Challenges for Amateur Pulsar Detection
Before attempting to detect pulsars it is wise to get an idea of the magnitude of the challenges faced by an amateur individual or group.
The information here does not deal with the more fundamental aspects of pulsars - this is left for education by other means. Google is your friend. Learn as much as you can about pulsars and make yourself familiar with their characteristics. Some good places to start are...
There is a lot to learn and some details can be challenging - but that's what makes the subject so interesting.
I recommend the "Handbook of Pulsar Astronomy" by D. Lorimer & M. Kramer as a mandatory companion for learning about pulsars and their detection. Without an understanding of the basic principles, attempts at detection will be difficult - or worse still, results may be gained that are thought to be valid, but in fact are not. More on that later.
A synopsis of the "Handbook of Pulsar Astronomy" can be read here.
The 'EME' Factor
The signals from pulsars are very weak and difficult to detect. Everything has to be spot-on, especially for observatories with small antennas. A strong understanding and experience of low-noise receiving systems is mandatory as well as the physics involved.
It should be noted that all the successful individuals/groups listed on this website's home page who are using dishes less than 10 m in diameter, or their equivalents, have prior experience and success at Earth-Moon-Earth (EME) activities. In addition - they are all amateur radio operators.
While these two criteria do not exclude the possibility of success by those not possessing them, it nonetheless gives a strong indication that detecting pulsars is not a 'plug-and-play' activity - and is actually more difficult than EME.
Strength of the Pulsar Signal
Not surprisingly the first pulsar parameter that should be considered is the strength of the pulsar radio signal. The signal strength is expressed as a flux density and has units of a jansky - named after Karl Guthe Jansky, an American radio engineer who first discovered radio waves emanating from the Milky Way.
(symbol Jy) is a non-SI unit of spectral flux
density equivalent to 10−26 watts per
square metre per hertz. As a yardstick, 10−26 watts per
square metre (1 Jy) would be about the same energy
received on Earth from a 100W light bulb (radiating
in all directions) at a distance of
more than 7 times that of the trip from Earth to
Neptune. Only the Vela pulsar exceeds 1 Jy by
any significant amount (5 Jy @ 400 MHz).
The challenges of detecting pulsars are clearly shown by the tables here which list the relative flux densities (in descending order) of a number of pulsars. It can be seen that after the first two pulsars (B0833-45, B0329+54) the signal strength drops away rapidly. As a result the usual candidate for first attempts is B0833-45 (Vela Pulsar) for Southern Hemisphere observers, and B0329+54 for Northern Hemisphere observers. Ironically, the first pulsar discovered (by Jocelyn Bell in 1967), B1919+21, would be too weak at 400 MHz and 1400 MHz for the usual amateur setup as shown by this extract from the ATNF Pulsar Catalogue.
-------------------------------------------------------------------------------------------------------------------------- # NAME PSRJ RAJ DECJ P0 DM W50 W10 S400 S1400 (hms) (dms) (s) (cm^-3 pc) (ms) (ms) (mJy) (mJy) -------------------------------------------------------------------------------------------------------------------------- 1 B1919+21 J1921+2153 19:21:44.8 +21:53:02.2 1.337302 12.44 30.900 40.800 57.00 6.00 --------------------------------------------------------------------------------------------------------------------------
At 400 MHz the 57 mJy signal strength of B1919+21 places it at #72 position in a list ordered by pulsar signal strength. At 1400 MHz the signal strength of 6 mJy places it at #89. The B1919+21 is nearly 88 times weaker at 400 MHz, and 183 times weaker at 1400 MHz than the strongest pulsar (B0833-45 - Vela). So, ironically, it is unlikely that the 'original' pulsar would be detected by amateurs unless the dish size is larger than about 15 metre - 20 metre in diameter; but advances are made all the time and so this requirement might be reduced.
It may seem odd that B1919+21 was the first pulsar discovered when it is well down in the list of signal strength. A number of factors might explain this. Firstly, nobody was looking for pulsars at that time (even though theoretically predicted years before) and the equipment had slow response times specifically to average out terrestrial noise for the usual observations. The facility at Cambridge which Jocelyn Bell operated was unusual for the time as it was looking for interplanetary scintillation (effect on radio waves from cosmic sources caused by the solar wind) which required a fast response time - a characteristic necessary to see a pulse. Also working in favour of the detection was the low frequency of observation (where signals are stronger) of 81.5 MHz and a very large antenna (about 4 acres of dipoles !). Serendipitously the dispersion measure for B1919+21 is low at 12.44, meaning that a wide bandwidth could be used without smearing the pulse.
As an introduction, in the above ATNF Pulsar Catalogue extract for B1919+21, we have, in left to right order...
