We want to observe M86, an elliptical galaxy in Virgo, using the G750M grating at a central wavelength setting of
λc=6768 å, the CCD detector and the
52X0.2 arcsec slit. Our aim is to calculate the H
α count rate in the central region of M86 and the expected signal-to-noise ratio per resolution element for an exposure time of 1 hour. M86 has an inhomogeneous surface brightness distribution in Hα and the line is well resolved with this grating. Let us consider a region with an H
α surface brightness of
Iλ = 1.16
â 10
-15 erg/s/cm
2/å/arcsec
2 (note the unit - it is not the entire Hα flux but the flux per unit wavelength interval) and a continuum surface brightness
Iλ = 2.32
â 10
-16 erg/s/cm
2/å/arcsec
2. To derive the H
α and continuum count rates from the source we use the formula from
Chapter 13:
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= 1.14 â 10 13 counts/s/pix λ/pix s per incident erg/s/cm 2/å /arcsec2.
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Nλpix = 4 and Nspix = 2 (1 resolution element).
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Using the equation given in Section 6.2.1 (
“Diffuse Source” on page 85), we get the count rate
C = 0.106 counts/s in H
α and C=0.021 counts/s in the continuum. Since we are interested in the properties of the H
α line, the H
α counts constitute the signal, while both the H
α counts and the continuum counts contribute to the noise.
The sky background is negligible in comparison to the source, but the dark current (4.5 â 10
–3 count/s/pix
â 8 pixels = 0.020 count/s) and the read noise squared (29 e
-/pix
â 8 pixels
â 3 reads = 465 counts, for
CR-SPLIT=3) are important here. Substituting the numbers into the equation for signal-to-noise, we get:
To increase our signal-to-noise or decrease our exposure time, we can consider using on-chip binning. Let us bin 2 pixels in the spatial direction so that Nbin = 2. To allow adequate sampling of our new binned pixels, we leave
Nλpix = 4, but set
Nspix = 4, so
Npix = 16 and then
C = 0.212 for Hα and C=0.254 for the sum of Hα and continuum. To compute the time to achieve a signal-to-noise of 12 using this configuration, we use the full expression for the exposure time given on page
90, generalized to treat the line counts (for signal) and total counts (for noise) separately, and determine that roughly 35 minutes are needed in this configuration:
We illustrate the calculation of the exposure time for the G230LB grating. P041-C is found to have a flux of 1.7
â 10
–15 erg/s/cm
2/å at 2300 å.
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= 1.7 â 10 14 counts/s/pix λ per incident erg/s/cm 2/å;
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TA = 0.86 for the aperture throughput, taken from Chapter 13;
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Nspix=3, since ~80% of the point source light is encircled within 3 pixels;
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Nλpix= 2, since two pixels resolve the LSF;
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Using the equation on page 84, we calculate a point source count rate of
C = 0.34 counts/s over
Npix = 6 pixels for GAIN=1.
Exposure times for the two remaining wavelength settings can be calculated directly as time = signal-to-noise2 / C since the read noise, detector background, and sky background are negligible. As above, 3 pixels are taken to contain 80% of the flux. The results are summarized in
Table 6.6.
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Nspix to encircle 80% of PSF
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C (counts/s from source over N λpix=2)
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The aim is to get a signal-to-noise ratio of 30 using the [O II] filter. We know that NGC 6543 is about 6 times fainter in [O II] than in Hβ, and its total flux at [O II] 3727 å is ~4.4
â 10
–11 erg/s/cm
2 contained within 1 å. Since the radius of the object is about 10 arcseconds, the average [O II] surface brightness is about 1.4
â 10
-13 erg/s/cm
2/arcsec
2/å.
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= 6.7 â 10 11 counts/s/pix/å per incident erg/s/cm 2/å/arcsec 2 as given in Chapter 14.
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We take Npix = 4 â 4 = 16, since a resolution element has radius of two pixels (see Chapter 14).
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To calculate the count rate we use the equation on page 88 for diffuse sources and determine a per-pixel count rate of 0.094 counts/s/pix or a count rate
C = 1.5 counts/s over 16 pixels. The background and the dark current can be neglected. To get a signal-to-noise of 30 we need ~10
3 counts, so the read noise can also be neglected and we can use the simplified expression to calculate exposure time (see page
90). We obtain 10
3 counts in ~667 seconds. To allow post-observation removal of cosmic rays we use
CR-SPLIT=2. We note that in each ~330 second exposure we predict a mean of ~31 counts/pix, and thus we are safely within the limits of the CCD full well.
