Wide Field and Planetary Camera 2 Instrument Handbook for Cycle 14
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6.1 System ThroughputChapter 6:
System Throughput and SNR / Exposure Time Estimation
6.2 On-Line Exposure Time Calculator
6.3 Target Count Rates
6.3.1 Count Rates for Stellar Sources
6.3.2 Count Rates for Power Law Sources
6.3.3 Count Rates for Emission Line Sources
6.4 Sky Background
6.5 Signal-to-Noise Ratio Estimation
6.5.1 Point Sources -- PSF Fitting
6.5.2 Point Sources -- Aperture Photometry
6.5.3 Extended Sources
6.6 Exposure Time Estimation
6.7 Sample SNR Calculations
6.7.1 Point Sources
6.7.2 Extended Sources
6.7.3 Emission Line Sources
6.8 Photometric Anomalies
6.9 Red Leaks in UV Filters
6.10 Long-term Photometric Stability
6.11 Short-term Time Dependence of UV Response
6.1 System Throughput
A decision on a suitable exposure time will require the combination of
- The overall spectral response of the system (Figure 2.4).
- The spectral transmission of the filters (Chapter 3 and Appendix A).
- The spectral energy distribution and spatial profile of the target.
- The point response function and pixel size of the instrument
(Chapter 5).- Criteria for specifying desirable charge levels.
When the transmissions of filters T(
) are combined with the overall system response Q(
These calibrations exist in the STScI Calibration Data Base, and are accessible with the STSDAS SYNPHOT package or with the XCAL software. The XCAL and SYNPHOT Users Guides should be consulted for further details.
The throughput calibration presented here is accurate to at least 10%-which is sufficient for planning observations, but not for the analysis of many programs. Investigators wishing to do photometry on WFPC2 images should refer to the HST Data Handbook for an explanation of the conventions used in determining WFPC2 zeropoints and should use the zeropoints given in Table 5.1 of the WFPC2 Data Handbook (Version 4.0, January 2002). For the most accurate and up-to-date calibrations, users should examine the on-line version of the Data Handbook to verify that no numbers of interest have changed since the last paper publication. A recent study has examined the issue of WFPC2 zeropoints (Heyer, et al. 2004, WFPC2 ISR 04-01) and is recommending using the zeropoints of Dolphin (2002, private communication) on his web site at
http://www.noao.edu/staff/dolphin/wfpc2_calib/In Table 6.1 the dimensionless efficiency and the mean wavelength for each filter are tabulated together with the effective width, the equivalent Gaussian dimensionless width, the maximum transmission, the derivative of the mean wavelength with respect to spectral index, the pivot wavelength, average wavelength, and wavelength of maximum transmission. The parameters are defined as follows. The dimensionless efficiency is
The mean wavelength is defined in Schneider, Gunn, and Hoessel (1993, ApJ 264, 337).
This rather unconventional definition has the property that the correspondingly defined mean frequency is just
. It is in some sense halfway between the conventional frequency mean and the wavelength mean.
The pivot wavelength is defined as
The average wavelength
is that defined in the simplest sense
The effective dimensionless Gaussian width is defined implicitly by
The effective width of the bandpass is
We note that all of the above integrals have been evaluated over the range
to
so as to avoid unrealistic contributions from imperfect blocking far from the bandpass. Where necessary, the integration range was further constrained to the range 1000Å to 11000Å.
Parameters
and
are the respective parameters at the peak throughput.
The parameter
is defined in section Count Rates for Power Law Sources.
The final two columns in Table 6.1 are defined as follows. In the next-to-last column me/sec is the zero-point magnitude for 1 e- s-1 (with AB
=0). The final column gives twfsky, which is the exposure time (in seconds) needed to make the sky noise equal to 5 e- RMS (i.e. ~read noise) in the WFC for a sky level of V=23.3 mag arcsec-2.
Table 6.1: System Efficiencies and Zeropoints. See Section 6.1 for definitions. 1QT d
/d
1All values have been computed using the WF3 chip, except for the Quad filters.
6.2 On-Line Exposure Time Calculator
We note that most of the calculations below are incorporated in the on-line WFPC2 Exposure Time Calculator (ETC) program, which is available on the WFPC2 WWW pages at:
http://www.stsci.edu/instruments/wfpc2/Wfpc2_etc/wfpc2-etc.html.To use this program, the user fills out an HTML form giving the target information (magnitude, color, and reddening), camera configuration (PC or WFC, desired gain setting, and filter), and either the exposure time or the desired signal-to-noise ratio. There are separate HTML forms for point sources, extended sources, point sources with a diffuse stellar background, and extended sources on a diffuse stellar background. After filling out the form the user then clicks on "calculate" and the program returns the resulting signal-to-noise ratio if the exposure time was specified, or vice versa. Examples of completed HTML forms and results are shown in Sample SNR Calculations. Note that clicking on any colored text on the HTML form will give a description of that item.
