6. GREAT

6.1 Specifications

6.1.1 Instrument Overview

The German REceiver for Astronomy at Terahertz Frequencies (GREAT) is a modular heterodyne instrument with multiple configurations (see below) that provide high resolution spectra (up to R = 108) in several frequency windows between 0.4900–4.7448 THz.

The front-end unit consists of two independent dewars that operate simultaneously. Each dewar contains one of the following channels:

upGREAT Low Frequency Array (LFA)
The LFA is a 14-pixel array with two polarizations (Horizontal and Vertical with 7 pixels each) with 6 pixels arranged in a hexagonal pattern around the central pixel. Both polarizations of the array (LFAH and LFAV) can be tuned to one simultaneous or two independent frequencies in the 1.835-2.007 THz range. Tuning the LFA polarizations to two independent frequencies is offered on a best effort basis. The [OI] 145 µm line can can only be observed in V polarization, which has an additional tuning range from 2.060 to 2.065 THz.

upGREAT High Frequency Array (HFA)
The HFA is a 7-pixel array arranged with 6 pixels in a hexagonal geometry around a central pixel. The local oscillator is a Quantum Cascade Laser with a narrow tuning range limited to the [OI] 63 µm frequency of 4.74477749 THz (with a tuning range of roughly + 250 km/s to -100 km/s in LSR velocity).

4GREAT
4GREAT consists of four co-aligned (to within a few arcsec) pixels at four different frequencies. The tuning ranges of these pixels are: 491-635 GHz (4G1), 890-1092 GHz (4G2), 1240-1525 GHz (4G3), and 2490-2590 GHz (4G4).

The GREAT instrument uses eXtended bandwidth Fast Fourier Transform Spectrometers (XFFTS) as backends. Each XFFTS has a bandwidth of 4 GHz and 16,384 channels, thus providing a resolution of 244 kHz (note that the useful bandwidth depends on the tuning and the atmospheric transmission). The beam size is close to the diffraction limit—about 14 x (1.9/ v THz) arcsec, where ν is the frequency observed in THz.

A detailed description of the GREAT instrument and its performance during the 2011 Basic Science flights can be found in the GREAT special issue of Astronomy & Astrophysics (Heyminck et al. 2012, A&A, 542, L1). The upGREAT arrays are described in Risacher et al. (2016, A&A, 595, A34) and Risacher et al. (2018, JAI 7, 1849914). 4GREAT is described in Duran et al. (in prep.). Additional information about the instrument can be found on the GREAT website.

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6.1.1.1 Design

Heterodyne receivers work by mixing the signal from a source on the sky at a given frequency νs with the signal from a local oscillator (LO) at a specified (and precisely controlled) frequency νLO. This signal mixing results in two frequency bands on the sky called the signal and image bands being folded into the intermediate frequency (IF) band in the few GHz range, which is amplified and then spectrally analyzed. These bands are located symmetrically on either side of the LO frequency νLO, and they are separated from that LO frequency by an intermediate frequency νIF = |ν– νLO|. GREAT operates in a double sideband mode (DSB), which means that both the image and signal bands are not separated and are equally sensitive to incoming radiation. By definition, the band containing the spectral line of interest is always the signal band and the opposite band is the image band. This signal band can be chosen to be either above (Upper Sideband, USB) or below (Lower Sideband, LSB) the LO frequency; see Figure 6-1. For sources rich in spectral lines, care has to be taken so that a spectral line in the image band does not overlap or blend with the line of interest in the signal band. This can be achieved by fine-tuning the local oscillator frequency.

4G1 and 4G2 use Superconductor-Insulator-Superconductor (SIS) mixing devices providing an IF band from 4 to 8 GHz. 4G3 and 4G4 and the upGREAT arrays employ Hot-Electron-Bolometer (HEB) mixers with IF bands of 0.1 to 4 GHz. The optics of GREAT generally utilize beam splitters to overlay the LO and sky signal. However - due to LO power limitations - 4G4 uses a diplexer, which reduces the usable IF band to 1.4 to 2.8 GHz.

Figure 6-1.

Figure 6-1. Schematics of a GREAT receiver channel. Only 4G4 uses a diplexer, all other channels have a beam-splitter. The second IF mixing process is optional and is only used when the mixer output range and the backend input range do not match.

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6.1.1.2 Configurations

The channels that are currently operational are listed in Table 6-1. Not every frequency in each tuning range has been checked, so there may be gaps where the LOs do not provide enough power to pump the mixer or regions where the receiver temperature deviates from the average values quoted in the table.

Table 6-1

GREAT Configurations

Front-End

Frequencies (GHz)

Lines of Interest

DSB6 Receiver Temperatures (K)

Main beam efficiencies

Front-End

Frequencies (GHz)

Lines of Interest

DSB6 Receiver Temperatures (K)

Main beam efficiencies

HFA 1

4744.77749

[OI] 63 μm

1250

0.63

LFAH 2

1835–2007

[CII] 158 μm, CO, OH, 2Π1/2,
12CH, 13CH

1000

0.69

LFAV 2

1835–2007
2060–2065

Same as LFAH,
plus [OI] 145 μm

4G4

2490–2590

OH 2Π3/218OH 2Π3/2

3300

0.57

4G3

1240–1395
1427–1525

[NII] 205 μm, CO, OD, HCN,
SH, H2D+

1100

0.70

4G2 3,4

890–984
990–1092

CO, CS

> 600
300

0.59

4G1 4

491–555
560–635

NH3, [CI] 609 μm, CO, CH

< 150

0.51

  1. One 7-pixel array. Quantum Cascade Laser LO with very small tuning range near the [OI] 63 μm line.

  2. Two 7-pixel arrays at two polarizations (H,V). Both polarizations can be tuned to the same frequency (LFAH range), or two separate frequencies on a best effort basis. The [OI] 145 μm line is only tunable with the V-array LO.<

  3. Strong degradation below 900 GHz.

  4. For 4G1 and 4G2, the gaps in the center of the LO tuning correspond to frequency ranges blocked by the low atmospheric transmission.

In Cycle 9, GREAT offers two configurations:

upGREAT LFA with upGREAT HFA
This configuration employs the Low Frequency Array (LFA) in one dewar and the High Frequency Array (HFA) in the second dewar. The LFA consists of two polarizations (H and V) with 7 pixels each, and the HFA consists of 7 pixels. Both polarizations of the array (LFAH and LFAV) can be tuned simultaneously in the 1.835-2.007 THz range (and specifically the [CII] line at 1900.536 GHz), or alternatively they can each be tuned to a different frequency on a best effort basis. The [OI]145 µm line can only be observed in V polarization.

