Some Worked Examples for Small Antennas at 400 MHz
While pulsar detection is an exciting activity - one can easily become literally 'starry-eyed', and so adherence to basic laws of physics provides a realistic benchmark of the possible pulsar candidates detectable with small dishes.
Two worked examples at 400 MHz for the strongest pulsars are done - J0835-4510 and B0329+54. The B0329+54 results are compared to published results obtained with a small 4 metre diameter dish.
A third work example for B0031-07 is done and the results compared to published results using a small 3 metre diameter dish for that pulsar.
For each example some parameters are set to nominal values: Observation time = 1 hour, system temperature = 100oK, S/Nmin = 4. These are selected to be typical amateur level values.
All examples assume an un-de-dispersed receiving system. For each pulsar, this limits the maximum bandwidth that can be used before loss due the dispersion (smearing) of the pulse becomes significant. This maximum bandwidth is calculated for the condition where the dispersion delay is half the W50 (50% power points) pulse width.
Note that the actual bandwidth that can be used might be further restricted by the maximum bandwidth of the data acquisition system employed. Some representative bandwidths for a number of SDRs can be found here.
All results are theoretical best results - practical results will be poorer due to imperfections in the receiving system.
Optimistically β has been assigned a value of 1.
The required antenna gain is calculated from the required aperture using:
where:
Ae = aperture of antenna in m2
λ = wavelength in metres
G = linear antenna gain
NOTE: This relationship between aperture and gain assumes the geometry of the antenna type employed is applicable at the frequency of use. For example, a parabolic dish size less than 5 λ in diameter sees efficiency, and therefore, gain, fall off significantly. A Yagi antenna or dipole array should be used in place of parabolic dishes less than 5 λ in diameter.
J0835-4510 (5 Jy @ 400 MHz, DM=68)
For this pulsar the maximum un-de-dispersed bandwidth for an observational frequency of 400 MHz is 120 kHz. The required antenna gain varies with bandwidth up to this limit.
Solving the ΔSmin equation above and plotting the required gain (calculated from required aperture) against bandwidth...
This specifies that an antenna with a gain of 15.7 dBi at the maximum un-de-dispersed bandwidth of 120 kHz is sufficient to detect J0835-4510 at 400 MHz - but remember, the equation gives the theoretical best result. In practice an antenna of at least 18 dBi gain would be required at a bandwidth 120 kHz to compensate for the inevitable additional losses. The minimum antenna gain required for smaller bandwidths is shown to the left of the vertical red ('Dispersion Bandwidth Limit') - rising to just over 21 dBi @ 10 kHz bandwidth.
B0329+54 (1.1 Jy @ 400 MHz, DM=27)
For this pulsar the maximum un-de-dispersed bandwidth for an observational frequency of 400 MHz is 1000 kHz. The required antenna gain (calculated from aperture) varies with bandwidth up to this limit.
Solving the ΔSmin equation above and plotting the required antenna gain (calculated from aperture) against bandwidth...
This specifies that an antenna gain of just over 14 dBi at 1000 kHz bandwidth is sufficient to detect B0329+54 at 400 MHz - but again remember, the equation gives the theoretical best result. In practice an antenna with a gain of 17 dBi to 19 dBi would be required. Once again the required antenna gains for smaller bandwidths is shown to the left of the 'Dispersion Bandwidth Limit'.
Examining the result for J0835-4510 shows that if the same bandwidth is used for B0329+54 - say 120 kHz (the maximum allowable un-de-dispersed bandwidth for J0835-4510 at 400 MHz) - a higher antenna gain is needed for B0329+54 compared to J0835-4510. This is to be expected as J0835-4510 is stronger than B0329+54. Given that J0835-4510 is not visible for many Northern Hemisphere observers (while B0329+54 is) this might be seen as a disadvantage for those northern observers, however, the antenna gain increase is a modest one - from about 16 dBi for J0835-4510 to about 19 dBi for B0329+54. But that result is at the bandwidth which is the maximum for J0835-4510. When observing B0329+54 the 'Dispersion Bandwidth Limit' is 1000 kHz - not 120 kHz. From the above graph it can be seen that widening the bandwidth up to the 'Dispersion Bandwidth Limit' of 1000 kHz for B0329+54 brings the required antenna gain back down to around 14 dBi.
The upshot of this is that for an un-de-dispersed system using the full allowable 'Dispersion Bandwidth Limit' bandwidths for both pulsars at 400 MHz, the required antenna gain for B0329+54 (the second strongest pulsar) is actually a little lower (by about 2 dB) than for J0835-4510 (the strongest pulsar).
The above analysis has been verified in practice by Andrea Dell'Imaggine (IW5BHY) who used a 4 metre diameter dish (~ 22 dBi gain @ 422 Mhz) to detect B0329+54 at a S/N which, by visual examination of the graphical results provided, seems to be > 10. (see here). Those results and the S/N obtained are in line with the pulsar radiometer equation.
B0031-07 (0.052 Jy @ 400 MHz, DM=11)
For this very weak pulsar the maximum un-de-dispersed bandwidth for an observational frequency of 400 MHz is 20000 kHz because of the low dispersion measure of about 11. The required antenna gain varies with bandwidth up to this limit.
The results are compared to the claimed detection of B0031-07 with 3 metre diameter dish at 408 MHz (~ 19 dBi gain).
Solving the ΔSmin equation above and plotting the required antenna gain (calculated from required aperture) against bandwidth...
This specifies that an antenna gain of 26.5 dBi is required (equivalent to a dish a little over 7 metres in diameter) is needed to detect B0031-07 at 400 MHz, but this is at the full allowable un-de-dispersed bandwidth of 20 MHz. In the published results the quoted bandwidth for the system is 75 kHz. This bandwidth would require an antenna gain of 38.6 dBi - equivalent to a dish diameter of 30 metres. Again, remember, the equation gives the theoretical best result. In practice a dish of about 35 metres diameter would be required using 75 kHz bandwidth.