Chapter 9:
Exposure-Time CalculationsIn this chapter...
9.2 Determining Count Rates from Sensitivities / 178
9.3 Computing Exposure Times / 184
9.4 Detector and Sky Backgrounds / 187
9.5 Extinction Correction / 192
9.6 Exposure-Time Examples / 193
9.1 Overview
In this chapter we explain how to use sensitivities and throughputs to determine the expected count rate from your source and how to calculate exposure times to achieve a given signal-to-noise ratio for your ACS observations taking all background contributions into account. At the end of this chapter you will find examples to guide you through specific cases.
9.1.1 The ACS Exposure Time Calculator
The ACS Exposure-Time Calculator (ETC) is available to help with proposal preparation at:
http://www.stsci.edu/hst/acs/software/etcs/ETC_page.html.
This ETC calculates count rates for given source and background parameters, signal-to-noise ratios for a given exposure time, or count rates and exposure time for a given signal-to-noise ratio for imaging, spectroscopic, and coronagraphic observations. A variety of apertures are now available, both circular and square, allowing the user to either select a radius in arcseconds or a size in pixels. The current default are a 0.2 arcseconds radius for both the WFC and the HRC, and a 0.5 arcseconds radius for SBC observations, which enclose approximately 80% of the PSF flux. Square and circular apertures are available between 0.1 and 2.0 arcseconds. For extended sources the S/N calculation is based on counts summed over one resolution element of 2 x 2 pixels, as the source size is assumed to be larger than the ACS resolution. A calibrated spectrum of your source can be provided directly to the Exposure Time Calculator. The ETC also determines peak per-pixel count rates and total count rates to aid in feasibility assessment. Warnings appear if the source exceeds the local or global brightness limits for SBC observations (see Section 7.2). The ETC has online help for its execution and interpretation of results. Alternatively, users can use synphot in STSDAS to calculate count rates and the wavelength distribution of detected counts.
9.2 Determining Count Rates from Sensitivities
In this Chapter, specific formulae appropriate for imaging and spectroscopic modes are provided to calculate the expected count rates and the signal-to-noise ratio from the flux distribution of a source. The formulae are given in terms of sensitivities, but we also provide transformation equations between the throughput (QT) and sensitivity (S) for imaging and spectroscopic modes.
Throughputs are presented in graphical form as a function of wavelength for the prisms and for the imaging modes in Chapter 10. Given your source characteristics and the sensitivity of the ACS configuration, calculating the expected count rate over a given number of pixels is straightforward, since the ACS PSF is well characterized. The additional required information is the encircled energy fraction (ef) in the peak pixel, the plate scale, and the dispersions of the grisms and prisms. This information is summarized in Table 9.1, Table 9.2, and Table 9.3 for Side 1. For updates please see the ACS Web page.
Table 9.1: Useful quantities for the ACS WFC.Table 9.2: Useful quantities for the ACS HRC.Table 9.3: Useful quantities for the ACS SBC.In each Table, the following quantities are listed:
- The pivot wavelength, a source-independent measure of the characteristic wavelength of the bandpass, defined such that it is the same if the input spectrum is in units of Fl or Fn:
- The integral ÚQlTl dl/l, used to determine the count rate when given the astronomical magnitude of the source.
- The ABmag zero point, defined as the AB magnitude of a source with a constant Fn that gives 1 count/second with the specified configuration.
- The sensitivity integral, defined as the count rate that would be observed from a constant Fl source with flux 1 erg/cm2/second/Å.
- The encircled energy, defined as the fraction of PSF flux enclosed in the default photometry aperture 0.2 arcseconds for the WFC and HRC, and 0.5 arcseconds for the SBC. These correspond approximately to 5 ¥ 5, 9 ¥ 9, and 15 ¥ 15 box-sizes respectively.
- The fraction of PSF flux in the central pixel, useful for determining the peak count rate to check for overflow or bright object protection possibilities.
- The sky background count rate, which is the count rate that would be measured with average zodiacal background, and average earthshine. It does not include the contribution from the detectors, tabulated separately in Table 3.1
Here, we describe how to determine two quantities:
- The counts/second (C) from your source over some selected area of Npix pixels, where a signal of an electron on a CCD is equivalent to one count.