Name - original designation made from RA and DEC - which sometimes are inaccurate (e.g. Vela - B0833-45)
PSRJ - more modern designation also made from RA and DEC - usually more accurate (e.g. Vela - J0835-4510)
RAJ - Right Ascension
DECJ - Declination
P0 - period of pulse (seconds)
DM - dispersion measure
W50 - width of the pulse measured between 50% power points of the pulse (ms)
W10 - width of the pulse measured between 10% power points of the pulse
S400 - flux density at 400 MHz in mJy
S1400 - flux density at 1400 MHz in mJy
There are over 2500 pulsars listed in the ATNF Pulsar Catalogue, but only a fraction of that number are in reach of the average amateur. However, armed with good knowledge of their target and a 25 metre dish the Astropeiler Group have managed to detect over 90 pulsars @ 1296 MHz - at least two of them binary pulsars.
Even so, the activities of the amateur radio astronomer is realistically restricted to detecting known pulsars - but of course, one can dream...
Nature of the Signal
The signal is essentially a narrow broadband noise pulse (usually about 5% duty-cycle) repeated at pulse periods from 11.8 seconds (0.085 Hz) down to 1.4 ms (716 Hz). As the flux density figure is averaged over the pulse period, those pulsars with smaller duty cycles (smaller W50 w.r.t. to P0, i.e., a narrower pulse w.r.t. to P0) will generally be easier to detect as their peak signals will poke their noses above the noise higher. Broadband signal energy from pulsars can extend from 10s of MHz up to microwave frequencies. But because the radio waves are generated by the synchrotron mechanism, the signal is stronger at lower frequencies.
Note that the signal strength is quoted in Janskys, where 1 Jy = 10−26 watts per square metre per hertz. To collect more energy from the pulsar we can receive energy over more square metres (bigger antenna) or more hertz (wider bandwidth). Unfortunately, there is a limit to the bandwidth unless complicated processing is done. This is due to dispersion, where the effects of the traversal of space cause lower frequencies of a pulse to arrive later than higher frequencies. Inconveniently the degree of delay over a given bandwidth increases rapidly the lower the observation receiving frequency - where the signal is much stronger. If the bandwidth chosen is too wide for the observation receiving frequency such that the effects of dispersion cause too much delay, the pulse is 'smeared' and is harder to detect. You can imagine if the delay over the chosen bandwidth is equal to the period of the pulse then the pulse completely disappears.
Without a doubt the most useful parameter for pulsar detection is aperture - the collecting area of your antenna. The bigger the antenna the easier it will be to detect a pulsar - an obvious statement for sure, but what is not so obvious is how quickly it becomes increasingly difficult as the size of the antenna is reduced. Reducing the diameter of a dish antenna by a factor of 2 (say, 6 metres to 3 metres) would need an increase in observation time by a factor of 16 to maintain sensitivity, or a factor of 16 increase in bandwidth. Fortunately these compensations can be combined as a product, e.g., 4 times the bandwidth combined with 4 times the observation time provides the factor of 16 required.
Each pulsar has its own period, pulse width, flux density and dispersion measure. Evaluating the ease, or otherwise, of detection of a particular pulsar requires taking all those into account.
Effects of Scintillation
In practice, the flux density quoted is an estimate for a number of pulsars, as another effect, called scintillation, can greatly enhance signal strength, but then at other times reduce signal strength to very low levels during the time of an observation. This can be the difference between successful detection and failure. As an example, the following figure shows the variation in signal strength over an observation period of 85 minutes of the B0329+54 pulsar...
Scintillation Effects on the B0329+54 Pulsar @ 1.38 GHz to 1.70 GHz
The frequency range of 1.38 GHz to 1.70 GHz is covered by 128 channels each of bandwidth 2.5 MHz - an individual bandwidth close to that of an RTL_SDR dongle (nominal maximum reliable bandwidth of 2 MHz). If the above observation was made by such a dongle set to 1.42 GHz, the above figure shows that the pulsar signal would have been almost entirely absent for the whole 85 minutes at that frequency. If, on the other hand, the dongle was set to receive at 1.495 GHz, then the signal would have been strong for nearly half the observation period.
Additionally, if, by good fortune, an observation was made for, say, only the first 10 minutes @ 1.4 GHz (during the time of the dark blob near the bottom left-hand corner of the figure) then the signal would be strong for the whole 10 minutes. Conversely, if an observation was made at 1.45 GHz for the same first 10 minute period in time, virtually nothing would have been seen at all. That is, depending on the current state of scintillation at the observation time and frequency, a signal strength may be observed well above the quoted mean flux density for B0329+54, or possibly no signal is seen at all. So scintillation can be your friend, but at other times your enemy.
Here below is that same signal strength time/frequency map when 'averaged' over the time and frequency channels...
