For heterodyne observations, there are three types of calibrations: temperature (chopper) calibration, telescope efficiency, and (if necessary) AOS frequency calibration. It is best to calibrate regularly.

• Temperature Calibration determines T_A^*, the antenna temperature corrected for atmospheric absorption and ambient temperature telescope losses, and measures the system temperature.
  UIP> cal
• Telescope efficiency. After pointing, measure the brightness of planets to determine the telescope efficiency.
  UIP> planet planet
  • Then with beam switching (secondary mirror wobbling)
    UIP> chop 2
  • or with position switching using an offset in arcsec
    UIP> oo 1 /step offset
    
• Frequency calibration is necessary for AOS5. It is done automatically each time AOS5 is restarted. Frequency calibration is not necessary for the FFT spectrometers.

REMEMBER Observers must decide how important the calibration is to them and how to deal with it.

Several authors (Tom Phillips, Ken [Taco] Young, and Todd Groesbeck) contributed to this description of the calibration procedure.

Heterodyne observations are calibrated with the chopper method first discussed for millimeter astronomy by Penzias and Burrus (1973 ARA&A 11, 51 ADS). Two measurements are made: one of the sky and one of an ambient temperature absorber (hot load or chopper) placed into the beam at a point between the receiver and the secondary mirror. As shown below, these measurements provide the information necessary to determine T_A^*, the antenna temperature corrected for atmospheric absorption and ambient temperature telescope losses, including hot spillover and blockage. Temperature calibration is performed with the Cal command. This is done automatically at the start of the next scan after changing sources, retuning the receiver, etc. The results of the most recent appropriate Cal measurement are automatically applied so the recorded data are calibrated on the T_A^* scale.

Define the following:

Several assumptions simplify the theory. The major ones are:

1. The hot load, ground, and air are the same temperature, T_h = T_ground = T_air. Often T_air may be less than T_h by about 20 K, but this is not too important. The observatory software uses T_h = 280 K.
2. The receiver sideband gains are equal, G_sig = G_sig.
3. All temperatures are large compared to the observing frequency, i. e., kT » hυ. The lowest temperature is likely to be T_c, which is about 70 K. The Rayleigh-Jeans approximation is valid, therefore, over the submillimeter range.

At the CSO, all the receivers are double sideband (DSB). Continuum (broad band) sources, such as calibrators, spillover, the atmosphere, and planets, contribute to the measured signal in both sidebands. Spectral lines, however, only contribute in one sideband.

Looking at the calibrator (hot load or chopper), the measured signal includes contributions from the receiver and the load,

• V_h = (G_sig+G_img) [T_RX + T_h ].

Looking at the sky (off position) and at the source and sky (on position), the measured signal include contributions from the receiver, the atmosphere, the ground (hot) spillover, the sky (cold) spillover, and the source,

or

Observations are made with some switching scheme, i. e., position switching. The observing procedure measures

• OO = (V_s - V_sky)/V_sky = αβγ [ G_sig T_s,sig e^-τ_sigA + G_img T_s,img e^-τ_imgA ] / V_sky ,

the calibration procedure measures

• Cal = (V_h - V_sky)/V_sky = α T_h [ G_sig e^-τ_sigA + G_img e^-τ_imgA ] / V_sky .

When the sideband gains are equal, G_sig = G_sig, the ratio OO/Cal is independent of the hot spillover efficiency, α, and of the system gain, G_sig + G_img. The data are recorded as SSB antenna temperature, corrected for atmospheric absorption and hot spillover,

• T_A^* = 2T_h × ( OO / Cal ) = 2βγ (T_s,sig e^-τ_sigA + T_s,img e^-τ_imgA ) / ( e^-τ_sigA + e^-τ_imgA ).

Individual spectral lines appear only in one sideband, so T_s,img = 0 and

• βγ T_s,sig [line] = C_SB T_A^* ,

where the sideband atmospheric correction ratio

• C_SB = {[1 + e^{(τ_sig-τ_img)A}]/2}.

The correction ratio may be calculated with an atmospheric model. Note the correction ratio depends on the observing zenith angle.

