[mmaimcal] distant Milky Way?

[Simon Radford]

So how far out could we see the Milky Way in CO? OK, here goes:

1) Take the array sensitivity goals listed in the slide from the
standard talk for a 25 km/s channel

freq   90    140    230    345    650   GHz
dS    2.1    2.2    2.6    3.5    8.6   mJy sqrt(s)

2) The approximate CO(1-0) luminosity and line width are
(Solomon, Downes, Radford, & Barrett 1997 ApJ 478, 144)

   L'co = 10^8.5 K km/s pc^2 and
delta V = 300 km/s.

In these units, you get the luminosity of any other 
CO line simply by multiplying by the intrinsic brightness 
temperature ratio of the lines. I'll make the (gross) assumption
that all lines of interest have the same brightness temperature.
This probably overestimates the luminosity of the higher
lines.

3) The line luminosity and line flux are related by
(Solomon, Downes, & Radford 1992 ApJ, 398, L29)

  L' = (c^2/2k) (S delta V) [freq(obs)]^-2 D_L^2(1+z)^-3.

We can invert to get the maximum distance of a detecatble
source,

X = D_L^2/(1+z)^3 = [L'/delta V] [freq(obs)]^2 /(c^2/2k) G dS.

Here G is the "significance" of the detection in each
channel, i. e., we don't want 1 sigma detections, but 
more like 5 sigma. So for G=5, we get

freq   90    140    230    345    650   GHz
X     2.5    5.8   13.2   22.0   31.8   x 10^4 Mpc^2 in 1 min
     19.3   44.7    102    171    246        in 1 hour
     54.7    126    298    482    697        in 8 hour

Note the scaling here. D_L^2 ~~ 1/dS ~~ sqrt(int time),
so long integrations are a tough row to hoe [D ~~ t^0.25].

4) The luminosity distance is a messy formula
(e. g., Weinberg 1972, p485, eq 15.3.24),

 D_L = (c/H0)/q^2 [zq + (q-1)(-1+sqrt(2qz+1))].

Here are tables of D_L and X = D_L^2/(1+z)^3

z \ q    0.05    0.25    0.5
0.01       40      40     40  Mpc
0.03      122     121    121
 0.1      419     414    409
 0.3     1370    1320   1280  D_L
   1     5810    5210   4860
   3    26900   20100  16000
  10   170000   90400  61400

z \ q    0.05    0.25    0.5
0.01      1.6     1.6    1.6  x 10^3 Mpc^2
0.03     13.5    13.5   13.4
 0.1      132     129    126
 0.3      851     799    743  X
   1     4260    3390   2740
   3    11300    6310   3990
  10    21800    6130   2830

5) Assume there is always a suitable CO line 
at any given observing frequency. Then
compare the tables for X in (2) and (4).
In 1 hour, at 230 GHz, we should be able to 
detect an (unresolved) Milky Way at 0.3 < z < 1, 
depending on q, at 5 sigma significance in a 10 
channel spectrum. 

Pulling out all the stops, i. e., an 8 hour 
integration and average all 10 channels, 
pushes us up to X = 9120 x 10^3 Mpc^2 at
230 GHz, which is a pretty interesting regime.
This was, no doubt, the basis of the claim
in various science reports that we could
see the Milky Way at z = 1. But it will be 
hard work.

Also, ultraluminous galaxies are 30-50 more
luminous, with intrinsic L' <= 10^10.2 K km/s 
pc^2. So these ought to be easier to detect.


Comments, please.

Simon