# Sensitivity Calculations

## Sensitivity Calculations

The likelihood of a positive detection is calculated using the standard radiometer equation. Most theoretical calculations are ‘ball park’ in the sense that they don’t include indeterminate factors – it’s just a question of how big the ‘ball park’ is. First - we can make some assumptions of system temperature, say, Tsys = 100 K. This may be conservative or optimistic depending on the transition frequency targeted - but is a nominal figure to begin with. Then we calculate the effective aperture of the antenna (Ae) using a conservative aperture efficiency of 50 %.  The spectral line resolution appropriate for the particular transition line width is calculated and used as the pre-detection bandwidth (Δf). For the term 't*n' the transit time is used for 't' and 'n' = 1. Note that in practice - during offline analysis - the total transit time (observation time) may be divided into smaller sections  (to more accurately track Doppler shift) and the results summed - but the product 't*n' remains the same. The term 'Ks' is normally used for accounting for losses in the observatory system used - but here it has been 'hijacked' for the use of entering the minimum S/N required for a postive detection. The value Ks  = 5 is used here.  Stacking successive days observations could improve this – but only if a ‘blind’ stack is done. That is, each observation would need to be corrected independently for terrestrial orbital Doppler and receiver frequency errors and then stacked.  Simply stacking the spectrum by aligning the maximum peak is not valid.

Values are plugged into the radiometer equation...

where...

ΔSmin = minimum detectable flux density (watts per square metre per hertz)

k = Boltzmann's constant (1.38064852 × 10-23 Joules • K-1)

Ks = factor for a particular observatory system, hopefully near to 1

Tsys = System noise temperature (K)

Ae = Effective aperture of the antenna (m2)

Δf = pre-detection bandwidth (Hz)

t = post-detection integration time (seconds)

n = number of records averaged

For each of the available size dish antennas available at HawkRAO calculations are done to compare system sensitivity to quoted observation flux densities for the stronger masers listed above that are accessible via the available sky view. Results obtained are used to determine  which transition and dish size would be best for initial activities.

NOTES:

## OH Masers - 1612 MHz

Only one dish antenna at HawkRAO is appropriate for 1612 MHz - a 3 m diameter TVRO mesh dish.

### Calculations for a 3 Metre Dish

First we can estimate the minimum transit time (where the source is between 3 dB points of the antenna beamwidth at DEC = 0) – where the diameter of the dish is 3 m and the frequency is 1612 MHz. The result for the transit is about 22 minutes (gain ~ 30 dBi, 3 dB beamwidth ~5.5o) for a source at DEC = 0o.  At other declinations the transit time will be longer and this is factored into the Smin calculation. Plugging values into the radiometer equation...and assuming Tsys = 100 K and Δf = 5.4 kHz (nominal spectral line width corresponding to a velocity resolution of about 1 km/s - which seems reasonable from an examination of the spectral profile of OH masers) gives an estimates for the various candidate sources of Smin ranging from 113 Jy to 146 Jy. The Smin results are colour-coded as  dark green  where Smin is less than the minimum observed flux density (indicating high probability of detection),  light green  where  Smin is less than the maximum observed flux density (indicating possible detection) and  grey  where  Smin is greater than the maximum observed flux density (indicating low probability of detection) or  Fmax Fmin (i.e., where there is only one observation and so a determination cannot be made).

Results seem to indicate that a fixed transit mode dish of 3 m should yield a successful detection of VY CMa OH and G357.3085-1.3330, possible detection of V0437 Sct, G331.5121-0.1025, G31.0123-0.2193 and VX Sgr, and unlikely detection of G357.3109-1.3370 and G328.2318+0.0394.

## Conclusion - OH Masers

The 3 m TVRO mesh dish appears to be suitable for a number of strong OH maser sources accessible from HawkRAO.

## Methanol Masers - 12.2 GHz

Calculations are done for three solid dish sizes - 0.65 m, 1.2 m and 1.5 m diameter. These are the dish sizes on hand at HawkRAO.

The Smin results are colour-coded as  dark green  where Smin is less than the minimum observed flux density (indicating high probability of detection),  light green  where  Smin is less than the maximum observed flux density (indicating possible detection) and unhighlighted where  Smin is greater than the maximum observed flux density (indicating low probability of detection) or  Fmax Fmin (where there is only one observation and so a determination cannot be made).