The ETC program handles sources with stellar spectra, power law sources, and emission line sources; point sources and extended sources; and sources superposed on a diffuse stellar background. The latest version (V4.0) includes calculations of exposure times and/or signal-to-noise ratios for point sources (plus background) using either the traditional "optimal PSF weighting" method or simple aperture photometry in a fixed aperture radius specified by the user. The latter option is more appropriate when comparing with the ACS ETC, which assumes the use of aperture photometry as a default.
In addition, the ETC allows for a flexible specification of the sky background. There are now three options. The first option uses a rough estimate of "average" or "high" or "low" sky background conditions. The second option estimates the sky background based on the position of the target and (optionally) an estimate for the heliocentric longitude of the target (sun angle). The last option allows the user to explicitly provide a value for the sky background, in magnitudes per square arcsecond. Finally, the program also returns advice on CR-SPLITing, use of CLOCKS=YES, and warnings about saturation, if appropriate. Results are typically accurate to a few percent.
While observers should familiarize themselves with the material below, most will find the ETC program faster and easier to use for actual calculations. The ETC program will also be updated to reflect any changes in instrument performance, so observers can be assured of up-to-the-minute information.
6.3 Target Count Rates
We now consider estimation of count rates for objects with stellar, power law, and emission line spectra.
6.3.1 Count Rates for Stellar Sources
To estimate the number of electrons collected from a point source of apparent visual magnitude V, one can use the equation:
where t is the exposure time in seconds, the QT integral is given in Table 6.1, and AB
where F
Table 6.2: AB
sky B0 A0 0.4 F0 0.1 G0 8.9 K0III 13 M0III 15 gE Sa Sbc Scd Ir I
Equation 6.1 may be trivially rewritten to give the count rate Robject in units of e- s-1 pixel-1 for a target with a stellar spectrum as:
6.3.2 Count Rates for Power Law Sources
If one knows the spectral index
where S
. It can be shown that
if the integrands are weighted by a source with spectral index
6.3.3 Count Rates for Emission Line Sources
The count rate in units of e- s-1 for a monochromatic emission line is given by
where F is the emission line flux in units of ergs cm-2 s-1, and
In cases where the width of the line approaches that of the filter, it will be necessary to convolve the line shape and filter bandpass using either the SYNPHOT or XCAL programs.
For example, H
6.4 Sky Background
The sky background can contribute significant Poisson noise in broad and medium band filters, and must be taken into account during noise calculations. The actual sky brightness depends on the heliocentric ecliptic coordinates (latitude and longitude) in a manner summarized in Table 6.3. The appropriate AB
If your observations are sky background limited, and signal-to-noise is a driver, consider the use of the special requirement LOW-SKY as described in the Call for Proposals or the Phase II Proposal Instructions. LOW-SKY has two effects:
- It causes the observation to be scheduled at the time of year when the zodiacal background light is no more than 30% greater than the minimum possible background value for the target, and
- It requires that the observation be made when the bright Earth limb is more than 40° from the OTA axis, which greatly reduces scattered light.
For many targets LOW-SKY will have minimal impact on the observing efficiency. Note, however, that targets in the Continuous Viewing Zone (CVZ) cannot be observed if LOW-SKY is specified. See Observing Faint Targets for more information.
Another option for reducing the sky brightness, is the special requirement SHADOW, which forces the observation to be made when HST is in the Earth's shadow. This usually has a large negative impact on the observing efficiency, and is recommended only when attempting to avoid geocoronal lines when observing far-UV emission lines (e.g. Ly
Table 6.4 shows approximate sky count rates for the WFC and PC1 for filters with significant sky count rates. An average sky brightness of V=22.9 mag arcsec-2 is assumed. Filters not listed in the table have sky count rates below that of the dark current, so the sky contribution will generally be unimportant. Values for other filters or sky brightnesses can be computed from Table 6.2, Table 6.1, Table 6.3, and Equation 6.2.
6.5 Signal-to-Noise Ratio Estimation
The signal-to-noise ratio (SNR) for a point source depends on both the Poisson noise of the object, and on noises associated with the background. Sources of background noise include "read noise" of the CCDs, and Poisson noise in the dark current, sky background, and any smooth galaxy light superposed on the target.
Two-Gyro Mode: At some future date HST may be operated with only two gyros, hence causing additional spacecraft jitter and degradation of the effective PSF. While this could potentially degrade the signal-to-noise ratio for point sources, we expect to see very little impact for WFPC2 due to its large pixel sizes. Please see the
Two-Gyro Mode Handbookfor additional discussion.The SNR obtained for photometry of a point source will depend on the analysis technique used. The optimum SNR will be obtained when the pixels of the point source PSF are weighted in proportion to their expected intensity by PSF fitting. Aperture photometry will tend to give lower SNR, especially for sources where the background is important, but nonetheless is widely used. We now consider both methods.
