This is the text of two email messages from Rick Perley dealing with how many sources we should expect to see in a single pointing with the expanded VLA. The numbers are based on counts from Jim Condon, also included below. This will be made pretty when I have time to make it so.
I don't have any equations for the cumulative counts relevant at
VLA levels, but I do have a simple table of numbers. At 1.4 GHz,
the cumulative source counts (sources per steradian stronger than
S) are given by column 5:
S(Jy) S**(5/2)n(S) Flat Steep Total N/sr
(differential) Spectrum
0.100E+04 78.909 14.031 64.878 0.0
0.631E+03 80.016 13.767 66.249 0.0
0.398E+03 81.507 13.561 67.946 0.0
0.251E+03 83.611 13.478 70.132 0.0
0.158E+03 86.639 13.600 73.039 0.0
0.100E+03 91.031 14.034 76.997 0.1
0.631E+02 97.380 14.916 82.464 0.1
0.398E+02 106.460 16.419 90.041 0.3
0.251E+02 119.201 18.737 100.463 0.6
0.158E+02 136.581 22.060 114.521 1.3
0.100E+02 159.376 26.505 132.871 3.0
0.631E+01 187.737 32.023 155.714 6.8
0.398E+01 220.635 38.284 182.352 16.0
0.251E+01 255.359 44.603 210.756 37.1
0.158E+01 287.398 49.986 237.412 85.1
0.100E+01 311.080 53.343 257.737 189.9
0.631E+00 321.058 53.839 267.219 409.0
0.398E+00 314.167 51.220 262.947 844.5
0.251E+00 290.721 45.920 244.801 1664.2
0.158E+00 254.473 38.888 215.584 3124.6
0.100E+00 211.235 31.242 179.993 5591.3
0.631E-01 166.980 23.937 143.042 9555.4
0.398E-01 126.345 17.593 108.751 15643.8
0.251E-01 92.012 12.477 79.535 24628.0
0.158E-01 64.864 8.588 56.275 37435.3
0.100E-01 44.526 5.774 38.751 55173.2
0.631E-02 29.964 3.821 26.144 79192.1
0.398E-02 19.947 2.517 17.430 111243.6
0.251E-02 13.316 1.682 11.634 153884.4
0.158E-02 9.114 1.177 7.937 211500.2
0.100E-02 6.603 0.896 5.707 292842.7
0.631E-03 5.235 0.764 4.471 416983.3
0.398E-03 4.590 0.718 3.873 626102.4
0.251E-03 4.329 0.705 3.625 1009557.2
0.158E-03 4.172 0.682 3.490 1741466.6
0.100E-03 3.921 0.627 3.294 3125746.0
0.631E-04 3.489 0.537 2.952 5631486.5
0.398E-04 2.908 0.429 2.479 9899998.0
0.251E-04 2.271 0.322 1.950 16719941.0
0.158E-04 1.675 0.228 1.446 26985924.0
0.100E-04 1.176 0.155 1.021 41660408.0
0.631E-05 0.793 0.102 0.692 61749244.0
0.398E-05 0.518 0.065 0.453 88290704.0
0.251E-05 0.329 0.040 0.289 122353744.0
0.158E-05 0.205 0.025 0.180 165041696.0
0.100E-05 0.125 0.015 0.110 217498592.0
0.631E-06 0.075 0.009 0.067 280917184.0
0.398E-06 0.045 0.005 0.040 356548064.0
0.251E-06 0.026 0.003 0.023 445710112.0
0.158E-06 0.015 0.002 0.014 549801344.0
0.100E-06 0.009 0.001 0.008 670309760.0
You can scale to other frequencies (at least over the range 300 MHz to
5 GHz) by assuming that flux is proportional to frequency ** -0.7.
The reference for this is ApJ, 287, 461.
In general, the fact that the average spectral index is -0.7 and the
primary beam area scales as frequency ** -2.0 implies that you get a
lot more sources per unit time by going to low frequencies.
Last week, Barry asked at what frequency we can
detect the maximum number of objects in a given length
of time. This is an attempt to answer the question.
