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3.1 Through-put Measurements and SNR

In order to provide an easy way of estimating exposure times, we provide in Fig. 8 the actual measured count rates (per Å, not per pixel) normalized to an mag star. We show both the case where the slit is essentially wide-open (8 ) and when the slit has a nominal size in modest seeing conditions. Although various gratings and grisms are shown, the basis for comparing different gratings should really be the efficiency plots in the previous section. But Fig. 8 should provide a good guide to what to expect at the telescope in terms of count rates near the grating blaze in medium and excellent seeing.

Thus, we might expect to obtain about 800 e/sec/Å at th mag with the 4-m RCSpec under moderate seeing conditions. If the seeing were spectacular, it might be as much as 1000 or 1300, but let's not get too optimistic at proposal writing time---we're after a realistic number. Thus at 18th magnitude, we would expect to obtain e/sec/Å. To obtain a SNR of 50 per 3Å resolution element would require 2500 e per 3Å, or 2500/1.5 1700s, or about half an hour.

Figure 8:   The measured count rates per Å for the RC Spectrograph (4-m), the CryoCam (4-m), and GoldCam (2.1-m), normalized to a 10th magnitude star observed near zenith.

If you are working near the sky limit, you will have to contend not only with photon-noise from your object, but photon-noise from the sky. We assume that you can determine the sky level to infinite precision (a good enough assumption if your object is small compared to the slit length [not always true with multislits], and if you have done a good job in matching the slit-function as described in Sec. 4.1), but there will still be root-Nish fluctuations in the sky over your spectrum.

Let us imagine that we wish to observe a V=21 mag object at 5000Å with the RC Spectrograph on a good, moonless night, with 1.4 seeing. We are using a 2 slit.

Figure 9:   The spectrum of the moonless Kitt Peak sky. In the ``optical" region (4000-6500) the continuum rises from 23 mag/arcsec2 to 21.5, with the major artifical source the NaD lines from street-lights (see Massey, Gronwall & Pilachowski 1990 PASP, 102, 1046). Further in the red, the sky spectrum is dominated by OH emission lines.

We show the spectrum of the Kitt Peak dark sky in Fig. 9. At 5000Å the sky is about 22.8 mag/arcsec. The number of square arcseconds in this example is 4 arcsec, if we assume that we will extract the object spectrum using a 3 pixel extraction aperture (3 pixels = 2 ). Thus on top of our V=21 object we have a mag source: the sky. We expect to obtain e/sec/Å from the sky, and, unfortunately, only e/sec/Å from the object. Note that we needed to use the ``full throughput" number for the sky, while we must accept the ``modest seeing" number for our object! To reach a SNR of 50 per 3Å resolution element, we have as ``signal" from our object, but our noise sources are two-fold: the photon-noise from our object () plus the photon-noise from the sky: (), where t is the integration time in seconds. We must add the two noise sources in quadature, and hence

or 20800s (5.8 hrs), a little more than double the 2.6 hrs had there been no sky contribution.

With some moonlight, the sky will be brighter in the blue; the following table gives a rough guide.

Sky Brightness (mag/arcsec²)
lunar age (days) U B V R I

3600Å 4300Å 5500Å 6500Å 8200Å
0 22.0 22.7 21.8 20.9 19.9
3 21.5 22.4 21.7 20.8 19.9
7 19.9 21.6 21.4 20.6 19.7
10 18.5 20.7 20.7 20.3 19.5
14 17.0 19.5 20.0 19.9 19.2

We have ignored a third source of noise in the above discussion, namely read-noise. The read-noise component will be , where p is the number of pixels in a spectral resolution element and integrated over the spatial profile, and R is the read-noise. In the example above we would expect p to be roughly 2.5 X 3 = 7 pixels. With T2KB, the read-noise R=3 e. The noise contribution from photon-noise of the object is ; the contribution from the photon-noise is , while the read-noise contribution will be , and thus we were justified in ignoring it, even though in practice we'd probably break the 5.8 hr exposure down into 6 one-hour integrations, increasing the read-noise by . But in the case of low signal-to-noise, the read-noise may be important, particularly with the 15 e read-noise of CryoCam.

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Next: 3.2 Summary
Previous: 3 Estimating Exposure Times
Updated: 02Sep1996