Sensitivity of Proposed Instrument
Position or Beam Switching
The sensitivity per pixel for the proposed instrument in either position or beam switching observing is calculated to be:
ΔT(K) = (9f'v't')^{-0.5},
where f' is frequency in THz, v' is velocity resolution in km/s, and t' is integration time in minutes.
For example, a 10 minute integration of the 14->13 CO line at 1495 GHz with 10 km/s resolution would have 27 mK rms sensitivity.
This calculation assumes a double-sideband receiver temperature of 1000K and a atomspheric transparency of 95%. Assumptions used for mean atmospheric temperature, telescope radiation temperature, etc., as well as the equations used to calculate sensitivity can be found in the
GREAT Observing Guide supplied by the Deutsches SOFIA Institut.
For the SOFIA telescope, the conversion of temperature to flux density is:
ΔS(Jy) = 750 * ΔT(K)
On-The-Fly Mapping
Assume a 1x8 array with pixels spaced two beamwidths apart, where beamwidth is 20". We will map a 5' x 5' area with the telescope slewing at a 5" per second rate. The spectrometer will record an average every 2s (or every 10"). This will complete one 5' row in 60 seconds (since we have a 1x8 array, this will give eight 5' rows with each row 40" apart). Then the telescope slews to a reference position, integrates on the reference for ~10 seconds, then moves back to begin the next row. We will assume this process takes a total of 15 seconds, including the 10 seconds of integration. So each row plus reference requires 75 seconds. After four rows or 5 minutes, we will have fully sampled a 5' x 5' area.
The 5' x 5' area will be mapped with a cell size of 10"x10". Since each 20" beamwidth is covered by 4 data cells, the sensitivity per beam is increased by a factor of two. For this reference case of a 5' x 5' map in 5 minutes, the sensitivity per beam is therefore ~0.5K for 1300 GHz. Additional passes can be made over the same area, increasing the sensitivity by sqrt(N) where N is the number of passes. Some additional time will be required for system temperature calibrations if many passes are made.
ΔT_{otf}(K) = (1.2Nf'v')^{-0.5} where N is number of 5 minute 5' x5' maps, f' is frequency in THz, and v' is velocity resolution in km/s.
A good approximation is that a 5' x 5' area can be mapped with 0.5K temperature resolution in 5 minutes with a 3 km/s velocity resolution.
Sensitivity Examples
H2D+ (1370 GHz) with 100 m/s velocity resolution
ΔT(K) = 0.12 K rms per pixel with 60 minutes of integration time
[NII] (1461 GHz) On-The-Fly 5' x 5' map with 3 km/s velocity resolution
In three hours, we can make 33 fully-sampled passes of a 5' x 5' area (this reserves 5 minutes every hour for system temperature measurement). For N = 33, this gives a sensitivity per beam of about 0.08 K.
1250-1500 GHz Full Spectral Scan of 5' x 5' area with 10 MHz spectral resolution (about 2.15 km/s velocity resolution at 1.4 THz)
Assuming 5 minutes for each 1.5 GHz with 5 minutes of every hour for system temperature measurement, in about 15 hours we can make a complete 1250-1500 GHz spectral scan with 10 MHz resolution of a fully spatially sampled 5' x 5' area with 0.5K sensitivity.