[tel-00726959, v1] Caractériser le milieu interstellaire ... - HAL - INRIA

IRAM-30m EMIR time/sensitivity estimator2. tracked observationsIRAM-30m EMIR time/sensitivity estimator3. on-the-fly observationstel-00726959, version 1 - 31 Aug 20121.4 The number of polarizationsHeterodyne mixers are coupled to a single linear polarization of the signal. Hence, heterodyne receivershave at least two mixers, each one sensitive to one of the two linear polarization of the incoming signal.Both mixers are looking at the same sky position. This implies that we have to distinguish between thetime spent on a given position of sky and the human elapsed time. Indeed, we will use the time spent ona given position of the sky when estimating the sensitivity, while we will give human elapsed time for thetelescope and the on and off times.If the mixers are tuned at the same frequency, the times spent on and off in the same direction ofthe sky will be twice the human elapsed time. We thus have to introduce the number of polarizationsimultaneously tuned at the same frequency, npol, which can be set to 1 or 2. It happens that for EMIR,the two polarizations are always tuned at the same frequency, i.e. npol = 2. The simplest way to take intoaccount the distinction between human time and sky time is to slightly modify the radiometer equationto take into account the number of polarizationTsysσ =ηspec√dν npol tsigThis equation implies that ton, toff, tonoff and ttel will be human times.1.5 Switching modes and observation kindsSwitching is done in two main ways.ton toffwith tsig = . (5)ton + toffPosition switch where the off-measurement is done on a close-by sky position devoid of signal. Wobblerswitching is a particular case.Frequency switch where the telescope always points towards the the source and the switching is donein the frequency (velocity) space.Moreover, there are two main observation kinds.Tracked observations where the telescope track the source, i.e. it always observes the same position inthe source referential. The result is a single spectra.On-The-Fly observations where the telescope continuously slew through the source with time to mapit. The result is a cube of spectra.In the following, we will work out the equations needed by the time/sensitivity estimator for eachcombination.2 Tracked observations2.1 Frequency switchedIn this case, all the time is spent in the direction of the source. However, the frequency switching alsoimplies that all this times can be counted as on-source and off-source times. ThusandIRAM-30m EMIR time/sensitivity estimatortonoff = ton = toff, (6)tsig = ton2 = toff2 = tonoff2 , (7)√2 Tsysσfsw =. (8)ηspec√dν npol ηtel ttel43. on-the-fly observationsIn addition, we must ensure that the user does not try to scan faster than the telescope can slew. To dothis, we need to introduce• The linear scanning speed, v linear , and its maximum value, v maxlinear .• The area scanning speed, v area , and its maximum value, v maxarea. When the scanning pattern is linear,then v area and v linear are linked throughv area = v linear ∆θ, (16)where ∆θ is the separation between consecutive rows. To avoid nasty signal and noise aliasingproblems, we must ensure a Nyquist sampling, i.e.3.2 Frequency switched∆θ =θ2.5 . (17)In frequency switched observations, the switching happens as the telescope is slewed. This is correct aslong as the switching time is much smaller than the time needed to slew a significant fraction of thetelescope beam.It is easy to understand thattonoff = t toton = t totoff , (18)andt beamsigThe velocity check can then be written as3.3 Position switchedt beamon = t beamoff = tonoffnbeam= tbeam on2, (19)= tbeam off = tonoff , (20)2 2nbeam√ 2 nbeam Tsysσfsw =. (21)ηspec√dν npol ηtel ttelAmap≤ varea. max(22)tonoff3.3.1 Two key points: 1) Sharing OFF among many ONs and 2) system stability timescaleWhen the stability of the system is long enough, we can share the same off for several independent onpositionsmeasured in a row (e.g. ON-ON-ON-OFF-ON-ON-ON-OFF...). The first key point here is thefact that the on-positions must be independent. The OTF is an observing mode where the sharing ofthe off can be used because the goal is to map a given region of the sky made of independent positionsor resolution elements. When sharing the off-position between several on, Ball (1976) showed that theoptimal off integration time ist optimaloff = √ n on/off ton (23)where n on/off is the number of on measurements per off. Replacing toff by its