ASTR 680Problem Set 1Due Thursday, February 12, 20091. Suppose that there is a population of standard candles in the universe. Regardless of theirredshift, they emit a rest frame pure power law with specific intensity Iν∝ ν−α, with a luminosityindependent of redshift, for a rest frame time T independent of redshift. On Earth, we observethese sources with detectors that have perfect sensitivities between received frequencies νminandνmax, and no sensitivity outside this range. Please calculate:(a) The dependence of the observed flux per solid angle on the redshift of the source. Flux here isenergy per time per area in the observed frequency range νmin< ν < νmax.(b) The dependence of the observed fluence per solid angle on the redshift of the source. Fluencehere is the energy per area, integrated over the observed duration of the event, in the observedfrequency range νmin< ν < νmax.2. Dr. I. M. N. Sane, an employee of your least favorite institution, has had a dazzling revelation.Just as the microwave background has given us unprecedented understanding of the universe at afew hundred thousand years, so will the thermal neutrino background tell us about the universe ata few seconds. He has proposed to the Astronomy Decadal Survey panel that a large water tankbe built to observe these neutrinos scattering off the electrons in the water. You have been askedto comment on the viability of this plan.(a) (3 points) We will assume that the neutrinos are relativistic, and distributed as a blackbody.Their background has an expected temperature of T = 1.9 K. Assuming for simplicity that allthe neutrinos are at the thermal peak E = 2.7kT , compute to within a factor of 5 the volume ofwater (at 1 g cm−3) needed so that we would expect 100 scatterings i n one year (so that we havegood statistics and can derive interesting quantities). Assume that all electrons in the water canpotentially scatter the neutrinos, and that any scattering will be detected. Find a b ody of waterwith approximately this volume.(b) (1 point) Give a simple argument that, in reality, your answer is a tremendous underestimateof the volume actually needed to detect electron scattering of these neutrinos.3. There has been a great deal of recent interest in how dark matter in our Galaxy might bedetected. One possibility that has been suggested is that dark matter might decay into electronsand positrons. Indeed, anomalies in the spectra of electrons and positrons have recently beenproposed as signatures of dark matter (but be skeptical of this...).(a) Suppose that the decay of a dark matter particle produces a 1 TeV electron. Ignoring itspossible interactions with other particles, calculate to within a factor of 3 how long it will be beforethe electron’s energy drops to half its initial value because of synchrotron radiation. Assume thatthe interstellar magnetic field is tangled (so that the pitch angle is random at any time) and has auniform strength of 3 microGauss.(b) If the electron undergoes a random walk, where the step size equals the radius of curvatureat any given time, then within a factor of three how large will be the net distance it travels bythe time it gets down to 500 GeV? In a random walk of N steps of length d, the net distance isroughly dnet≈ d√N . Here, though, the stepsize changes as the energy do es, so you may need tofactor that in.4. Compare the total energy ever emitted in gravitational waves with the total e mitted in starlight.To do this, assume that gravitational wave energy is dominated by mergers of supermassive blackholes (SMBH); that a typical current SMBH has 10−4of the total mass of the stars in a galac ticdisk; and that in their lifetimes most SMBHs have one merger with a comparable-mass SMBH,and in doing so emit 5% of their mass-energy in gravitational waves. For the starlight, you willneed to look up galaxy luminosities and multiply by the age of the universe, so please give me theluminosity and tell me the
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