EXPOSURE TIME CALCULATOR User Manual - v. 2.3.2


This manual provides a description of the formulae and computations performed by the Single-Dish Exposure Time Calculator (ETC) considering the different observing modes and observing strategies. Specific characteristics related to observations with Medicina or SRT are mentioned in the text when appropriate.

WHAT'S NEW IN VERSION 2.3.2

 Changed computations for OTF Cross Scan to match the DISCOS observing system requirements. Now one Cross Scan is considered as composed by two crosses (i.e. four subscans) instead of one, see On-The-Fly Cross Scan computations for details. Improved MathJax support.

WHAT'S NEW IN VERSION 2.3.1

 Improved documentation regarding how the ETC deals with multi-feed receivers.

WHAT'S NEW IN VERSION 2.3

WHAT'S NEW IN VERSION 2.2
Basic Formulae
System Temperature
Receiver Gain
Multi-feed Receivers

Radiometer Formula + Position Switching computations
    Input parameters
    Position Switching computations
    Output Parameters

On-The-Fly Cross Scan computations
    Input parameters
    Output parameters

On-The-Fly Map computations
    Input parameters
    Output parameters

Confusion Noise computations (contribution by V. Vaccca)


BASIC FORMULAE

 The basic formula for the ETC is the so-called radiometer formula:
$$t_{TOT} = \left(\frac{T_{sys}}{G \; s_{TOT}}\right)^2 \frac{1}{totBand}$$
where:
\(T_{sys}\) = system temperature (\(^{\circ}\)K)
\(G\) = gain (K/Jy)

and:
\(s_{TOT}\) = \(s_{TOT}^{input}/RealWorld\) = input sensitivity (converted to Jy) corrected by the \(RealWorld\) factor.

\(RealWorld\) is a numerical factor to take into account the observed discrepancy between the expected and measured sensitivity depending on the bandwidth. Observational data from Medicina and SRT show that, for large bandwidths (empirically, larger than 250 MHz), the actual sensitivity is twice the theoretical one from the radiometer formula. The term \((s_{TOT}/RealWorld)\) is the expression of the following statement: if one wants to know how much time it takes to actually reach a measured sensitivity \(s_{TOT}\), the ETC must compute how much time it takes to reach a theoretical sensitivity which is RealWorld times better than \(s_{TOT}\).

The term \(totBand\) has different meanings depending on the observing mode:
Continuum (no polarimetry):     \(totBand = \Delta \nu \; N_{IF}\)
Spectroscopy (no polarimetry):     \(totBand = \Delta \nu_{ch} \; N_{IF} \; Rebin\)
Spectropolarimetry:     \(totBand = 2 \; \Delta \nu_{ch} \; Rebin\)

where:
\(\Delta \nu \) = instrument bandwidth (Hz).
\(\Delta \nu_{ch} \) = spectroscopic channel bandwidth (Hz).
\(N_{IF}\) = number of output IF
\(Rebin\) = multiplying factor to simulate the spectral rebinning that may be applyied during the data processing.

In the following computations, \(totBand\) assumes the value corresponding to the user-selected observing mode.


SYSTEM TEMPERATURE

 System temperature for receivers working below 22 GHz is considered to be a function of the source elevation only. The user gives in input to the ETC the elevation at which the observation will be conducted (see next paragraphs) and the ETC selects among tabulated values the \(T_{sys}\) best matching the user-selected elevation.
When observing at 22 GHz we consider also a dependence on atmospheric conditions through the formula:
\(T_{sys} = T_{ric} + \eta_{f}T_{atm}(1 - e^{\frac{-\tau}{sin(El)}}) + (1 - \eta_{f}) T_{gnd}\)

where:
\(T_{ric}\) = receiver temperature (K)
\(\eta_{f}\) = feed efficiency, characteristic of each telescope
\(T_{atm}\) = atmospheric brightness, expressed as a temperature (K)
\(T_{gnd}\) = ground level temperature (K).
\({\tau}\) = atmospheric opacity
\(El\) = source elevation at the epoch of the observation.