Signal Strength Averaged Over Time and Frequency Channels
...which corresponds, presumably, to the level of the quoted flux density of 203 mJy of B0329+54 @ 1400 MHz. It can be seen in the previous un-averaged graphic ("Scintillation Effects on the B0329+54 Pulsar @ 1.38 GHz to 1.70 GHz") that at times the flux density received is much higher than the average (and also at other times, much lower than the average). By fortuitously picking an observation time during a 'scintillation boost', the received flux density can be significantly higher than the quoted average flux density of 203 mJy. The B0329+54 pulsar is particularly co-operative in this regard as the 'scintillation boosts' can last for 20 to 30 minutes (see the big black blobs in the un-averaged graphic above).
From this graph of long-term scintillation observations (Figure 7: Long-Term Scintillation Observations of 5 Pulsars at 1540 MHz) it can be seen there is a large variation in flux density around the nominal quoted value of 203 mJy...
B0329+54 Long-Term Scintillation at 1540 MHz
...regularly falling to low values around 100 mJy (sometimes for over a month) and at other to rising to 300 mJy. Note that there was a period (near MJD 52100) where S1540 rose to just over 500 mJy. It should remembered that shorter variations (over a few ten's of minutes) will ride on top of this longer term rise - meaning at the time of observation the flux density can be even higher.
The bottom-line to this phenomena is that it may take a number of tries before a good 'scintillation boost' is captured to allow successful detection of B0329+54 @ 21cm wavelengths.
Note that scintillation occurs at faster rates at lower frequencies as this figure of an observation for B0329+54 @ 408 MHz shows...
Scintillation Effects on the B0329+54 Pulsar @ 408 MHz
At this observation frequency and using a bandwidth of, say, 2 MHz (i.e., 407 MHz to 409 MHz), the peaks and valleys in signal strength over that bandwidth and, say, an observation time of 90 minutes, would average out to be close to the quoted mean flux density - i.e. - no scintillation boost to be had. It would be expected, excluding other effects, that the results would be fairly similar from observation to observation.
This is likely the reason behind the reports from amateurs that more consistent results for B0329+54, from day to day, are obtained at 400 MHz when compared with 1400 MHz.
On the other hand, the use of smaller bandwidths, say, 100 kHz, at 400 MHz would see at least a partial return to the situation where the result would vary depending on frequency chosen and time chosen - as was the case when observing at 1.5 GHz. In this case it would be expected that results from observation to observation could vary quite significantly.
For other pulsars, say B0835-4510 (Vela), the scintillation peaks and troughs coming and go much quicker (in a matter of a few seconds), so over that same 20 to 30 minutes they average out and, therefore, there is no 'scintillation boost' to be had.
Effects of Dispersion
On its way from the pulsar to receivers on Earth, the pulsar signal passes through the Inter Stellar Medium (ISM). One of the effects (in addition to scintillation effects) is dispersion. Essentially this effect causes signals at lower frequencies to arrive later in time w.r.t. to higher frequencies. For continuum signals this effect is hidden because there is no temporal information markers to detect the delay. In contrast, pulsar radio band signals start their journey aligned in time (those moments in time when the radiation beam is pointing in our direction) independent of frequency. By the time the reach our receivers the passage through the ISM has introduced a significant delay between lower and higher frequencies. The journey of the pulsar signal involves different paths and path lengths for each pulsar - imparting varying degrees of dispersion on different pulsars. This unique pulsar signal characteristic is given as the 'dispersion measure' or DM.
A general equation which gives the relative delay between two frequencies (assuming the difference between the frequencies is small compared to the absolute frequencies) is as follows...
...where the delay is in milliseconds and frequencies are in MHz.
The effect of this delay is to smear the pulse shape when received over a wide bandwidth. This means that the pulse signals over the bandwidth don't line up in time and so do not add together to improve the S/N in the folded profile.
For the same DM value, observation frequency and bandwidth, a pulsar with a narrow pulse width (PW50) is going to be affected more compared to one with a wider pulse.
Vela Dispersion @ 430 MHz
To counteract the effects of dispersion a process of de-dispersion can be applied to the data. Coherent de-dispersion applies a filter whose response is the inverse of the ISM before detection; incoherent de-dispersed applies a time correction across data split into separate frequency channels (to line up the pulses in each channel in time) after detection. Incoherent de-dispersion is the simpler of the two techniques to both understand and implement. However, is not as accurate as the coherent technique (which implements continuous 'perfect' de-dispersion) because the time delay corrections for incoherent de-dispersion are implemented on individual channels in a step-wise fashion and there can still be some residual dispersion across the channel bandwidths.
Comparing two of the strongest pulsars: Vela (PW50=2.1 ms; DM=68) and B0329+54 (PW50=6.6 ms; DM=27) at 400 MHz, the bandwidth at which the delay is equal to PW50 (at which point smearing starts to cause loss in sensitivity) is 240 kHz for Vela and 1900 kHz for B0329+54. Using bandwidths above those respective values will require de-dispersion of the data.