### Calculations for a 0.65 Metre Dish

First we can estimate the transit time where the diameter of the dish is 0.65 m and the frequency is 12.2 GHz. The result for the transit is about 13 minutes (gain ~ 35 dBi, 3 dB beamwidth ~3.3o). Plugging values into the radiometer equation (accounting for source declination) and assuming Tsys = 150 K and Δf = 8.1 kHz (nominal spectral line width corresponding to a velocity resolution of about 0.2 km/s - which seems reasonable from an examination of the spectral profile of G351.42+0.64) gives estimates of Smin = 3687 Jy to 4471 Jy (depending on source declination) – indicating no candidates are detectable by the 0.65 m dish.

### Calculations for a 1.2 Metre Dish

Again we estimate the transit time where the diameter of the dish is 1.2 m and the frequency is 12.2 GHz. The result for the transit is about 7 minutes (gain ~ 41 dBi, 3 dB beamwidth ~1.8o). Plugging values into the radiometer equation (accounting for source declination) and assuming Tsys = 150 K and Δf = 8.1 kHz gives estimates of Smin = 1788 Jy to 1474 Jy (depending on source declination) – indicating no candidates are detectable by the 1.2 m dish.

### Calculations for a 1.5 Metre Dish

Again we estimate the transit time where the diameter of the dish is 1.5 m and the frequency is 12.2 GHz. The result for the transit is about 6 minutes (gain ~ 43 dBi, 3 dB beamwidth ~1.4o). Plugging values into the radiometer equation (accounting for source declination) and assuming Tsys = 150 K and Δf = 8.1 kHz gives estimates of Smin = 1236 Jy to 1012 Jy (depending on source declination) – indicating no candidates are detectable by the 1.5 m dish.

The first table of results below are from conservative input values assumptions for declinations (0o), Tsys (150 K) and velocity resolution (0.2 km/s). The results for all dish sizes are all classified as low probability of detection.

However - based on actual experience by others a second analysis is done with somewhat relaxed input values. The next table of results are for more relaxed values - where Tsys = 100 K and velocity resolution (0.85 km/s).  Under these conditions both the 1.2 m and 1.5 m dishes show high probability of detection for two and three sources respectively, while detection with the 0.65 m dish remains  at a low probability of detection.

## Conclusion - Methanol Masers

Both 1.2 m and 1.5 m dishes appear to be suitable for at least two Methanol 12.2 GHz masers accessible at HawkRAO assuming more relaxed input values as indicated by actual experience by others.

## H2O Masers - 22.2 GHz

Again calculations are done for three solid dish sizes - 0.65 m, 1.2 m and 1.5 m diameter. The assumed input values are Tsys = 200 K and Δf = 14.8 kHz (0.2 km/s velocity resolution) and 3 dB beamwidths of 1.8o, 1.0o and 0.8o respectively. The nominal transit time for each beamwidth has been adjusted according the declinations of each candidate source.

The Smin results are colour-coded as  dark green  where Smin is less than the minimum observed flux density (indicating high probability of detection),  light green  where  Smin is less than the maximum observed flux density (indicating possible detection),  light gray  where  Fmax Fmin (i.e., where there is only one observation and so a determination cannot be made)  or unhighlighted where  Smin is greater than the maximum observed flux density (indicating low probability of detection) .

It should be noted that G208.9928-19.3847 (ORION-KL) has been known to flare to over 7 MJy - which would seem to indicate that it would be an interesting object to observe long-term and might be detectable at times with a 0.65 m dish.  Also it might be possible that G351.2450+0.6644 at DEC = -36o (11700 Jy from just one observation) may be detectable.

## Conclusion - Water Masers

Although these results are just theoretical estimates, it seems that a number of water masers (22 GHz) are detectable from HawkRAO. The uncertainty in many estimates is due to the 'flaring' nature of water masers and the propensity for only such flare flux densities to be published.  Some masers have been regularly monitored and so their 'quiet phase' flux levels are indicated - while others with high flux densities have only one observation done - which means that the flux level for their 'quiet phases' are not known. An example is G351.2450+0.6644 - where the one published value of 11700 Jy would be detected in all three dish sizes - but if that was a 'flare' value, then currently it might be too 'quiet'. This uncertainty means the only way to find out if such candidates are detectable is to actually attempt detection.