Opacity and \(T_{gnd}\) are tabulated values varying with season (Spring, Summer, Autumn, Winter) and are selected on the basis of the ETC user input parameters. We use \(\eta_{f}=0.986\) and \(\eta_{f}=0.9\) respectively for Medicina and SRT. For Medicina we use tabulates values also for \(T_{atm}\), while for SRT the following approximated formula is used:
\(T_{atm} \approx 0.683 T_{gnd} + 77.919\)



RECEIVER GAIN

 Receiver gain is computed differently for SRT and Medicina telescopes. For Medicina, gain is computed by means of the polynomial formula: \(G(x) = DPFU \times (ax^2 + bx +c)\) K/Jy, where x is the source elevation and DPFU, a, b, and c are tabulated values for each receiver.

In the case of SRT, gain is approximately constant at any elevation, thus each receiver is characterized by a single gain value which is used by the ETC without the need to apply the above formula.


MULTI-FEED RECEIVERS

ETC computations are done always considering one feed, even if a multifeed receiver (like the K-band one at SRT and Medicina) is selected. Multi-feed receivers offer a variety of observing strategies (from nodding to continuous on-the-fly mapping etc.) that can be hardly represented in a simple and effective way in the Exposure Time Calculator. It is therefore left to the user the adjustment of the results to take into account the adopted multi-feed observing strategy.


RADIOMETER FORMULA + POSITION SWITCHING COMPUTATIONS

INPUT PARAMETERS

Computations in this case are done by simply applying the radiometer formula to the user-selected values.
In addition, an example case of Position Switching observation is evaluated, assuming to perform an ON-OFF-OFF-ON cycle. The following Figure illustrates the geometry of a Position Switching observation.



The ON position corresponds to the on-source one, while the OFF observation is executed at a nearby position on the empty sky. The OFF position is assumed to be at a distance of 5 beamsizes from the ON one. An ON-OFF-OFF-ON sequence is assumed to minimize the time requested to move the telescope between consecutive positions.

ETC computations are done always considering one feed, even if a multifeed receiver (like the K-band one at SRT and Medicina) is selected.

The user provides in input:

CASE A: GIVEN \({\bf s_{TOT}}\) COMPUTE \({\bf t_{TOT}}\)
The radiometer formula is used to compute the exposure time needed to reach a given sensitivity:
$$t_{TOT} = \left(\frac{T_{sys}}{G \; s_{TOT}}\right)^2 \frac{1}{totBand}$$


CASE B: GIVEN \({\bf t_{TOT}}\) COMPUTE \({\bf s_{TOT}}\)
By reverting the radiometer formula:
$$ s_{TOT} = \left(\frac{T_{sys}}{ G}\right) \; \sqrt{\frac{1}{totBand \; t_{TOT}}} $$

POSITION SWITCHING COMPUTATIONS

An example case for Position Switching observations is also computed as a particolar case for the radiometer formula mode.
For Position Switching computations we assume that the four exposures in the cycle will be combined as [(ON\(_1\)-OFF\(_1\)) + (ON\(_2\)-OFF\(_2\))]/2 to get the final result. We also assume that the time spent on each position is the same (\(t_{ON} = t_{OFF}\)), that measurements are independent and that they have identical uncertainties.
The r.m.s. on the frame resultant from one (ON-OFF) subtraction is:
$$ s_{ON-OFF} = \sqrt{s_{ON}^{2} + s_{OFF}^2} = \sqrt{2} \times s_{ON} $$
The r.m.s. on the frame resultant from the average of the two (ON-OFF) subtractions, i.e. the r.m.s. of one ON-OFF-OFF-ON cycle is, according to the error propagation for the mean:
$$ s_{cycle} = \sqrt{\frac{1}{2} \times s_{ON-OFF}^{2}} = \sqrt{\frac{1}{2} \times (\sqrt{2} \times s_{ON})^{2}} = s_{ON} $$


