Building high resolution spectrographs for 8 m telescopes while keeping large aperture size on the sky is very difficult if matching the aperture size to the detector pixel size and high spectral resolution are also requirements of the instrument. It is important to revisit the above mentioned requirements and determine whether any of them may be relaxed somewhat thereby making the design and construction of the instrument that much easier to achieve.
OH-suppression spectroscopy requires resolution of at least R > 2500. Compromising on the spectral resolution of the instrument is just not a possibility for the near-IR. If the instrument's performance is not to be affected significantly by detector dark current limitations, then keeping the number of pixels per object-resolution element as small as possible is clearly desirable. This strongly argues in favour of matching the aperture size to as few detector pixels as possible. Furthermore, the requirements of a high multiplex gain for multi-object spectroscopy pushes the design in this same direction.
The only variable remaining is the aperture size on the sky. Galaxies are not point sources even at high redshift. While HDF images of high redshift galaxies show a lot of structures present in them, it is not clear how much this structural make-up can be exploited by ground-based spectroscopy as this structure will be largely washed out by the seeing.
Two instrument design studies have explored this question: AUSTRALIS (Taylor et al. 1996) and VIRMOS (Le Fèvre et al. 1996). Unfortunately, these studies fail to elucidate the question as they reach somewhat different conclusions. VIRMOS concludes that a slit width of 0.8" is optimum, while AUSTRALIS has almost twice that value with a 1.5" circular aperture diameter. It is unlikely that the difference lies in the shape of the apertures: rectangular for VIRMOS, and circular for AUSTRALIS. Even more puzzling is the fact that the galaxy used in the VIRMOS simulation has I = 21, while the galaxies used for the AUSTRALIS study have 24 < I < 25. One would expect a larger aperture size to be required for a brighter galaxy, and not the other way round. The AUSTRALIS design study gives a fairly good description of the procedure and parameters used for the simulation. The VIRMOS report only shows results for a single galaxy, says nothing about the procedure and parameters used in the calculations, nor does it give any idea of the extent of the simulation work carried out. What was the spectral resolution of the instrument (R = 500 or R = 2500)? Was the simulation carried out for the visible and for the infrared?
It is to clarify this unsatisfactory state of affair
that it was decided that a comprehensive study of the optimum
aperture size for CIRPASS in both instrument configurations (slit
and fiber) had to be carried out. This document gives a detailed
description of the procedure and parameters used for the simulation
and presents the results obtained.
2. Sample Selection
The galaxy sample used for this study is the HAWAII
sample of Hubble Deep field (hereafter HDF) IR observed galaxies
found at "http:///www.ifa.hawaii.edu/~cowie/hdf.html"
(Cowie et al. 1996). This table was cross-referenced with the
main photometry database from Williams et al. (1996) so as to
identify their pixel co-ordinates on the HDF field. Each object
in the list was visually inspected on the HDF F814W image. Stars,
faint galaxies situated in the halo of bright objects, and bright
galaxies (J < 18.5 magnitude) were rejected from the sample.
The final sample consists of 170 galaxies distributed as follows:
22 objects brighter than J = 20 magnitude, 84 galaxies in the
J magnitude range 20 to 22, and 64 galaxies fainter than J = 22
magnitude (the faintest galaxy in the sample has J of 25.2).
3. Image Processing
The version 2 drazzled F814W HDF images were pulled
from the website archive and converted into IRAF image format.
To simulate atmospheric seeing, the images were processed by applying
a gaussian filter with a FWHM corresponding to 0.4" and 0.8"
seeing. All data reduction proceeded on all three sets of images:
no seeing (original), 0.4" seeing and 0.8"seeing.
4. Galaxy Curves of Growth
The first step in the process is the determination
of the galaxy brightness profiles, their curves of growth. Since
we are working with the visible light images, an assumption is
made that the galaxy brightness profiles at J are the same as
those observed at 814 nm. This should be a very reasonable expectation
as J is not that much redder than the images used for the curve
of growth determination. This is one of the factor that led us
to work with J rather than H magnitudes. The other factor is the
lack of H band magnitude measurements as the IR photometry was
carried out at J and with a notched H+K filter.
