region. The potential is $V(r)$ is zero everywhere for $0<r<r_{N}$ and infinite otherwise. What are the energy levels of this system for the following states

1. The states with zero angular momentum,
2. The states with arbitrary angular momentum (ASTR 500)

Hyperfine Transition

Calculate the energy and wavelength of the hyperfine transition of the hyodrgen atom. You may use the following formula for the energy of two magnets near to each other

$E=-{\frac {{\vec {\mu }}_{1}\cdot {\vec {\mu }}_{2}}{r^{3}}}$

We are looking for an order of magnitude estimate of the wavelength.

Density and Ionization

Calculate the ionized fraction of pure hydrogen as a function of the density for a fixed temperature. You may take $U(T)=g_{0}=2$ and $U^{+}(T)=g_{0}^{+}=2$.

Problem Set 7 (November 6)

Lyman Alpha

Calculate the oscillator strength for the Lyman-α transition in hydrogen.

Line Width

Line radition is emitted from an optically thin, thermal source. Assuming that the only broadening mechanisms are Doppler and natural broadening, show that the observed half-width of the line is independent of the temperature up to some critical temperature and then increases a the square root of the temperature. Estimate the critical temperature for the Lyman-α line of hydrogen.

Problem Set 8 (November 15)

The Number of Levels

I fit a Morse function to the potential of H₂⁺. The parameters were

$E_{n,0}=-0.065{\frac {e^{2}}{a_{0}}},B_{n}=0.07{\frac {e^{2}}{a_{0}}},\beta _{n}=0.7a_{0}^{-1},R_{0}=2.5a_{0}$

How many vibrational levels does H₂⁺? How many rotational levels does each vibrational level typically have?

Broadening

Consider an electrically neutral mediu, of diatomic molecules in thermal equilbrium at temperature $T$. Each molecule contains a nucleus of mass $M_{p}$ and a nucleus of mass $2M_{p}$ at an equilibrium separtion $r_{0}$.

1. Estimate $r_{0}$ in terms of fundamental constants.
2. Estimate the cross section $\sigma _{c}$ for collisions between molecules.
3. It is experimentally observed that as a function of mass density $\rho$ of the meidum, the line width at low density is a constant independent of density and above a critical density it increases as $\rho ^{2}$. Find the critical density in terms of fundamental constants.

Problem Set 9 (November 22)

Bondi Solution

Generate a picture like the figure in the lecture notes for the Bondi solution to spherical accretion. Show $v/c_{s}$ versus $r/r_{c}$ for the actual solution to the differential equation or Bernoulli equation. Use $\gamma =7/5$.

Problem Set 10 (December 1)

Show that the entropy of the fluid increases as it passed through a shock. Hint: the equation of state of an isentropic fluid is $P=K\rho ^{\gamma }$ where the value of $K$ increases with increasing entropy.