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Calibration
What is calibration?
When we use our radio telescope, we are generally trying to determine how much radio energy an object out in space is emitting. (The amount it emits helps us learn what it is made of, its temperature, and other things.) When we point our telescope at the object, sensitive electronic devices report how much power they are receiving. The amount of power received in our instrument, however, is not the same as the power emitted by the object in space! When we calibrate our instrument, we are learning how to translate the power measured in our detectors into the amount of energy emitted by the target.
    There are many things that reduce the amount of power we detect - for example, the metal dish and mirrors that reflect energy into our receivers are not perfect, and some of the energy is scattered away or absorbed. Can you think of other things that cause our telescope to not receive exactly as much energy as the object in space emits? Those of you who have done the "Bucket of Noise" activity might remember that not all of the fuzz-balls "emitted" actually hit the telescope. And those of you who have studied the inverse square law might be able to describe how the farther away something is, the less energy we receive.
    In addition to things that reduce the amount of power we detect, there are even some things that put extra energy into our detectors, causing us to see more than is coming from the object in space. For example, you may have learned that everything emits radio waves through a process called thermal radiation, also referred to as blackbody radiation. This means that the Earth's atmosphere, as well as all the pieces of our telescope and electronics, are emitting energy which our detectors see. Can you think of other things that might emit energy which our telescope picks up?
    When we calibrate the telescope, we are determining the effect of all these other factors; some of which have increased the power we receive, some which have decreased it. We then know how to convert the readings from our power meters into a measurement of just the energy coming from the object in space. That is the process of calibration.

Why do we have to re-calibrate our telescope every time we observe?

As a matter of fact, we actually have to re-calibrate the telescope several times an hour. Why isn't one time good enough? There are two reasons. The first is that some of the effects we are trying to account for depend on the position of the telescope. Since the telescope is moving all the time as it tracks a source across the sky, these effects are constantly changing. The second reason we need to re-calibrate is that certain effects are changing over time. Even if the telescope were not moving, we would have to account for these changes.
    What are some of the things that change with the position of the telescope? One thing that changes is the shape of our telescope. Surprising, but true! The DSS-12 telescope weighs around 2 million pounds, and the big dish sags and stretches in different ways as it points to different parts of the sky. The change in shape alters how effectively the dish reflects energy into our receivers. Another thing that changes is the amount of energy we pick up from the ground. When we try to observe an object low in the sky, you can imagine that more energy from the surrounding ground or mountains makes it into our receivers than when we are observing something straight up.
    What are some of the things that change over time, even if the telescope is not moving? The main thing that changes is the performance of our electronics, particularly the amplifiers used to boost the faint whisper of energy we get from space. (If the amplifiers are a problem, why do we use them? The reason is that without them, the signals are too weak to be detected!) Other things that change are the wind here on Earth (which can influence the shape of the antenna and our ability to point it accurately), and the amount of energy emitted by the Earth's atmosphere. (Think about a cloud moving across the sky. When it passes in front of our telescope, we might get more signal from the cloud, and less from the object in space.)
    Because of all these things that change, we must constantly re-calibrate our instrument if we wish to get an accurate measure of the amount of energy coming from space.

What exactly are the steps involved?

There are many things done to properly calibrate our instrument. Here are the main steps involved in creating the data plots you see on these web pages.
  • Doing cross scans
  • Observing a flux calibrator
  • Doing mini-calibrations
  • Correcting for antenna efficiency
Each of these is described in more detail below. Putting it all together, though, it goes something like this:
  • Whenever we observe something, make cross-scan measurements. These are "on-source" minus "off-source" measurements, which allow us to eliminate most unwanted signals from the sky, the Earth, and our own equipment. (In the plots you see on this web page, the subtraction has already been made, so you don't see the raw numbers.)
  • Very rarely (perhaps once a year), observe astronomical sources of known brightness all over the sky, in order to determine how the telescope's antenna efficiency changes from place to place on the sky.
  • Observe a flux calibrator (an astronomical object of known brightness) at least once a day to determine the end-to-end system gain. This is a factor that converts the power we measure in our detectors (measured in micro-Watts) to a flux coming from space (in Jansky). It is valid only for one place on the sky, however, and for one moment in time.
  • Make frequent (roughly every 20 minutes) mini-cal measurements to track changes in how our receivers respond to signals.
  • Make observations of the unknown source you wish to measure (using cross-scans, of course!). Apply the following correction factors to convert the power measured in our receiver to a flux from the sky:
    • Multiply by the end-to-end system gain found in step 3 to convert micro-Watt to Jy.
    • Apply a correction based on the mini-cal data (step 4) to account for changes in our electronics between the time the flux calibrator was observed and the time our target was observed.
    • Apply a correction factor based on our antenna efficiency data (step 2) to account for changes in the antenna performance between the location of the source we are looking at and the location of the flux calibrator used to get the end-to-end gain.
Once you've done these things, you know the flux from your target! Sometimes we like to use different units than Jy to measure how bright our target is. For information on that, please refer to our Brightness Units F.A.Q. entries.