4.1 Aperture Photometry
The aperture photometry extraction was carried out
using the task APPHOTX in IRAF. The aperture diameters selected
correspond to 0.2", 0.4", 0.8", 1.0", 1.2",
1.4", 1.6", 1.8", and 2.0". A further 3"
diameter aperture was used to set the zero-point of the photometric
calibration as this was also the aperture size used to perform
the J photometry by the Hawaii group. They corrected their magnitude
to a 6" aperture but that correction factor, being unknown,
was ignored in calibrating the zero-point of our photometry. The
sky was determined in an annulus with an inner diameter of 4"
and a 2" width. The resulting curves of growth are presented
in Figures 1a-c. Comparing the three plots the increased loss
of light at small aperture size, as the seeing degrades, is obvious.
Furthermore, the effect of the seeing washing out the detailed
of the galaxy inner structures can be seen in the smoothness of
the curves of growth shown in Fig. 1c as compared to the undegraded
curves of growth presented in Fig. 1a.
4.2 Slit Photometry
The slit photometry is trickier to do than aperture photometry as there is no automated photometry package to perform this task. Furthermore, unlike aperture photometry for which a single parameter - diameter - is required to fully specify the aperture property, in the case of slit photometry, we need two parameters to characterise the slit geometry - length and width. Strictly speaking the optimal slit length is dependent on the galaxy observed, its shape and orientation with respect to the slit . In order to keep the problem tractable, we opted for a single value of slit length. The length of the slit was selected to be equivalent to the largest diameter aperture used in the previous section: 2". The slit widths selected correspond to 0.20", 0.36", 0.52", 0.68", 0.84", 1.00", 1.16", 1,32", 1.48", 1.64", 1.80", and 1.96". These values were selected so as to have an odd number of pixels across the slit. Furthermore, the slits were always kept aligned along the HDF detector columns. So there is no preferred orientation of slits versus galaxy orientations as would be the case for a multi-slit instrument.
The photometry was performed by calculating the means in boxes of the appropriate size with the task IMEXAMINE in IRAF. The sky was determined over a square region 8" on a side with the central inner square region 4" on a side taken out of it. The magnitudes observed in the 1.96" slit was set to the J magnitudes given by the Hawaii group. Comparing to the aperture photometry, the slit photometry calibration zero-point is about 0.2 mag brighter.
Because of the crudeness of the sky determination (no sigma clipping, no hi-lo rejection criterion applied, no sky skewness applied, etc ), it was found necessary to correct the sky measurements so as to obtain physically meaningful curve of growths. In some cases, the steepness of the curve of growth at large slit width exceeded that of the corresponding circular aperture curve of growth. This cannot be as the loss of light associated with a slit is affected in one dimension only, while a circular aperture is affected two dimensionally. This is the result of an underestimate in the sky determination. At the opposite end of the spectrum is the unrealistic case of a galaxy whose curve of growth faded with larger slit width. This is due to an overestimate of the sky value. The correction was performed by assuming that the galaxy image is small enough that the magnitude observed for a 1.96" slit width is the same as that for a 1.8" slit width. This correction worked well for most galaxies. There remained about 25 odd galaxies for which a "manual" re-adjustment of the sky was necessary to obtain believable curves of growth. Fortunately, 95% of these occurred in the original undegraded images whose analysis is included here mainly for illustration and completeness purposes. The seeing degradation applied to the image had the salutary effect of smoothing out the galaxy and sky structures that were affecting the statistics used in constructing the curves of growth. The main outcome of the sky correction is to underestimate the size of bright galaxies (which are likely to have curves of growth extending beyond 1.8") and/or underestimate the effect of seeing when that seeing approaches values close to 1.8". This means that determinations of the optimal slit width for galaxy spectroscopy given below are likely to be slightly underestimated.
The resulting curves of growth are presented in Figures
2a-c. As in Figs 1a-c, the effect of light loss due to seeing
is apparent, as is the smoothing out of galaxy inner structures
by the seeing. Also, as expected, the slit geometry is less sensitive
to light loss at small width than a circular shape is. This is
the result of circular apertures gaining light as the second power
of the diameter, while slits gain light as the first power of
5. S/N Calculations
With the galaxy curves of growth determined, calculating the S/N ratios achieved by a spectrograph is only a matter of choosing instrument parameters. The curves of growth data having already been fed into Excel spreadsheets, the S/N ratios were calculated by programming Excel macros to process the data appropriately. At a given aperture diameter or slit width, the corresponding galaxy magnitude, taken from the curve of growth, was fed into the S/N ratio calculation template. The process was repeated until the S/N ratios of all galaxies for all different aperture diameters/slit widths were determined.
