Exposure Time Calculators

Exposure time calculators are typically used in one of two ways:
• If you know what level of accuracy you need, they tell you how long your exposure needs to be to get to that level of accuracy.
• If you want to expose for a certain amount of time, they tell you how accurate your photometry will be.

Philosophy/Goals while using the Calculators:

The goal of this exercise it to get you reasonably correct exposure times for understanding observing constraints. DO NOT OVERTHINK THE DETAILS. You want to get a rough idea of how long an observation will take, how good the data will be, and whether you should be using a big telescope or a small telescope. If you find yourself worrying about differences at 50% level, stop; your answer is good enough.

Using the Exposure Time Calculator:

• At CTIO, use either "4-m PF CCD Mosaic Imager", or "0.9-m Cass Direct CCD"
• At KPNO, use either "4-m PF CCD Mosaic Imager", or "*0.9-m CCD Mosaic Imager"
• Set magnitude to the faintest thing you need to observe, then set signal-to-noise ratio to what level of accuracy you want to get to.
• Leave seeing at 1.1 arcsec (typical seeing), leave airmass at 1.2 (this is a reasonable average), leave CCD summing factor at 1x1.
• Set lunar phase to be dark time (new moon, phase=0), gray time (partial moon, phase=7), or bright time (full moon, phase=14)
• Hit "Execute"

So let's say I wanted to do a study of the color-magnitude diagram of the globular cluster M13 down to an apparent magnitude of 22, and I wanted to get photometric accuracy of 0.02 mag, which is a signal-to-noise of 50. Let's compare what we get for the CTIO 4m and the KPNO 0.9m:

CTIO Results:
```Database: ctio.db  Telescope: 4mpf        Detector: MOSAIC2
Sum: 1   Arcsec/pixel: 0.27  Pixels/star: 24.0
Seeing: 1.1  Airmass: 1.20  Phase: 7.0

Filter    Time     Mag     SNR    Star Sky/pix    Noise contributions
Star     Sky     CCD

U 8618.86    22.0    50.0 36942.6 21168.7  192.20  712.78   30.17
B  200.26    22.0    50.0  9377.8  1049.7   96.84  158.72   24.64
V  185.98    22.0    50.0 11108.2  1568.4  105.40  194.01   24.63
R  290.22    22.0    50.0 18847.8  5109.9  137.29  350.20   24.71
I 1778.95    22.0    50.0 39347.9 24137.1  198.36  761.11   25.77

```
```Database: kpno.db  Telescope: 0.9m        Detector: MOSAIC1_1
Sum: 1   Arcsec/pixel: 0.42  Pixels/star: 10.0
Seeing: 1.1  Airmass: 1.20  Phase: 7.0

Filter    Time     Mag     SNR    Star Sky/pix    Noise contributions
Star     Sky     CCD

U 91881.2    22.0    50.0 37988.2 53807.7  194.91  733.54   34.31
B 3297.32    22.0    50.0  9489.2  2625.5   97.41  162.03   16.54
V 3662.80    22.0    50.0 11286.8  3939.3  106.24  198.48   16.66
R 6113.75    22.0    50.0 19293.2 12929.6  138.90  359.58   17.39
I 18605.7    22.0    50.0 40434.6 61311.9  201.08  783.02   20.73
z 96909.3    22.0    50.0 185198.5 1353296.6  430.35 3678.72   35.05

```
So what does this tell me? The "Time" column is the one you are most interested in-- that gives you the time it would take (in seconds) to get an image that gives you S/N=50 at 22nd magnitude in each filter. We can get an image really fast with the 4m, in fact, so fast that its probably better to use the 0.9m. Remember, you always want to use the smallest telescope that lets you do your science efficiently. So for this I'd use the KPNO 0.9m telescope.

