# SCUBA-2 Integration Time and Sensitivity

The amount of elapsed time required to map a certain field size, to a certain depth, at a given atmospheric transmission is described in the equations below. For users who simply want a quick time estimate we advise using the ITC gui which is explained here.

### Elapsed time

The elapsed telescope time (seconds) to map a given field size to a given 1-sigma depth (mJy) is described in the relations below.

Mapping Mode Time elapsed 450 microns (s) Time elapsed 850 microns (s)
Daisy  $\frac{1}{f}\bigg[\bigg(\frac{689}{T_{450}}-118\bigg)\frac{1}{\sigma_{450}}\bigg]^{2}$ $\frac{1}{f}\bigg[\bigg(\frac{189}{T_{850}}-48\bigg)\frac{1}{\sigma_{850}}\bigg]^{2}$
900″ $\frac{1}{f}\bigg[\bigg(\frac{1483}{T_{450}}-254\bigg)\frac{1}{\sigma_{450}}\bigg]^{2}$ $\frac{1}{f}\bigg[\bigg(\frac{407}{T_{850}}-104\bigg)\frac{1}{\sigma_{850}}\bigg]^{2}$
1800″ $\frac{1}{f}\bigg[\bigg(\frac{2904}{T_{450}}-497\bigg)\frac{1}{\sigma_{450}}\bigg]^{2}$ $\frac{1}{f}\bigg[\bigg(\frac{795}{T_{850}}-203\bigg)\frac{1}{\sigma_{850}}\bigg]^{2}$
3600″ $\frac{1}{f}\bigg[\bigg(\frac{6317}{T_{450}}-1082\bigg)\frac{1}{\sigma_{450}}\bigg]^{2}$ $\frac{1}{f}\bigg[\bigg(\frac{1675}{T_{850}}-428\bigg)\frac{1}{\sigma_{850}}\bigg]^{2}$
7200″ $\frac{1}{f}\bigg[\bigg(\frac{12837}{T_{450}}-2200\bigg)\frac{1}{\sigma_{450}}\bigg]^{2}$ $\frac{1}{f}\bigg[\bigg(\frac{3354}{T_{850}}-857\bigg)\frac{1}{\sigma_{850}}\bigg]^{2}$

The 450 and 850 transmission factors (Τ450, Τ850) and the sampling factor (f) in the above equations are described below:

### Atmospheric transmission

For a given air mass (AM) and opacity (classically defined at 225GHz, τ225GHz), the transmission can be calculated using the following relations:

$T_{450}=exp(-AM\times26(\tau_{225GHz}-0.01196))$

$T_{850}=exp(-AM\times4.6(\tau_{225GHz}-0.00435))$

The opacity (τ225GHz) ranges for various weather Grades are given below:

• Grade 1: Less than 0.83mm PWV.  (τ225GHz < 0.05)
• Grade 2: 0.83 – 1.58 mm PWV.  (0.05 < τ225GHz < 0.08)
• Grade 3: 1.58 – 2.58 mm PWV.  (0.08 < τ225GHz < 0.012)
• Grade 4: 2.58 – 4.58 mm PWV.  (0.12 < τ225GHz < 0.2)
• Grade 5: More than 4.58 mm PWV.  (τ225GHz > 0.2)

For a source at a given declination (δ, measured in degrees), a representative air mass near transit can be derived using:

$AM=\frac{1}{0.9cos\big[\frac{\pi}{180}(\delta-19.823)\big]}$

### Sampling factor f

Elapsed times are derived using the basic reduction parameters in SMURF and use default map pixels of 2″ and 4″ at 450 and 850 microns, respectively. A change to this default pixel size is taken into account by the sampling factor (f) when estimating the elapsed time. The sampling factor f is simply defined as:

f = ( pixel size requested / default pixel size )2

At 450 and 850 this becomes:

f450 = ( pixel size requested / 2 )and f850 = ( pixel size requested / 4 )2

For Point-source detections the S/N can be dramatically improved by applying a matched-beam filter which utilizes the full flux in the beam rather than just the peak value at the position of a source. This will shorten required observing times to reach a certain S/N typically by factors of f = 8 (450μm) and f = 5 (850μm).

### SCUBA-2 Confusion limit

The confusion limit depends on a number of factors including Galactic cirrus emission, the extra galactic background as well as the beam size. While dependent on assumptions and location the derived values are close to

• 850 microns = 0.7 mJy/beam
• 450 microns = 0.5 mJy/beam

The sensitivity of a SCUBA-2 blank field will be limited by a un-reduceable noise level of this order.
The implies that going below a detection limit of ~ 2 mJy in a blank field will give a high risk of false detections. See for instance Chen et.al Ap.J. 762, 81 (2011) ”Faint Submillimeter Galaxy Counts at 450 micron”. The situation is better for detection of a source at a known position. The lower probability that a random peak coincide with your know source increases the chance it is a real detection. However, the confusion limit still needs to be taken into account when estimating the position and flux error.