optimal value in eq. 5, weobtain√tonTsys1tsig =1 and σ =1 + √ . (24)1 + √ non/offηspec√dν npol ton non/offWe thus see that the rms noise decreases when the number of independent on per off increases. It seemstempting to have only one off for all the on positions of the OTF map. However, the second key point ofthe method is that the system must be stable between the first and last on measurement. To take thispoint into account we must introduce2.2 Position switchedIn this case, only half of the time is spent in the direction of the source. Thusand2.3 Comparisonton = toff = tonoff2 , (9)tsig = ton2 = toff2 = tonoff4 , (10)σpsw =2 Tsys. (11)ηspec√dν npol ηtel ttelFor tracked observations, position switched observations results in a noise rms √ 2 larger than frequencyswitched observations for the same elapsed telescope time. In other words, frequency switched observationsare twice as efficient in time to reach the same rms noise than position switched observations.However, time efficiency is not the only criteria of choice. Indeed, with the current generation ofreceivers (before march 2009), the IF bandpass is much cleaner in position switched than in frequencyswitched observations. Frequency switched is thus really useful only when the lines are narrow so that theIF bandpass can be easily cleaned out through baselining with low order polynomials.3 On-The-Fly observations3.1 Additional notions and notationsThe On-The-Fly (OTF) observing mode is used to map a given region of the sky. The time/sensitivityestimator will have to link the elapsed telescope time to cover the whole mapped region to the sensitivityin each independent resolution element. To do this, we need to introduce• Amap and Abeam, which are respectively the area of the map and the area of the resolution element.The map area is a user input while the resolution area is linked to the telescope full width at halfmaximum (θ) byηgrid π θ2Abeam =4 ln(2)where ηgrid comes from the fact that the OTF data is gridded by convolution. When the convolutionkernel is a Gaussian of FWHM equal to θ/3 (the default inside the GILDAS/CLASS software), it iseasy to show that(12)ηgrid = 1 + 1 ≃ 1.11. (13)9• The number of independent measurement (nbeam) in the final map which is given by• The on and off time spent per independent measurement, t beamonthen be writtennbeam = Amap . (14)Abeamt beamsig = tbeam on t beamofft beam on + t beamoff• The on and off time spent to map the whole map, t toton and t totoffin a way which depends on the switching scheme.IRAM-30m EMIR time/sensitivity estimator5and t beamoff. The associated t beamsig can. tonoff is deduced from ttot on(15)and t totoff3. on-the-fly observations• The concept of submap, which is a part of a map observed between two successive off measurements.• Asubmap, which is the area covered by the telescope in each submap.• nsubmap the number of such submaps needed to cover the whole map area.• tstable, the typical time where the system is stable. This time will be the maximum time betweentwo off measurements, which is noted tsubmap.• ncover, the number of coverages needed either to reach a given sensitivity or to exhaust the acquisitiontime.• t coveroncoverage.and t coveroff are the times spent respectively on and off per independent measurement and perWe note that the number of on per off (n on/off) is a purely geometrical quantity. This implies that the timespent off is linked to the time spent on by Eq. 23 both in each individual coverage and when averaging allthe coverages.3.3.2 Relation between tonoff and t beamsigBy construction• The number of submaps is the area of the map divided by the area of a submapnsubmap =Amap . (25)Asubmap• The number of on per off is the number of independent resolution elements in each submapn on/off = Asubmap . (26)Abeam• The number of independent resolution elements in the map is the product of number of submaps bythe number of on per offnbeam = nsubmap n on/off. (27)• The submap area is the product of the area velocity by the time to cover itAsubmap = v area tsubmap. (28)• The time to scan a submap is the sum of the n on/off independent on integration timetsubmap = n on/off t coveron . (29)• The relations between times per coverage and times integrated over all the coverages aret beamon = ncover t coveron and t beamoff = ncover t coveroff with t coveroff = √ n on/off t coveron . (30)• Using the last two points, it is easy to derivet beamsig = ncover t coversig =ncover tsubmapn on/off + √ . (31)n on/off67