Determination of the Power Pattern

The antenna attached to a car radio detects radio signals in much the same way as the antenna of a radio telescope, but there are several differences. The most important one perhaps is that the reflecting dish of the radio telescope gives the antenna considerable directionality by focusing radio waves from a particular direction onto the horn. Unfortunately, a radio telescope is not perfectly directional, because it is still sensitive to radiation from directions other than the pointing direction. Careful design of the reflecting dish can minimize the contributions from sources outside the desired field although they cannot be eliminated. The power pattern is the measure of the response of a telescope to a point source as a function of angle. A hypothetical power pattern (in one angular dimension) is shown below.


A two-dimensional representation of a hypothetical power pattern.


The power pattern is normalized at its most sensitive direction (ideally, this will be along the physical axis of the antenna). The largest lobe is called the main beam and the smaller lobes are called side lobes. The full width half maximum (FWHM) of the main beam defines the resolution of the telescope so it is important to know the size of the telescope's main beam.

Since the response of the telescope is a convolution of the telescope's beam with a source's signal, a bright and isolated point source should be used to measure the shape of the main beam. Using such an object allows one to be confident that all of the flux from the source is contained in the beam shape and that the flux measured is from that single object. Thus, one is effectively convolving a delta function with the telescope's beam. When the point source is in the most sensitive part of the main beam, the measured signal will be highest. As the point source is moved away from the centre of the main beam, the measured signal will decrease. Therefore, by moving the point source through the main beam, the shape of the main beam is mapped out.

In our case, pointing and time limitations prevented us from making a detailed map of the telescope's main beam, but it was still possible to measure its width by fixing the telescope on the celestial meridian and allowing Taurus A, a bright, compact supernova remnant, to drift by.

Since the pointing of the telescope is primitive (and it is possible that the feed was mounted slightly off axis), it is impossible to be certain that the transit occurs through the centre of the beam. Fortunately, this will not affect the size of the beam width provided the main beam is a symmetrical gaussian. Any slice through such a shape will have the same FWHM. This can be shown by examining the equation for a gaussian:

G(r) = A e-ar2

where A is the amplitude of the gaussian and a is related to the FWHM of the gaussian. Rewriting r as its x and y components and setting y to an arbitrary value c gives:

G(r) = A e-a(x2+c2) = G(r) = A e-ac2 e-ax2 = A' e-ax2

This is simply the equation for a slice through the two dimensional gaussian. While the amplitude of this slice is smaller than A, the FWHM of the gaussian slice is unchanged since it is dependent only on the value of a.


A cylindrically symmetric gaussian. Any vertical slice through such a
gaussian will have the same FWHM.


The graph below shows the telescope's response as Taurus A drifted through the main beam. In order to measure the width of this profile, it was necessary to remove the effect of the changing baseline, caused by variations in the receiver gain with temperature. This was done by linearly interpolating between the baseline levels on either side of the profile and subtracting.


The Taurus A transit measurement from April 4, 1996.


It was then possible to fit a gaussian to the resultant beam profile. The profile contains considerable noise as well as two high peaks. The right-hand peak was attributed to interference. However, since the left-hand peak was present in all the drift scans performed, it was attributed to a side lobe. Therefore, the gaussian was fit to only the main part of the profile.


A gaussian fit to the baseline-subtracted beam profile.


The FWHM of the gaussian fit is 0.132 hours which is converted to degrees using the formula:

FWHM(°) = FWHM(hours) * 15°/hour * cos (declination)

giving a beam width of 1.85°. This is only slightly larger than the 1.7° diffraction limit of the telescope for a uniformly illuminated aperture, given by the relation:

resolution = 1.22 * wavelength / diameter.


Measurement of System Temperature

The system temperature should ideally be measured for each observing run, both before and after. It is a measure of the brightness of the sky, but also requires the measurement of "cold" and "hot" loads (liquid nitrogen and ambient temperatures in our case) as in the determination of the receiver temperature (see the Front-End section).



Measuring the system temperature with a "hot" load, and a "cold" load.