4GREAT with upGREAT HFA
This configuration implements 4GREAT in one dewar and the HFA in the second dewar. The 4GREAT pixels are coaligned on the sky to within a few arcseconds, and operate simultaneously.

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6.1.2 Performance

6.1.2.1 Sensitivities

The GREAT sensitivities and integration times are calculated with the SOFIA Instrument Time Estimator (SITE). This section presents the background for these calculations as well as some worked-out examples. Because heterodyne receivers are calibrated by measuring the receiver temperature Trx with a hot and a cold load, the logical intensity unit for a heterodyne observation is temperature expressed in Kelvin (K). Typically, measurements are expressed either with antenna temperature TA* or main brightness temperature Tmb.The asterisk for the antenna temperature TA* refers to values corrected for atmospheric transmission, telescope losses, and rearward spillover (see Kutner & Ulich, 1981, ApJ, 250, 341) . Similarly, the noise (ΔTA * or ΔTmb) is also expressed in units of temperature. The sensitivity, or signal-to-noise ratio, of the observations is given by the ratio between the source temperature and the root mean square (rms) temperature of the spectrum. In order to calculate these quantities, we first estimate the single sideband (SSB) system temperature Tsys which includes the noise from the thermal emission of the telescope and the sky, and refers to the brightness temperature scale on the sky:

(Eq.6-1)
Tsys = 2 x [Trx + ηtel x Tsky + ηtel x Ts + Ttel ] /( ηtel x ηsky) ,

where Trx is the dual sideband (DSB) receiver temperature;
Tsky is the radiation temperature of the sky;

Ts is the continuum temperature of the source, which is usually negligible unless observing an extremely bright object such as Jupiter, Saturn, or Sgr B2, and is therefore set to zero in the GREAT time estimator;

Ttel is the radiation temperature of the telescope;
ηsky is the atmospheric transmission;
ηtel is the efficiency of the telescope, which includes ohmic losses and spillover.

All temperatures in Equation 6-1 are Rayleigh-Jeans equivalent "radiation" temperatures, and not physical temperatures.

The factor 2 in Equation 6-1 assumes that the noise temperature is the same in both signal and image band, which is true for the HEB and SIS mixers used by GREAT. The atmospheric transmission ηsky can be estimated using the transmission code ATRAN for the altitude, observing frequency, and airmass of the planned observations. The GREAT time estimator in SITE automatically accounts for the attenuation of the signal in the atmosphere by calling ATRAN. The radiation temperature Tsky depends on ηsky and the ambient temperature Tatm of the sky:

(Eq. 6-2)
Tsky = Tatm,RJ x (1 - ηsky) ,

where Tatm,RJ is the Rayleigh-Jeans (RJ) equivalent mean temperature of the atmosphere. The relationship between Tatm and Tatm,RJ is given by:

(Eq. 6-3)
Tatm,RJ = c2 x B(ν, Tatb) / (2 x kB x ν2),

where B(ν, Tatm) is the flux density derived from Planck's law of blackbody radiation for a temperature Tatm at a frequency ν, c is the speed of light, and kB is the Boltzmann constant. As an example, assuming an ambient temperature Tatm of 220 K (a typical temperature of the atmosphere at 41,000 ft), we find Tatm,RJ = 177.5 K at 1.9 THz.

Likewise, the telescope temperature Ttel is related to ηtel by the following equation:

(Eq. 6-4)
Ttel = TTA,RJ x (1 - ηtel) ,

where TTA,RJ the Rayleigh-Jeans (RJ) equivalent temperature of the telescope assembly. For a physical temperature TTA of 230 K for the telesope assembly, and using Equation 6-3, we find TTA,RJ = 187.4 K at 1.9 THz. If the telescope efficiency ηtel is 0.92, then the radiation temperature Ttel of the telescope is 14.8 K.

As an example, let us calculate the system temperature for the [CII] fine structure line at 157.74 μm (1.9005369 THz). In this example, we assume typical observing conditions at the beginning of a flight. Specifically, we suppose an altitude of 39,000 ft and a source elevation of 30 degrees. For a standard atmospheric model this corresponds to a sky transmission ηsky of 0.76, which gives Tsky = 42.6 K. For a receiver temperature Trx = 1000 K, Equation 6-1 therefore predicts a single sideband system temperature Tsys = 3015 K from the background contribution of the sky alone.

Now we are ready to calculate the sensitivity. The rms antenna temperature ΔTA*, which is corrected for the atmospheric absorption and telescope losses, is given by the following equation:

(Eq. 6-5)
ΔTA* = (2 x Tsys x κ ) x (t x Δν)-0.5 ,

where k is the backend degradation factor, t is the total integration time on each pair of on and off positions, and Δν is the frequency resolution of our spectra. Strictly speaking, Δν is the noise bandwidth which can be slightly different than the frequency resolution depending on the design of the spectrometer. For this example, we suppose the Full Width at Half Maximum (FWHM) of the line to be only a few km/s wide, and so we choose a velocity resolution of 1 km/s (or 6.3 MHz) for the calculation. Since the GREAT backends have a much higher resolution (244 kHz), we can easily bin the spectrum to our desired velocity resolution. For an integration time of 240 seconds, and assuming a backend degradation factor k of 1, we find a rms antenna temperature ΔTA* of 0.155 K.

The relationship between the antenna temperature TA* and the brightness temperature TR* is expressed as:

(Eq. 6-6)
TR* = TA/ ηfss ,

where ηfss is the forward scattering efficiency, which is ηfss = 0.97 for GREAT. A rms antenna temperature ΔTA* = 0.155 K therefore corresponds to a rms brightness temperature ΔTR* = 0.160 K. Please note that the GREAT time estimator uses the brightness temperature TR*, and not the main beam brightness temperature Tmb = TR* x ηfss / ηMB, where ηMB is typically 0.65 for the LFA and 0.63 for the HFA resulting from the large central blockage in the folded Nasmyth optics of the SOFIA telescope.