6.5.1 Point Sources -- PSF Fitting
In the bright target limit, Poisson noise sets the SNR and
where S is the number of detected photons, and Robject is given by the above Equations 6.2 through 6.4, and t is the exposure time.
In the background limited case (e.g. read noise, dark current, or sky noise limited) the SNR is a function not only of the expected number of detected photons S from the source but also of the average effective background count rate B in each pixel, the point spread function
, and the weights used to average the signal in the pixels affected by the source. It is easy to show that the signal-to-noise ratio for optimal weights (which are proportional to the point spread function) is given by:
where sharpness is effectively the reciprocal of the number of pixels contributing background noise. The summation is tabulated for a few representative cases in Table 6.5. To estimate the signal-to-noise, multiply the signal-to-noise obtained, assuming all the flux is in one pixel, by the square root of the value in the table.
We note that PSF fitting is equivalent to convolving the image with the PSF, and then measuring the peak counts for stellar objects. Also, the location of the star on the pixel grid will be impossible to know in advance of the observation (i.e. pixel center vs. pixel corner in Table 6.5). In general, the lower "pixel corner" values should be used, so as to insure adequate SNR.
The average effective background counts per exposure and per pixel can be expanded to include various sources:
where terms include the read out noise of the CCD (readnoise), the dark current (Pdark), sky background count rate (Psky), and the count rate of any diffuse background light from astrophysical sources (Pbackground). Herein we will use "P" to represent count rates per pixel, and "R" to represent the total counts for an object. The exposure time is represented by t.
For example, Table 2.2 lists the faintest V magnitude star, V=28.19, measurable with a signal-to-noise ratio of 3 in a 3000s integration in F569W in the Wide Field Cameras. The calculation to check this goes as follows. The efficiency of the filter is 0.02343 from Table 6.1. The sky background in each pixel is 23.3+5=28.3, assuming an ecliptic latitude of 90° from Table 6.3, and the pixel area correction for the WFC given in that section. The total sky background collected per pixel in 3000 seconds is given by Equation 6.1 as 84.1 electrons. Note that the AB
If, instead, the peak count rate comes out much greater than the background, the observation is photon noise limited, and the signal-to-noise should be computed as the square root of the signal S in electrons.
In principle, one should also include contributions in the signal-to-noise for flat fielding uncertainties, noise in the bias and dark calibration files, and quantization noise. Flat fielding errors will be of order 1%, and will limit SNR in the large-signal limit. Noise in the bias and dark calibration files will be unimportant in most pixels, although these could become important if many (>10) non-dithered frames of the same field are combined.
Quantization noise can be estimated as
(i.e.,
in the 7 e- DN-1 channel, and
in the 14 e- DN-1 channel). In nearly all situations it can be ignored. In the weak signal case, the quantization noise is effectively included in the read noise values given throughout this Handbook; in the strong signal case it is very small compared to the Poisson noise and can be ignored.
A generalized equation for estimating point source signal-to-noise ratio per exposure is given below (Equation 6.6). It is exact in both the bright and faint object limits, and is a reasonable approximation to the intermediate case. Pbackground represents any generalized source of diffuse background light (e.g. galaxy on which target is superposed). Table 6.6 gives rough values for some of the parameters, along with references for more accurate values.
Note that in this formulation, sharpness-1 is the equivalent number of pixels the weighted signal is integrated over. In the event that multiple exposures are taken (e.g. to remove cosmic rays), the signal-to-noise ratio for the final averaged image is approximately given by:
where N is the number of images averaged.
Table 6.6: Parameters for Point Source SNR Estimation - PSF FittingRobject object count rate e- s-1 Equation 6.1, 6.2, or 6.3 Pdark dark count rate e- s-1 pixel-1 0.004 Table 4.2; Eqn 4.1 on page 90 Psky sky count rate e- s-1 pixel-1 Table 6.4 Table 6.2, Table 6.1, Table 6.3; Eqn 6.1 Pbackground count rate from background light (if any) e- s-1 pixel-1 Table 6.2, Table 6.1; Eqn 6.1 read noise e- ATD-GAIN=7 use 5.31 ATD-GAIN=15 use 7.5 Table 4.4 sharpness WFC use 0.11 PC1 use 0.06 Table 6.5 t exposure time s
1ATD-GAIN defaults to 7 unless otherwise specified on Phase II proposal.
6.5.2 Point Sources -- Aperture Photometry
When aperture photometry is used, one must consider the fraction of the object counts encircled by the aperture, as well the background noise in the aperture. In the bright target limit the SNR is given by
where S is the number of detected photons, f(r) is the fraction of the total counts encircled by the aperture with radius r, and Robject is target count rate. Representative values of f(r) are given in Table 6.7; values for other aperture sizes and filters can be estimated from Figure 5.3, or Figure 5.4.
In the faint target limit the noise contributed by background counts determines the SNR
where B represents the effective background counts per pixel, and r is the aperture radius in pixels.