- The peak counts/second/pixel (Pcr) from your source, which is useful for avoiding saturated CCD exposures, and for assuring that SBC observations do not exceed the bright-object limits.
We consider the cases of point sources and diffuse sources separately in each of the imaging and spectroscopy sections following.
9.2.1 Imaging
Point Source
For a point source, the count rate, C, can be expressed as the integral over the bandpass of the filter:
- A is the area of the unobstructed 2.4 meter telescope (i.e., 45,239 cm2)
- Fl is the flux from the astronomical source in erg/second/cm2/Å
- h is Planck’s constant
- c is the speed of light
- The factor l/hc converts ergs to photons.
- QlTl is the system fractional throughput, i.e., the probability of detecting a count per incident photon, including losses due to obstructions of the full 2.4 meter OTA aperture. It is specified this way to separate out the instrument sensitivity Ql and the filter transmission Tl.
- ef = the fraction of the point source energy encircled within Npix
pixels.- Sl is the total imaging point source sensitivity with units of
counts/second/Å per incident erg/second/cm2/Å.The peak counts/second/pixel from the point source, is given by:
Where:
Again, the integral is over the bandpass.
If the flux from your source can be approximated by a flat continuum (Fl = constant) and ef is roughly constant over the bandpass, then:
We can now define an equivalent bandpass of the filter (Bl) such that:
Where:
The count rate from the source can now be written as:
In Tables 9.1 to 9.3, we give the value of
for each of the filters.
Alternatively, we can write the equation in terms of V magnitudes:
where V is the visual magnitude of the source, the quantity under the integral sign is the mean sensitivity of the detector+filter combination, and is tabulated in Tables 9.1 to 9.3, and ABn is the filter-dependent correction for the deviation of the source spectrum from a constant Fn spectrum. This latter quantity is tabulated for several different astronomical spectra in Tables 10.1 to 10.3 in Chapter 10.
Diffuse Source
For a diffuse source, the count rate, C, per pixel, due to the astronomical source can be expressed as
:
Where:
Emission Line Source
For a source where the flux is dominated by a single emission line, the count rate can be calculated from the equation
where C is the observed count rate in counts/second, (QT) is the system throughput at the wavelength of the emission line, F(l) is the emission line flux in units of erg/cm2/second, and l is the wavelength of the emission line in Angstroms. (QT)l can be determined by inspection of the plots in Chapter 10. See Section 9.6.4 for an example of emission-line imaging using ACS.
9.2.2 Spectroscopy
Point Source
For a point source spectrum with a continuum flux distribution, the count rate, C, is per pixel in the dispersion direction, and is integrated over a fixed extraction height
in the spatial direction perpendicular to the dispersion:
Where:
For an unresolved emission line at
with a flux of
in
erg/second/cm2 the total counts recorded over the Nspix extraction height is:
These counts will be distributed over pixels in the wavelength direction according to the instrumental line spread function.
In contrast to the case of imaging sensitivity
, the spectroscopic point source sensitivity calibration (
) for a default extraction height of Nspix is measured directly from observations of stellar flux standards after insertion of ACS into HST. Therefore, the accuracy in laboratory determinations of
for the ACS prisms and grisms is NOT crucial to the final accuracy of their sensitivity calibrations.
The peak counts/second/pixel from the point source, is given by:
Where:
9.3 Computing Exposure Times
To derive the exposure time to achieve a given signal-to-noise ratio, or to derive the signal to noise ratio in a given exposure time, there are four principal ingredients:
- Expected counts C from your source over some area.
- The area (in pixels) over which those counts are received (Npix).
- Sky background (Bsky) in counts/pixel/second.
- The detector background, or dark, (Bdet) in counts/second/pixel and the readnoise (R) in counts of the CCD.
- Section 9.4 provides the information for determining the sky-plus-detector background.
9.3.1 Calculating Exposure Times for a Given signal-to-noise
The signal-to-noise ratio,
is given by
:
Where:
- C = the signal from the astronomical source in counts/second, or electrons/second from the CCD. The actual output signal from a CCD is C/G where G is the gain. You must remember to multiply by G to compute photon events in the raw CCD images.