For some projects, it may be necessary to instead convert the antenna temperature TA* into a flux density Sν, such as to compare the line intensity with the continuum level of the source. The flux density Sν can be obtained from the antenna temperature TA* using the following standard relation:

(Eq. 6-7)
Sν = 2 x kB x ηfss x TA/ Aeff ,

where kB is the Boltzmann constant, and Aeff is the effective collecting area of the telescope. Aeff is related to the geometrical surface area Ag of the telescope by the aperture efficiency ηa such as Aeff = ηa x Ag. For a main beam efficiency ηMB of 0.67 and assuming a Half Power Beam Width (HPBW) of 14.1± 0.3 arcsec, we find an aperture efficiency ηa of 0.55 ± 0.02. For the 2.5 m telescope of SOFIA, the measured flux density for a [CII] line can be expressed as:

(Eq. 6-8)
S (Jy) = 992 x TA(K) or, within errors, S(Jy) ~ 1000 x TA(K) .

We can also use Equation 6-8 to convert the units of line intensities from Jy to W/m2, which may be a more familiar unit for the far-infrared community. If we assume that the [CII] line we are observing has a Gaussian profile with a Full With Half Maximum (FWHM) of = 5 km/s, or 31.8 MHz, then the integrated line intensity is given by 1.065 x Tpeak x Δν with Δν = 31.8 MHz. If we take Tpeak equal to our rms antenna temperature ΔTA* = 0.155 K, we find using Equation 6-8 that a four minutes integration corresponds to a one sigma brightness limit of 5.2 x10-17 W/m2 for a line width of 5 km/s at a 1 km/s resolution. For a detection experiment, the resolution could even be degraded to 2 km/s instead. In this case, we gain a square root of 2 and our one sigma detection limit becomes 3.7 x 10-17 W/m2.

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6.1.2.2 Examples

Single point observation
When writing a proposal, investigators often already know what sensitivity is needed given the expected width and brightness of the line to be observed, and they are instead interested in calculating the required integration time. For example, we can calculate the integration time needed to detect the [CII] 158 µm line toward T Tauri, a young low-mass star. Based on Herschel PACS observations, Podio et al. (2012, A&A, 545, A44) find a line intensity of 7.5 x 10-16 W m-2. However, the line width was not resolved due to the ~240 km/s resolution of PACS. In this case, the high velocity resolution of GREAT would determine if the line is dominated by an outflow or by the circumstellar disk, or both. If we assume the line is outflow dominated with a FWHM of 20 km/s (or 127.2 MHz), and using Equation 6-8, we find a peak antenna temperature TA* of 0.56 K or a brightness temperature TR* of 0.58 K. Assuming a SNR of 10 and a resolution of 1 km/s are needed for this project, we can now calculate the required integration time.

If these observations are taken at an altitude of 41,000 ft and at a target elevation of 40 degrees, the ATRAN transmission tool predicts an atmospheric transmission ηsky of 0.86 integrated over the received bandpass. According to Equation 6-2, the sky temperature Tsky is therefore 24.9 K for an ambient temperature Tatm of 220 K. For a receiver temperature Trec of 1000 K, Equation 6-1 then provides a system temperature Tsys of 2623 K. To achieve a signal to noise ratio of 10 for the observed line, we then need to integrate until we reach a rms antenna temperature ΔTA*of 0.056 K. We can now solve for the integration time t using Equation 6-5, where we set the frequency resolution Δν to be 6.338 MHz (or 1 km/s). In this case, we find the required integration time t to be 1385 s (or 23.1 minutes). Depending on the sideband chosen as the signal band, SITE provides two comparable solutions, 1465 s and 1456 s, which differ slightly due to the atmospheric transmission in each sideband and the star's Local Standard of Rest (LSR) velocity of ~25 km/s.

Since the PACS data show the emission in T Tauri to be compact, the GREAT observations could be achieved with a single pointing in Dual Beam Switching mode (DBS; see Section 6.2.1.1) using the default chop throw of 60 arcsecond. The overheads for both the DBS and Total Power (TP) modes are estimated to be 100%, and they are automatically included when setting up the observations in USPOT. The duration of the observations would therefore be 48.5 minutes.

Since this example applies specifically to [CII] 158 µm line measurements toward an object with low LSR velocity, both LFA polarizations would contribute equally to the observations. This effectively halves the required observing time to be 24.3 minutes instead (including overheads). However, please note that this additional step is only valid for lines in the 1.835-2.007 THz range when both polarizations of the LFA are tuned to the same line. Prospective investigators should contact the SOFIA help desk if they have any question on how to use both LFA polarizations in their sensitivity calculations.

On The Fly mapping
Sensitivity calculations are done a bit differently in the case of an On The Fly map (OTF), where data is acquired as the telescope is continuously scanning the sky.

Let's take the example of a 5 x 5 arcmin2 map to measure extended [CII] 158 µm line emission in a star-forming region. For OTF maps using the LFA, we typically sample the beam every 6 arcsec (slightly oversampled compared to the ~14 arcsec HPBW of a LFA pixel). The number of map points Non for a single 5 arcmin-wide scan would therefore be 50. If the on-source exposure time ton per point is 0.5 s, then the total on-source time for each line is 25 s. In Total Power mode, the off-source time toff is then ton x Non0.5, which in this case would be 3.5 s. The integration time for each line is therefore 28.5 s, and the total integration time to complete all 50 lines of the entire map is 23.8 minutes. Assuming overheads of 100%, which are automatically added when designing observing plans in USPOT, the total observing time for this 5 x 5 armin2 map in Total Power mode would be 47.6 minutes.

In Dual Beam Switching mode, the off-source time is equal to the on-source time, and so the total observing time (including overheads) for the entire map would be 83 minutes instead. The Dual Beam Switching mode is rarely used for OTF maps due to its lower observing efficiency relative to the Total Power mode, and is typically reserved for observations of faint lines (TR* < 0.5 K) with large FWHMs (> 30 km/s) where a stable receiver baseline is crucial.

For an OTF map, each map point is covered once by each pixel of the array. For the LFA and the HFA, this effectively means that the integration time per point is multiplied by a factor 7. In addition, if both LFA polarizations are tuned to the same frequency, they can be combined to further multiply the integration time per point by a factor 2. The effective on-source integration time per point Ton can therefore be estimated with:

(Eq. 6-9)
Ton = ncycle x npix x npol x ton ,

where ncycle is the number of times the map is repeated, npix is the number of pixels contributing to the map position, npol is the number of array polarizations tuned to the observed line, and ton is the on-source exposure time.

