[an error occurred while processing this directive]

Background Sources with the VLA Expansion Project
(Rick Perley 24nov98)


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.


Message #1: Source counts
(Jim Condon, 24nov98)

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.

Message #2: Number of sources per primary beam per unit time
(Rick Perley, 24nov98)

  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.  


Message #3: Using background sources for polarization studies
(Rick Perley, 24nov98)

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.  
   

VLA Expansion Project Science
VLA Expansion Project Home Page


Last modified 08 August 2001

mrupen@nrao.edu