I obtained from Jim Condon a useful integral
source count, made at 20cm. To within a factor of a bit
better than two, the integral count in sources/steradian
between 0.5 microJy and 20 mJy can be written:
N(>S) = 5E5 S^(-.9)
with S being the flux density in mJy at 20cm.
Jim says that it is adequate to convert this to
other frequencies between 300 MHz and 5 GHz simply by
scaling with a -0.7 spectral index. Doing this, and
converting to (sources /sq.arcmin) gives :
N(>S) = 0.11 (Lambda)^(0.63) S^(-0.9)
where Lambda is the wavelength in meters, and S is
the flux density at the wavelength in question. The count
is in sources/sq. arcmin.
To convert to sources/primary beam, we multiply by
the solid angle of the VLA antenna: 1.1(Lambda/D)^2 ster.
Doing this gives:
Nb(>S) = 2300 (Lambda)^(2.63) S^(-0.9)
where again the wavelength is in meters, and the
flux density in mJy at that wavelength, and the count
is in sources/sq. arcmin.
To compare to the confusion limit, we need the
mean source separation, which is the inverse sq. root of
the integral source count per sq. arcmin:
Theta_sep = 3.0 S^(0.45) (Lambda)^(-.32) arcminutes.
We can now put this into VLA terms by inserting
the expected sensitivity after a given length of time. I ran
two estimates, both based on a 12 hour integration. The first
is the count and separation at the 1-sigma level, the second
at the 5-sigma level. Here's the table:
Freq Lambda rms Number in beam Theta_sep Theta_res
(MHz) (m) (mJy) 1-sigma 5-sig 1-sig 5-sig
(thousands) (arcminutes) (arcmin)
------------------------------------------------------------------
75 4 2.4 41 9.5 2.9 5.9 0.4
150 2 .11 106 25 .88 1.8 0.2
240 1.25 .036 82 19 .59 1.2 0.12
370 0.81 .021 43 9.6 .57 1.3 0.081
550 0.55 .012 25 6.0 .64 1.0 0.055
830 0.36 .0086 11 2.6 .49 1.0 0.036
1450 0.21 .0017 12 2.8 .28 .58 0.021
3000 0.10 .00088 3 0.71 .27 .55 0.010
6000 0.05 .00068 0.62 0.14 .30 .61 0.005
-------------------------------------------------------------------
The subject of confusion is confusing to me. I have a
faint recollection that about 50 beam/source is a standard,
corresponding to a mean separation of 7 resolution elements.
The senstivities quoted are based on wide bandwidth
capabilities -- fairly optimistic. The system temperatures
assumed, on the other hand, are fairly pessimistic at 1.2 Ghz
and above. We should be able to do better, leading to better
sensitivities, and higher counts. I doubt the maximum will be
shifted far from its current position -- the expansion of the
solid angle of the primary beam is hard to beat.
I have calculated the expected background areal density
of polarized sources, using the same number counts I sent around
earlier. To give quantitative estimates, I assumed:
1) The average background source is 2% polarized at all
frequencies.
2) We need a 3-sigma detection of the polarized flux to
get a reasonable measure of the p.a. of the electric vector.
3) We integrate for 12 hours, full continuum sensitivity.
Below are the number of objects in the primary beam at
the 150 sigma level (2% at 3-sigma), and the mean separation between
sources, in arcminutes.
Freq. Wavelength 150sigma Nbeam Theta_sep
MHz Meters mJy arcmintues
---------------------------------------------------------------
370(!) 0.81 3.15 468 5.4
550 0.55 1.80 280 4.7
830 0.36 1.29 124 4.6
1450 0.21 0.26 127 2.7
3000 0.10 0.13 34 2.5
6000 0.05 0.10 7 2.8
10000 0.03 0.12 1.5 3.5
----------------------------------------------------------------
The conclusions are pretty obvious -- this type of
experiment is only for L and S bands. Above these, the primary
beam is getting too small. And above X-band, the sensitivity
to non-thermal sources is not competitive with L and S bands.
Below L-band, the loss of sensitivity from the prime focus
(if it is really as bad as some think), and the declining
intrinsic polarization, make the areal density much lower.
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