Required Minimum Antenna Gain for Detection of the 100 Strongest Pulsars

The above worked examples can be expanded and summarised into a list.  Note once again these gain figures are the most optimistic cases and calculated from aperture - the real world will require gains above these values to overcome other losses.

Antenna gains are colour-coded in green for gains ≤ 20 dBi and red for gain > 20 dBi @ 400 MHz, with an antenna gain figure of 30 dBi used as the borderline for 1400 MHz - antenna gain figures chosen to be within reach of the capabilities of experienced amateurs.

The maximum un-de-dispersed bandwidths listed in the table have been limited to 20 MHz for 400 MHz and 100 MHz for 1400 MHz as being nominal upper practical limits for amateur radio astronomy environments.

Note that the antenna gain values are conditional on the observing bandwidth used being the maximum un-de-dispersed bandwidth as listed in the table.  For bandwidths less than the maximum listed, the required antenna gains need to be increased by the square root of the ratio.  For example, if the bandwidth is decreased by a factor of 10, the gain needs to be increased by 5 dB.

Note also that the antenna gain values have been calculated using parameters set to nominal values: Observation time = 1 hour, system temperature = 100oK, S/Nmin = 4.  These are selected to be typical amateur level values.  Systems which have different parameters will need appropriate adjustments to the predicted antenna gains.

To find antenna gains for lower system temperatures, decrease the required linear antenna gain by the ratio w.r.t. 100oK, e.g. dropping the system temperature from 100oK down to 50oK, divide the required linear antenna gain by 2.  This is equivalent to a drop of 3 dB in required antenna gain.

Doubling the observation time from 1 hour to 2 hours gives 1.414 times better S/N, so divide the required linear antenna gain by 1.414.  This is equivalent to a drop of 1.5 dB in required antenna gain..

Comparison to Actual Results

It is interesting to note the comparison of the calculated minimum required antenna gain with the published results of the Astropeiler Stockert Group using a 25 metre dish @ 1400 MHz.  When the bandwidth in the calculation is set to the published value (55 MHz) instead of the calculated maximum un-de-dispersed bandwidth, all pulsars reported as being detected by the group are predicted by the radiometer equation to require less than a 15 metre diameter dish (i.e., expected to require not more than about a 20 - 25 metre dish).  The Astropeiler Stockert Group use de-dispersion.

So it seems that the radiometer equation agrees reasonably closely with practical results there.

Comparison to Joe Taylor's Analysis of K5SO's Results for B0329+54

It is even more interesting to take the results of Joe Taylor's (K1JT) analysis of Joe Martin's (K5SO) results for B0329+54 (J0332+5434).

Joe Taylor's analysis returns a S/N of 112 for K5SO's system on B0329+54.

Setting the required S/N in the above "Minimum Antenna Gain" calculations to that result of 112 obtained by Joe Taylor and plugging in the K5SO's system bandwidth (250 kHz) and observation time (6 hours) returns a minimum antenna gain of 28 dB.  This would require a dish size of 7.6 metre @ 436 MHz.  This is pretty close to K5SO's actual dish size of 8.6 metre - just over 1 dB gain difference..

So - the pulsar radiometer equation used here agrees with both K5SO's published results, and Joe Taylor's analysis.