NOTE: dispersion results in a delay differential between signal frequencies - it does not affect the period (as can be seen in the graphic above).
A ground-based observatory is not operating in a non-inertial frame of reference - that is, from an outside point of view it is moving at a variable velocity. Depending on the longitude, latitude and time of day, the relative velocities of ground-based observatories in the direction of an observed object are different for different locations. These different velocities cause a doppler shift on the observed pulse period of the pulsar which is unique to each observatory at any point in time. Pulse periods thus measured by an ground-based observatory are termed 'topocentric'.
In order to 'normalise' the time of arrival of pulses (which also means the stated period of the pulse) to facilitate exchange of data between observatories, topocentric observations are corrected to an inertial frame of reference. This frame of reference is taken to be the Solar System Barycentre - the centre-of-mass point around which the Sun and its planets revolve. This inertial frame of reference actually moves slowly in time as the planets orbit (and so is not a 100% true-blue inertial frame of reference) with the bulk of the 'wobble' coming from the effects of Jupiter (1/1000th the mass of the Sun and 2.5 times the combined mass of all the other planets).
For amateur pursuits this 'wobble' can be ignored as it takes many years and only causes ~0.05 ppm doppler shift.
It is important to note that the period quoted in pulsar data (e.g. the ATNF Pulsar Catalogue) are 'barycentric'.
In the pursuit of detecting known pulsars, the observational data received will be 'topocentric'. In order to convert predictions expressed in barycentric form into the topocentric equivalent for the purpose of epoch folding at the correct period to reveal a pulse, other orbital effects must be taken into account. The annual orbit around the Sun of the Earth produces significant doppler shift on the pulse period of a pulsar, depending on the orientation the observed pulsar w.r.t to the axis of the Earth's orbit around the Sun. The magnitude of the diurnal doppler shift is dependent on the product of the cosine of the declination of the pulsar and the cosine of the latitude of the observatory. For Vela the annual doppler shift caused by the Earth's orbit around the Sun has a range of ~ ±48 ppm. The diurnal doppler shift for Vela for NRARAO (latitude -34) is ~ ±1 ppm. There is a small monthly doppler caused by the Earth/Moon barycentre, but this only amounts to ~ ±0.02 ppm and can be ignored for amateur purposes.
Failure to take these doppler effects into account can prevent finding the correct folding period - hence a failure to detect a pulsar signal. Most amateurs (and professionals) use some version of TEMPO to calculate the current topocentric period of the observed pulsar for their individual observatories.
The degree of linear polarisation of signals varies from weak to strong between different pulsars. The Vela (B0835-4510) signal is almost 100% linearly-polarised. In contrast, B0329+54 is moderately linearly-polarised. During the peak of the pulse the degree of linear polarisation for B0329+54 is only around 25%. The post-pulse shows a higher degree of linear polarisation at around 75%.
What this means for the strongly linearly-polarised Vela signal is that if, at any instant, the orientation of a linearly-polarised antenna is orthogonal to the incoming signal then a loss of many dBs occurs. For B0329+54 the cross-polarisation loss is about 1 dB due to the low degree of linear polarisation.
Note that the relative angle between an observatory linearly-polarised antenna arranged for 'horizontal polarisation' and a linearly-polarised signal fixed in orientation from interstellar space varies with time and location on Earth due to geometrical effects as well as (primarily at lower frequencies) Faraday rotation. Therefore, the term 'horizontal polarisation' w.r.t. to extraterrestrial objects is largely meaningless (see 'spatial polarisation' in EME terms).
However - the orientation, or the position angle (PA), of the radiation from a pulsar is not fixed in orientation, but swings through a typical PA S-curve during the pulse on-time. In the case of Vela, the PA swings through more than 90 degrees over a few milliseconds. The worse-case scenario for Vela is where, by an unfortunate combination of the aforementioned factors existing at the time of the observation, the incoming signal swings to be cross-polarised with the linearly-polarised antenna right at the time of the peak of the pulse. Much attenuation of the signal will occur. The best case is the reverse situation where the incoming signal polarisation lines up with the antenna polarisation at the peak of the pulse. Therefore, a receiving system which has only one linear polarisation might see a large variation in received signal from Vela at different observation times.
For B0329+54, because the degree of polarisation is much less, the above polarisation effects are much milder and will most likely be hidden by the much larger variations in signal strength due to scintillation for that pulsar. Using linear polarisation is not critically important for this pulsar - however, the local orientation (horizontal, vertical or somewhere in-between) might be advantageously adjusted to be the least sensitive to local RFI.
To counteract the effects of cross-polarisation on the strongly linearly-polarised Vela signal, a two channel system with orthogonal polarisations can be used, as when cross-polarisation occurs in one channel, in-line polarisation occurs in the other.