Back to our previous example of a 5 x 5 arcmin2 OTF map of [CII] 158 µm line emission, the effective integration time per point Ton would therefore be Ton = 1 x 7 x 2 x 0.5 s = 7 s. Assuming typical observing conditions (i.e., an altitude of 41,000 ft and an elevation of 30 degrees), the GREAT time estimator calculates a rms temperature ΔTR* of 0.43 K for a resolution of 1 km/s. This means that any [CII] line emission larger than 4.3 K in the map will be detected at a 10-sigma level or higher.

Finally, in contrast to classic OTF maps, each map point in an Array OTF or a Raster map are only covered by a single pixel of the array instead (i.e., npix = 1 in Equation 6-9). To reach comparable effective Ton integration times, Array OTF or Raster maps need longer exposure times ton per point or multiple repetitions.

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6.2 Planning Observations

 

This section describes the GREAT instrument and the available observing modes. It also contains additional information to estimate observing times in individual modes. 

If the frequency of interest is not listen in Table 6-1, please contact the GREAT team to ensure the feasibility of the observations. There may be gaps where the broadband Local Oscillators do not provide enough power to pump the mixers.

Note: Allan variance affects the capabilities of GREAT and requires special attention when planning observations, see Section 6.2.1.2 for details.

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6.2.1 Observing Modes Overview

Two observing modes are currently offered: Total Power (TP) and Beam Switching (BSW), the latter of which is available as either Single Beam Switching (SBS) or Dual Beam Switching (DBS).

Total Power Mode
In Total Power mode, the telescope alternates between the target and a nearby reference position that is free of emission. The GREAT software then produces a spectrum from the difference between measurements on the target (ON) and reference (OFF) positions. The optimal integration time in this mode depends on the stability of the receiver and the rate of atmospheric fluctuations. Typically, the recommended integration times on the ON position are < 30 s for the LFA and < 20 s for the HFA. Shorter integration times provide a better cancellation of atmospheric fluctuations, but at the cost of a reduced observing efficiency due to overheads. The ON-OFF cycle is repeated until the required sensitivity is reached. 

Total Power mode is most often used to maximize the observing efficiency of On The Fly maps, as well as for single point observations in crowded regions with significant extended emission. Reference positions that are far from the target position may result in poor baselines due to temporal fluctuations in the sky background. If a reference position cannot be identified within 30 arcmin of the target, an intermediate reference position can be selected instead even if it potentially has some emission from the line of interest. This intermediate position will then be measured against a faraway reference position with no line emission. Any contamination in the original spectrum due to line emission in the intermediate reference position can then be accounted for and removed.

Beam Switching Mode (BSW)
In Single Beam Switching (SBS) mode, the telescope is pointed at an intermediary position between the target and reference positions while the secondary mirror is chopped. The main parameters for this mode are the chop throw, which is the angular distance between the target and reference position, and the chop angle relative to the chosen reference frame (i.e., sky coordinates or telescope assembly). 

In Dual Beam Switching (DBS) mode, there are instead two reference positions placed symmetrically around the target position. In addition to chopping the secondary mirror like for the SBS mode, the telescope nods between two intermediary positions so that the target is measured against each reference position alternatively.

While using either beam switching modes, the secondary mirror typically chops at a rate between 1.0 and 2.5 Hz. The maximum chop throw allowed is 5 arcmin due to the physical limitations of the secondary mirror. Because the time between the ON and OFF Positions is short (~1 s), this mode typically results in better sky cancellation, and better baseline stability. 

Beam switching modes are most often used for point or compact sources for which there is a nearby reference position clear of emission. Smaller chop throws between 60 and 180 arcsec are usually preferred to limit pointing errors and beam distortions (coma). 

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6.2.1.1 Astronomical Observation Templates (AOTs)

GREAT offers five standard Astronomical Observation Templates (AOTs) in USPOT: Single Point, Raster Mapping, On The Fly (OTF) Mapping, OTF Array Mapping, and OTF Honeycomb Mapping. Each AOT can  be run in either Total Power or Beam Switching modes.

Single Point
The Single Point AOT is used to observe a single position on sky. It is typically used to observe faint sources that require long integration times.

Raster
Raster mapping is essentially a collection of single point observations used to create small maps where a relatively long integration time per map point is needed.

On the Fly
In an OTF map, the telescope scans along a series of rows while the back-ends are continuously integrating the incoming signal. An average is recorded after the telescope has moved a specific distance on the sky. Typically, the size of these steps is half of the beam width. Using default mapping parameters (see Section 6.2.3), each point in the central region of an OTF map will be covered at least once by each pixel of the array (either the HFA or the LFA). The size of this central region changes as a function of the mapping parameters, and it can be assessed easily with USPOT's visualization options (see Section 6.2.3.2).

In Total Power mode, each row scan is preceded by a measurement of the reference position. The integration time tOFF on this reference position before each scan is tOFF = NON0.5 x tON, where NON is the number of steps in the row and tON is the on-source exposure time per point. In this mode, the size of a scan row is limited by the stability of the receiver and by atmospheric fluctuations. The recommended time to complete a single row is under 30 seconds for the LFA and 20 seconds for the HFA. In addition, the typical on-source exposure times per point are between 0.3 s and 2.0 s. Large maps (> 10 arcmin for the LFA and > 3 arcmin for the HFA) should therefore be divided into smaller sub-maps to ensure receiver stability during observations. 

In Beam Switching mode, the secondary mirror chops while the telescope is scanning along each row. No dedicated reference position is needed, and so the telescope can immediately start the next row. However, the times spent integrating on the source and reference positions are equal, and so the observing efficiency of the Beam Switching mode is still lower than for the Total Power mode. This mode is rarely used for OTF maps, but it can nonetheless be useful for small maps of compact regions where a stable baseline is needed for the detection of the line observed. 

OTF Array
An OTF Array map is created in a similar fashion to a regular OTF map; the telescope scans along a series of rows while the back-ends are continuously integrating the incoming signal. The main difference is that the distance between scan rows is not constant, and is instead designed so that each map point is covered by a single pixel of the array (assuming default mapping parameters; see Section 6.2.3.3). OTF Array maps are typically a combination of several scan "blocks", and are often used to observe relatively compact regions expected to have bright line emission.

OTF Honeycomb 
An OTF Honeycomb map is the preferred solution for compact objects that are comparable in size to the array used (either the LFA or HFA). Instead of scanning rows, the telescopes follows a small 25-points hexagonal pattern in order to fully-sample the gaps between each array pixel on the sky (see Section 6.2.3.4). This avoids the issue of wasted integration time outside the region of interest, which can be an issue for OTF and OTF Array maps depending on the size and geometry of the target.