Doing Cross-Scans
We have discussed elsewhere that everything emits radio waves, not just objects in the sky. The Earth's atmosphere emits some, as do the ground and mountains surrounding our antenna, and even our own equipment. Some energy from all of these other sources enters our detector along with the object we are trying to measure. Doing cross-scans is one way we can eliminate this effect. In a cross-scan, we basically are just comparing how much power our instrument measures when it is pointed straight at our target - let's use Jupiter as an example - and how much power it measures when pointed a little bit off of the target at an empty patch of sky. We assume that the amount of unwanted energy we measure when pointed at empty sky just to the side of Jupiter is the same amount of unwanted energy we pick up from the direction of Jupiter. So if we subtract the "side" measurement from the measurement we got looking at Jupiter, we are left with the power from Jupiter. (Those of you who have done the "Bucket of Noise" activity may remember discussing this.) Many of you have probably done cross-scans by hand during your observing run. Did you take "baseline" measurements and then "peak" measurements, and subtract the two? The baseline was the "side" measurement made when the antenna was pointing slightly away from your target. The "peak" was the on-source measurement!    
     In summary, to eliminate unwanted radio emissions from our measurement, we usually scan across our target, and use the difference between "on-source" and "off-source" measurements as our signal. We do these cross-scans on our Flux Calibrators as well as the objects we are studying.

Cross-Scan Details
When measuring an object on the sky, we scan our antenna across the object. We start well away from the target, where we think there is just empty sky, move across the position where we think the object is, and then back onto empty sky.    
     Can you explain the main features of this plot? Why is there a peak in the middle? Why does the signal drop off smoothly to either side of the peak? Why is the signal relatively low and unchanging at the beginning and end?
     We do cross-scans in both the declination (abbreviated dec) direction and in the cross-declination (abbreviated x-dec, or xdec) direction. For each cross-scan we do a least-squares fit in order to determine the baseline level, and the location, width, and height of the signal in the center. (We fit a straight line to the baseline, and a Gaussian to the part where we see our target.) With our current data access tool, you cannot see the raw data, all you can access is the fit to the data. The peak height of the Gaussian above the baseline is the amount of extra power we receive when pointed at the source, over what we get from the nearby sky. That is the power measurement that is plotted, which gets converted to flux or brightness temperature.
     The other parameters of the fit are used in different ways. For example, if the location of the peak of the Gaussian is not at the center of the scan, that indicates that the antenna isn't pointing exactly where we think it is. We use this information to adjust our pointing next time we scan across the source.

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.

Doing Mini-Cals
"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:

  • The zero-level of the power meter. This is what the power meter registers when no power is input to it.
  • Sky (we use the sky as a low-power signal).
  • Sky +noise-diode (adding a little extra signal with the diode helps us determine how linear our system really is).
  • Hot-Load (this is a target whose temperature is known).
  • Hot-Load + noise-diode (again, adding the noise diode helps us determine the system linearity).

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.

Mini-Cal Details
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.


Putting it all Together
If the cross-scans give us a power measurement of Power_Source (in micro-Watts) when looking at our target source, then we can apply the calibration equations to determine the flux from the source, Flux_Source, in Jy:

          Flux_Source = Power_Source * EndToEndGain * TimeCorrection * PositionCorrection where quantities for either X- or S-band are used.
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