Course:ASTR406/2023W/Assignments
In-Class Tutorial Assignments
Tutorial on Relativity and Electrodynamics (14 September)
You can work on this together with whatever books that you want. I want to see how much E&M and special relativity you know to gauge what I have to teach you. You will be able to bring your completed tutorial to your subsequent examinations in this class (along with a one-sided 8.5x11 "cheat sheet").
Solutions to the tutorial can be found in the chapters 2 and 3 of Astrophysical Processes.
Unit 1 Tutorial (19 October)
There will be a few multiple choice questions that you should do on your own (without notes or books). You can work on the long answer questions together with whatever books that you want, the same rules as the E&M tutorial. You will be able to bring your completed tutorial to your subsequent examinations in this class (along with a one-sided 8.5x11 "cheat sheet").
Homework Assignments
Homework Assignment One (Due 2 October)
Surface Emission from the Crab Pulsar:
The neutron star that powers the Crab Pulsar can be assumed to have a mass of 1.4M⊙ and a radius of 10 km with constant internal density and an effective temperature of 106 K. The frequency of the Crab Pulsar is 30 Hz and its period increases by 38 ns each day. Compare the power from the surface emission to the power lost as the neutron star spins down. The total power of the Crab Nebulae is about 75,000 times that of the Sun. What is the likely source of this power?
Velocity Field
Show that if a charge is not accelerating, the electric field vector points to the current (not the retarded) position of the charge.
Synchrotron
A particle of mass m, charge q, moves in a plane perpendicular to a uniform, static, magnetic field B.
(a) Calculate the total energy radiated per unit time, expressing it in terms of the constants already defined and the ratio γ = 1/(1 − β2)1/2 of the particle’s total energy to its rest energy. You can assume that the particle is ultrarelativistic.
(b) If at time t = 0 the particle has a total energy E0 = γ0mc2, calculate the time at which it will have energy E = γmc2 < E0.
Collapsing Gas
Consider a sphere of ionized hydrogen plamsa that is undergoing spherical gravitational collapse. The sphere is held at uniform temperature, T0, uniform density and constant mass M0 during the collapse and has decreasing radius R0. The sphere cools by emission of bremsstrahlung radiation in its interior. At t = t0 the sphere is optically thin.
(a) What is the total luminosity of the sphere as a function of M0, R(t) and T0 while the sphere is optically thin?
(b) What is the luminosity of the sphere as a function of time after it becomes optically thick in terms of M0, R(t) and T0?
(c) Give an implicit relation in terms of R(t) for the time t1 when the sphere becomes optically thick.
(d) Draw a curve of the luminosity as a function of radius.
Solutions to homework assignment one can be found in the appendix of Astrophysical Processes.
Homework Assignment Two (Due 4 November)
Flux
Calculate from the Euler equation and the continuity equation, at what velocity does the flux (ρV ) reach its maximum for fluid flowing through a tube of variable cross-sectional area? At which velocities does the flux vanish? You can consider the flow to be adiabatic.
Sound Wave
Show that for a linear sound wave i.e. one in which δρ ≪ ρ that the velocity v of fluid motion is much less than c_s. Estimate the maximum longitudinal fluid velocity in the case of a sound wave in air at STP in the case of a disturbance which sets up pressure fluctuations of order 0.1%.
Bomb
The figure shows shocked air heated to incandescence about two milliseconds after the detonation of a nuclear bomb. The height of the device was 90 meters. What was the approximate yield of the device?
Star
We are going to do some dimensional analysis to understand stars with a polytropic equation of state, polytropes. Consider a star of mass, M, and radius, R. Construct by dimensional analysis a characteristic pressure and a characteristic density from these quantities and Newton's constant, G. A polytropic equation of state is a power-law relationship between pressure and density, P = K ρα. Substitute the characteristic pressure and density into the polytropic equation of state to derive a mass-radius relation. Which values of α have special properties? What are they?
Calculate from dimensional analysis the typical mass of a neutron star. Use the characteristic density and pressure of a star that you just derived. Neutron stars have relativistic neutrons so the pressure is about the density times c2. Use this to derive a relationship between the mass and radius of the star. A relativistic degenerate gas has a density of one particle in a cube a Compton wavelength on a side. Combine this with the result from mass-radius relationship to solve for the mass of the star.
Homework Assignment Three (Due 25 November)
Galactic Supermassive Black Hole
Andrea Ghez's group at UCLA constructed this beautiful movie of the centralmost arcsecond of our Galaxy. The edge of the box measures on arcsecond on the sky.
Use the movie to estimate the mass of the black hole at the center of our Galaxy.
How many Schwarzschild radii does the closest star approach the black hole?
How big would the black hole look on the sky to the hapless inhabitants on a planet orbiting this star? Would it be as big as the moon, Jupiter, Mars?
You have probably assumed something about the orbit of one of the stars. What did you assume? How does the mass of the black hole change if you vary this assumption? What could Andrea's team do to tighten the estimate of the black hole mass?
A Simplified Accretion Disk
This is a simplified model for an accretion disk. It is simpler than the model outlined in the chapter but it will give the right order of magnitude for things. We are also using Newtonian gravity.