Like OTF Array maps, OTF Honeycomb maps can be tiled together to map larger regions. However,  OTF and OTF Array maps are recommended for regions larger than a few times the array size in order to obtain a more homogeneous noise map by covering each map point with different array pixels.

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6.2.1.2 Spectroscopic Stability Limitations

As mentioned in Section 6.2.1, the spectroscopic stability (e.g., Allan variance) sets limits to the operation of a heterodyne instrument like GREAT.

Beam Switching modes typically operate with a chop frequency around a few tenths of a Hz to a few Hz, which is relatively fast, and so the observing limitations are mainly set by the atmospheric stability. As such, on-source integration times of 30 s for the LFA and 20 s for the HFA are typically implemented to achieve optimal spectroscopic stability. A Dual Beam Switching mode is recommended, when possible, in particular for projects aiming to measure weak and/or broad lines. Some performance decrease has been observed when using a Single Beam Switching mode instead, but it is a viable alternative for crowded regions.

For a Total Power mode, the situation is more complicated and multi-dimensional. Observing line widths larger than a few hundreds MHz and using a reference position farther than 30 arcmin from the source position may negatively impact the stability of the spectroscopic baseline. These impacts depend on the frequency observed, the weather conditions, the stability of the sky background, the elevation of the source, and the arrays used. Please contact the SOFIA help desk if your Total Power observations require a faraway reference position and/or if the line of interest is expected to be weak and broad.

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6.2.2 Estimation of Exposure Times

Estimations of exposure times for GREAT can be made using the SOFIA Instrument Time Estimator (SITE). SITE is a web-based tool that calculates either the signal-to-noise ratio for a given line brightness and integration time, or the integration time needed to reach a certain RMS noise level for either one point on the sky or per map position for an OTF map in Total Power mode. These integration times do not include tuning, chopping, slewing, and other observatory overheads. The total time, including all overheads, is determined in USPOT after entering the time calculated by SITESITE is also useful to determine which sideband is the best to use as the signal band by taking the atmospheric transmission into account. System temperatures for the line in the USB or LSB are given, as well as a plot showing the line locations for either Local Oscillator tuning compared to the atmospheric transmission.

The time estimator calculates the time required to reach an RMS brightness temperature ΔTR* for a line at a frequency ν by solving the standard radiometric formation for a single point observation. As a reminder, the relation between the brightness temperature TR* and the antenna temperature TA* is:

TR= TA*/ ηfss ,

where ηfss is the forward scattering efficiency which is equal to 0.97 for GREAT. The antenna temperature TA* is corrected for ohmic losses and rear spillover. The RMS temperatures ΔTR* and ΔTA* follow the same relation. The RMS antenna temperature ΔTA* is obtained from:

ΔTA= (2 x Tsys) / (t x Δν)0.5 ,

where Tsys is the single sideband system temperature, t is the integration time, and Δν is the requested frequency resolution. This formula applies when tON=tOFF, which is the case for single point Total Power observations and for all Beam Switching observations.

For On The Fly maps in Total Power mode, the RMS antenna temperature is instead: 

ΔTA= Tsys (1 + (1 / Non)0.5)0.5 / (t x Δν)0.5 ,

where t is the on source exposure time per point, and Non is the number of on-source positions between measurements of the reference position.

SITE uses the most recent receiver temperatures measured by the GREAT team, and it calls the program ATRAN to estimate the atmospheric transmission for a given altitude, telescope elevation, and water vapor overburden. This transmission is used to calculate Tsys assuming an ambient temperature of 220 K for the atmosphere and a telescope assembly temperature of 230 K.

GREAT is a dual sideband receiver, which means that it simultaneously observes in two frequency bands: the upper sideband (USB) and the lower sideband (LSB). The transmission plot in SITE shows the locations of each sideband relative to the observed line, depending on which sideband is used as the signal band. The sidebands for the LFA, HFA, 4G3, and 4G4 are separated by +/- 1.5 GHz, while for 4G1 and 4G2 they are separareted by +/- 5.5 GHz for 4GREAT bands 1 and 2. The required integration times are calculated for both sidebands. The lowest value provided by SITE should be typically used for the time estimate in a proposal. Differences in the calculated integration times for the LSB and USB are due to the frequency dependence of the atmospheric transmission.  

Description of SITE Input Parameters

Type of Observation
Select "Single Point or Beam Switch OTF/Raster Map" for single point observations in either total power or beam-switched mode, and for OTF/Raster maps observations in beam-switched mode. Select "TP OTF/Raster Map" for OTF/Raster maps in total power mode.

Rest Frequency
Enter the rest frequency (in THz, using 7 decimal places) of the line you wish to observe. The current tuning ranges for the GREAT receivers are listed in Table 6-1.

Frequency or Velocity Resolution
Enter the frequency (in MHz; select the "MHz" radio button) or velocity (in km/s; select the "km/s" radio button) resolution that is requested in the final spectrum.

Line Width
Enter the frequency (in MHz; select the "MHz" radio button) or velocity (in km/s; select the "km/s" radio button) window that will be used to calculate the atmospheric transmission. Modifying this parameter may be important if the line you wish to observe falls close to a narrow atmospheric feature.

Total Power Map Parameters
For OTF maps: Enter the number of ON positions (steps or dumps) in each OTF scan row in the Non field, or have the time estimator calculate this value for you. If you choose to have the estimator calculate it, you should enter the dimensions of the map (in arcsec) and select a "Map Type" option (Classical OTF or Array OTF). For a Classical OTF map, the "Map Size" refers to the area mapped by the central pixel only. For an Array OTF map, the "Map Size" refers to the area that will be fully-sampled (i.e., the array width is added to the length of each scan). The Array OTF map should only be selected if the frequency falls within the tuning range of the LFA or HFA (see table above).

Using these inputs, the calculator evaluates scanning in both x- and y-directions, and selects the direction that has fewer scan lines. It then estimates Non using the length of the scans and a frequency-based receiver stability time. The step sizes assumed for each frequency band are: 3 arcsec for the HFA, 6 arcsec for the LFA, 25 arcsec for 4G1, 12 arcsec for 4G2, 8 arcsec for 4G3, and 5 arcsec for 4G4. Note that there are many ways to configure a mapping observation, and the calculated value of Non is only one of many possible values. For an OTF Honeycomb, Non should be set to 25. 