- G = the gain is always 1 for the SBC, and ~1, 2, 4, or 8 for the CCDs, depending on GAIN.
- Npix = the total number of detector pixels integrated over to achieve C.
- Bsky = the sky background in counts/second/pixel.
- Bdet = the detector dark current in counts/second/pixel.
- R = the readnoise in electrons; = 0 for SBC observations, 5.0 and 4.7 for WFC and HRC respectively
- Nread = the number of CCD readouts.
- t = the integration time in seconds.
This equation assumes the optimistic (and often realistic) condition that the background zero point level under the object is sufficiently well known (and subtracted) to not significantly contribute; in crowded fields this may not be true.
Observers using the CCD normally take sufficiently long integrations that the CCD readnoise is not important. This condition is met when:
For the CCD in the regime where readnoise is not important and for all SBC observations, the integration time to reach a signal-to-noise ratio
, is given by:
If your source count rate is much brighter than the sky plus detector backgrounds, then this expression reduces further to:
i.e., the usual result for Poisson statistics of
.
More generally, the required integration time to reach a signal-to-noise ratio
is given by:
9.3.2 Exposure Time Estimates for Red Targets in F850LP
At wavelengths greater than 7500 Å (HRC) and about 9000 Å (WFC) ACS CCD observations are affected by a red halo due to light scattered off the CCD substrate. An increasing fraction of the light as a function of wavelength is scattered from the center of the PSF into the wings. This problem affects particularly the very broad z-band F850LP filter, for which the encircled energy depends on the underlying spectral energy distribution the most. In the currently available ETC, the treatment of such an effect has been ameliorated but not solved. The encircled energy fraction is calculated at the effective wavelength which takes into account the source spectral distribution. This fraction is then multiplied by the source counts. (The effective wavelength is the weighted average of the system throughput AND source flux distribution integrated over wavelength). However, this does not account for the variation in enclosed energy with wavelength.
As a consequence, in order to obtain correct estimated count rates for red targets, observers are advised to use the synphot package in IRAF/STSDAS for which the proper integration of encircled energy over wavelength has now been incorporated. To quantify this new synphot capability, we compare the ETC results with synphot for a set of different spectral energy distributions and the observation mode WFC/F850LP. In Table 9.4, the spectral type is listed in the first column. The fraction of light with respect to the total integrated to infinity is listed in the other two columns, for the ETC and synphot calculations respectively. These values are derived for a 0.2 arcseconds aperture for the ETC calculations and synphot.
Table 9.4: Encircled energy comparison for WFC/F850LP.The ETC results are off by 3% (O star), 2% (M star), 2% (L star), and 1% (T star). If this small effect is relevant to particular observations, then the synphot software package can be used. To learn how to use the synphot tool, we refer to the instructions provided in the April 2003 STAN, http://www.stsci.edu/hst/acs/documents/newsletters/stan0302.html, and in ACS ISR 2003-08.
9.4 Detector and Sky Backgrounds
When calculating expected signal-to-noise ratios or exposure times, the background from the sky, and the background from the detector must be taken into account.
9.4.1 Detector Backgrounds
See Table 3.1 for readnoise and dark current characteristics of the detectors, including variations by amplifier and GAIN for the CCDs.
9.4.2 Sky Background
The sources of sky background which will affect ACS observations include:
The background in counts/second/pixel for imaging observations can be computed as:
Where:
The image of the sky through a disperser is not uniform, since some wavelengths fall off the detector for regions of sky near the edge of the field of view (FOV). Since the ACS grism spectra are of order 200 pixels long, the regions of lower sky will be strips at the long and short wavelength edges of the FOV. The maximum width of the strips from where the signal starts to decline to the edge, where the signal is down by roughly 2x, is about half the total length of a spectrum of a point source, i.e., roughly 100 pixels in the case of a sky background with a continuum of wavelengths. In the case of the HRC, the sky for the dispersed mode will not have the low background strips, since the FOV is not masked to the detector size. These small strips of lower sky background in the SBC and the WFC are ignored in the following formulae. Furthermore in the SBC and the WFC, since the spectra do not lie along the direction of the anamorphic distortion, the plate scales of
and
above must be replaced by the plate scales
and
in the orthogonal spatial and dispersion directions, respectively. Interior to the strips, a point on the detector sees a region of sky over the full wavelength coverage of the disperser. Thus, for spectroscopic observations:
For a monochromatic sky emission line at
like Lyman-a, which will dominate the background through the LiF prism:
where
The total sky background is:
Figure 9.1 and Table 9.7 show “high” sky background intensity as a function of wavelength, identifying the separate components which contribute to the background. The “shadow” and “average” values of the Earthshine contribution in the ACS Exposure Time Calculator correspond, respectively, to 0 and 50% of the “high” values in Figure 9.1 and Table 9.7.