If the atmospheric transmission in the two sidebands is equal, τ_sig = τ_img and C_SB = 1, independent of zenith angle, and the recorded data are properly calibrated.

If the atmospheric transmission in the two sidebands is unequal, τ_sig ≠ τ_img, then for proper calibration the recorded data should be multiplied by C_SB ≠ 1.

If the atmospheric transmission in the image sideband is better than in the signal sideband, τ_sig > τ_img and C > 1. In this case the atmospheric contribution to the system noise in the image sideband is less than the contribution in the signal sideband.

For continuum sources, the source temperature is the same in both sidebands, T_s,sig = T_s,img = T_s, so

• βγ T_s [continuum] = T_A^* / 2 .

This is independent of the relative atmospheric transmission in the two sidebands.

The system temperature, T_sys, indicates the system noise level corrected for atmospheric absorption and hot spillover. Among other things, it is used for weighting scans when summing them. The system temperature includes contributions in both sidebands from the receiver, from the atmosphere, and from hot spillover. For line (SSB) observations,

• T_sys = (e^τ_sigA / α) {[T_rx + α(1 - e^-τ_sigA)T_atm + (1 - α)T_spill ] + [T_rx + α(1 - e^-τ_imgA)T_atm + (1 - α)T_spill]} .

If the air temperature and spillover temperature are the same, T_atm = T_spill = T_h, then

• T_sys = (e^τ_sigA / α) [2T_RX + (2 - αe^-τ_sigA - αe^-τ_imgA)T_h] .

The system temperature can be determined from a calibration measurement,

• Cal = [α (e^-τ_sigA + e^-τ_imgA) T_h] / [2T_RX + (2 - αe^-τ_sigA - αe^-τ_imgA)T_h] ,

so

• T_sys = {[1 + e^{(τ_sig-τ_img)A}]/2} × 2T_h / Cal = C_SB 2T_h / Cal .

The observatory software assumes the atmospheric transmission in the two sidebands is equal, τ_sig = τ_img, i. e., C_SB = 1, and reports

• T_sys = 2T_h / Cal .

The receiver temperature, T_RX, and Y factor may be determined from measurements of two loads, hot (ambient) and cold (LN₂).

• V_h = (G_sig+G_img) [T_RX + T_h ] ,
• V_c = (G_sig+G_img) [T_RX + T_c ] ,
• Y = V_h / V_c = [T_RX + T_h ] / [T_RX + T_c ] , and
• T_RX = (T_h - Y T_c) / (Y - 1) .

They may also be determined by substituting a cold load for the sky when making a (manual) calibration.

• Cal = (V_h - V_c)/V_c ,
• T_sys = 2T_h / Cal ,
• Y = V_h / V_c = 1 + Cal = 1 + 2 T_h / T_sys , and
• T_RX = (T_h - Y T_c) / (Y - 1) .

For a piece of Eccosorb drenched in LN₂, it is usually assumed that T_c = 80 K.

To correct for cold spillover and for the coupling of the beam to the source, it is necessary to measure the telescope efficiency, βγ. Then the source temperature may be determined,

• T_s = T_A^*/βγ .

The actual value of βγ, and what you may wish to call it, depends on the size of the source. There are three typical cases: sources large, similar, or small compared to the beam.

1. For large sources, those a few times the beam size, determine βγ from measurements of the full moon, which fills the main beam and the near side lobes,
   βγ (Moon) = T_A^*(Moon)/2T_Moon ,
   where T_A^*(Moon) is the observed (SSB) lunar brightness and T_Moon is the intrinsic lunar temperature. Probably you will still want to call data calibrated this way T_A^*, but note that they are calibrated with respect to the Moon.
   
   
2. For sources comparable to the beam size, determine βγ from measurements of a planet. Usually this called the main beam temperature scale, T_mb. The best planets for this are Jupiter and Mars, with Venus possibly OK. Note the planet temperatures depend on observing frequency (cf. Butler 2012 ALMA memo 594). Mars is the most reliable but it does have absorption lines in its atmosphere. For critical observations, measure the telescope efficiency at different zenith angles. For a planet of uniform temperature, calculate
   βγ (Main beam) = (T_A^*(Planet)/2T_Planet) × [1 - exp(-(D/Θ)² ln(2))]^-1 ,
   where T_A^*(Planet) is the observed (SSB) planetary brightness and T_Planet is the intrinsic planetary temperature, D is the planet diameter, and Θ the half power full beamwidth. This calculation assumes that the main beam is gaussian. The beam_eff.astro and beam_eff.class scripts perform this calculation in Astro and Class, respectively.
   