Circular polarisation (CP) might be another solution for strongly-polarised pulsars (such as Vela). The equal response to all polarisations will eliminate cross-polarisation attenuation variations. A linearly polarised antenna might, at times, produce a signal strength 2 dB or so better than a CP antenna during the best inline-polarisation conditions, but for a significant percentage of the observations, will produce a signal strength similar to, and in some cases, much worse than the CP antenna. It is probably a philosophical argument as to which case is the better.
In general, as the relative degree of linear polarisation varies across pulsars, trying to optimise the receiving system in terms of polarisation options is probably not an overly useful exercise compared to other system factors.
Getting to know your target pulsar and understanding what each characteristic means is an important aspect of successful amateur pulsar detection efforts.
Method of Detection
As mentioned previously, the signal from a pulsar appears in the receiver as short pulse of broadband noise. The pulsar signal is extracted by measuring the broadband noise power from the receiver. This can be done by a number of ways, e.g., a simple analogue circuit RMS power detector (which responds fast enough to catch the narrow pulsar pulse signal), or capturing RF data and performing power detection in software (e.g., RTLSDR IQ data converted to power: P=I2+Q2).
Although the signals from pulsars are weak, the time between the pulses is stable to an extraordinary degree. Some pulsars have pulse period stabilities better than the accuracy of the best terrestrial clocks. For radio astronomers who have antennas large enough to detect single pulses, it is possible, after receiving a pulse, to predict the arrival time of subsequent pulses many hours - or even days - later.
For those observers whose antennas are of a more modest size - say a dish less than 20 metre in diameter - it is not generally possible to receive single pulses of sufficient S/N to be useful. For those scenarios use can be made of the extreme regularity of the pulses to dig them out of the noise. This introduces a critical requirement for receiving pulsars - data must be acquired with high temporal accuracy. Many amateurs use atomic clock standard disciplined data acquisition.
The usual method for amateurs of detecting known pulsars is by synchronous averaging ("Handbook of Pulsar Astronomy" - Lorimer & Kramer - page 165) of many pulses to reduce noise. This method goes by a few names such as "folding" or "coherent integration". Note: it is the author's opinion that the term 'coherent' should not be used in this context because 1. it is not used in prominent references by those who would know and 2. it will confuse the distinction in descriptions of processes where it is a correct usage of the term (see "Handbook of Pulsar Astronomy" - Lorimer & Kramer - Section 5.2 'Incoherent De-dispersion'; Section 5.3 'Coherent De-dispersion').
Any process of detecting pulsars is governed by the general radiometer equation ("Radio Astronomy" - Kraus)...
ΔSmin = minimum detectable flux density (watts per square metre per hertz)
k = Boltzmann's constant (1.38064852 × 10-23 Joules • K-1)
Ks = factor for a particular observatory system, hopefully near to 1
Tsys = System noise temperature (Ko)
Ae = Effective aperture of the antenna (m2)
Δf = pre-detection bandwidth (Hz)
t = post-detection integration time (seconds)
n = number of records averaged
The above equation is a general relationship for continuum measurements and so is modified for the pulse-like signal from a pulsar (Eqns. A1-14 & A1-21 - "Handbook of Pulsar Astronomy" by D. Lorimer & M. Kramer)
= minimum detectable flux density (watts per square metre per hertz
- averaged over the
period of the pulse)
β = factor for imperfections in the observatory system, usually near to 1 (values >1 makes the system less sensitive)
kb = Boltzmann's constant (1.38064852 × 10-23 Joules • K-1)
S/Nmin = required minimum linear S/N for validation of result (professional require at least 6, amateurs ≥ 4)
Tsys = System noise temperature (Ko)
Ae= antenna aperture (m2)
np = number of polarisations (usually 1 for amateurs, generally a maximum of 2)
tint = integration time (observation time in seconds)
Δf = pre-detection bandwidth (Hz)
W = width of pulse (seconds)
P = period of pulse (seconds)
Effect of Parameters
The smaller the value of Smin the weaker the pulsar that can be detected. Therefore the larger the aperture of the antenna or the narrower the pulse w.r.t. the pulse period, the weaker the pulsar that can be detected. Note that also the longer integration time and the wider the pre-detection bandwidth the weaker the pulsar that can be detected - but by a factor of a square root - i.e., an increase of 4 in either of these only decreases the weakest signal detectable by a factor of 2. The most effective changes to the receiving system are aperture (bigger is better) and Tsys (lower is better) - which both affect the weakest pulsar signal detectable directly.
The linear factor S/Nmin is the lowest S/N that still provides a solid validation of the result. Professional astronomers require an S/Nmin of around 8 and amateur astronomers should aim for this, but certainly a minimum of 4.
The important thing to remember for amateurs is that the above equation is derived from the laws of physics and represent the ideal theoretical case. Practical implementations are always poorer than this. No magic process will produce a result better than the laws of physics and so if a result is obtained which is better than the above equation (and you still want your results to have scientific validity) then it should be regarded as invalid - to do otherwise is folly.