“Extremely high” Earthshine corresponds to twice the “high” value. For the zodiacal sky background, the values in Figure 9.1 and Table 9.7 and the ETCs correspond to the high value of mv = 22.1 from Table 9.5, while the “low” and “average” zodiacal light is scaled to mv = 23.3 and 22.7, respectively.
Figure 9.1: High sky background intensity as a function of wavelength.The zodiacal contribution (ZL) is at ecliptic latitude and longitude of 30×,180×, and corresponds to mv = 22.1 per square arcseconds. The Earthshine (ES) is for a target which is 38× from the limb of the sunlit Earth. Use Figure 9.2 to estimate background contributions at other angles. The daytime geocoronal line intensities are in erg/cm2/ second/arcseconds2 (see Table 9.6).Background Variations and LOW-SKY
In the ultraviolet, the background contains bright airglow lines, which vary from day to night and as a function of HST orbital position. The airglow lines may be the dominant sky contributions in the UV both for imaging-mode and spectroscopic observations. Away from the airglow lines, at wavelengths shortward of ~3000 Å, the background is dominated by zodiacal light, where the small area of sky that corresponds to a pixel of the high resolution HST instrumentation usually produces a signal that is much lower than the intrinsic detector background. The contribution of zodiacal light does not vary dramatically with time, and varies by only a factor of about three throughout most of the sky. Table 9.5 gives the variation of the zodiacal background as a function of ecliptic latitude and longitude. For a target near ecliptic coordinates of (50,0) or (-50,0), the zodiacal light is relatively bright at mv = 20.9, i.e. about 9 times the faintest values of mv = 23.3. Deep imaging applications must carefully consider expected sky values!
On the other hand, Earthshine varies strongly depending on the angle between the target and the bright Earth limb. The variation of the Earthshine as a function of limb angle from the sunlit Earth is shown in Figure 9.2. This figure also shows the contribution of the moon, which is typically much smaller than the zodiacal contribution, for which the upper and lower limits are shown. For reference, the limb angle is approximately 24× when the HST is aligned toward its orbit pole (i.e., the center of the CVZ), this limb angle corresponds to "Extremely High" Earthshine. The Earthshine contribution shown in Figure 9.1 and Table 9.7 corresponds to a limb angle of 30×, High Earthshine.
Figure 9.2: Background contributions in V magnitude per arcseconds2 due to the zodiacal light, Moon, and the sunlit Earth, as a function of angle between the target and the limb of the Earth or Moon.The two zodiacal light lines show the extremes of possible values.For observations taken longward of 3500 Å, the Earthshine dominates the background at small (< 22×) limb angles. In fact, the background increases exponentially for limb angles < 22×. The background near the bright limb can also vary by a factor of ~2 on timescales as short as two minutes, which suggests that the background from Earthshine also depends upon the reflectivity of the terrain over which HST passes during the course of an exposure. Details of the sky background as it affects ACS, as well as STIS, are discussed by Shaw, et al. (STIS ISR 98-21). The impact of Earthshine on ACS observations is discussed by Biretta, et al., (ACS ISR 03-05).
Table 9.5: Approximate zodiacal sky background as a function of ecliptic latitude and ecliptic longitude (in V magnitudes per square arcseconds).
