[mmaimcal] some more high z CO rantings
Bryan Butler
bbutler at aoc.nrao.edu
Fri Feb 19 15:13:01 EST 1999
for your enjoyment,
here are my calculations for CO detectability at high redshift.
they are more pessimistic than lorne avery's numbers.
take:
nu_obs^2 (1+z)^3
Sco = 3.08e-8 ---------------- L'co
dv dL^2
and calculate it for different lines, at different redshifts, given an
excitation model.
take L'co = 5e8 for the 1-0 line
this is slightly higher than the solomon & rivolo number, but close
to the 1/10th 10214 number (intrinsic - not lensed), and also matches
pretty well with COBE results, i think.
take dv = 300 km/s
take Ho = 75 km/s/Mpc
take q0 = 0.5
now, what to use for the relative line radiation temperatures? harvey
has done a careful calculation, and gets the following:
(a) T_B
z 1-0 2-1 3-2 4-3 5-4 6-5 7-6
0 5.910 4.564 2.423 0.470 0.094 0.199 0.030
1 4.401 3.839 2.408 0.633 0.087 0.100 0.027
2 3.444 3.159 2.254 0.826 0.118 0.050 0.012
3 2.906 2.716 2.099 0.978 0.203 0.048 0.014
4 2.583 2.431 1.966 1.072 0.306 0.063 0.020
5 2.448 2.316 1.921 1.165 0.415 0.096 0.029
(b)
z 1-0 2-1 3-2 4-3 5-4 6-5 7-6 8-7
0 16.753 15.398 10.254 4.677 1.309 0.452 0.076 0.010
1 14.965 14.522 9.980 4.694 1.342 0.443 0.079 0.010
2 13.403 13.314 9.385 4.633 1.406 0.409 0.077 0.011
3 11.929 12.055 8.652 4.479 1.477 0.410 0.076 0.011
4 10.579 10.828 7.889 4.261 1.535 0.439 0.086 0.013
5 9.361 9.684 7.152 4.011 1.570 0.480 0.104 0.016
(c)
z 1-0 2-1 3-2 4-3 5-4 6-5 7-6 8-7
0 26.655 35.218 33.268 30.913 28.593 26.216 23.414 18.780
1 25.018 33.799 32.524 30.579 28.468 26.189 23.431 18.821
2 23.340 31.879 31.049 29.604 27.890 25.904 23.345 18.901
3 21.787 29.907 29.382 28.293 26.936 25.280 23.015 18.889
4 20.321 28.040 27.685 26.860 25.773 24.409 22.431 18.710
5 18.925 26.245 26.032 25.378 24.498 23.373 21.649 18.350
where (a) is a typical cold cloud (like what most of the MW emission
comes from), (b) is a weak PDR, and (c) is a strong PDR. now, assume
that some fraction of the emission comes from the cold material
(call it fcold), and some fraction from the strong PDR's (ignore the
weak PDR's) (this fraction is obviously 1-fcold). then, calculate the
luminosity for a given transition and redshift as:
L'co = Lco(1-0) * fCO
fCO = fCO'(transition,z) / fCO'(1-0,0)
fCO'(transition,z) = fcold * fCO'cold(transition,z)
+ (1-fcold) * fCO'hot(transition,z)
where fCO'cold and fCO'hot are interpolated from harvey's calculated
numbers. then, choosing fcold, you can simply calculate the expected
flux density as a function of transition and z. for fcold = 0.9, which
might be a good number for the milky way, this comes out as:
z=.2 1 2 3 4 5
1-0 1.621 0.201 0.122 0.102 0.095 0.094
2-1 0.175 0.024 0.015 0.013 0.012 0.012
3-2 0.290 0.042 0.029 0.026 0.025 0.025
4-3 0.332 0.050 0.037 0.035 0.035 0.036
5-4 0.431 0.063 0.045 0.043 0.043 0.045
6-5 0.587 0.084 0.059 0.055 0.055 0.055
7-6 0.680 0.100 0.072 0.067 0.067 0.068
8-7 0.704 0.104 0.075 0.072 0.073 0.075
for fcold = 0.0, and L'co = 5e9 (maybe a more typical ULIRG?), you get:
z=.2 1 2 3 4 5
1-0 16.646 2.333 1.568 1.394 1.330 1.297
2-1 2.454 0.350 0.238 0.213 0.204 0.200
3-2 5.235 0.758 0.521 0.470 0.453 0.446
4-3 8.666 1.267 0.884 0.805 0.781 0.773
5-4 12.540 1.843 1.301 1.197 1.171 1.166
6-5 16.565 2.441 1.739 1.617 1.597 1.601
7-6 20.141 2.972 2.133 2.004 1.997 2.018
8-7 21.102 3.118 2.255 2.147 2.175 2.234
