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Homework Help: Comoving redshift and area of sphere

  1. Nov 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Standard candles may be used to measure the "luminosity distance", using DL =
    (L/4F)1/2, where L is the source's intrinsic luminosity, and F is the observed
    flux. Inthis problem you will relate the luminosity distance to the previously discussed angular
    diameter distance DA. Consider a source at redshift z, which emits light isotropically. In its rest frame, suppose it emits a pulse of total energy Ee over a short time interval Δte. Suppose that today, at z = 0, we have a telescope with collecting area ΔA. For simplicity, assume that the universe is spatially at, with ##\Omega##curv = 0, so that the angular diameter distance is related to the comoving distance by DA = aDco.

    The pulse of light isotropically emitted at redshift z is now, at z = 0, spread out
    over the surface of a sphere. Assuming that this source is at comoving distance
    Dco from us, what is the comoving area of that sphere today? What is the
    proper physical area of that sphere today
    ? Assuming that there is no intervening
    absorption or scattering, what fraction of the photons emitted by the source
    impinge on our telescope with collecting area ΔA? Express your answer in terms
    of Dco.

    2. Relevant equations

    3. The attempt at a solution

    I think that I need to do an integral somewhere with z going from z to 0. However, at first I just thought that the the surface area (SA) would just be ##4\pi(\frac{aD_{co}}{2})^2## but Dco is just suppose to be the distance from me, the observer, to the source of the emitted energy.

    Anyway, I know that SA=##4 \pi r^2##. I just need to find 'r' from z → 0. I know that ##D_{co}=\int\frac{c dt}{a(t)}##. a has a time dependence so I can't just replace it with a-1=1+z. If I do I just get ##\int_z^0c(1+z)dt##. Which would be ##-cz(1+z)##
    I don't believe a negative distance would properly describe this situation.

    Also, the second question I bolded has me confused. Isn't comoving distances suppose to describe how things actually are? How are the two bolded questions different at all?
  2. jcsd
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