In the generalized case the SNR per exposure for aperture photometry is given approximately by:
where the parameters are summarized in Table 6.8.
Table 6.7: Encircled Energy for Representative Filters. Encircled energy values are normalized to unity at large radius.
Radius(r)
Table 6.8: Parameters for Point Source SNR Estimation - Aperture Photometry.
Robject object count rate e- s-1 Equation 6.1, 6.2, or 6.3 Pdark dark count rate e- s-1 pixel-1 0.004 Table 4.2; Eqn 4.1 on page 90 Psky sky count rate e- s-1 pixel-1 Table 6.4 Table 6.2, Table 6.1, Table 6.3; Eqn 6.1 Pbackground count rate from background light (if any) e- s-1 pixel-1 Table 6.2, Table 6; Eqn 6.1 readnoise e- ATD-GAIN=7 use 5.31 ATD-GAIN=15 use 7.5 Table 4.4 f(r) encircled energy Table 6.7 Figure 5.3 or Figure 5.4 r aperture radius pixels t exposure time s
1ATD-GAIN defaults to 7 unless otherwise specified on Phase II proposal.
6.5.3 Extended Sources
The calculations for extended sources are nearly identical to those for point sources. The easiest procedure is to compute the SNR per detector pixel, and then adjust this value if the total SNR is required for an area encompassing many pixels.
In general, one will have the target magnitude or flux per square arcsecond. To compute the flux per pixel for the PC one merely multiplies the flux per square arcsecond by 0.00207, or instead, adds the value 6.7 to the magnitude per square arcsecond to get the necessary magnitude per PC pixel. For the WFC, one either multiplies the flux per square arcsecond by 0.00993, or adds 5.0 to the magnitude per square arcsecond. Equations 6.2, 6.3, and 6.4 can be rewritten including these factors as below.
PC Camera
For the PC camera, sources with stellar spectra, and V surface brightness per square arcsecond
we have a count rate in e- s-1 pixel-1 of
For power law sources where
is the target flux in units of ergs cm-2 s-1 Hz-1 arcsec-2 we have
And finally for emission line sources where
is the flux in ergs cm-2 s-1 arcsec-2 we have
where the emission line wavelength
WFC Cameras
For the WFC cameras and stellar sources with V surface brightness per square arcsecond
we have a count rate in e- s-1 pixel-1 of
For power law sources where
is the target flux in units of ergs cm-2 s-1 Hz-1 arcsec-2 we have
And finally for emission line sources where
is the flux in ergs cm-2 s-1 arcsec-2 we have
where the emission line wavelength
SNR
The generalized SNR per pixel per exposure for an extended source is then obtained simply by setting the sharpness to unity in Equation 6.5:
Table 6.9: Parameters for Extended Source SNR Estimation.Pobject object count rate e- s-1 pixel-1 Equations 6.9 to 6.12 Pdark dark count rate e- s-1 pixel-1 0.004 Table 4.2; Eqn 4.1 on page 90 Psky sky count rate e- s-1 pixel-1 Table 6.4 Table 6.2, Table 6.1, Table 6.3; Eqn 6.7 (PC) or 6.10 (WFC) Pbackground count rate from background light (if any) e- s-1 pixel-1 Table 6.2, Table 6.1; Eqn 6.7 (PC) or 6.10 (WFC) readnoise e- ATD-GAIN=7 use 5.31 ATD-GAIN=15 use 7.5 Table 4.4 t exposure time s
1Default value is ATD-GAIN=7.
Since many observations of extended sources are for galaxies in broad-band filters, a few rules of thumb can be useful. Saturation is seldom a concern, except in very bright spots such as the inner core of ellipticals and of some bulges. Count rates for spiral galaxies range typically from 2 to 0.01 e- pixel-1 s-1 (and lower) for filters such as F555W, F606W, F702W, and F814W; the lower end of the range corresponds roughly to the de Vaucouleurs D25. Count rates are significantly lower in blue and UV filters. Spiral structure can typically be traced reasonably well with total exposures of 3000 seconds or longer in the above filters.
For galaxies of very small angular size at redshifts of cosmological interest, the image may cover a small number of pixels; thus the detection of such objects follows rules similar to those of point sources. However, the fraction of light falling in the central pixel is smaller for most galaxies than it is for true point sources. The approximate magnitude difference between the light falling in the central pixel and the entire galaxy is plotted in Figure 6.1 for a typical giant elliptical galaxy, as a function of redshift. For other types of galaxies, a morphological term can be added to the values (for example, 0.6 magnitudes for lenticulars, 0.7 for S, 0.8 for Sab, 0.9 for Sbc, 1.2 for Scd, and 1.8 for Irr). These values must be increased by an additional 1.7 magnitudes for the PC.
Figure 6.1: Giant Elliptical Galaxy.