Observations of the faintest objects may need the special requirement LOW-SKY in the Phase II observing program. LOW-SKY observations are scheduled during the part of the year when the zodiacal background light is no more than 30% greater than the minimum possible zodiacal light for the given sky position. LOW-SKY in the Phase II scheduling also invokes the restriction that exposures will be taken only at angles greater than 40× from the bright Earth limb to minimize Earthshine and the UV airglow lines. The LOW-SKY special requirement limits the times at which targets within 60× of the ecliptic plane will schedule, and limits visibility to about 48 minutes per orbit. The use of LOW-SKY must be requested and justified in the Phase I Proposal.
The ETC provides the user with the flexibility to separately adjust both the zodiacal (low, average, high) and Earthshine (shadow, average, high, extremely high) sky background components in order to determine if planning for use of LOW-SKY is advisable for a given program. However, the absolute sky levels that can be specified in the ETC may not be achievable for a given target; e.g., as shown in Table 9.5 the zodiacal background minimum for an ecliptic target is mv = 22.4 which is still brighter than both the low and average options with the ETC. By contrast, a target near the ecliptic pole would always have a zodiacal = low background in the ETC. The user is cautioned to carefully consider sky levels as the backgrounds obtained in HST observations can cover significant ranges.
Geocoronal Emission and Shadow
Background due to geocoronal emission originates mainly from hydrogen and oxygen atoms in the exosphere of the Earth. The emission is concentrated in the four lines listed in Table 9.6. The brightest line is Lyman-a at 1216 Å. The strength of the Lyman-a line varies between about 2 and ~30 kilo-Rayleighs (i.e., between 6.1x10-14 and 6.1x10-13 erg/second/cm2/arcseconds2 where 1 Rayleigh = 106 photons/second/cm per 4p steradian) depending on the position of HST with respect to the day-night terminator and the position of the target relative to the Earth limb. The next strongest line is the OI line at 1304 Å, which rarely exceeds 10% of Lyman-a. The typical strength of the OI 1304 Å line is about 1 kilo-Rayleighs (which corresponds to about 2.85x10–14 erg/second/cm2/arcseconds2) on the daylight side and about 75 times fainter on the night side of the HST orbit. OI 1356 Å and OI 2471 Å lines may appear in observations on the daylight side of the orbit, but these lines are ~10 times weaker than the OI 1304 Å line. The width of the lines also vary with temperature, the line widths given in Table 9.6 are representative values assuming a temperature of 2000 ×K.
Except for the brightest objects (e.g., planets), a filter or prism mode which does not transmit at Lyman-a should be employed. To minimize geocoronal emission the special requirement SHADOW can be requested. Exposures using this special requirement are limited to roughly 25 minutes per orbit, exclusive of the guide-star acquisition (or reacquisition), and can be scheduled only during a small percentage of the year. SHADOW reduces the contribution from the geocoronal emission lines by roughly a factor of ten while the continuum Earthshine is set to zero. SHADOW requirements must be included and justified in your Phase I proposal (see the Call for Proposals).
9.5 Extinction Correction
Extinction can dramatically reduce the counts expected from your source, particularly in the ultraviolet.Figure 9.3 shows the average
Av /E(B–V) values for our galaxy, taken from (Seaton, MNRAS, 187, 73P, 1979). Large variations about the average are observed (Witt, Bohlin, Stecher, ApJ, 279, 698, 1984).Extinction curves have a strong metallicity dependence, particularly in the UV wavelengths. Sample extinction curves can be seen in Koornneef and Code, ApJ, 247, 860 1981 (LMC); Bouchet et al., A&A, 149, 330 1985 (SMC); and Calzetti, Kinney and Storchi-Bergmann, ApJ, 429, 582, 1994, and references therein. At lower metallicities, the 2200 Å bump which is so prominent in the galactic extinction curve disappears; and Av/E(B–V) may increase monotonically at UV wavelengths.
Figure 9.3: Extinction versus wavelength.9.6 Exposure-Time Examples
In the following you will find a set of examples for the three different channels and for different types of sources. The examples were chosen in order to present typical objects for the three channels and also to present interesting cases as they may arise with the use of ACS.