A) If the user-selected input sensitivity is intended to be reached in one complete ON-OFF-OFF-ON cycle of Position Switching, i.e. \(s_{cycle} = s_{TOT}\), \(t_{cycle}\) is computed as:
$$ t_{cycle} = 2 \times (t_{ON} + t_{OFF} + t_{shift}) + t_{prep} $$
The value of \(t_{ON}=t_{OFF}\) is computed by obtaining \(s_{ON}\) from the above equation for \(s_{cycle}\) and applying the:
$$ t_{ON} = \left(\frac{T_{sys}}{G}\right)^2 \frac{1}{s_{ON}^2 \times totBand} $$
The other terms in \(t_{cycle}\) are: \(t_{prep}\), which is the instrument setup time, {\bf currently set to zero for Medicina and SRT telescopes (as of Dec. 2014)} and \(t_{shift}\), the slewing time needed to move from one position to the next. The value for \(t_{shift}\) is computed by means of the uniformly accelerated motion formula assuming that the OFF position is 5 beamsizes away from the ON position:
$$ t_{shift} = \sqrt{(5 \times HPBW) \; \frac{2}{MaxAcc}} $$
The value of the maximum acceleration is set inside the ESCS/Nuraghe CommonDataBase (CDB). For Medicina and SRT we consider the maximum acceleration \(MaxAcc\) to be \(0.4 \,\, {\rm deg/ sec}^2\) and \(0.25 \,\, {\rm deg / sec}^2\) respectively.

B) In the opposite case in which the user-selected input exposure time is intended to be used for the execution of a complete ON-OFF-OFF-ON cycle of Position Switching, \(t_{cycle} = t_{TOT}\). The value for \(s_{ON}\) to be inserted in the \(s_{cycle}\) equation is evaluated as:
$$ s_{ON} = \left(\frac{T_{sys}}{G}\right) \; \sqrt{\frac{1}{totBand \times t_{ON}}} $$
where \(t_{ON}\) is obtained by reverting the equation for \(t_{cycle}\):
$$ t_{ON} = [ t_{cycle} - (2 \times t_{shift}) - t_{prep}]/ 4 $$


OUTPUT PARAMETERS

When selecting the radiometer formula computations, the following parameters are given in output:

ON-THE-FLY CROSS SCAN COMPUTATIONS

 OTF Cross Scans acquired with the DISCOS observing system at the INAF radio telescopes are by default composed by two crosses, i.e. four subscans. Each scan is thus composed by four orthogonal subscans, to form two crosses. The source is assumed to be coincident with the intersection of the cross. Data acquisition is continuously performed in short time intervals during the scan. To reach a given sensitivity over a position many consecutive Cross Scans may be required.
An example of the Cross Scan geometry is shown in the following Figure, where two consecutive crosses are executed over the same position (the small displacement between the blue and red crosses is for visualization purposes) and each arrow represent a subscan.


ETC computations are done always considering one feed, even if a multifeed receiver (like the K-band one at SRT and Medicina) is selected.

INPUT PARAMETERS

For OTF Cross Scan computations the user provides in input:

IMPORTANT NOTES:

1) In our computations, the sensitivity associated to a single subscan coincides with the one calculated, using the plain radiometer formula, considering an integration time equal to the sampling interval. This way, it practically represents the RMS noise that will characterize the single subscan. When the user provides the desired final sensitivity, the ETC computes how many subscans must be integrated in order to reach it, subsequently estimating the total required observing time - as it includes intra-subscan and inter-subscan dead times (known a priori). Conversely, if the user provides a time interval and wants to know to which sensitivity it corresponds to, such time is intended as the total duration of the observation, and the ETC computes how many subscans can be executed in that time range, estimating the sensitivity reached by integrating them. Those users who plan to employ the CAP (Cross-scan Analysis Pipeline) tool, made available by INAF, must take into account that it separately performs flux density measurements on each scan direction (e.g. separately integrating RA scans and Dec scans, then computing the weighted average of the two measurements). This way, the actual integration time associated to the final measurement is half the total time, and the sensitivity drops by a \(\sqrt{2}\) factor. Cross-scans are necessary in order to accurately assess the pointing offset and correct the flux density measurements accordingly.

2) Due to the presence of polarized RFI or other sources of asymmetry between the LCP and RCP data, computations conservatively take into account a single IF. If the data quality allows for the integration of LCP and RCP acquisitions, thus doubling the integration time, the actual sensitivity will improve by a factor of \(\sqrt{2}\). As far as Total Power acquisitions are concerned, the single-IF scenario is not infrequent.

3) Due to the need to make some approximation (f.i. the Cross Scan geometry requires the total number of subscans to be a multiple of four), the effective total sensitivity or the effective total time are re-computed, starting from the desired ones and taking into account the observing setup. Effective values may slightly differ from the input values.

