[mmaimcal] more high z musings

[Bryan Butler]

all,

sorry to keep bugging you with my inept fumbling around with cosmology 
concepts, but i've now spent enough time on this exercise that i'm 
(_gasp_) trying to really understand what's going on.  in that vein, 
i'm trying to reproduce the derivation of equation (1) from solomon et 
al. 1992 (which is the same as equation (1) in solomon et al. 1997).  
this equation relates the luminosity of CO to the flux density:

                             nu_rest
  Lco = 1.04E-3 Sco dV dL^2  -------
                             (1 + z)

where Lco is the CO line luminosity in Lsun (i.e., in Watts), Sco dV 
is the velocity integrated flux density in Jy km/s, nu_rest is the 
rest frequency of the transition in GHz, and dL is the luminosity 
distance in Mpc.  now, according to the paper, this equation derives 
from conservation of energy via:

  nu_rest L(nu_rest) = 4 pi dL^2 nu_obs S(nu_obs)

where L(nu_rest) is monochromatic luminosity (in this equation, it 
must be in W/Hz for units to work out, i guess) and S(nu_obs) is 
observed flux density.  they reference weinberg's 1972 book for this,
so i thought that i'd go back to weinberg to see where this comes from.

on p. 453, there is the following (eqn. 14.7.10):

              P(nu_obs R(t0)/R(t1)) R(t1)
  S(nu_obs) = ---------------------------
                     R^3(t0) r1^2

where S(nu_obs) is flux density, power per unit antenna area per unit 
frequency interval (W/m^2/Hz or Jy), at a fixed frequency, P is the 
_intrinsic power_, the power emitted per unit solid angle and per unit 
frequency interval (W/Hz/sr).  now, the definition of z is (eqn 14.3.6):

    R(t0)
z = ----- - 1
    R(t1)

so that nu_obs R(t0)/R(t1) = nu_obs (1+z), which is just nu_rest, the 
rest frequency of the transition.  so,

              P(nu_rest) R(t1)
  S(nu_obs) = ----------------
                R^3(t0) r1^2

from the definition of z, R(t1)/R(t0) = 1/(1+z), so

                  P(nu_rest) 
  S(nu_obs) = ------------------
              (1+z) R^2(t0) r1^2

now, the definition of the angular size distance is (eqn. 14.4.17):

  dA = R(t1) r1

so, with some algebra (since we know that R(t0) = (1+z) R(t1)), we get:

               P(nu_rest) 
  S(nu_obs) = ------------
              (1+z)^3 dA^2

if you want it in terms of luminosity distance instead, the luminosity 
distance and the angular size distance are related via (eqn. 14.4.22):

           dL
  dA = ---------
        (1+z)^2

so

              P(nu_rest) 
  S(nu_obs) = ----------
              (1+z) dL^2

now, how is P related to "luminosity"?  P is power per unit solid angle
per unit frequency interval, so:

       L
  P = ----
      4 pi 

where L is the power emitted per unit frequency interval (W/Hz).  so,

                 L(nu_rest)
  S(nu_obs) = ---------------
              4 pi (1+z) dL^2

this is very similar to the starting point of solomon et al., but 
differs in the (1+z) being in the denominator instead of the numerator,
i.e., substituting (1+z) = nu_rest/nu_obs, i get:

  4 pi dL^2 nu_rest S(nu_obs) = nu_obs L(nu_rest)

instead of what solomon et al. show:

  4 pi dL^2 nu_obs S(nu_obs) = nu_rest L(nu_rest)

pressing on with what i have, if you define the total luminosity over 
the line as:

  L' = integral of L(nu_rest) over the line ~ L(nu_rest) del_nu_rest

then,

                         L'
  S(nu_obs) = ---------------------------
              4 pi (1+z) del_nu_rest dL^2

and if you substitute del_nu_rest = nu_rest del_v_rest / c, then,

                           L' c
  S(nu_obs) = ----------------------------------
              4 pi (1+z) nu_rest del_v_rest dL^2

or, inverting:

       4 pi S(nu_obs) del_v_rest nu_rest (1+z) dL^2
  L' = --------------------------------------------
                         c

which, using the same units as solomon et al. for the quantities, 
agrees with their equation (1) except for the (1+z) term, which is of 
course still on the wrong side of the divisor.

i've been through the algebra on this about a half a dozen times now,
and it all seems right to me.  why do i end up with a different answer
than solomon et al.?


	-bryan