All the successful amateur groups/individuals listed on the 'Home' page are in compliance, according to the data they have supplied, with the pulsar radiometer equation.
Which Observation Frequency ?
Note that while the pulsar radiometer equation has a term for the observation bandwidth, there is no term for observation frequency. Nonetheless, the observation frequency is very important because the signal strength of pulsars is inversely proportional to frequency - that is, pulsars are generally weaker at higher frequencies.
However, different pulsars 'roll-off' in signal strength at the higher frequencies at different rates - specified by what is called the "Spectral Index" (Spi)...
F1, F2 = observation frequencies
SF1, SF2 = flux densities at F1 and F2
Large negative values indicate a fast roll-off in signal strength at higher frequencies. Smaller negative values indicate a flatter roll-off, where signal strengths are largely maintained at higher frequencies.
For example - the pulsar B0531+21 (J0534+2200) has a flux density of 550 mJy @ 400 MHz, but 'drops like a rock' @ 1400 MHz with only 14 mJy (nearly 16 dB drop) - giving an Spi = -2.9.
In contrast, B1641-45 (J1644-4559) has a flux density of 375 mJY @ 400 MHz and largely maintains that at 1400 MHz @ 310 (less than a 1 dB drop) - giving an Spi = -0.15.
Why go to higher frequencies if the signal strengths are lower ? The usual reasons are...
The observatory might have evolved from an EME station and already has 1296 MHz capability
There may be too much RFI @ 400 MHz at the observatory's location
Bottom line - choosing the operating frequency is a result of finding the best compromise considering the spectral index of the target pulsar and the local environment. Generally speaking, the more 'urbanised' the location of the observatory, the more likely that higher frequencies will yield a better chance of success.
Note that for many pulsars there a 'turn-over' point in the usual increase of flux density with decreasing observation frequency.
That is, for those pulsars, there is a frequency where the flux density is a maximum and falls away at both higher and lower frequencies. For the first pulsar detected (PSR 1919+21) this peak is around 70 MHz - making it a good candidate for the Cambridge antenna array operating at 81.5 MHz. For B0329+54 the peak is about 390 MHz - making 408 MHz the prime spot to detect this pulsar if signal strength was the only determining factor, i.e., no local RFI.
The Vela Pulsar (B0833-45) is the strongest pulsar given as 5000 mJy @ 400 MHz and 1100 mJy @ 1400 MHz - but it has a 'turn-over' @ 600 MHz (according to published literature), so 400 MHz is not actually the strongest flux density frequency. Using the quoted Spi of -2.4, it is estimated that the peak @ 600 MHz is close to 7000 mJy, falling away to 5000 mJy @ 400 MHz - about 1.5 dB lower. This might not seem to be much, but it would require half the observation time at 600 MHz to get the same S/N as at 400 MHz (all else being equal).
Some Worked Examples for Small Antennas at 400 MHz
While pulsar detection is an exciting activity - one can easily become literally 'starry-eyed', 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 - B0835-4510 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:
= 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.
B0835-4510 (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 B0835-4510 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 B0835-4510 shows that if the same bandwidth is used for B0329+54 - say 120 kHz (the maximum allowable un-de-dispersed bandwidth for B0835-4510 at 400 MHz) - a higher antenna gain is needed for B0329+54 compared to B0835-4510. This is to be expected as B0835-4510 is stronger than B0329+54. Given that B0835-4510 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 B0835-4510 to about 19 dBi for B0329+54. But that result is at the bandwidth which is the maximum for B0835-4510. 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 B0835-4510 (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.
Unfortunately, this result shows the "408 MHz, 3 metre dish and 75 kHz bandwidth" claims are dubious (where 'dubious' is being polite) as the published results are nearly 20 dB above the radiometry equation - a factor of 100 !!!
Even more unfortunate is the fact that these dubious results have been promoted in various places in good faith and have been accepted as valid. As a service to those interested in good science, a list has been compiled showing where this dubious result has been propagated. That list can be found here. You have been warned... :-)
This is NOT a personal attack on an individual, it is a criticism of the science and should not be painted as otherwise.
Those who would support the above dubious claims (notably all of which have no practical experience in pulsar detection), often cite the fact that "synchronous detection has gain" to explain the discrepancy between these dubious results and the radiometer equation. Unfortunately those individuals are unaware that that technique has been applied to pulsar detection almost as far back as the discovery of the first pulsar itself - and is not a new technique. Other references are made to the 'fast folding algorithm' as the magic technique which produces results which violates the radiometer equation. Once again, this technique has been around for nearly half a century (see Section 6.3.1 Fast Folding Analyses - "Handbook of Pulsar Astronomy" by D. Lorimer & M. Kramer). Neither of these techniques has caused the radiometer equation to be altered - simply because the equation already covers the effects of these techniques.