For Raster maps: Enter the number of on positions that will be used for each reference position in the Non field. You may ignore the Map Size and Map Type fields.

Signal to Noise Ratio / Integration Time
If the SNR radio button is selected, enter the desired signal to noise ratio in this field and the estimated line strength in the Brightness Temperature field. The time estimator will calculate the integration time required to reach this SNR. If the Integration Time radio button is selected, enter the integration time (in seconds) for your observation. If your observation is a Total Power OTF map or a Total Power raster map, enter the effective on-source exposure time per map point. Otherwise, enter the ON+OFF integration time. The time estimator will calculate the 1-sigma rms sensitivity (in units of TR*) based on the input integration time.

Brightness Temperature TR* (K)
Enter the estimate of the peak brightness temperature of your line. This field is only used if the SNR radio button is selected (see above). As is the case for other heterodyne receivers that use hot and cold loads to measure the receiver temperature, the intensity units are Kelvin (K). The intensity scale used in the online tool is brightness temperature TR*. This relates to the measured antenna temperature as TR* = TA* / ηfss, and the main beam temperature (corrected for losses in the side lobes) as TMB = TA* / ηmb . The main beam efficiency has been measured from planetary observations, and has been determined to be 0.70 for the LFA and 0.63 for the HFA. For the latest 4GREAT main beam efficiencies, please contact the Help-Desk. A detailed description of the GREAT intensity calibration is given in Section 6.1.2.2, which also contains worked examples for different observing modes and unit conversions.

Source Velocity
Enter the source velocity (in km/s) in the Local Standard of Rest (LSR) reference frame.

Observer Velocity
Enter the velocity of the observatory with respect to the LSR on the date of the observation. If this is unknown, you may either leave the default (0 km/s) or enter the date, time, coordinates, and location for your observation to have the time estimator calculate the observer velocity automatically. Note that if your desired line rest frequency falls close to or in an atmospheric absorption feature, you may still be able to observe the line if you choose the right time of the year as the source is blue- or red-shifted relative to the atmospheric feature.

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6.2.3 On The Fly Technical Details

The three On The Fly (OTF) mapping options are OTF mapping (also referred to as Classic OTF), OTF Array mapping, and OTF Honeycomb mapping. Each OTF Astronomical Observation Template (AOT) can be used with the two instrument configuration of GREAT: LFA H/V and HFA, and 4GREAT and HFA. However, please keep in mind that OTF Array and OTF Honeycomb maps are not designed to be used with the single pixel of 4GREAT. Otherwise, the parameters for OTF observing schemes are flexible and can be easily modified in USPOT depending on the needs of the project. The recommended parameters presented in this section should perform well for most mapping projects. 

Any OTF AOT can be executed in Beam Switching (BSW) or Total Power (TP) mode. Some things to consider when deciding between these two modes are:

Efficiency
Maps obtained using BSW modes need an equal amount of integration time on the ON and OFF positions. In contrast, Total Power maps spend less time on the OFF position, and are therefore more efficient.

Baseline Stability
BSW observations generally have better baseline stability. If the source is expected to have faint and/or wide lines, then BSW modes should be preferred. If the source is expected to have bright and/or narrow lines, then the baseline quality is less of an issue.

Reference positions
For a BSW map, there needs to be an emission-free region on the sky that is the same size as the OTF map and within a few arminutes of the map. Finding such a reference position is often a challenge, especially in crowded regions or region with significant extended emission. For Total Power maps, only a single emission-free pointing is needed as a reference position, and it can be significantly farther away from the source (up to 30 arcmin, although baselines will be better for closer reference positions).

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6.2.3.1 Coordinates and Array Geometry

Because upGREAT maps can be rotated relative to standard sky coordinates (e.g., J2000), they have their own coordinate system defined as x and y. With a map rotation of 0 degrees, the +x axis is aligned with the +RA axis, and the +y axis is aligned with the +Dec axis (Fig. 6-2). The map rotation angle increases in the counter-clockwise direction, and this parameter can be set anywhere between -360° to +360°. For OTF maps, scans in the x-direction are parallel to the x-axis, and scans in the y-direction are parallel to the y-axis.

Figure 6-2.

Figure 6-2. Left, The x/y map coordinate system is aligned with RA/Dec (J2000) for a map angle of 0°. Right, The x/y coordinate system at a map angle of +30°. The angle is measured in the counter-clockwise direction.

 

The upGREAT Low Frequency Array (LFA) and High Frequency Array (HFA) are arranged in a hexagonal pattern with a central beam. The spacings between the beams are approximately 2 beam widths (31.7 arcsec for the LFA, 13.8 arcsec for the HFA). For efficient mapping, the array is typically rotated by 19.1 degrees relative to the scan direction, resulting in a projected pixel spacing of 10.4 arcsec for the LFA and 4.6 arecsec for the HFA (see Fig. 6-3). The array can be rotated independently of scanning direction for maximum flexibility during observation planning.

Figure 6-3.

Figure 6-3. Left, The configuration of an upGREAT array. The numbers 0-6 mark the pixel numbers of the array, and the separation between the pixels, r, is 31.7 arcsec for the LFA, and 13.8 arcsec for the HFA. Right, An upGREAT array, rotated by -19.1° (pixel 1 is labeled) with arrows indicating the scan direction. The projected pixel spacing perpendicular to the scan direction is 10.4 arcsec for the LFA, and 4.6 arcsec for the HFA.

When creating a mapping strategy, observers will have to weigh many factors such as the area to be mapped, the required integration time per point, and of course the scientific objectives. Examples given in Section 6.2.3.2, Section 6.2.3.3, and Section 6.2.3.4 aim to provide guidelines

Figure 6-4. 

Figure 6-4. Comparison of the footprints for each GREAT array. From top to bottom: 4G4, 4G1, LFA, and HFA. This example shows an on-sky visualization of each array using USPOT and Single Point AOTs in Dual Beam Switching mode. The red stars identify the central position for each array, and the blue circles represent their on-source footprints. The green and magenta circles respectively show the two reference positions.

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6.2.3.2 OTF Mapping

The classic OTF mapping can be used effectively with any of the GREAT arrays. For an OTF mapping, each array pixel creates a fully-sampled rectangular map based on the step size along the scan row and the spacing between each row. For the upGREAT arrays (LFA and HFA), the result is seven overlapping rectangular maps (see Section 6.2.3.2a and Section 6.2.3.2b). Classic OTF mapping is currently the only available OTF AOT that can create fully-sampled maps using the single-pixel 4GREAT array.

