AY121: Radiative Processes in Astrophysics: Homework Assignment 7 Due Thursday 11/27 Homework (20 points) Pulsar Pulse Dispersion (8 points) For a pulsar whose pulses are observed to be delayed by 1.13 sec between 400 and 300 MHz when they arrive at an earth-based radio telescope: (a: 2 points) What is the the value of the dispersion measure DM? (b: 3 points) If this pulsar is known to be 1800 pc from Earth, what is the average electron density < ne > along the line of sight to the pulsar? (c: 3 points) What is the minimum frequency at which radiation can propagate through this density of ISM. The Sunyaev-Zeldovich Effect (12 points) The Sunyaev-Zeldovich, or SZ, effect is the net statistical gain of energy for CMB photons that are passing through a galaxy cluster, due to inverse-Compton scattering off the free electrons in the hot X-ray intracluster gas. The fractional change in intensity of the CMB radiation is given by ∆Iν = −2τc (1) Iν where τc , the Compton scattering optical depth through the region of hot gas, is given by Z kTe τc = σT ne dl. (2) me c2 (a: 2 points) The negative fractional change in intensity given by Equation 1 implies we are looking at the CMB radiation in which region of the blackbody spectrum? Why? (b: 3 points) A typical value for the magnitude of the decrement in the CMB spectrum is ∆Iν /Iν ∼ 10−4 . Assuming the gas temperature and electron density are constant along the line of sight through the cluster, find the size Lc of the cluster as a function of Te and ne . (c: 3 points) The hot intracluster gas is producing X-rays through thermal bremsstrahlung emission. Combining observations of this thermal bremsstrahlung with our expression for Lc we found above, we can solve for the true size of the cluster. 1 (i) Start by writing the expression for the emissivity of optically thin bremsstrahlung emission. (ii) We measure the luminosity of the Coma cluster IGM between 0.5 and 10 keV to be 5 × 1044 erg s−1 . Use this information with your expression from (ii) to find the number density of electrons, ne , as a function of Te and Lc . Note: Assume a spherical geometry for the cluster, assume the IGM is composed only of hydrogen, and set gf f = 1. (d: 2 points) We can determine the temperature of the gas Te from the bremsstrahlung spectrum – specifically, the ”knee,” or turnover, in the spectrum at hν ∼ 10 keV. Now combine the results of part (c) with (b) to find the size of the Coma cluster, Lc . (e: 2 point) The Coma cluster has an angular diameter of ∼ 50 arcminutes. What is the distance to the Coma cluster? The cluster is also known to be at redshift z ∼ 0.023. You now have enough information to determine the Hubble constant (in units of km s−1 Mpc−1 ) using z= H0 r. c (3) In this way, the SZ effect, in combination with X-ray observations of the hot intracluster gas, is used to determine the sizes and distances to clusters of galaxies, and thereby measure the Hubble constant. The value of using the SZ effect to measure the Hubble constant is that the observed decrement in CMB radiation is independent of the redshift of the cluster, and therefore the measurement of H0 does not rely on a hierarchy of distance indicators. 2