6.6 Exposure Time Estimation
In many instances one desires a certain SNR, and wishes to solve for the corresponding exposure time. Given the SNR, Equations 6.6, 6.7, or 6.14 can be solved for the exposure time, t. Since there are time-dependent and time-independent noise sources, quadratic equations are obtained. For example, we may solve Equation 6.6 for the point source exposure time:
where the term A contains the time-independent noise sources
and B contains the time-dependent noise sources
Equations for aperture photometry (6.7) and extended sources (6.12) can be solved with similar results. Parameters are as described in Table 6.6, Table 6.8, and Table 6.9. We again note that the on-line WFPC2 Exposure Time Calculator program provides an easy method for these calculations.
6.7 Sample SNR Calculations
Below we give further examples of SNR calculations. Appendix B also contains SNR plots for a wide range of representative cases.
6.7.1 Point Sources
Simple Star, Manual Calculation, PSF Fitting
We begin with the simple example of a V=20 star of spectral class G0. We want to observe with the PC using filter F555W. The star is somewhere near the ecliptic pole. We want to know the SNR for a 1200s CR-SPLIT exposure. Default ATD-GAIN=7 is used. We plan to use PSF fitting to analyze the data.
First we estimate the count rate for our target. Consulting Equation 6.2, Table 6.1, and Table 6.2 we have:
in units of e- s-1. Next we fill out Equation 6.6. To keep things simple we just use values from Table 6.6, and get the sky count rate from Table 6.4. There is no background light (i.e. no superposed galaxy), so Pbackground=0. The exposure time t=600 for each exposure of the CR-SPLIT:
The SNR for the total 1200s exposure, i.e. both halves of the CR-SPLIT, would simply be:
At these high SNR levels, it is likely that flat fielding would limit the photometric accuracy, rather than the noise. If we have a look at the terms in the SNR equation, we can see that the Poisson noise dominates; the term containing the sharpness and background noise sources is unimportant.
Just for fun, let us see what happens if we keep everything the same, but give the target V=25. Now we have Robject=0.74 e- s-1, and:
We see that now the term with the background noise (in particular, the read noise) limits the SNR. For the full 1200s exposure the SNRtotal=19.3.
Simple Star, Manual Calculation, Aperture Photometry
What if we now want to observe this same V=25 star, but we plan to reduce the data by measuring counts in a 0.5" radius aperture? We now use Equation 6.7 instead, consult Table 6.7 for the encircled energy f(r), and note that 0.5" corresponds to r=11.6 PC pixels:
Apparently using aperture photometry with a 0.5" radius aperture reduces the SNR by a factor ~4 as compared to PSF fitting, for this background limited case.
Simple Star, SNR Plots, PSF Fitting
We now repeat the first calculation above for the V=20 star using the SNR plots in Appendix B. We look up the G0 spectral class and F555W filter (5500Å) in Table B.1, and obtain AB
.
Simple Star, On-Line Calculator, PSF Fitting
The above calculation for a V=20 G0 star may also be performed using the WFPC2 Exposure Time Calculator program, which is available on the WFPC2 WWW pages at:
Figure 6.2: Sample Fill-out Form for WFPC2 On-Line Exposure Time Calculator..http://www.stsci.edu/instruments/wfpc2/Wfpc2_etc/wfpc2-etc.html
To use this program, access the above address with Netscape, or a similar program. Once in the WFPC2 area, select the "Software Tools" page, and then the "ETC" page. For the first example above, choose the "Point Source" form and complete it as shown in Figure 6.2 for the 600s sub-exposure. Then click the "calculate" button and after a few seconds the result is displayed (Figure 6.3). The answer, SNR=208, is comparable to that obtained by the manual calculation above for the 600s sub-exposure (SNR=209). Alternatively, one can input the total exposure time (1200s), and then use the result farther down the output page for "No. Sub-Exposures = 2" (see Figure 6.4), thereby obtaining SNR=291 for the total 1200s CR-SPLIT exposure.
Figure 6.3: Sample Results from WFPC2 On-Line Exposure Time Calculator.
Figure 6.4: Sample Results on CR-SPLITing from WFPC2 On-Line Exposure Time Calculator Results Page.
Star Superposed on Galaxy, Manual Calculation
We now consider a B=25 point source of spectral class B0, which is superposed on an elliptical galaxy with
We begin by computing the total count rate for the target. Using Table Table 6.2 we see that this target will have V=25.31. From Table 6.1 we obtain the filter efficiency and mean wavelength. Interpolating by mean wavelength in Table 6.2 we obtain AB
in units of e- s-1. Next we consider the background light from the superposed galaxy. We set
For the sky background, we note that Table 6.4 has no entry for F300W, so that the sky must be unimportant. If we wanted to calculate it anyway, as a check, we would use Table 6.3 for the sky brightness, Table 6.2 for the sky's AB
For the sharpness function we will use "pixel corner" values (least optimistic choice) from Table 6.5. Using read noise and dark current from Table 6.6, and Equation 6.6 for point source SNR:
for this single exposure. The SNR for multiple 40 min. exposures would be simply 17.9(N1/2), where N is the number of exposures.