Let’s divide the accretion disk into a series of rings each of mass dm. What is the total energy of a ring at a distance r from the central black hole of mass M?
Let’s say that the ring shrinks by a distance dr. What is the change in the energy of the ring (dE/dr)? As the ring shrinks mass is moving toward the black hole.
Divide both sides the answer to (b) by dt to get an equation for the energy loss rate per radial interval.
What is the energy loss rate per unit area?
Let’s assume that this energy is radiated at the radius where it is liberated. Using the blackbody formula what is the temperature of the surface of the disk?
Let’s assume that the disk extends from an outer radius rA to an inner radius r0. What is the total luminosity of the disk if the accretion rate is dm/dt? What and where is the peak temperature of the disk? What and where is the minimum temperature of the disk?
Sketch the spectrum from the accretion disk on a log-log plot. You can use temperature units for the energy axis (i.e. kTmax and kTmin). To do this you will have to think about the peak flux from a blackbody at a particular temperature and the size of the disk that radiates at Tmax and Tmin.
The accretion rate is determined by the evolution of the orbit of the black hole with its companion, so it doesn’t know about the Eddington limit of the black hole. What do you suppose happens if the rate that matter falls onto the disk exceeds the Eddington limit?
What major bit of physics has been left out of this analysis?
The Eddington Luminosity
There is a natural limit to the luminosity a gravitationally bound object can emit. At this limit the inward gravitational force on a piece of material is balanced by the outgoing radiation pressure. Although this limiting luminosity, the Eddington luminosity, can be evaded in various ways, it can provide a useful (if not truly firm) estimate of the minimum mass of a particular source of radiation.
Consider ionized hydrogen gas. Each electron-proton pair has a mass more or less equal to the mass of the proton (mp) and a cross section to radiation equal to the Thompson cross-section (σT). The radiation pressure is given by outgoing radiation flux over the speed of light. Equate the outgoing force due to radiation on the pair with the inward force of gravity on the pair.
Solve for the luminosity as a function of mass.
The mass of the sun is 2 x 1033 g. What is the Eddington luminosity of the sun?
Solutions to Homework Assignment Three
Homework Assignment Four (Due 9 December)
1. Galactic GRBs
We are going to explore the galactic model for GRBs a bit.
Suppose that GRBs lie in a uniform spherical distribution of radius R, and we lie a distance R0 away from the center. Furthermore suppose that we can see every GRB in the volume and we have seen a total of N GRBs. How many GRBs will we have seen in the hemisphere of sky toward the center of the distribution versus the opposite hemisphere?
Using the results from (a), what is the difference in the number of GRBs in each hemisphere to lowest order in R0/R?
Let's compare this result with observations. The one-sigma error in the difference between number of objects in the hemispheres is given by approximately by N1/2. If you have observed N objects without detecting an difference between the two hemispheres at the two-sigma level, what is the minimum acceptable value of R/R0?
At the time of the Paczynski-Lamb debate BATSE had observed about 600 bursts and R0 was assumed to be 10kpc, what was the minimal acceptable value of R at the two-sigma level? Does this distance sound familiar.
2. Holes vs. Stars
I have claimed in class that accretion produces a large fraction of the light in the universe. You are going to see if this holds water.
It turns out that the masses of black holes in the centers of galaxies is well correlated with the mass of the bulge of the galaxy (if it is a spiral galaxy) or the entire galaxy if it is an elliptical: MBH ≈ 0.016 Mbulge.
Let's take a bulge of 108 solar masses. If the black hole was built up by accretion over the age of the universe, what would its average luminosity be? Let's assume that it is a Schwarzschild hole.
The mass-to-light ratio of the bulges of galaxies is given by Mb/Lb = 0.0776 (Lb/Lsun)0.18 Msun/Lsun. What is the luminosity of the stars in bulge?
3. Thinking about Instruments
You will probably have to surf the net a bit or use things you have learned from other courses to work these out, but the equations will be rather simple once you have them.
The black hole in the center of our Galaxy has a mass of 106 solar masses. Let us assume that it is a maximally rotating (a=M) Kerr black hole. How big is its horizon? How big is its ergosphere?
What angle does the horizon of the central black hole subtend in the sky?
I would like to build a telescope that can resolve the central black hole. What is the angular resolution of a telescope as a function of the wavelength of the light and the diameter of telescope. You can look up the formula, use dimensional analysis or the Heisenberg uncertainty principle.
What is the diameter of the telescope if you use 2 GHz radio waves?
What is the diameter of the telescope if you use 1 keV X-ray photons? Scaling from Chandra, what is the focal length of the telescope?
4. Accretion-Disk Efficiency
Let's assume that an accretion disk extends from infinity down to some rA.
Using Newtonian gravity, how much energy is released per unit mass as material spirals to the inner edge of the disk?
Now using general relativity, redo the calculation. Assume that the central object is a non-rotating black hole. The energy released per unit mass is given by 1-ut where uα is the four-velocity of material in the disk and you are using the Schwarzschild metric.
In general relativity, an accretion disk can only extend down to R=6M around a non-rotating black hole. What is the efficiency of accretion onto such a black hole?