But if I was looking at very distant, faint, and small galaxies at high redshift, they might be as faint as 25th magnitude. But maybe I dont need magnitudes quite that accurate, lets go to a signal-to-noise of 10 and see what we get:

CTIO Results:
```Database: ctio.db  Telescope: 4mpf        Detector: MOSAIC2
Sum: 1   Arcsec/pixel: 0.27  Pixels/star: 24.0
Seeing: 1.1  Airmass: 1.20  Phase: 7.0

Filter    Time     Mag     SNR    Star Sky/pix    Noise contributions
Star     Sky     CCD

U 81022.7    25.0    10.0 21912.1 198999.7  148.03 2185.40   59.30
B 1479.86    25.0    10.0  4372.6  7756.9   66.13  431.47   25.56
V 1454.80    25.0    10.0  5482.5 12268.7   74.04  542.63   25.54
R 2542.72    25.0    10.0 10419.1 44769.5  102.07 1036.57   26.30
I 16794.6    25.0    10.0 23438.4 227871.8  153.10 2338.57   34.71

```
```Database: kpno.db  Telescope: 0.9m        Detector: MOSAIC1_1
Sum: 1   Arcsec/pixel: 0.42  Pixels/star: 10.0
Seeing: 1.1  Airmass: 1.20  Phase: 7.0

Filter    Time     Mag     SNR    Star Sky/pix    Noise contributions
Star     Sky     CCD

U 865915.9    25.0    10.0 22589.1 507100.2  150.30 2251.89   95.25
B 24760.9     25.0    10.0  4496.1 19716.0   67.05  444.03   22.20
V 29013.4     25.0    10.0  5641.0 31203.3   75.11  558.60   23.15
R 53882.8     25.0    10.0 10728.7 113953.2  103.58 1067.49   28.10
I 176050.8    25.0    10.0 24140.5 580146.4  155.37 2408.62   45.12
z 961370.7    25.0    10.0 115921.2 1.3425E7  340.47 11586.7  100.23
```
Looking at those numbers, the times are unworkably long with the KPNO 0.9m, so this is a project for which you'll need a 4-meter telescope.

Signal to noise considerations:

We usually think of magnitude uncertainties -- how big of a magnitude error can I have. To turn this into a signal-to-noise, remember that for small errors (<0.2 mag), the relative flux uncertainty is approximately equal to the magnitude uncertainty. And the relative flux uncertainty (error/flux) is simply the inverse of the signal-to-noise (flux/error). So if I want 0.1 mag accuracy, that's a signal-to-noise of 10. If I want 0.02 mag accuracy, that' a signal-to-noise of 50. The best you'll be able to do is a S/N=100, and the worst you should consider is a S/N=3. Stay within those ranges.

Filters:

The calculators use UBVRI filters. If you are interested in using ugriz filters, that's fine, just adopt the UBVRI times (i.e., use U for u, B for g, V for r, etc). This is not a great assumption, but again will get you reasonable estimates.

Galaxies vs Stars (THIS IS REALLY IMPORTANT):

The exposure time calculator assumes you are doing photometry of individual stars, i.e., point sources. So it works well for giving you an estimate of exposure time to image a star of a given total magnitude. It also works pretty well if your galaxies are very small (say you are observing galaxies at very high redshift where their angular sizes are very small).

But if you are looking at extended sources (like nearby galaxies, which are spread over many pixels), the calculation will give you the wrong answer.  If you want to measure a large object down to a given limiting surface brightness (in mag/arcsec^2) -- for example, measure the surface brightness of a nearby galaxy, it's more complicated, and the calculator won't give accurate values. Instead, do the following: figure out how low in surface brightness you want to go (for example, the faint outskirts of galaxies might be at a surface brightness of mu(V)=27 mag/arcsec^2). Take the surface brightness limit you want to achieve, and subtract 1.5 from that number to use as the magnitude in the calculator.

So if I wanted to image faint galaxies with a surface brightness of mu(B)=26 mag/arcsec^2, I'd enter 26 - 1.5 = 24.5 as my target magnitude. (This is a TOTAL fudge factor that will get you in the right ballpark, but you should never do this on a real observing proposal!)

How faint in surface brightness do you need to go? If you are just thinking about the relatively bright parts of big galaxies, you'd probably want to be going down to a surface brightness of mu(V)=25 mag/arcsec^2 or so. If you were interested in the faint outskirts of galaxies, or looking for faint tidal tails around galaxies, or studying low surface brightness galaxies, you'd be wanting to get data that goes down to mu(V)=26 or 27 mag/arcsec^2. You can get a feel for this by looking at papers that have done what you are interested in doing, and seeing how faint their data goes.

Note: Observing at surface brightnesses fainter than 27 mag/arcsec^2 requires special techniques, and the calculators do not give proper estimates for such faint light levels. Don't do a project that requires data that deep.