9.6.1 Example 1: WFC Imaging a Faint Point Source
What is the exposure time needed to obtain a signal-to-noise of 10 for a point source of spectral type F2 V, normalized to V = 26.5, when using the WFC, F555W filter? Assume a GAIN of 1 and a photometry box size of 11 x 11 pixels, and average sky values.
The ACS Exposure Time Calculator (ETC) gives a total exposure time of 4410 seconds to obtain this S/N in a single exposure. Since such an exposure would be riddled with cosmic rays and essentially useless, it is necessary to specify how many exposures to split the observation into. ACS WFC observations generally should be split if the exposure time is larger than about 5 minutes, but for multi-orbit observations, splitting into 2 exposures per orbit is generally sufficient.
For a typical object visibility of 53 minutes, after applying the requisite overheads, there is time for two 1200 seconds exposures per orbit. The required exposure time can thus be reached in 4 exposures, but re-running the ETC using CR-SPLIT=4 raises the required exposure time to 5303 seconds (because of the extra noise introduced by the four extra readouts). To achieve the required exposure time would require CR-SPLIT=5, or three orbits.
Using the pencil and paper method, Table 9.1 gives the integral QTdl/l as 0.0775, and the ABn correction term can be retrieved from Table 10.1 as 0.040. According to Figure 5.9, a circular aperture of radius 0.3 arcseconds (which has an area of 116 pixels, close to the 121 pixel box specified) encloses about 90% of the light from a star. The count rate is then 2.5x1011*0.0775*0.9*10-0.4(26.5+0.040) = 0.423 counts/second, which agrees with the ETC-returned value of 0.42. The exposure time can then be found by using the equation
to give t = 4172 seconds, which is close to the ETC-derived value of 4410 seconds. We have inserted the background rate from Table 9.1 (Bsky = 0.055) and Table 9.5 (Bdet = 0.0032), and assumed that the noise on the background is much greater than the readout noise.
Note that this can be greatly shortened by specifying a smaller analysis box (for example, 5 x 5) and using LOW-SKY. Dropping the aperture size to 5 x 5 at average sky which still encloses 81% of the light requires 1532 seconds. Including both the smaller 5 x 5 box and LOW-SKY (Zodiacal = LOW, Earthshine = AVERAGE), using the ETC gives the required exposure time as only 1306 seconds (using CR-SPLIT=1), or 1540 seconds with CR-SPLIT=2. The LOW-SKY visibility per orbit is 47 minutes, which allows a total on-target exposure time of 2000 seconds in one orbit with CR-SPLIT=2.
Note also that the count rate from WFPC2 would be 0.167 electrons/second, a factor of 2.5 lower.
9.6.2 Example 2: SBC Objective Prism Spectrum of a UV Spectrophotometric Standard Star
What is the peak count rate using the PR110L prism in the SBC for the HST standard star HS2027+0651 (V = 16.9) that was used for the STIS prism calibration (this spectrum is not in the ETC list, therefore we quote below the flux which could be found by dearchiving the STIS spectrum)?
The sensitivity peaks in the 1500 Å to 1600 Å region. To find the count rate at 1537 Å, inspection of Figure 6.22 gives the sensitivity of 9.9x1014 counts/second per erg/cm2/s/Å. Multiplying by the stellar flux of 5.3587x10-14 gives 53.0 counts/second, summed in the cross dispersion direction. For the fraction of light in the central pixel e = 0.31, the brightest pixel at 1437.6 Å is 17.6 counts/second/pixel, well below the bright object limit.
The SBC has no readout noise, and the dark current rate is negligible, while the main sky contribution for PR110L is from Lyman-a. For daytime Ly-a intensity of 20kR = 6.1x10-13 ergs/cm2/second/arcseconds2, S¢ = 1.7x1014, and d, the dispersion in Å/pixel, is 2.58. Therefore, the background count rate is 6.1x10-13*1.7x1014*0.0322/2.58 = 0.041 counts/second/pixel. This value varies somewhat over the field, as the plate scale varies from the nominal 0.032 arcseconds/pixel. For faint source spectroscopy, it is better to use PR130L, which is on a CaF2 substrate to block Ly-a.