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6.2.3.2a LFA OTF mapping

In this example, we design an OTF map in Total Power mode to measure [CII] 158 µm line emission in NGC 1333 using the LFA as the primary array. The final map is composed of 14 fully-sampled maps of the central region combining both the LFAH and LFAV polarizations. The step size in the x-direction and the y-direction are set to 6 arcsec. Both the number of steps per row and the number of rows are both set to 60, which provides a 360 arcsec-wide square map (see Fig. 6-5). The array rotation is set to -19.1 degrees.

The on-source exposure time per point is 0.5 s, and so each row scan is completed in 30 s. The time spent on the off-position before each scan is therefore 3.9 s, and so the on+off time for each scan is 33.9 s. For 60 rows, the total integration time for the map is 33.9 minutes, and the total observing time including overheads is approximately 67.8 minutes. For a single coverage of the map, the effective integration time per point for the central region will be Ton = 1 x 2 x 7 x 0.5 s = 7 s. Using the GREAT time estimator in SITE, we find a RMS temperature ΔTR* of 0.43 K assuming a resolution of 1 km/s, a line width of 10 km/s, an altitude of 41,000 ft, and an elevation of 40 degree. 

Figure 6-5.

Figure 6-5. Overlay of an OTF map designed for the LFA+HFA configuration. The primary array is the LFA. Each red rectangle shows the region mapped by one of the LFA pixels. The footprint of the array is shown by the red circles and marks the starting position of the map. The central region of the map is covered once by each pixel of the array. In this configuration, the HFA will produce a significantly under-sampled map due to the large step size (6 arcsec). 

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6.2.3.2b 4GREAT and HFA OTF mapping

In this example, we design an OTF map in Total Power mode of the same region as previously. This time we plan to measure the [OI] 63 µm line with the HFA and the 12CO J=22-21 line with 4G4. All 4GREAT receivers observe simultaneously, but we choose 4G4 for this example because it has the closest beam size (9.9 arcsec) to the HFA (5.2 arcsec). The final results will be 7 fully-sampled [OI] maps of the central region with the HFA and one over-sampled CO map with 4G4. The step size in the x-direction and the y-direction are set to 3 arcsec. Both the number of steps per row and the number of rows are both set to 60, which provides a 180 arcsec-wide square map (see Fig. 6-6). The array rotation is set to -19.1 degrees.

The on-source exposure time per point is 0.3 s, and so each row scan is completed in 18 s. The time spent on the off-position before each scan is therefore 2.3 s, and so the on+off time for each scan is 20.3 s. For 60 rows, the total integration time for the map is 20.3 minutes, and the total observing time including overheads is approximately 40.6 minutes. For a single coverage of the map, the effective integration time per point for the central region will be Ton = 1 x 1 x 7 x 0.3 s = 2.1 s for the HFA and Ton = 1 x 1 x 1 x 0.3 s for 4G4. Using the GREAT time estimator in SITE with a resolution of 1 km/s, a line width of 10 km/s, an altitude of 41,000 ft, and an elevation of 40 degree, we find a RMS temperature ΔTR* = 0.9 K for [OI] 63 µm and ΔTR* = 5.4 K for 12CO J=22-21. 

Figure 6-6.

Figure 6-6. Overlay of an OTF map designed for the 4G+HFA configuration. The primary array is the HFA. Each red rectangle shows the region mapped by one of the HFA pixels. The footprint of the array is shown by the red circles and marks the starting position of the map. The central region of the map is covered once by each pixel of the HFA, as well as by the single pixel of 4GREAT. In this configuration, the resulting 4GREAT maps will be significantly over-sampled due to the small step size (3 arcsec). USPOT can only show the HFA visualization when using the 4G+HFA instrument configuration.

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6.2.3.3 OTF Array Mapping

The basic unit of the upGREAT array mapping scheme is referred to as a block, which consists of a single or multiple scans of the same length, in the same direction (Fig. 6-7). For both the LFA and the HFA, the projected pixel spacing (after rotating the array by -19.1 degrees) is such that a single scan results in an under-sampled map. To create a fully sampled map, it is necessary to make at least one more scan to fill in the gaps between pixels. The default behavior is to make a second scan, creating a fully sampled map and completing the block. It is possible, however, to scan only a single time (creating an under-sampled map), or more than two times (creating an over-sampled map), depending on the goals of the project. 

Figure 6-7.

Figure 6-7. Left, A block that consists of a single scan of the rotated array. The gaps between the scans indicate an under-sampled map. Right, A block that consists of two scans. This block is fully sampled in the direction perpendicular to the scan direction. The second scan fills in the gaps in the single-scan block.

A single map can consist of any number of blocks, and can scan in the x- or y- direction, or both. The parameters of the x- and y-direction scans are independent, but can be used in concert to create fully-sampled maps of a region in both directions (Fig. 6-8). Scanning in both directions helps to minimize striping effects that can be caused by the different characteristics of individual array pixels.

Figure 6-8.

Fig. 6-8. Left: A map composed of six blocks scanning in the x-direction. There are three blocks along the scan direction, and two blocks perpendicular to the scan direction. The region within the two vertical white lines shows the inner coverage region. Right: A map composed of six blocks scanning in the y-direction. There are two blocks along the scan direction, and three blocks perpendicular to the scan direction. The region within the two horizontal white lines shows the inner coverage region. Maps scanning in both directions can be set up so that the central regions of each map align.

To calculate the size of the inner coverage region along the scan direction:

inner coverage = [(block length) * (# of blocks in scan direction) – 1] * (array size)

where block length is in units of the array size.

To calculate size of the inner coverage region perpendicular to the scan direction:

inner coverage = (# of blocks perpendicular to scan direction) * (array size)

Because of the flexibility of this mapping scheme, there can be multiple ways to observe the same region. For example, the two setups in Fig. 6-9 fully cover the same area. Some important factors to consider to differentiate thse options are the desired integration time per point, the step size, and the duration of a single scan.

Figure 6-9.

Figure 6-9. Left: A map consisting of two blocks, each one 2*(array width) long. Right: A map consisting of a single block that is 4*(array width)-long. The total inner coverage for both maps is the same.