4) Given the observing constraints at the telescopes, sensitivity and time are computed considering always a minimum number of one Cross Scan (i.e. 4 subscans).


CASE A: GIVEN \({\bf s_{TOT}}\) COMPUTE \({\bf t_{TOT}}\)
The total time \(t_{TOT}\) needed to reach the desired sensitivity \(s_{TOT}\) at the source position is given by:
$$ t_{TOT} = (2\,(2\,subscanDuration + deadTime)) \; nCrossScans $$
Where:
\(nCrossScans\) = total number of Cross Scans that must be executed to reach the desired sensitivity;
\(deadTime\) = time which is not spent acquiring data during the execution of each one of the two crosses composing a Cross Scan.

To evaluate \(t_{TOT}\) we must compute: \(deadTime\), \(subscanDuration\) and \(nCrossScans\).

- Evaluate deadTime:
The parameter \(deadTime\) is the time which is not spent acquiring data during the execution of each one of the two crosses composing a Cross Scan. It is given by the sum of two terms: \(interSubscanTime\) and \(intraSubscanTime\). The first term is given by the duration of the acceleration/deceleration ramps taking place before/after the actual subscans, and is determined as a function of the scanning speed and of a given fraction of the maximum acceleration allowed by the mount, specific for each telescope.
For Medicina and SRT we consider the maximum acceleration MaxAcc to be \({\bf 0.4^{\circ} / s^2}\) and \({\bf 0.25^{\circ} / s^2}\) respectively, and we use one tenth of the maximum acceleration to compute the ramps (as the system itself does, to guarantee the maximum precision):
$$ rampTime = \frac{speed}{0.1 \, MaxAcc} $$
The execution of each subscan implies to perform 2 ramps, so:
$$ interSubscanTime = \frac{2\, speed}{0.1 \, MaxAcc} $$
The term \(intraSubscanTime\) is an indicative intra-scan slewing time (i.e. performed at full \(MaxAcc\)), which takes into account the change in position among subscans. By reverting the formula for the uniformly accelerated motion (\(s = \frac{1}{2} \, a \, t^2\)) and considering that \(s\) (the intra-scan path) in this case is the hypotenuse of a triangle whose sides are the two half-subscans:
$$ intraSubscanTime = \sqrt{\frac{\sqrt{2} \; subScanLength}{MaxAcc}} $$
It is now possible to estimate the overall dead time for one cross in a Cross Scan: it is given by twice the \(interSubscanTime\) (two sub-scans form one Cross Scan) plus once the \(intraSubscanTime\) (the telescope changes position between subscans once per Cross Scan).
$$ deadTime = 2 \; interSubscanTime + intraSubscanTime $$
- Evaluate subscanDuration:
The duration of a single subscan is easily computed as:
$$ subscanDuration = subscanLength \; / \; scanSpeed $$
- Evaluate nCrossScans:
First compute \(t_{TOT,sample}\) = time necessary to reach the desired sensitivity \(s_{TOT}\) by using the radiometer formula (inclusive of the Real World factor if appropriate):
$$t_{TOT,sample} = \left(\frac{T_{sys}}{G \; s_{TOT}}\right)^2 \frac{1}{totBand}$$
Given the sample interval, the number of subscans to be performed to reach the requried sensitivity at the source position is:
$$ nSubscans = \frac{t_{TOT,sample}}{sampleInterval} $$
The number of complete Cross Scans is:
$$ nCrossScans = (nsubscans)/4$$
This value is rounded up to the upper integer.