now, combine this with the best estimates for the noise (via the
technique that i described in my previous email on this), and you get
the following tables of max detectable z for the antenna size/number
combinations listed (the noise here is calculated for a 50 km/s
channel and 4 hour integration, and a 5-sigma detection is required):
******************** old MMA default ********************
36 10m antennas
trans max detectable z nu
1-0 0.2950 89.013
2-1 0.5800 145.910
3-2 0.6900 204.613
4-3 0.6900 272.805
5-4 0.4400 400.186
6-5 0.1000 628.612
7-6 0.2150 663.911
8-7 0.1300 815.752
******************** MMA/LSA 10m ************************
64 10m antennas
trans max detectable z nu
1-0 0.3900 82.929
2-1 0.7850 129.153
3-2 0.8050 191.577
4-3 1.0700 222.725
5-4 1.1150 272.467
6-5 1.0150 343.163
7-6 0.2550 642.750
8-7 0.1450 805.065
******************** MMA/LSA 12m ************************
48 12m antennas
trans max detectable z nu
1-0 0.4050 82.044
2-1 0.8050 127.722
3-2 0.8050 191.577
4-3 1.1250 216.960
5-4 1.1900 263.136
6-5 1.0500 337.304
7-6 0.2600 640.200
8-7 0.1450 805.065
*********** with japanese ($600M total)? 10m ************
100 10m antennas
trans max detectable z nu
1-0 0.4800 77.886
2-1 0.8650 123.613
3-2 1.4300 142.303
4-3 1.3300 197.872
5-4 1.6050 221.216
6-5 1.8300 244.337
7-6 1.9150 276.724
8-7 0.3850 665.559
*********** with japanese ($600M total)? 12m ************
75 12m antennas
trans max detectable z nu
1-0 0.4900 77.363
2-1 0.8650 123.613
3-2 1.5000 138.318
4-3 1.3550 195.771
5-4 1.6900 214.226
6-5 1.9550 234.001
7-6 2.0850 261.475
8-7 0.3900 663.165
************* pipedream (gives 10000 m^2)? **************
60 15m antennas
trans max detectable z nu
1-0 0.5350 75.095
2-1 1.2500 102.461
3-2 1.6700 129.512
4-3 1.8750 160.362
5-4 1.8850 199.746
6-5 2.2950 209.855
7-6 2.5650 226.270
8-7 2.4500 267.188
and, as a contrast, a possible ULIRG (Lco = 5e9, fcold=0):
48 12m antennas
trans max detectable z nu
1-0 0.7050 67.608
2-1 2.4150 67.507
3-2 4.0450 68.542
4-3 >5.000 <76.84
5-4 >5.000 <96.05
6-5 >5.000 <115.25
7-6 >5.000 <134.44
8-7 >5.000 <153.63
so, can we make a general scaling with how the maximum z (zmax) relates
to the collecting area of the array (Aarray)? the following were
calculated for increasing numbers of antennas:
K = Nant*D^2/(No*D^2) zmax K/zmax
--------------------- ---- ------
1 .575 1.74
1.5 .730 2.05
2 .910 2.20
2.5 1.12 2.23
3 1.43 2.10
4 1.98 2.02
6 2.85 2.11
8 3.53 2.27
it's roughly linear (scaling between collecting area and max z), with
a mean value of about 2.1. this was done with No = 40, and D = 8m,
so the scaling looks something like:
Aarray
zmax ~ --------
4200
if you change the "detection" requirements to: 75 km/s channels,
6 hour integrations, and 4-sigma detections, then this scaling becomes:
Aarray
zmax' ~ --------
2200
remember that this is only for detection of a single line. if we
require that 3 lines must be available for detection, then the revised
requirements are:
Aarray
zmax ~ --------
5000
Aarray
zmax' ~ --------
2700
note that i've only gone up to the 8-7 transition, and the result may
be modified slightly by the ability to detect the higher transitions
at higher z, but i think it won't change by much.
better weather should improve things. remember that all of these
numbers were calculated with "typical" opacities. at night in winter,
it should get better, e.g...
-bryan
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