Star Superposed on Galaxy, On-Line Calculator
The above calculation could also be performed with the on-line WFPC2 Exposure Time Calculator. One would select the "Point source + stellar background" form, and complete it as in Figure 6.5, and then click on "calculate." Figure 6.6 shows some of the results.
Figure 6.5: Point Source + Stellar Background Fill-out Form for WFPC2 On-Line Exposure Time Calculator. SNR is calculated for B=25 star (class B0) superposed on an elliptical galaxy (gE) with
Figure 6.6: Sample Output from WFPC2 On-Line Exposure Time Calculator.
6.7.2 Extended Sources
In general, the signal-to-noise level for extended sources can be computed by comparing the expected signal, S, in each pixel, determined from Equations 6.8 through 6.13, to the noise N=(S+B)1/2, where B is the equivalent background, determined in a manner similar to that for point sources. Unlike for point sources, the calculation does not, in a first approximation, involve the sharpness of the point spread function. For example, let us consider the observation of a source with a V surface brightness of 24 mag arcsec-2, assuming the F569W filter, WFC camera, and sky background V=23.3 mag arcsec-2. The signal-to-noise estimate goes as follows. The signal in each WFC pixel is 24.0+5.0 = 29.0 magnitude. By Equation 6.11, the total signal collected from the source in a 3000 second integration is S = 44.1 electrons, neglecting the small AB color correction. The sky signal per pixel is 84.1 electrons. The dark current is ~12 electrons. The total equivalent background is thus B = 84.1+5.32+12 = 124.2 electrons, larger than the signal detected, thus the noise is background-dominated. The noise is N=(S+B)1/2= 13.0 electrons, and the signal-to-noise per pixel expected in this case is 3.4. Similar calculations can be carried out for other filters; for observations in narrow-band filters and in the UV, the sky background signal will usually be unimportant. For very long observations of faint objects, other noise terms, such as flat field uncertainty, and errors in dark (and possibly bias) subtraction, must be considered more carefully.
If the scale of features in the target is larger than one pixel, the signal-to-noise can sometimes be improved by smoothing the observed image or - if read noise is a significant contributor - by reading the image out in AREA mode (see CCD Orientation and Readout).
6.7.3 Emission Line Sources
The signal-to-noise ratio calculation for point-like or extended emission-line sources is similar to that for continuum sources. However, the details of the calculation are different, because of the units used for the line flux, and because the flux is in a narrow line. The integrated filter efficiency is not relevant for the signal calculation; what matters is the total system throughput QT at the wavelength of the line, which can be determined from inspection of the figures in Appendix A. For lines near the center of the filter bandpasses the QTmax values from Table 6.1 can be used. The total signal expected for a point source of line strength F, expressed in erg s-1 cm-2, is S=2.28x1012
If the source is extended, the expected signal per arcsecond must be multiplied by the effective pixel area: 0.0099 arcsec2 for the WF, 0.0021 for the PC. For a line flux of, say, F = 10-15 erg s-1 cm-2 arcsec-2, this corresponds to 16 electrons in 1000 seconds for a WFC pixel. The noise is now dominated by the background, and the single-pixel signal-to-noise ratio is 16/(33 + 16)1/2 ~ 2.3.
Extended Line Emission Source, Manual Calculation
We now consider a detailed example of a planetary nebula observed on the PC with the F502N filter. The nebula has a diameter of 5" and a total flux F=4x10-13 erg s-1 cm-2 in the [OIII] 5007Å line. We want to estimate the SNR for an 1800s exposure, which will be CR-SPLIT.
First we must estimate the flux per square arcsecond. Using the nebula diameter, the average brightness is I
Next we estimate the SNR for each 900s sub-exposure using Equation 6.14 and Table 6.9. For this narrow filter the sky background can be ignored. We presume there is no background light from astrophysical sources:
Hence SNR=3.1 per pixel for each 900s sub-exposure. The SNR per pixel for the total 1800s is
The SNR for the entire nebula is this SNR per pixel times the square root of the number of pixels in the image, or ~460. In actuality, uncertainties in the photometric calibration and flat fields, would limit the SNR to ~100.
Extended Line Emission Source, On-Line Calculator
The above example could be calculated with the "Extended Source" form of the ETC program. The fill-out form would be completed as shown in Figure 6.7.
Figure 6.7: Extended Source Form for WFPC2 On-Line Exposure Time Calculator. Here the target is a galactic [OIII] 5007 line emission source and is observed on PC with filter F502N. SNR is computed for 1800s exposure.