9.6.3 Example 3: WFC VIS Polarimetry of the Jet of M87
What signal-to-noise ratio is reached in three one orbit exposures (~2400 seconds each) for M87, when using the WFC, F555W for one orbit each in the three VIS polarizers? Gain is 2, box size is 5 x 5 pixels, CR-SPLIT=2, and average sky.
If the M87 jet region has mV = 17 magnitudes/arcseconds2, using the ETC with a flat continuum spectral distribution and an exposure time of 2400 seconds (CR-SPLIT=2), gives S/N = 131.6 for an observation with each VIS polarizer filter (which is an average of the polarizer at the 3 available position angles 0×, 60×, and 120×). If the polarization P is 20%, then P*S/N = 26.3, so using
from Chapter 6, sP/P = 0.031, or sP = 6.2x10-3, which is the error on the fractional polarization. The error on the position angle should be ~1.0×using the formula, again from Chapter 6, of
9.6.4 Example 4: SBC imaging of Jupiter’s Aurora at Lyman-alpha
What signal-to-noise ratio is reached in a one orbit exposure (2000 seconds) observing Jupiter’s aurora in Ly-a using the SBC and F122M filter?
The equation from the Section , “Emission Line Source,” on page 183 can be used to calculate the expected count rate. The aurora is variable, up to ~100kR. The value of (QT) for the SBC+F122M filter at 1216 Å is 0.0009, from inspection of Figure 10.106 on page 238. For a surface brightness of 40kR = 1.22x10-12 erg/cm2/second/arcseconds2 (See “Geocoronal Emission and Shadow” on page 191 for conversion), the total counts per pixel are given by the following calculation:
2.23x101 2* 0.009 * 1.22x10-12 * 1216 * (0.032)2 * 2000 = 61.0.
The background contributions are the detector dark of 1.2x10-5 counts/pixel/second (which can be ignored in this case) and a sky background which is dominated by geocoronal Lyman-a. During the daytime, the geocoronal background is 20kR, or 30.5 counts, while at night the background drops to one tenth of this, or 3.05 counts.
Finally, we calculate the signal-to-noise ratio S for a 2 x 2 pixel resolution element: in the daytime,
= 12.7, while at night,
= 15.2
9.6.5 Example 5: Coronagraphic imaging of the Beta-Pictoris Disk
In the final example we shall consider the case where we are trying to determine the S/N achieved on the Beta Pictoris disk, assuming a disk surface brightness of R magnitude of 16 arcseconds2 at a distance of 6 arcseconds from the central star with a V magnitude of 3.9, for an exposure time of 1000 seconds with an F435W filter. Assume that the star and disk have an A5 V-type spectrum. Using the ACS Exposure Time Calculator and considering the case for the 3.0 arcseconds occulting mask:
- Disk count rate = 4.98 e-/second for a 2 x 2 aperture (including 47.5% throughput of coronagraph) Sky count rate = 0.010 e-/second/pixel, Detector dark rate = 0.015 e-/second/pixel
- In 1000 seconds, this gives 4,980 e-/2 x 2 aperture in the disk region.
- Central star count rate = 3.63x108 e-/second for a 101 x 101 aperture (101 x 101 aperture used to estimate total integrated flux)
- At a distance 6 arcseconds from the central star, the fraction of flux per square arcsecond in the PSF wings is 2.6 x 10-6.
BPSF = 3.63 x 1011 * 2.6x10-6 = 943.8 e- per square arcsecond. The counts collected in 4 pixels are 4 x 0.0272 x (943.8 = 2.752.- The S/N in a 2 x 2 box is then
.
9.7 Tabular Sky Backgrounds
We provide Table 9.7 of the “high” sky background numbers as plotted in Figure 9.1. See the text and the caption in Figure 9.1 for more details. These high sky values are defined as the earthshine at 38° from the limb and the high zodiacal light of mv = 22.1 magnitude/arcseconds2.
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is the total point source sensitivity in units of counts/second per incident erg/second/cm
is the fraction of the point source energy within 
is the fraction of energy contained within the peak pixel.
is the point source sensitivity for the imaging mode.
and 
are the plate scales along orthogonal axes.
is the monochromatic intensity of a line at wavelength L in 
