# How to Determine Parameters

## Determining Parameters and Sensitivities

You can determine observation parameters and estimate sensitivities in the following way. See Fig. 1-1 in "Introduction" to understand meanings of variables to appear below.

#Derivation of equations is shown in "Miscellaneous Information".

At first you should set fundamental parameters listed below:

- The dimension of the mapping area
l
_{1}["] l_{2}["] (l_{1}along the scan, l_{2}across the scan). - Time to be taken in a scan row (on-source)
t
_{scan}[s] : scan speed on the celestial sphere then becomes v_{scan}["/s] = l_{1}/t_{scan}.- t
_{scan}should be chosen so that the antenna does not move more than 1/3 or 1/4 of HPBW during the data dump (integration) time t_{dump}[s], 0.1 (ACG) or 0.2 (WHSF) sec. - See Hints on Choosing Parameters.

- t
- Set the number of on-source scans per one OFF N
_{scan}^{SEQ}; separation between the scan rows l ["]; and the grid spacing of a map to be made d ["] d ["]. Now the number of scan rows is written as N_{row}= l_{2}/ l +1 .- We recommend to set l to be about 1/3 of HPBW, and d to be 1/3 -- 1/2 of HPBW.
- See Hints on Choosing Parameters.

Using these values, parameters and the sensitivity are determined as follows. You can calculate them using a tool (otfaste.pl).

- An overhead time per one scan row t
_{OH}[s] is written as t_{OH}= 6 + 8/N_{scan}^{SEQ}. - Determine an optimal OFF-point integration time t
_{OFF}[s] as

.

Here =4.3 (It depends on the type and parameters of convolution function. With the default parameters, it becomes 4.3 for Bessel*Gauss, 1.2 for Sinc*Gauss, 6.3 for Gauss, 1.0 for Pillbox, 10.2 for Spheroidal. See Miscellaneous Information: Derivation of Observation Parameters and Sensitivity). - The total time spent to run a observe table including OFF, R-SKY, antenna slew, etc. is estimated to be

.

Here f_{cal}is an overhead to obtain R-SKY calibration data (if you use 1 minute at every 15 minutes, f_{cal}=16/15).- If t
_{OBStot}becomes larger than the interval between pointing observations, you have to reduce t_{OBStot}by, e.g., divide the mapping area into multiple observation tables.

- If t
- The total on-source time is

. - The effective integration times of ON and OFF for a map grid are written as, respectively,

and

.

- Using the system noise temperature T
_{sys}[K] and the frequency resolution of the map to be made B [Hz], the rms noise temperature of the map is estimated as

.

Here_{q}is the quantization efficiency of the spectrometer: it is 0.88 for the MAC (ACG), 0.60 for the WHSF.

Compared with the common position-switch ("step-and-integrate") observations, higher observing efficiency is expected to be acheived in OTF observations due to the following facts:

- The ratio of on-source time to the total time spent
_{ON/OBS}= t_{ONtot}/t_{OBStot}becomes larger because not only the dead time is reduced, but also optimum OFF point integration time is relatively shorter (t_{OFF}< t_{scan}). - In general t
_{cell}^{ON}< t_{cell}^{OFF}, thus T_{A}^{*}nearly equals to T_{A}^{*}(ON) (for position-switch observations, T_{A}^{*}T_{A}^{*}(ON) ).

See Miscellaneous Information: FAQ section: an example of comparison between OTF and position-switch is shown. In case of the limit of dead time -> 0, the efficiency in time can get 4 times higher than the position-switch observations.

## Hints on Choosing Parameters

- As t
_{scan}becomes long (v_{scan}becomes small), nominal observing efficiency increases. The upper limit of t_{scan}is defined by the following conditions:- As time between OFF integrations becomes longer, baseline gets poor.
- When the time spent to cover the entire map becomes too long, the quality of the obtained map is no longer "uniform" (the uniformity is one of main advantages of OTF observations).

- On the other hand, the lower limit of t
_{scan}is defined by a condition that the sampling along the scan should be frequent enough. At least Nyquist sampling is required. We recommend to sample more frequently than 1/3--1/4 of the telescope beam in order to avoid beam smearing: i.e., t_{dump}v_{scan}< (1/3--1/4) HPBW. - Spacings between scan rows l should be smaller than the Nyquist sampling rate. Taking the antenna jitters etc. into account, oversampling is recommended (about 1/3 of the HPBW). It is also recommended to set l to be equal to or smaller than the grid spacing d.
*The resultant map has so-called scanning effects if scans along only one direction are coadded. The scanning effect will be significantly reduced by combining orthogonal scans.*The so-called*PLAIT*algorithm described by Emerson & Gräve (1988) is implemented in the data reduction software. See Data Reduction section. [The*PRESS*method by Sofue & Reich (1979), which is applicable to single-direction-scanned maps, is also impremented for those who could not obtain orghogonal scans.]

## An Example of Parameters

Assuming to use the MAC (ACG) with parameters
l_{1}=l_{2}=300 ["],
t_{scan}=30 [s],
l=7.5 ["],
d=7.5 ["],
T_{sys}(SSB)=500 [K], and
B=1 [MHz]

% perl otfaste.pl.txt ### OTF sensitivity estimation for ASTE ### backend type (1:MAC, 2:WHSF): 1 length of mapping area (along the scans) l1 [arcsec]: 300 length of mapping area (perp. to the scans) l2 [arcsec]: 300 time for scan tscan [s]: 30 number of ONs per OFF Nscan(SEQ): 1 separation between scans (delta)l [arcsec]: 7.5 map grid d [arcsec]: 7.5 system noise temperature Tsys [K]: 500 resolution bandwidth B [kHz]: 1000 --- Summary of parameters of your observation --- Backend type : MAC Length of mapping area (along the scans) l1 [arcsec]: 300.00 Length of mapping area (perp. to the scans) l2 [arcsec]: 300.00 Time for scan tscan [s]: 30.00 Number of ONs per OFF Nscan(SEQ): 1 Separation between scans (delta)l [arcsec]: 7.50 Map grid d [arcsec]: 7.50 System noise temperature Tsys [K]: 500.00 Resolution bandwidth B [kHz]: 1000.000 Scan speed (1.0" per 0.1s sample) ["/sec]: 10.0 Effective integrtion time of ON [sec]: 3.31 Effective integrtion time of OFF [sec]: 12.00 Overhead time per one scan [sec]: 14.0 Total on-souce time [min]: 20.5 Total time per one map [min]: 40.8 Observing efficiency eta(ON/OBS) : 0.50 Optimal OFF-point integration time [sec]: 12.0 -> 12 Rms noise temperature of one map [K]: 0.353

The optimal OFF-integration is 12 seconds.
This table will take about 40 minutes to run and
T_{A}^{*}=0.35 [K]
is obtained. Effective spatial resolution becomes 24"
(see Miscellaneous Information: Convolution in "Make Map").

#FYI: Redirection of the standard input to a file (sample) may make recalculation much easier: e.g., perl otfaste.pl.txt < otfaste_input.txt. If you execute this script as perl otfaste.pl.txt tex, you can see an example of results of sensitivity estimation.