We have selected "[OIII] 5007" on the emission line menu, and have left the redshift (z) set to zero. The PC and F502N filter are selected. Note we have entered the exposure time as 1800s. Scrolling down through the output page we find a table of SNR for various CR-SPLITings of the exposure (See Figure 6.8). "No. Sub-Exposures = 2" gives the answer we want, SNR=4.6 per pixel.
Figure 6.8: Sample Results on CR-SPLITing from WFPC2 On-Line Exposure Time Calculator Results Page.
Line Emission Point Source w/ LRF, Manual Calculation
In this example we consider an unresolved source of H
Since the redshift is significant, we cannot observe with the F656N filter. Instead we will use the Linear Ramp Filter (LRF). The observed wavelength will be 8007Å. From Table 3.7 on page 58 we see that this will be observed using the FR868N filter on CCD WF3. Combining the LRF transmission from Figure 3.3 on page 54 and the "WFPC2 + OTA System Throughput" from Figure 2.4 we estimate QT=0.054. We compute the count rate using Equation 6.4.
To estimate the SNR we use Equation 6.6, which assumes that PSF fitting will be used to analyze the image. Since the filter is narrow, we will ignore the sky emission. We use Table 6.6 for the WFC sharpness and also the read noise.
which is for an un-split 2400s exposure. The Poisson noise and background noises contribute nearly equally. For three such exposures over three orbits
Line Emission Point Source w/ LRF, On-Line Calculator
The above calculation can be performed using the ETC program. The "Point Source" form is used. "Emission Line" source and the "H 6563" line are selected; the redshift is set to 0.22. The program will automatically choose between PC and WFC, depending on the LRF setting. The least optimistic case of placing the object on a "pixel corner" is selected. The filter "LRF" is selected from the filter menu, and 8007Å is given for the central wavelength. The exposure time is specified as 2400s. (See Figure 6.9 for example of completed form.)
Figure 6.9: Point Source Form for WFPC2 On-Line Exposure Time Calculator. The target is an unresolved galaxy (z=0.22) nucleus with H
The result is SNR=13.1 for the un-split 2400s exposure (Figure 6.10), which is comparable to the manual calculation of SNR=14.
Figure 6.10: Sample Output for WFPC2 On-Line Exposure Time Calculator.
6.8 Photometric Anomalies
There are two photometric anomalies resulting from nonlinearities of the WFPC2 detectors. The first is due to the imperfect charge transfer efficiency (CTE) of the detectors, which causes sources at high row and column numbers to appear fainter because the charge is transferred over a bigger fraction of the chip. This anomaly is increasing with time, especially for faint sources, presumably as a consequence of on-orbit radiation damage. We have developed correction formulae which appear to reduce the impact of this anomaly to about 1-3% for faint sources. The second, called "long vs. short", causes sources to have a lower count rate - and thus appear fainter - in short exposures than in longer exposures, and appears independent of the position on the chip. The most likely explanation is that this effect is due to an overestimate of the sky measurement in the short exposure due to the presence of scattered light around bright stars. For further discussion, see Photometric Anomalies: CTE and Long vs. Short.
We also note the F1042M filter has an anomalous PSF which can impact aperture photometry. See PSF Anomaly in F1042M Filter.
6.9 Red Leaks in UV Filters
The presence of significant red leaks in the UV filters, together with the much greater sensitivity and wavelength coverage in the red part of the spectrum, can make UV observation and calibration difficult. Observers must sometimes be prepared to take additional frames at red wavelengths, in order to estimate the contribution of red leak to the UV counts. The counts contributed by red leak can be a significant noise source, and must also be taken into account during SNR and exposure time estimation. See "Red Leaks in UV Filters" on page 68 for detailed information. Note that the SYNPHOT synthetic photometry package can be used to estimate counts due to red leak for particular filter / target combinations.
6.10 Long-term Photometric Stability
The long-term photometric stability of WFPC2 has been evaluated by examining the photometric monitoring data collected over the lifetime of the instrument. Our primary standard, GRW+70D5824, has been observed roughly every four weeks, before and after decontamination procedures, both in the far UV and in the standard photometric filters. Early observations were taken monthly in both the PC and WF3. Later observations (since Cycle 6) were on a rotating schedule, where observations are taken in a different chip each month. Overall, a baseline of over ten years is available for the PC and WF3, and about eight years in WF2 and WF4. The data have been analyzed and reported by Baggett and Gonzaga (1998); here we summarize their main conclusions.
Overall, the WFPC2 photometric throughput, as measured via our primary standard, has remained remarkably stable throughout. Its long-term behavior in filters longward of F336W is characterized by small fluctuations (2% peak-to-peak) which appear to have no specific pattern, and there is no significant overall sensitivity trend. Aside from contamination corrections, which are only significant shortward of F555W, the same photometric zeropoints can be applied to non-UV data throughout the life of WFPC2.