In the visible, the aim is to get a signal-to-noise of about 100 at λ = 4861 å, with the
G430M grating at a central wavelength setting of
λc = 4961 å, the CCD detector, and the
52X0.1 arcsecond slit. In the UV, the aim is to get a signal-to-noise ratio of about 20 at the C IV ~1550 å line with the
G140M grating at a central wavelength setting of
λc = 1550 and the
FUV-MAMA detector. To increase our signal-to-noise ratio in the UV, we use the
52X0.2 arcsecond slit for the
G140M spectroscopic observations.
NGC 6543 has an average Hβ surface brightness of
S(Η
β) ~ 8.37
â 10
-13 erg/s/cm
2/å/arcsec
2 at 4861 å and has a radius of about 10 arcseconds.
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= 1.62 â 10 12 counts/s/pix λ/pix s per incident erg/s/cm 2/å/arcsec 2 for G430M;
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Nλpix = Nspix =2 since 2 pixels resolves the LSF and PSF;
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Using the equation for diffuse sources on page 85, we derive a per-pixel count rate of 1.4 counts/s/pix and a count rate integrated over the four pixels of
C = 5.4 counts/s at 4861 å from the astronomical source. The sky background and the detector background are much lower. To allow cosmic ray removal in post-observation data processing, we use
CR-SPLIT=3. To achieve a signal-to-noise of 100, we require a total of roughly 10,000 counts, so read noise should be negligible, even over 4 pixels and with
NREAD=3. We calculate the time required to achieve signal-to-noise of 100, using the simplified equation on page
90, and determine that we require roughly 30 minutes.
At a count rate of ~1 counts/s/pix for 600 seconds per CR-SPLIT exposure, we are in no danger of hitting the CCD full-well limit.
The C IV flux of NGC 6543 is ~ 2.5 â 10
-12 erg/s/cm
2/arcsec
2 spread over ~1 å. The line, with a FWHM ~ 0.4 å, will be well resolved in the
G140M configuration using the
52X0.2 slit.
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= 5.15 â 10 9 counts/s/pix λ/pix s per incident erg/s/cm 2/å/arcsec 2 for G140M at λ=1550å using the 0.2 arcsecond wide slit.
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We take Nλpix = Nspix = 8, since the line emission is spread over the ~8 pixels of the slit width in dispersion, and we are willing to integrate flux along the slit to improve the signal-to-noise ratio.
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Using the equation for diffuse sources on page 85, we determine a per-pixel peak count rate of ~0.013 counts/s/pix and a count rate over the 64 pixels of
C = 0.82 counts/s at 1550 å from the astronomical source. The sky and detector backgrounds are still negligible, and the read noise is zero for the MAMA detector so we can use the simplified equation for exposure time on page
90 directly. We determine that we require ~7 minutes.
The aim here is to do high-resolution echelle spectroscopy of an O5 star in the LMC (such as Sk –69° 215) at 2500 å, using the E230H grating at a central wavelength of
λc = 2513 å and using the
0.2X0.09 arcsecond slit. The aim is to get a signal-to-noise ratio of about 50 from photon statistics. We will assume that the exact UV flux of the star is unknown and we need to estimate it from the optical data. This calculation of the stellar flux at 2500 å involves 2 steps:
We assume that it is an O5 V star with V = 11.6 (its exact spectral type is slightly uncertain). The expected
B –
V value from such a star is –0.35, whereas the observed
B –
V is –0.09; we thus get
E(
B –
V) = 0.26 mag.
We assume all the extinction to be due to the LMC, and use the appropriate extinction law (Koornneef and Code, ApJ,
247, 860, 1981). The total visual extinction is then
R â E(
B –
V) = 3.1
â 0.26 = 0.82, leading to an unreddened magnitude of
V0 = 10.78. The corresponding flux at 5500 å (using the standard zero point where
V = 0 corresponds to F(5500 å) = 3.55
â 10
-9 erg/s/cm
2/å) is F(5500 å) = 1.73
â 10
-13 erg/s/cm
2/å.
The model atmosphere of Kurucz predicts F(2500 å) / F(5500 å) = 17.2 for an O5 star, which leads to a flux of F(2500 å) = 2.98 â 10
-12 erg/s/cm
2/å at 2500 å for the unreddened star. Reddening will diminish this flux by a factor of 10
-0.4xA(2500 å), where the absorption at 2500 å can be determined from the extinction curve; the result in this case is A (2500 å) = 0.3. Thus the predicted flux of this star at 2500 å is 9.0
â 10
-13 erg/s/cm
2/å.
