Assignment #1 AY 21, Winter 2010Due Monday 1/25/2010, in class.(1) Consider an infinite, static Universe containing 0.01 galaxies Mpc−3, each of which has beenshining with a constant luminosity of 1010L⊙since its birth 10 billi on years ago (previouslybeing completely dark). What is the energy flux at the earth’s surface due to these galaxies?Compare it with that of the sun (Earth–Sun distance = 1.5 × 1013cm). Is it dark at night?Will it always be so? If not, when will night be like day?(2) Assuming that galaxies in a particular cluster have luminosities distributed according to theSchechter luminosity function,Φ(L)dLL∗= Φ∗LL∗αe−L/L∗dLL∗, with α = −1.a) Calculate the ratio of the total cluster luminosity to the luminosity of just those galaxiesbrighter than some luminosity Lmin.b) Suppose that the above luminosity function applies to field galaxies, and that all galaxies up toa given flux limit are observed in a small area of sky (in practice, this i s how it is done). Whatwill be the luminosity distribution of the observed sample? What fraction of the observedgalaxies will be within a factor of 2 of the characteristic L∗luminosity?(3) One of the tools used by astronomers to assess whether a population of objects is distributeduniformly with distance is called the V /Vmaxtest. For each object in a sample, one formsVi/Vmaxby considering the ratio of the spherical volumes enclosed by the actual observeddistance of object i, divided by the volume enclosed by the maximum distance at which thesame object could be placed and still appear in the sample. For uniformly distributed objects,hV/Vmaxi=0.5.Suppose the density of galaxies falls off like the inverse 1.8 power of distance about an observeras the correlation function ξ suggests it will, on average. What mean value of V/Vmaxwillobservers determine for galaxies in a magnitude–limited sample? How many galaxies will berequired in a sample to demonstrate that the galaxy distribution is not uniform using the meanV/Vmaxtest?1(4) One of the ways that the presence of “dark matter” is inferred in spiral galaxies is through thegalaxy rotation curves, where the rotational velocity as a function of distance from the centerof the galaxy is plotted, as in the example below:a) The very surprising result is that the rotational velocity remains essentially constant wellbeyond where most of the starlight is found; if one were to assume that spiral disks werethin disks undergoing Keplerian rotation, and that their physical sizes were identical to theirapparent optical sizes, what would you expect the rotation curve to look like? Draw a plot ofvrversus R, extending beyond the optical size Ropt. What dependence of mass with radius isimplied by a “flat” rotation curve?b) Suppose that the observed rotation speed of a 1010L⊙galaxy is 220 km s−1and its optical“edge” is at Ropt= 10 kpc. What is the mass–to–light ratio i nside this radius, i n solarunits? Suppose that the mass distribution continued out to R = 100 kpc with the same radialdependence; what is M/L for the galaxy now?c) Suppose now that all galaxies have this same value of M/L, that they obey a Schechterluminosity function wi th Φ∗= 2 × 10−2Mpc−3, and α = −1 (assume that L∗= 1010L⊙).What is the total mass density (in g cm−3) due to galaxies? The “critical” density for a closedUniverse, as we will discuss later in the course, i s ρc= 2 × 10−29g cm−3. Comment on thedifference between ρgaland
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