   
3. Small sources can be calibrated using the main beam efficiency or, more traditionally, using the aperture efficiency, which is very similar except that it is assumed that the beam is the diffraction function of a uniformly illuminated, perfect telescope, rather than a gaussian. In that case, slightly higher temperatures will be obtained and the result is called brightness temperature, T_b. This is not a good unit because, in its original definition, it requires a correction for the Planck factor also. This makes sense for very hot radio sources, but not for low temperature molecular sources. To avoid this difficulty, we should either invent yet another temperature unit or use Janskys. I suggest you use Janskys.

For line observations :

The efficiencies vary with the receiver and, strictly speaking, with frequency for each receiver. It is best to measure the efficiency yourself, but approximate values are:

We will use many of the same quantities and some of the results of the previous section. In addition, let us define the following quantities:

[η_hot) ≡ α.]

As noted, we assume a plane-parallel model for the atmosphere, so for a given zenith angle we have
τ = τ_z A.

We will be comparing measurements at different zenith angles rather than on- and off-source measurements. The entire passband of the receiver is treated as a single channel (the total power output is used), and we assume that the contribution of any source to the total power is negligible in comparison with the contributions of the receiver, spillover, and the atmosphere.

From the previous section, we have

and
V_sky = G [T_RX + (1 - η_hote^-τ )T_h ].
If we now form the quantity

we see that A is a linear function of airmass with slope τ_Z. With T_h and T_c known, η_hot can be found from the intercept as
η_hot = (1 - T_h/T_c) e^-(intercept).
T_spillover can then be calculated from its definition. Note that for any measurement of V_sky we may define a T_equiv by requiring
V_sky = G [ T_RX + T_equiv ].
T_spillover can then also be thought of as the T_equiv that would be measured for an airmass of zero.

If V_c (or T_c) is unknown, τ_z can be found by defining

where now S' is linear in airmass with a slope of τ_z. If T_RX is known, this intercept can be used to find η_hot. However, it is more likely that T_RX will be determined from hot and cold load measurements of V_h and V_c at known temperatures as follows:
Y = V_h/V_c = (T_RX + T_h)/(T_RX + T_c)
and
T_RX = (T_h - Y T_c)/(Y - 1) .
(This is of course unrelated to doing a skydip except insofar as measurements are typically made of V_h and V_c.)

Whether S or S' is used, it is necessary to make more than one measurement in order to find the slope and the intercept. The usual way of doing this is outlined below.

N. B. This section is obsolete. The UIP SKYDIP macro will position the telescope to successive airmasses.

A skydip is usually performed at the CSO using the 1M, 1.5M, etc., macros under the UIP. Issuing one of these macros causes the telescope to go to the corresponding zenith angle for 1, 1.5, 2, 3, 4, or 5 airmasses. The total power must be recorded at each of these positions (some day it may be automated, but not yet) and for the hot and cold loads. A utility program called TAUPLOT exists at the CSO which will do all of the necessary calculations and produce pretty plots. This may be run by simply typing tauplot at either the UIP> prompt or the $ prompt. The program will then prompt you for an airmass and voltage, and will repeat the prompt after the values are entered. A simple carriage return will cause the program to go on, and you are prompted for V_h and then V_c. Default values for T_h and V_c will be used unless other values are specified at the next prompt.

IMPORTANT NOTE - The program is set up under the assumption that G is positive (it checks that V_sky is always less than V_h). This means that although the total power voltage may be displayed as negative, the absolute value of this should be used. Furthermore, it is often more convenient to use integral values though this is not necessary. Thus, if the meter reads -0.142, a value of 142 should be entered.

Once the data entry is complete, the program calculates τ_z, the intercept, η_hot, T_spillover, Y, and T_RX, as well as equivalent temperatures T_equiv for all of the measured V_sky. These are displayed on the screen, and you are asked if you wish to plot the results. The default is to produce first a screen plot and then hardcopy, but you may specify otherwise if you desire. All of the calculated values are also displayed on the plot.