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= 2.9 â 10 11 counts/s/ pix λ per incident erg/s/cm 2/å for E230H;
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TA = 0.659 for the aperture throughput;
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εf = 0.8 for the encircled energy;
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Nλpix= 2, since two pixels resolve the LSF;
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Nspix= 3, since 80% of the point source light is encircled by 3 pixels;
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Using the equation for point sources on page 84, we determine a total count rate from the star of
C = 0.3 counts/s over 6 pixels. From
Chapter 13 we see that ~22 percent of the point source flux will be contained within the peak pixel. Thus the peak per pixel count rate will be approximately 0.3
â 0.22 / (0.8
â 2) = 0.045 counts/s/pix and well within the local linear counting regime. We can use the information that we register ~0.3 counts/s for every two pixels in the dispersion direction to estimate the global count rate (over the entire detector) as follows. Each order contains ~1024 pixels, and the
E230H grating at the central wavelength setting of 2513 å covers 33 orders (see
Chapter 13). A rough estimate of the global count rate is thus ~33
â 512
â 0.3/ 0.8 ~6400 counts/s and we are well within the linear range.
Consider a case where the aim is to image a faint (V = 28), A-type star with the clear filter and the CCD detector. We want to calculate the integration time required to achieve a signal-to-noise ratio of 5. The count rate from the source is 0.113 counts/s distributed over about 25 pixels using the information in
Chapter 14. If we assume the background to be “typical high” (
Table 6.3), the count rate due to the background integrated over the bandpass is ~ 0.15 counts/s/pix or 3.8 counts/s in 25 pixels (and the detector dark rate is 35 times lower). We will need to be able to robustly distinguish cosmic rays if we are looking for faint sources, so we will use
CR-SPLIT=4. We use the
STIS ETC to estimate the required exposure time to be 8548 seconds. To reproduce the numbers given by the
ETC, we use the equation on page
89:
Alternately, we could have requested LOW-SKY (see
Section 6.5.2), since these observations are sky-background limited. In that case the sky background integrated over the bandpass produces ~0.035 counts/s/pix
to which we add the detector dark current to get a total background of 0.039 counts/s/pix. Using the full equation for exposure time again, we then determine that we require only ~60 minutes. This option is preferable to perform this experiment. To check the S/N, we use the equation on page
89:
Suppose the aim is to do TIME-TAG observations of a flare star such as AU Mic, in the hydrogen Lyman-
α 1216 å line (see
Section 11.1.3). We wish to observe it with the
G140M grating, the MAMA detector and a 0.2 arcsecond slit. AU Mic has
V = 8.75, the intensity of its Ly-
α line is about 6 (
± 3)
â 10
-12 erg/s/cm
2/å, and the width (FWHM) of the line is about 0.7 (
± 0.2) å. We will assume that during bursts, the flux might vary by a factor of 10, so that the line flux may be up to 60
â 10
-12 erg/cm
2/s/å.
AU Mic is an M star and its UV continuum is weak and can be neglected.
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= 2.30x 10 12 counts/s/pix λ per incident erg/s/cm 2/å;
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Derive Nλpix= 14 since the line FWHM is ~ 0.7 å and the dispersive plate scale for G140M is 0.05 å/pix;
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Plugging these values into the point source equation on page 84, we get
C = 927 counts/s
over
10
â 14 pixels, or ~1160 counts/s
from the source during a burst (taking
εf = 1.0). This is well below the MAMA
TIME-TAG global linearity limit of 30,000 counts/s and the continuous observing limit of 26,000 count/s. The line is spread over 14 pixels in dispersion and roughly only 10% of the flux in the dispersion direction falls in the peak pixel; thus the peak per-pixel count rate,
Pcr, is roughly 927 / (14
â 10) = 7 counts/s/pix, and we are not near the MAMA local linearity limit.
For a TIME-TAG exposure, we need to determine our maximum allowed total observation time, which is given by 6.0
â 10
7/
C seconds or roughly 1079 minutes = 18 hours. For Phase II only, we will also need to compute the value of the
BUFFER-TIME parameter, which is the time in seconds to reach 2
â 10
6 counts, in this case 2157 seconds (=2x10
6/927).