Ay121: Radiative Processes (Fall Term 2014/15) Homework Assignment #5, 2014/11/07. Due in class on Thursday 2014/11/13. Homework (20 points): Remember: The Honor Code applies – work in groups, but make your own detailed write up without skipping steps. No looking at solutions of any kind! 1. Thermal Bremsstrahlung from Galaxy Clusters (7 points, inspired by Niel Brandt, PSU) The Coma cluster of galaxies is pervaded by a hot intergalactic medium which emits thermal Bremsstrahlung X-rays at a temperature of 108 K (what is this in keV?). The luminosity between 0.5 and 10 keV is 5 × 1044 erg/s (the emission in this energy band dominates the total Bremsstrahlung luminosity). The radius of the cluster is 2 × 1024 cm. Assume that the emission is optically thin. (a) (3 points) Assuming a uniform distribution of gas, calculate the electron number density, the total mass of the plasma (in M⊙ ), the total thermal energy, and estimate cooling time (assuming that there is no source of heat balancing losses). Feel free to consult Readhead 2.1, but note that this situation is simpler than the isothermal sphere model. (b) (3 points) Compute the mean thermal velocity of the protons in the hot IGM using MaxwellBoltzmann statistics. If the thermal velocity of the protons in the gas is equal to the root mean square velocity of the galaxies in the cluster (a velocity equipartition), estimate the total mass of the cluster (in M⊙ ). Remember the viral theorem and the assumption of constant density that we made above. If you get stuck (and only then), consult: http://en.wikipedia.org/wiki/Faber-Jakson_relation. How does your result compare to the literature value of the Coma cluster mass? 2. Thermal Bremsstrahlung from Orion (inspired by J. Bechthold, UofA) (13 points) The Orion nebula (M42) is a typical HII region with hot O and B stars ionizing the surrounding gas. Assume that the nebula is spherical and has a radius of 12 lightyears. The temperature in the nebula is T = 8000 K, the electron density is ne = 6000 cm −3 , and assume that the gas is pure hydrogen. In all calculations assume a Gaunt factor gff = 1. (a) (1 point) Orion is a strong source of radio waves. What are the frequency and wavelength ranges of radio? Show that this is in the Rayleigh-Jeans (RJ) part of the thermal spectrum and that one can safely use the RJ approximation. (b) (2 points) In the RJ part of the spectrum, it is possible to express the specific intensity in terms of the brightness temperature Tb . Derive the general solution of the transfer equation (for thermal emission) for the brightness temperature. Do this by just re-writing the transfer equation in terms of Tb . The result is Tb (τν ) = Tb (0)e−τν + T (1 − e−τν ) . (1) (c) (2 points) Ignoring any absorption, what is the luminosity of the thermal Bremsstrahlung from Orion in erg/s and in solar luminosities? At what frequency does the intensity drop off? (This feature is called the “knee” of the spectrum). (d) (2 points) Calculate the optical depth to free-free absorption through the center of Orion. Remember that you may work in the RJ approximation. Above what frequency does Orion become optically thin? (e) (2 points) The angular diameter of Orion is ∼35 arcminutes. From this, derive the distance to Orion. What is the flux density you expect at 15 GHz in ergs/s/cm−2 /Hz/ster, and in Janskys? How does this compare to the observed flux density at 15 GHz of 4400 Jy. (f) (2 points) Calculate the radio spectrum you expect and plot log Fν vs. log ν, use units of Jansky. Plot also the brightness temperature as a function of frequency.