CASE B: GIVEN \({\bf t_{TOT}}\) COMPUTE \({\bf s_{TOT}}\)
The parameter \(t_{TOT}\) is the overall observing time and thus includes also \(deadTime\). The actual sensitivity \(s_{TOT}\) is computed starting from the effective time spent on Cross Scans, i.e. excluding \(intraSubscanTime\) and \(interSubscanTime\). First we need to compute how many Cross Scans can be executed in the given \(t_{TOT}\):
$$ nCrossScans = \frac{t_{TOT}}{singleCrossTotalTime} $$
where \(singleCrossTotalTime\) is the total time to execute one Cross Scan (including its overheads):
$$ singleCrossTotalTime = 2\, (2 \, subscanDuration + deadTime) $$
and \(deadTime\) is computed as in Case A. The value of \(nCrossScans\) is rounded to the upper integer. The sensitivity associated to an integration time equal to the sampling interval can be computed using the radiometer formula:
$$s_{subScan} = \left(\frac{T_{sys}}{ G}\right) \; \sqrt{\frac{1}{totBand \; \times \; sampleInterval}} $$
Then the sensitivity \(s_{TOT}\) at the source position is obtained from the instantaneous sensitivity and the number of subscans in a Cross Scan:
$$ s_{TOT} =\frac{1}{\sqrt{4 \times nCrossScans}} \; s_{subScan} $$

OUTPUT PARAMETERS

When selecting the On-The-Fly Cross Scan computations, the following paramenters are given in output:

ON-THE-FLY MAP COMPUTATIONS

 Each map is composed by a sequence of back-and-forth On-The-Fly subscans. Each subscan is shifted with respect to the previous one in order to fully cover the desired sky region.
An example of On-The-Fly Map geometry is shown in the following Figure, where each arrow represent a subscan.


ETC computations are done always considering one feed, even if a multifeed receiver (like the K-band one at SRT and Medicina) is selected.

INPUT PARAMETERS

For OTF Map computations the user provides in input:

IMPORTANT NOTES:

1) if the user provides a sensitivity and wants to know the time necessary to reach it, the sensitivity is intended to be reached over an HPBW by performing OTF map observations for a duration equal to the computed time. That means time is not on-source, but is the overall duration of the observation including "dead time" (a priori known).
Accordingly, if the user provides a time and wants to know to which sensitivity it corresponds, time is intended as the total time needed to perform all the map observations necessary to reach the given sensitivity over an HPBW, including "dead time".

2) Due to the need to make some approximation (f.i. the requirment that at least 1 map is to be executed), the effective total sensitivity or the effective total time are re-computed, starting from the desired ones and taking into account the observing setup. Effective values may slightly differ from the input requested values because of the above approximations.


CASE A: GIVEN \({\bf s_{TOT}}\) COMPUTE \({\bf t_{TOT}}\)
The time \(t_{TOT}\) can be expressed as:
$$ t_{TOT} = n_{map} * t_1 $$
where:
\(n_{map}\) = number of maps to be combined to reach \(s_{TOT}\).
\(t_1\) = time needed to complete one map given the observing parameters, including dead time.

To evaluate \(t_{TOT}\) we must compute: \(n_{map}\) and \(t_1\).

- Evaluate \({\bf n_{map}}\):
The expression for \(s_{TOT}\) can be written as:
$$ s_{TOT} = \frac{s_1}{\sqrt{n_{map}}} $$
Where \(s_1\): sensitivity reached over one HPBW in a single map assuming that each beam size is sampled with \(linesPerHPBW\) points (i.e. \(linesPerHPBW\) scans are needed to sample one HPBW). It can be easily computed by means of the sensitivity per HPBW reached in a single subscan:
$$ s_1 = \frac{sensitivityHPBW_{singlesubscan}}{\sqrt{linesPerHPBW}} $$
where the formula for \(sensitivityHPBW_{singlesubscan}\) is the same as in the OTF Cross Scan case.
By substituting \(s_1\) and \(s_{TOT}\) in the above formula, the number of maps \(n_{maps}\) can be evaluated.

- Evaluate \({\bf t_1}\):
The time needed to complete 1 map, including dead time, is:
$$ t_1 = (t_{singleScan} + deadTime) \; linesPerMap $$
where:
$$ t_{singleScan} = mapSize \; / \; scanSpeed $$
$$ linesPerMap = \frac{mapSize}{HPBW} \; linesPerHPBW $$
and \(deadTime\) in this case is:
$$ deadTime = interSubscanTime + intraSubscanTime $$
where now, differently to what happend in the OTF Cross Scan case, the path to be performed in uniformly accelerated motion is exactly given by the transverse distance between two subscans (no more need to build a triangle). The parameter intraSubscanTime is thus:

\(intraSubscanTime = \sqrt{\frac{HPBW}{linesPerHPBW} \; \frac{2}{MaxAcc}}\)

Note that in the formula for \(t_1\) we make an approximation which has a negligible effect on the computation: the correct one would be \(t_1 = (t_{singleScan} + interSubscanTime) \, linesPerMap + intraSubscanTime \, (linesPerMap-1)\) since the last subscan does not require additional turning time once completed.