In contrast, the UV photometric throughput of WFPC2 has changed measurably over the years. In most cases, the throughput has increased slowly, perhaps as a result of continuing evaporation of low-level contaminants. In F170W, the best-characterized UV filter on WFPC2, the clean throughput (immediately after a decontamination) has increased in the PC by about 12% from 1994 to 1998. Not all UV filter / detector combinations show this behavior; some combinations show a modest decline in throughput (e.g. 3% in F255W). Baggett and Gonzaga (1998) report the details of the secular throughput changes for the filters we monitor.
Finally, the contamination rates - the rate at which the camera throughput declines after a decontamination, due to the gradual buildup of contaminants on the cold CCD windows - have generally decreased since installation of WFPC2, possibly also because the environment has become cleaner with time. (This excludes brief periods of increased contamination just after servicing missions.) For example, the contamination rate in F170W in the PC has decreased from ~0.56%/day to ~0.45%/day. See Short-term Time Dependence of UV Response for additional discussion of the UV response variations.
Baggett and Gonzaga (1998) suggest a number of ways users can correct long-term changes in WFPC2 photometry. While these changes are generally small, users wishing to achieve high-precision photometry, especially in the UV, should follow their recommendations.
6.11 Short-term Time Dependence of UV Response
The UV throughput of the WFPC2 degrades in a predictable way after each monthly decontamination. The photometric calibration given in System Throughput is applicable at the start of each cycle, and measurements taken at other times must be corrected to account for the change in sensitivity since the last decontamination. In addition, a long-term change in sensitivity is present for the F160BW and F170W filter observations on the PC, and may be present to a lesser degree at other wavelengths.
Figure 6.11 shows the photometric monitoring data for the standard star GRW+70D5824 (a white dwarf classified DA3; B-V = -0.09) in the WF3 and PC1 for the set of filters which are routinely monitored. Only data after April 24, 1994, when the CCD operating temperatures were lowered from -76°C to -88°C, are shown. Figure 6.11 shows that the effect of contamination on the F675W and F814W filter observations is essentially negligible. However, at UV wavelengths contamination effects are readily apparent; the upper envelope of points indicate measurements made shortly after a decontamination, while the lower envelope are data taken shortly prior to a decontamination. Contamination effects are largest for the F160BW filter where they cause a 30% - 40% modulation in throughput. Table 6.10 shows the monthly decline in throughput based on this data. The values in parentheses are based on similar observations of the globular cluster
Cen (NGC 5139; mean B-V ~ 0.7 mag). In general, the values derived from the
Table 6.10: Change in WFPC2 Throughput Over 30 Days1.
1Values in parentheses are from the
A slight difference between the throughput declines for GRW+70D5824 and
Figure 6.12 and Figure 6.13 show the throughput decline for the F170W filter in all four chips as a function of days since the last decontamination. The contamination rate is remarkably constant during each decontamination cycle, and can be accurately modeled by a simple linear decline following the decontaminations, which appear to return the throughput to roughly the nominal value each month. While the contamination rates are similar for the three WF chips, the values for the PC are significantly lower.
In addition to the monthly changes in throughput there is evidence for a long-term variation in the F170W data on the PC, where the throughput has increased at the rate of approximately 3.3% ± 0.2% per year. This is evident in Figure 6.11, but is much clearer in the top panel of Figure 6.12 where lines are fitted separately to the epoch ~1994 (dotted line) and ~1998 data (solid line). The effect is most evident in Figure 6.14 where only data taken 4 days or less after a decontamination are shown. The F160BW filter shows an even stronger trend but with larger uncertainties (i.e., an increase of 9.0% ± 1.7% per year). The WF chips do not show this effect, nor do the observations on the PC at longer wavelengths. One possible explanation of the throughput increase is that WFPC2 was flown with some initial contaminant on the PC1 optics which is slowly evaporating on-orbit. The pre-launch thermal vacuum test gave evidence of elevated contamination in PC1, which is consistent with this hypothesis.
A second long-term effect is also apparent in Figure 6.12 and Figure 6.13. In all four CCDs the line fitted to the later data show a shallower slope, which indicates a slower throughput decline. The decline rate is reduced by 19% (PC) to 30% (WF4) over the four-year interval between the dotted and solid lines in each panel. This is likely caused by contamination slowly escaping the camera.
ISRs WFPC2 96-4 and WFPC2 98-3 describe detailed results of this monitoring (available from our WWW site). Observers are advised to consult the STScI WFPC2 WWW page for the latest information at the following address:
Figure 6.11: Photometric Monitoring Data for WFPC2.http://www.stsci.edu/instruments/wfpc2/wfpc2_top.html
Figure 6.12: Post-decontamination Throughput for F170W Filter in PC and WF2.
Figure 6.13: Post-decontamination Throughput for F170W Filter in WF3 & WF4.
Figure 6.14: Change in Throughput vs. Time.1
1Only data taken 4 days or less after a decontamination are shown. Data taken 0 to 60 days after service missions are also excluded. The fit is to data prior to MJD 51100.
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