With a microwave absorbing foam over the horn, voltages for the hot and cold loads were measured either by the computer (with the A/D converter discussed in the astronomy Calibration section) or on a chart recorder. Then the foam was removed from the feed, and the telescope was pointed at the sky. This produced a voltage output as well, which was converted to a temperature using the conversion factor generated by the prior two measurements.

The conversion factor between voltage and temperature based on the measurements of April 10, 1996 is:

V = (0.0242V/K)*T + 1.94V.

Since the sky voltage was 2.3V, the sky brightness (a.k.a. the system temperature) was estimated to be 14.9K. The sky brightness temperature at 4GHz is expected to have a value of about 10K, meaning that our telescope is detecting about a factor of 50% too great a flux. This could be the result of noise estimates which have not been accounted for, such as the absorbing coefficient of the sky.

An estimate of oscillation about the sky signal, delta Trms is necessary, since only objects which exceed that oscillation in strength will be visible above the sky background signal. Delta Trms was estimated from the data of April 4, 1996 of Taurus A. Since data were taken over a period of many hours, there exists long stretches in which no signal above the sky is observed to be present. Extracting one such hour's worth of data and averaging the errors in the signal for each measurement (as recorded when the A/D converter was in use) yielded a reasonable value of the standard deviation about the mean. The value obtained was 11mK.

Thus, sources which can produce signals above the sky of less than 11mK would have no chance of being detected, since they would be mired in noise. The choice of what voltage above delta Trms must be reached to be considered a detection is up to the discretion of the observer.


Determination of Aperture Efficiency

The aperture efficiency is the ratio between the effective collecting area (effective aperture) of the dish and the geometrical area of the dish:

nap = Aeff/Ageom

This is a measure of the efficiency of the telescope, and thus a fundamental calibration parameter needed for meaningful results.

The effective aperture, Aeff, is a component of the scaling factor between the true flux or brightness temperature of the sky, and that which is actually reflected by the antenna and picked up by the feed, i.e. the antenna temperature. The antenna temperature is given by the following:


where I is the source intensity, k is Boltzmann's constant, P the power pattern discussed above, and cos(theta) represents the projection of the solid angle onto a plane normal to the pointing direction. The integral is over the solid angle of the source where the intensity off the source is assumed to be zero.

The flux density of a source modified by the power pattern of the antenna is given by the following expression:

.

If a point source is observed, or in our case, a source with an angular size much less than the size of our beam, we can determine the aperture efficiency. Due to the small angular size of the source, we assume that the flux from the source will fall entirely within the telescope's main beam. For this calibration we used Taurus A; with an angular size of 7' X 5', this easily falls within our beam size of 1.85°. Taurus A was chosen as the calibrator source due to the number of drift scans we obtained for it. By comparing the amplitudes of Taurus A's drifts, we were able to choose the best observation, i.e. that during which the source passed closest to the centre of the beam. When the source lies at the centre of the main beam, cos(theta)=1, and the power pattern will be at its maximum value of one. Therefore the flux density simplifies to:

.

We then derive a simple relationship between the antenna temperature and the flux density of the point source:

Tant = (A effSv)/2k.

By observing a source with a known flux density at the observation frequency, we then easily calculate the aperture efficiency.

From the baseline subtracted gaussian fit, the signal amplitude of Taurus A was measured to be 21.5mV. Ideally we should have system temperature calibrations specific to the observation of Taurus A. However, because the telescope was in "automatic" mode during the drift of Taurus A across the telescope, we were not present to do the calibration and instead used the same calibration as that given in system temperature section above. Using this calibration, the amplitude corresponds to an antenna temperature of 0.9K.

The flux density of Taurus A was obtained from the Catalogue of Galactic Supernova Remnants, 1995 July Version compiled by D. A. Green. At 1GHz, Sv = 1040Jy with a spectral index of 0.3. At 4GHz this corresponds to 686Jy, resulting in an effective aperture of 3.6x104cm2 . Using an aperture diameter of 308cm to calculate the geometrical aperture, we find the aperture efficiency to be 48%. This is lower than typical values of 60% from professional telescopes. A large loss of efficiency is attributed to the feed being off-axis from the focal point of the dish, as well as a loss due to unequal illumination of the dish. Further work will be required to better align the feed.