Map size for pointlike sources is:
$$ mapSize = HPBW + 2\,(mapEdge \times HPBW) $$
While for extended sources:
if \(size_x>HPBW\) or \(size_y>HPBW\):   \( mapSize = [max(size_x,size_y)] + 2\,(mapEdge \times HPBW) \)
if \(size_x \le HPBW\) and \(size_y \le HPBW\):   \( mapSize = HPBW + 2\,(mapEdge \times HPBW) \)

Finally, the Signal to Noise ratio over 1 HPBW area on the combined nmap is computed as:
pointlike sources:   $$\frac{S}{N} = \frac{Flux}{s_{TOT}} $$
extended sources:   $$\frac{S}{N} = \left(\frac{Flux \times HPBWarea}{sourceArea}\right) \frac{1}{s_{TOT}}$$
where:
$$ sourceArea = \pi \times 0.5 \, size_x \times 0.5 \, size_y $$
$$ HPBWarea = \pi \times (0.5 \, HPBW)^2 $$

NOTE that the formula for extended sources gives an approximated result when \(size_x\) or \(size_y\) is sensibly smaller than \(HPBW\) i.e. the source is elongated and it actually does not fill "well" the beam. Results are accurate if the extended source completely fills the beam in both directions.


CASE B: GIVEN \({\bf t_{TOT}}\) COMPUTE \({\bf s_{TOT}}\)
The parameter \(t_{TOT}\) is the overall observing time and thus includes also \(deadTime\). The actual sensitivity \(s_{TOT}\) must be computed using the effective time spent on the map, i.e. excluding \(intraSubscanTime\) and \(interSubscanTime\). First we need to compute how many maps can be executed in the given \(t_{TOT}\), rounded to nearest integer (and at least equal to 1):
$$ nMaps = {\rm (nearest \;integer)}(t_{TOT} \; / \; t_1) $$
where \(t_1\) is the time needed to complete one map given the observing parameters, including dead time, and is computed as in Case A.
The sensitivity \(s_{TOT}\) is thus computed as:
$$ s_{TOT} = \frac{s_1}{\sqrt{nMaps}} $$
Where \(s_1\) is the sensitivity reached over one HPBW in a single map, computed as in Case A.


OUTPUT PARAMETERS

When selecting the On-The-Fly Map computations, the following paramenters are given in output:

CONFUSION NOISE COMPUTATIONS (contribution by V. Vacca)

 Currently, confusion noise computations are implemented for SRT only.
The rms width of the point-source confusion amplitude distribution calculated by Condon (1974) at the generic frequency \(\nu\) is:
$$\sigma_{c,\nu} = \left(\frac{q^{3-\gamma}}{3-\gamma}\right) ^{\frac{1}{\gamma-1}} \left(\frac{k \Omega_{b}}{\gamma -1}\right) ^{\frac{1}{\gamma-1}} \left(\frac{\nu}{\nu_{0}}\right) ^{-\alpha}$$
Where \(q\) is the signal-to-noise ratio under which we expect confusion, \(\Omega_{b}\) the beam solid angle, \(\alpha\) the spectral index (with \(S = S_{0} \nu^{-\alpha}\)), and \(k\) and \(\gamma\) are the parameters of the power-law differential source counts calculated at the frequency \(\nu_0\):
$$ n_{\nu_{0}}(S) = k S^{-\gamma}$$
Note that for the ETC computations we assumed q = 5.

To calculate the rms confusion at various frequencies the following power-law differential counts have been used: REFERENCES:
Bondi, M., Ciliegi, P., Zamorani, G., et al. 2003, A&A, 403, 857.
Condon, J. J. 1974, ApJ, 188, 279.
Davies, M. L., Franzen, T. M. O., et al. 2011, MNRAS, 415, 2708.
Owen, F. N., Morrison, G. E., Klimek, M. D., & Greisen, E. W. 2009, AJ, 137, 4846.
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