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6.2.3.3a LFA OTF Array mapping

In this example, we design an OTF Array map in Total Power mode to measure [CII] 158 µm line emission in NGC 1333 using the LFA as the primary array. The final map is composed of 2 horizontal and 2 vertical blocks to cover the region of interest (see Fig. 6-10), which are covered by both polarization of the LFA. The parameters in USPOT for this setup are 1 block in the scan direction and 2 blocks perpendicular to the scan direction for both the x and y scans. Each block has a width of 3 arrays, the array is rotated by -19.1 degrees, and the number of lines per block is 2 (which gives a fully-sampled map). 

The on-source exposure time per point is 0.5 s, and so each row scan is completed in 21 s. The time spent on the off-position before each scan is therefore 3.2 s, and so the on+off time for each scan is 24.2 s. For each block of two rows, the total integration time for the map is 48.4 s, and the total observing time including overheads is approximately 96.8 s. The total observing time for all four blocks is therefore 6.5 minutes. For a single cycle of the OTF Array map, each position in the central region of the map is going to be covered once in each scan direction. The effective integration time per point for the central region will therefore be Ton = 1 x 2 x 2 x 0.5 s = 2 s. Using the GREAT time estimator in SITE, we therefore find a RMS temperature ΔTR* of 0.8 K assuming a resolution of 1 km/s, a line width of 10 km/s, an altitude of 41,000 ft, and an elevation of 40 degree. 

Figure 6-10.

Figure 6-10. Overlay of an OTF Array map in the LFA+HFA configuration. The primary array is the LFA. Each red rectangle shows one of the array blocks described in Section 6.2.3.3. In this configuration, the HFA will produce a significantly under-sampled and inhomogeneous map due to the large step sizes and tiling based on the LFA array size. 

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6.2.3.4 OTF Honeycomb Mapping

The OTF Honeycomb mapping AOT can be used to map an area comparable in size to one of the GREAT receiver arrays (HFA or LFA). The resulting map is fully-sampled at the resolution of the respective array. The OTF Honeycomb mappint is optimized for such small maps by avoiding not fully covered edges common to the other two OTF modes.

In this OTF mode, the telescope moves while integrating not in a straight line, but instead by spiraling in a 25-point hexagon pattern filling in the space between the pixels of the array. Figure 6-11 shows this pattern for all pixels, thus illustrating how all pixels combined completely fill the mapping area.

Figure 6-11.

Figure 6-11. Honeycomb pattern for the LFA with all pixels showing how the pointings mesh together. The units of the axes are in arcseconds. The patter for the HFA will be a factor of 2.7 smaller since the pattern scales with the array size.

When executing an OTF honeycomb map with the HFA in tha LFA+HFA configuration, the LFA will produce 7 small oversampled maps centered around each LFA pixel locations, with gaps between each of these small maps. Alternatively, when executing an OTF honeycomb map with the LFA, the HFA will produce a sparse map the size of the LFA mapping area.

When planning an OTF honeycomb map, one needs to select the Time Per Point so that one OTF scan, the 25-point pattern, does not take longer than 20 or 30 seconds, for an HFA or LFA map respectively. Each location in the mapping area will be covered only once by one pixel (two for the LFA, if both polarization of the LFA are tuned to the same frequency). Repeat the 25-point pattern (the cycle parameter in USPOT) as often as necessary to reach the integration time per map position estimated with SITE.

The OTF honeycomb pattern can also be tiled, though ideally do not tile more than a few tiles at once; larger maps are covered more efficiently when using the other OTF modes. Each honeycomb tile requires its own AOR, and tiling can be done by offsetting the next tile from the first by moving the AOR center by the following vector (±2.5, ±0.866). The unit of this separation vector is in array sizes, and the angular size is obtained by multiplying it with 31.7 arcsec for the LFA, and 13.8 arcsec for the HFA. Rotate the vector by 60 degrees for adjacent tile positions and/or extend the map in the same way from a new AOR center. Below is an illustration of such a tiling with eight tiles. In practice, please consider the other OTF maps modes for such a large mapping area.

Figure 6-12.

Figure 6-12. How a honeycomb map is tiled: Each symbol is a different pixel, each color is a different tile. The snowflake-shaped outline is how each AOR is visualized in USPOT. The left and bottom axes show the size of the tiling for the LFA, and the right and top axes show the size of the tiling for the HFA. The offset of each honeycomb tile is given in Table 6-2.

Table 6-2 

Example of Honeycomb Tiling Offsets

 

LFA

HFA

 

LFA

HFA

Tile 0

(0”,0")

(0”,0")

Tile 1

(-79.25”,+27.45")

(-34.50”,+11.95")

Tile 2

(-15.85”,+82.36”)

(- 6.90”,+35.85”)

Tile 3

(+63.40”,+54.91”)

(+27.60”,+23.90”)

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6.2.3.4a LFA OTF Honeycomb mapping

In this example, we design an OTF Honeycomb map in Total Power mode to measure [CII] 158 µm line emission in NGC 1333 using the LFA as the primary array. The final map is composed of the 4 individual tiles defined in Fig. 6-13 to cover the region of interest (see Fig. 6-13), which are covered by both polarization of the LFA. The parameters in USPOT for each tile are a pattern angle of 0.0 degrees and the Target Offset RA and Target Offset Dec values defined in Table 6-2 for the LFA. 

The on-source exposure time per point is 1.0 s, and so each tile is completed in 25 s. The time spent on the off-position before each scan is therefore 5.0 s, and so the on+off time for each scan is 30.0 s. The total observing time including overheads is approximately 120 s. The total observing time for all four tiles is therefore 8 minutes. For a single cycle of the OTF Honeycomb map, each position in the central region of the map is going to be covered once during the scan. The effective integration time per point for the central region will therefore be Ton = 1 x 1 x 2 x 1.0 s = 2 s. Using the GREAT time estimator in SITE, and using the Total Power OTF option with a Non of 25, we find a RMS temperature ΔTR* of 0.8 K assuming a resolution of 1 km/s, a line width of 10 km/s, an altitude of 41,000 ft, and an elevation of 40 degree. 

Figure 6-13.

Figure 6-13. Overlay of 4 OTF Honeycomb tiles in the LFA+HFA configuration. The primary array is the LFA. Each "hexagon" one of the tiles described in Section 6.2.3.4. The yellow circles represent the corners of the pattern drawn by the central pixel of the LFA. In this configuration, the HFA will produce an under-sampled and inhomogeneous map due to the large step sizes and tiling based on the LFA array size. 

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