If for some reason, V_c is not measured, the program may be run using the A' formalism outlined above by not entering a value at the V_c prompt. In this case, τ_z is the only value calculated and is displayed with a message to that effect.

N. B. This section is obsolete.

Frequency calibration of the AOS backends at the CSO is done using a multiplied crystal oscillator to generate a comb with peaks separated by 100 MHz for the 500 MHz AOS or 10 MHz for the 50 MHz AOS. These allow one to determine the relationship between the IF frequency υ_IF and channel number of the AOS. (This relationship is assumed to be linear and experience shows this to be so.) When the LO command is issued, the LO frequency υ_LO is set so that the desired line frequency will correspond to the center of the passband, or approximately channel 512. This will take into account velocity and frequency offsets as well as the correct υ_IF. A default relationship between channel number and frequency is then written into the observation headers. However, because the AOS exhibits some variation in response to temperature changes, the calibration comb should be used to determine the precise relationship as follows:

• Start by using the FCAL macro to record a scan showing the response of the AOS to the comb. Then use CLASS to look at this scan using units of channel numbers.
  Determine the channel numbers corresponding to the peaks of the comb. As a peak may cover more than one channel, it may be necessary to estimate where the center of the peak occurs. Note that as displayed in CLASS, channel N extends from N-0.5 to N+0.5 and that for the purposes of these calculations, non-integral channel numbers should be used, e. g., a peak may occur at channel number 308.7.

• Next use the number of channels separating the peaks to find the width of each channel in MHz. These values should be roughly the total backend width (500 MHz or 50 MHz) divided by the total number of channels (1024), or about 0.49 MHz or 0.049 MHz. The channel width in velocity is determined from the channel width in frequency and is calculated for the desired line frequency.

• Finally determine the channel number which actually corresponds to the desired line frequency. For the 500 MHz AOS, this is the channel location of the central peak. For the 50 MHz AOS, this is the channel location halfway between the two central peaks. This should not vary by more than a few channels from the default channel location of 511.5 for both of the backends.

If after determining the correct values for the reference channel number (corresponding to the desired line frequency) and the width of the channels you desire to change from the default values to the correct values, you can do this within CLASS for data taken in the IRAM format. At the present time, there is no provision made for changing the default values that will be written as you observe, so corrections must be made after the fact. As you will want to write out the scans after they have been corrected, this means you must first open a new data file for writing. See the CLASS manual for a description of the FILE OPEN command if you are not sure how to do this.

Variables exist corresponding to the reference channel location and the channel spacing in both frequency and velocity which are used by CLASS in plotting, etc. The names of these variables are REFERENCE, FREQ_STEP, and VELO_STEP. The command EXAMINE variable_name can be used to display the current values, while the command LET variable_name = value can be used to change the values. Unfortunately, these new values do not become effective until the scan is written out to a data file and the new scan then read back in. Alternatively, the command MODIFY can be used (with conveniently different variable names, of course). To change the reference channel, type MODIFY RECENTER value; to change the channel frequency spacing, type MODIFY WIDTH value. Using MODIFY has some advantages: the values become effective immediately, and changing the frequency spacing automatically causes the velocity spacing to be changed correctly. However, with either LET or MODIFY the scan must be written to a data file or the changes will be lost. This can be done using either WRITE or UPDATE. Again, see the CLASS manual for more details on these commands.

Some conditions for which having the correct values is essential would include: adding together spectra where a line has been observed in different parts of the passband; observing weak or narrow lines which might be lost if any smearing were to occur because of AOS drifts; observing lines near the edge of the passband, or very wide lines. Note that because all values are calculated from the reference channel at around 512, errors in the channel width value become progressively more important as you move toward the edge of the passband.

At the present time, the frequency calibration comb is switched in only upon the user's request (or if the antenna is sent on a very long slew). The FCAL macro in the UIP program does this; to do this in a command file you must give the actual series of commands which correspond to the FCAL macro. Type DEFINE/LIST FCAL to see this series of commands.