It is difficult to know whether the continuation of the defence and/or publication without comment by some individuals of the above dubious claims is an act of ignorance or an act of laziness - or even perhaps they are not too fussed about ensuring good information is supplied to those seeking to educate themselves.
A quote from Joe Taylor (K1JT)...(reproduced with permission)
"The claimed pulsar detections I have seen, attributed to Van Prooyen, are simply not credible. The graphs I've seen don't show anything like the expected pulse shapes or fading patterns. And why make claims for relatively weak pulsars PSR B0031-07 and PSR B1237+25, but show nothing for PSR B0329+54, which is far easier to detect with amateur-level equipment? That pulsar should be the first target for an amateur radio astronomer in the northern hemisphere."
Joe Taylor has been jointly awarded a Nobel Prize in Physics for the discovery of binary pulsars - so he probably knows a bit more about pulsar detection than most...
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 rates K5SO's results as proven as being successful (and he should know...). His 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 - as it should, as all are based on the science.
As the key to digging the pulsar signal out of the noise is to accurately synchronously average the pulse signal over a period of time (typically an hour or so), the accuracy of the data sampling timing must be high. As a general guide, timing accuracies of better than 1 ms over the observation time is required. For an observation time of 1 hour this translates to better than 0.1 ppm stability for the data sampling timing. Using a soundcard, for example, with the large variations in quality between models, without modification with a more accurate clock, is not recommended.
Usually data is recorded and then processed "off-air" by software. As the actual observed period of the pulse is the intrinsic pulsar signal period modified by doppler effects due the observer's moving point of reference in space (Earth's orbit around the Sun and rotation on its own axis), some sort of search mechanism is required in the "off-air" software. A number of test synchronous integrations will need to be done to find the best integration period (which produces the best S/N).
General Description of Equipment
The best benchmark for describing the level of equipment required is an EME (Earth-Moon-Earth) station. Individuals who have an EME station for 70 cm or 23 cm have about 75% of the work already done for pulsar detection. Some upgrading of frequency stability of data sampling may have to be done - but all the problems of low noise front ends, antenna construction and pointing have already been solved. So a successful pulsar detection station could be described as an EME station plus - where the 'plus' comes from the extra knowledge required about the nature of the pulsar signal, coupled with the need for high data sampling accuracy. From an amateur radio operator's point of view, pulsar detection is extreme DX.
To date, all the successful amateurs listed on the home page of this site have observatories that are either derived from EME station experience/operation, or associated with former professional installations.
However, do not be discouraged as there are a number of amateurs working on establishing how small an observatory station can be and still detect at least a few of the strongest pulsars. Early verified results indicate that a 3 metre to 6 metre fixed dish may be sufficient. (the author of these pages is trying to succeed with a 3 metre TVRO dish).
Verification of Results
In all scientific pursuits the key is proof. Results without proof are not science - merely conjecture.
The Scientific Method
If it is desired that your results are to be regarded as scientific, then it surely follows that they must be subject to the scientific method. To define what the scientific method is one need not go any further than what is returned when typing in "scientific method definition" into Google search - which returns this (a screenshot is shown below)...
A number of significant points are mentioned: systematic observation and measurement and testing - and most significantly "criticism is the backbone of the scientific method".
So what are the key responsibilities when presenting results in a scientific manner ? The following is a broad outline - details are left for individual self-education.
Adherence to Science-based Principles
All results must be consistent with the known laws of physics. There are many texts which contain these laws and develop equations which describe the limits of the process. If you think you are on the level of Einstein or other such luminary then your 'new law of physics' needs to be proven independently before being splashed about w.r.t. pulsar detection.
Application of Statistics
If the results are marginal we are in the realm of human subjectivity. If one wants to see a result, there will be one. If one is unbiassed, there may not be. To sort this out the results must be subjected to some sort statistical analysis and presented as some level of probability of being valid - but not a definite, unequivocal result.
Of course, if the results are like this...
...or like this...
..or like this...
... where the S/N is high, then correlation with the actual period of the pulsar lends strong evidence of the validity of the result and S/N statistics are not needed. Note that these results are very similar to professional results...
...the original graph of Jocelyn Bell's discovery of the first pulsar where a clear periodicity is visible (note: an increase in signal strength caused a downward deflection of the chart recorder needle).
Traps for the Unwary
Be warned, you will see data presented in otherwise reputable publications as valid results similar to this...
..where there is claimed to be a periodicity by selecting conveniently spaced pulses out of an array of pulses. These types of results fail any sort of objective test of scientific validity based on the details presented.
It is instructive to note that the above graph was produced mathematically by applying a random function in a spreadsheet '=5*(IF(RAND()>0.5;RAND();0))' and is NOT a copy of the original graphics in question - despite the strong similarity with the claimed 'pulsar' results...
