RADIO ASTRONOMY CALCULATIONS


INTRODUCTION

The question as to whether a radio source, such as Cygnus A or the Sun, is detectable with a small radio telescope often arises in the study of radio astronomy. Despite the fact that a quantitative determination of the signal to noise ratio is reasonably straight forward, it is sometimes quite difficult to resist the temptation to point the antenna at a particular source and see if an increase in radio noise can be detected. From experience, we can say that this almost never works! The reason for this, as it turns out, is that when using antennas with relatively small collecting area and wide beamwidth, the ratio of the noise power available at the antenna terminals from the radio source under study, is often only a small fraction of the total power available, which is made up predominantly of additional contributions (often unavoidable) from the sky noise background, the receiver electronics and unfortunately external radio interference. It can be very helpful if the signal to noise ratio is calculated before observations are made. These results can then be used to guide design of the post-detection circuitry. It is worth remembering that such calculations require at least some knowledge of source flux densities and background sky noise. Neither of these were known for sure when these pioneering observations were made in 1947.


DERIVATION OF THE SIGNAL TO NOISE RATIO

The noise power Psrc available at the receiver input due to the radio source is:

The signal to noise ratio of Psrc to Psys is then given by (eqn.1)/(eqn.2) :

and substituting G λ2 / 4 π = A in (3) where G is the antenna gain and λ is the wavelength in metres we get:


CYGNUS A AT 100 MHZ

For a simple radio telescope at 100 MHz with the parameters:

The same calculation can be repeated for the Bolton & Stanley (1948) experiment to detect Cygnus A at 100 MHz, using the following parameters:

Interestingly, on page 68 of Bolton & Stanley (1948) paper (1), there is a statement (para.4) “At rising, the constant component on 100 MHz is about one twentieth of the total noise received’’ which tends to broadly confirm the above approximate calculation.

The significant reduction of the system noise Tsys in the first example experiment essentially accounts for the improvement in the signal to noise ratio between this data and that of Bolton & Stanley. Despite this, it is doubtful whether the presence of Cygnus A in the antenna beam would be readily detectable aurally in either case because of the small signal to noise ratio (<5%). The same could be said about the detection of the Quiet sun which by coincidence has a similar flux density to Cygnus A at 100MHz (104 Jy). By contrast, even with a small radio telescope, radio emission from the disturbed Sun should be readily detected at 100MHz, being orders of magnitude more intense (109 Jy).

So how did Bolton & Stanley achieve such clean records. Only limited information is given in the paper about the post-detector processing. However, it is clear that a current amplifier was used to increase sensitivity to small signals and to provide a means of offsetting the receiver noise component of detector output. It is not clear what the frequency response of this amplifier or the recording milliammeter was. On the other hand, they do provide the fluctuation level (ΔTflc) as a fraction of the total receiver noise (Ttot); (for convenience here, we have expressed this as a ratio of noise temperatures rather than flux densities). This ratio, which appears in Table 1 of Bolton & Stanley, implies that a significant degree of post-detection averaging has been applied.

To see how much, we can write:

where τ is the effective integration time and ΔTflc/Ttot is the fractional noise temperature level.

Substituting Δf=1.5MHz and solving for τ gives 2.67 secs. This integration time seems very short but one needs to keep in mind that although the fringe period is near 10 minutes in fig. 1(A) of Bolton & Stanley, there are large very brief fluctuations (caused by scintillations) with periods as little as 6 seconds superimposed on the sea interferometer fringe recordings (see fig.1(A), page 60 and section 6 (iv), page 84). Since these were of great interest at the time, a short integration time was clearly required to avoid excessive damping of these scintillations.

We can conclude from the foregoing that post-detection integration is required (and was used by Bolton & Stanley) to achieve a large improvement in the signal to noise ratio by a factor of 2000 giving a signal to noise ratio of (0.024x2000) ~50. The wide bandwidth of the receiver (1.5MHz) coupled with a calculated post-detector integration time of 2.67 secs. can account for this improvement.


IMPROVEMENTS ON A SIMPLE RADIOMETER

One improvement would be to add a second identical yagi antenna and to implement a phase switched interferometer. The phase switched interferometer has a number of very significant advantages over total-power designs such as the sea interferometer. Prominent amongst these, is the fact that the sky background can be largely rejected at reasonable antenna separations, so that baseline stability is significantly improved. This means that the fringes generated by weak discrete radio sources stand out much more on recordings. In addition, local incoherent radio frequency interference (RFI) is strongly rejected.

The bandwidth will probably need to be much reduced compared to the Bolton & Stanley receivers to avoid RFI. We have successfully used 250 kHz ceramic filters to achieve this but it may be possible to make software adjustments in an SDR receiver as long as the filter can be made with very steep skirts. To make it possible to tune over a range of frequencies (often required to avoid adjacent channel interference), I have found that the half wavelength antenna phase switch, that is usually implemented by switching a half wave length of transmission line in and out of the connecting cable from one antenna can be more conveniently implemented using an integrated circuit balanced modulator (see figure below). These are available fairly cheaply and can be used up to several hundred MHz, and perhaps more.


A phase switch implemented with an integrated circuit balanced modulator

A low noise, moderate gain, head amplifier located at each antenna is essential to reduce the effect of receiver noise so that system noise is dominated by the sky background (see eqn. 3). An antenna separation of 10 wavelengths is probably the minimum requirement so (at the very least) 2x15m cables are needed at 100MHz between the antenna phase switch and the head amplifiers. In addition, processing software is required (possibly using a RPi processor) to provide switching waveforms to the antenna phase switch and the post-detection demodulator. Finally, software is also needed to integrate the demodulator output, and for calibration and display of observed fringe data.

There are many very fine books and scientific papers written about radio astronomy that make excellent reading, but nothing is as satisfying and exhilarating as exploring it yourself with equipment you have fashioned with your own hands. This is the spirit with which fledgling astronomers, John Bolton and Gorden Stanley, using equipment they had assembled on the cliff edges at Dover Heights in 1947, located the precise position in the sky of Cygnus A. And, (in the words of Woody Sullivan (3)) “their discovery was made in a most unlikely manner, for they did their astronomy not with glass lenses but with rods of metal”.


REFERENCES

1 Bolton, J.G. and Stanley, G.J. (1948). ‘Observations on the variable source of cosmic radio frequency radiation in the constellation of Cygnus’. Aust. J. Scient. Res. 1, 58-89.

2 Wilkinson, Malcolm H. and John A. Kennewell, ‘Optical Imaging of Bright Radio Objects’. Proceedings of the 23rd Australian Space Research Conference, Sydney, December 2-4, (2024).

3 Orchiston, W., Robertson, P. and Woodruff T. Sullivan. ‘Golden Years of Australian Radio Astronomy, An Illustrated History’. Springer Series on Historical and Cultural Astronomy, Series Editors W.Orchiston, M. Rothenberg, C. Cunningham, page v. (2021).




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