Observing a Flux Calibrator
The most straightforward way to calibrate is to look at something whose brightness you know. We will refer to these reference objects as "flux calibrators". For example, we happen to be pretty sure that 3C48 emits 3.313 Jy at X-band. If we point our antenna at it and measure 10 micro-Watts in our power meter, then we would guess that anything else we look at that shows a power reading of 10 micro-Watts is also emitting a flux of about 3.313 Jy. We might also assume that if we measure 20 micro-Watts, the thing we're looking at is probably emitting 6.626 Jy. (This assumption is also referred to as the "linearity assumption." This means that if the flux from a source changes by a certain percentage, the power we measure changes by the exact same percentage. It's a pretty good approximation for DSS-12.) If looking at a known object tells us everything we need to know, why are there other steps to calibration? It's because the performance of the antenna changes with time and position on the sky (see the "Why do we have to re-calibrate our telescope every time we observe?" discussion.) So when we look at a flux calibrator, we determine how our system performs at that instant in that part of the sky. We have to make adjustments to estimate how our system is performing a few minutes later looking at a different part of the sky.
Flux Calibrator Details
Flux calibrators are astronomical sources whose flux we believe we know very well. (See the "Parameters" listing below.) Since we know the flux of these targets, and we can measure the power received in our detector, we can determine the end-to-end system gain with this equation:
EndToEndGain = (Flux_Cal / SizeCorrection) / (Power_Cal)
EndToEndGain has units of Jansky per micro-Watt, and tells us how to convert a power reading in our detector to the amount of flux coming from an object on the sky. It is only valid for one place in the sky, and one instant in time.
Flux_Cal is the known flux (in Jy) coming from the source.
SizeCorrection is a correction factor we apply to Flux_Cal. If a source is larger than our antenna beam, we do not pick up all the flux it emits. The fraction that we do receive depends on the size of the source and the size of our beam. We have determined this correction factor for each of our Flux Calibrator sources, and it is listed on the Parameter Values page.
Power_Cal is the power, in micro-Watts, measured by our detectors when doing a cross-scan of the calibrator.
"Mini-Cal" is short for mini-calibration. A mini-cal allows us to determine how the hardware and electronics in our receiver and detectors are performing. (It does not tell us about how the antenna is performing. We discuss that under "antenna efficiency" below.) Doing a mini-cal involves taking several measurements of things whose brightness we know. In that sense it is similar to looking at a flux calibrator - the difference is that instead of pointing the whole antenna at a calibrator, we either insert a known target right in front of our receiver (we actually have a small target - called a hot-load - that is mounted on a swing arm and can move in front of our receiver), or we inject power into our electronics using something called a noise diode. The advantage of doing it this way is we can do a mini-cal anytime we want, wherever the antenna happens to be pointed. The disadvantage is that we are not testing the entire system. Because the test signals enter the system at our receiver, they never pass through the big dish antenna or the subreflector. That means they do not tell us how those parts of our instrument are working.
For those of you wanting some more specifics, a mini-cal consists of 5 measurements:
Every time we do a mini-cal, we determine a quantity we call the "Receiver Gain." It was the first plot shown on the main page. It tells us how to convert the power measurements made in our power-meters to a brightness temperature entering the receiver. (Note that it does not tell us the brightness entering the antenna!)
In summary, to track changes in our electronics that happen in between our looks at a flux calibrator, we do mini-cals. Typically they are done about every 20 minutes.
The EndToEndGain is only valid at the moment we observed our flux calibrator. To adjust this to the time we were observing our target, we use the Mini-Cals. Remember, the mini-cals tell us how our electronics are performing at any instant, and they give us a quantity we call the Receiver Gain. We do a mini-cal every 20 minutes or so, and it gives us a Receiver Gain in units of Kelvin per micro-Watt. To determine the receiver gain at the time we observed our flux calibrator, we do a linear interpolation between the Receiver Gain found just before and after observing the flux calibrator. Let's assume we measured a Receiver Gain of G1 at time T1, a few minutes before looking at our flux calibrator. Let's also assume we have another gain measurement of G2 taken at T2, a few minutes after looking at our calibrator. If the calibrator was observed at time Tcal, we estimate the Receiver Gain at time Tcal with this linear interpolation:
ReceiverGain_Cal = (G2 - G1) / (T2 - T1) * (Tcal - T1) + G1
Similarly, the Receiver Gain when looking at our target is given by: ReceiverGain_Source = (G4 - G3) / (T4 - T3) * (Tsource - T3) + G3 where G4 and T4 are for a mini-cal just before observing the source, G5 and T5 are from the mini-cal after observing the source, and Tsource is the time at which the source was observed.
We therefore can correct the EndToEndGain for changes in our receiver over time with this equation:
TimeCorrectedGain = EndToEndGain * ReceiverGain_Source / ReceiverGain_Cal
Another way of saying this is that the time correction to the EndToEndGain is a multiplicative factor given by:
TimeCorrection = ReceiverGain_Source / ReceiverGain_Cal
Correcting for Antenna Efficiency
We previously described how the shape of the antenna changes as it points to different parts of the sky, and this alters how effectively it collects radio waves. We describe this with a multiplicative factor we call the antenna efficiency. The only way to determine this reliably is to look at flux calibrators on the sky. Fortunately for us, the antenna efficiency does not change much over time - at a particular position on the sky today, the antena efficiency should be pretty much the same next month. Therefore, periodically we track a bunch of trusted flux calibrators all over the sky. This allows us to make a map of the antenna efficiency as a function of antenna position.
Antenna Efficiency Details
From observations of our Primary Flux Calibrators, we have determined that the Antenna Efficiency at S-band, as a function of elevation, is given by this equation:
AntennaEfficiencySband = As + (Bs * Elev) + (Cs * Elev*Elev) where Elev is the elevation in degrees, As = 0.589412 Bs = -0.001069 Cs = 1.240e-05
Since we know the declination of our antenna when we observed the flux calibrator, we can use the above equation to calculate the antenna efficiency for the location of the calibrator. We'll refer to that quantity as AntEffSband_Cal.
Similarly, we know the declination when we observed our target source, and can calculate the antenna efficiency at the location of our source, and we call it AntEffSband_Source.
We can then use the ratio AntEffSband_Cal / AntEffSband_Source to correct for changes due to the changing position of the antenna:
TimeAndPositionCorrectedGain = TimeCorrectedGain * AntEffSband_Cal / AntEffSband_Source
Another way of saying this is that the position correction to the EndToEndGain is a multiplicative factor given by:
PositionCorrection = AntEffSband_Cal / AntEffSband_Source
The antenna efficiency at X-band is given by an equation with different coefficients:
AntennaEfficiencyXband = Ax + (Bx * Elev) + (Cx * Elev*Elev) where Ax = 0.158, Bx = 0.0034, Cx = -6.1e-6. and this can be used to determine the TimeAndPositionCorrectedGain (or the PositionCorrection) for X-band, just as in the above S-band example.