Post-Detection Processing 'Magic'
In an effort to improve the sensitivity of a receiving system to pulsar signals there has been claims of magical results with special post-detection processing. The claimed results far exceed what is physically possible according to the pulsar radiometer equation. One of the traps when applying filtering/processing without fully understanding the nature of the signal is that it is possible to manufacture signals from random noise (see figures above and below).
All sorts of convincing (to those not familiar with the fundamentals) results can be produced. The following figure has been produced by applying 'special post-detection processing' to random noise...
'Special Post-detection Processing' applied to random noise
Here we see a pretty convincing result, complete with pulse to pulse variability - but this result is entirely manufactured by processing random noise.
In addition to many terrestrial signals convincingly mimicking pulsar signals, poor use of post-detection processing can manufacture false results out of the noise.
One robust method of sorting the wheat from the chaff is repeatability. All the successful amateurs listed on the home page have demonstrated repeatability. As pointed out in professional literature on the subject there are many terrestrial sources which can mimic quite closely the signal from a pulsar. Pulsars appear regularly in a fixed position on the cosmic sphere, while terrestrial sources will likely have a random existence. By repeating the results the terrestrial sources can be identified. The author has several hundred results which look like a Vela pulsar signal, but fail the repeatability and S/N test. Despite the convincing nature of these results, the author does not claim success, merely that they are encouraging results.
Most pulsars spend some portion of each day below the horizon. How large that portion is depends on the pulsar's declination combined with the latitude of the observatory. For the author's NRARAO location and Vela, the time below the horizon is 6 hours every day.
A simple 'exclusion test' is to acquire data when the pulsar is below the horizon and verify an absence of signal.
Another exclusion test could be done by driving the antenna as if tracking the pulsar, but adjust the declination such that the pulsar is out of the 3 dB beamwidth of the antenna - say in the first null in the beam pattern. This a good test as it includes all possible sources of RFI (tracking motors/electronics and terrestrial RFI) while excluding the pulsar which lies outside the beamwidth. There should be an absence of pulsar signal.
Yet another exclusion test could be done by acquiring 24 hours of data from the antenna pointed at the correct declination of the pulsar, but fixed in position (drift-scan mode) so that the pulsar passes through the beamwidth of the antenna. Then by processing successive slices (say 1 hour duration) of the 24 hours of data it can be demonstrated that the pulsar signal only arises when the pulsar is passing through the antenna beamwidth. An animated graphic of this would be a powerful verification of results.
Such efforts to try and disprove the results increase the validity of the results.
Ultimate Responsibility for Verification of Results
Ironically there is a greater responsibility for the burden of proof required from the amateur than the professional. For the professional, the system used for the observations has been scrupulously characterised in terms of spurious responses (RFI surveys, etc) and any results will be vigorously peer-reviewed. Amateurs can escape, if they are so inclined, with little censure applied to claims of 'successful' results.
Further Details About the Importance of the Scientific Method
For further discussion about the scientific method, with a practical example, see here - The Scientific Method - a cautionary tale (published in the SARA Journal - "Radio Astronomy" - March/April 2015).
Location of Observatory - RFI and Ground Noise
One of the advantages a professional radio astronomer has is that, generally speaking, the observatory is located in area which is protected against radiofrequency interference (RFI). If the amateur observatory is not located in the wilderness, far from civilisation, the effects of neighbours' plasma screens and the myriad of other electronic devices which radiate radio frequency energy will have a significant negative effect on efforts to receive faint pulsar signals from light-years away. The characteristic of amateur observations which can counter this is that he/she is not usually limited in observation time. For the location of NRARAO the best times for observing Vela are the months from January to July when it transits between the local hours of midnight and dawn. This reduces the amount of RFI due to the perpetrators being asleep...
Nonetheless, the minimum strength pulsar detected can be severely compromised by RFI. Not only due to intermittent carriers (which can easily identified and avoided), but more insidiously, the general noise floor level can be raised - which is not easily identified unless calibrated measurements are made. This can be designated Trfi.
In addition, ground noise from buildings, trees and other objects add to the system temperature through sidelobe and spillover responses of the antenna, further compromising the sensitivity. This can be designated Tsidelobe.
In practice, in usual amateur observatory environments, the sensitivity radiometry equation becomes...
...which can be the reason why there is a lack of success with small antennas.
Some Information About 'Folding'
The commonest way to dig pulsar signals out of the noise is by "synchronous averaging" or "epoch folding". A link to a description of the technique can be found in 'Technical Topics'.
Trying to detect a pulsar is an exciting project. It presents challenges on many fronts and exercises many disciplines. While more difficult to achieve than even EME success, the feeling of seeing signals from fantastic objects light years away (extreme DX in amateur radio parlance) is, I am told, exhilarating. I hope to experience that exhilaration in the coming year (2017) at NRARAO.