The instrument parameters selected are shown in Table
1. The sky level selected represents the suppressed sky level
between the OH lines. Estimates of the sky level at J range from
s arcsec2 mm
to 1000 g/m2
s arcsec2 mm.
The value of 500 g/m2
s arcsec2 mm
was selected as it is close to the measured value of 580 g/m2
s arcsec2 mm
measured by Maihara et al. (1993) measured at 1.65 mm,
and atmosphere models predict a slightly lower OH-suppresssed
background at J than at H. The detector dark current and readnoise
values are based on the current performances achieved by Hodapp
et al. (1996). The spectral resolution of the instrument is set
at R = 3000 and the S/N ratios are calculated for a single resolution
element (i.e. no rebinning to lower resolution). Optical aberration
considerations are included by adding an extra pixel each in the
spatial and the spectral directions. For example, if the sky flux
is calculated over a 2 x 2 pixel region, the dark current and
readnoise are calculated over a 3 x 3 pixel region due to the
effect of optical aberrations. The total instrument throughput
was set to 30% from the throughput estimates made during the CIRPASS
feasibility study. This number includes telescope, detectors QE,
instrument optics, grating efficiency etc
, but excludes
slit or aperture losses as these are inherent to the curves of
growth. The total integration time is taken to be 4 hours (14400
seconds) made up of four 1 hour exposures.
5.1 Variable f-ratio systems
We calculated the S/N ratios for two possible instrument configurations. The first option is to match the aperture diameter/slit width to two detector pixels. The pixel size is taken to be 18.5 mm, which is the right value for the HAWAII detector. In this configuration, the f-ratio of the camera is variable with aperture diameter/slit width. For reference, an aperture diameter/slit width of 0.8" translates into an f/1.2 camera on an 8 m telescope. The system requires a faster (slower) camera for larger (smaller) aperture diameters/slit widths. In the case of circular aperture, the number of detector pixels used per object-resolution element remains constant throughout the entire set of aperture diameters. In the case of slits, however, the slit has a fixed length of 2" while the pixel scale keeps on changing along with the slit width. Therefore, the narrower the slit gets, the finer the pixel scale becomes, and the number of pixels required to sample the full 2" length of the slit increases accordingly. This leads to the situation where the detector dark current starts having a non-negligible effect on the instrumental performances for the narrowest slits (0.2", 0.36" ). We are ignoring for the moment any technical feasibility considerations associated with the speed of the camera.
The resulting S/N ratio curves are shown in Figures
3a-c and Figures 5a-c for circular apertures and slits respectively.
The galaxy sample was broken down into three magnitude bins: J
< 20, 20 < J < 22, and J > 22. The f/ratio of the
camera system corresponding to the different aperture/slit width
scales is indicated on the upper x-axis.
5.2 Fixed f-ratio systems
The second instrument configuration is to fix the pixel scale to a given value and let the aperture diameter/slit width cover a variable number of pixels. Because our previous design study of fast IR cameras for CIRPASS has shown that an f/1.2 system is about as fast a camera as is technologically achievable, we fixed the f-ratio of the system to f/1.2. This corresponds to a pixel scale of 0.4" for an HAWAII detector pixel size on an 8 m telescope and matching of the aperture size onto two pixels.
The resulting S/N ratio curves are shown in Figures
4a-c and Figures 6a-c for circular apertures and slits respectively.
The galaxy sample was broken down into three magnitude bins: J
< 20, 20 < J < 22, and J > 22.
6. Discussion of the Results
Inspection of Figures 3 - 6 reveals the following:
! The bright galaxy sample (J < 20) requires somewhat larger aperture sizes than the fainter sample does. This is exactly what is expected.
! In cases of exceptional seeing (0.4") or no seeing at all, the optimum aperture sizes are distributed over quite a broad range. The effect of average seeing (0.8") is to smooth out the sample and make all the galaxy images more similar to each other.
! Slit instrument performances are less sensitive to aperture size than circular aperture instrument. This is seen in the slightly broader shape of the S/N ratio curves for slits than for circular apertures. This is expected because of the shallower galaxy curves of growth for slits than for circular apertures.
! There is a steep drop in S/N ratios at narrow widths for slit instruments with variable camera f-ratio due to the onset of dark current limiting the instrumental performances.
! The results
for both circular apertures and slits are comparable. By "eye-balling"
Figures 3 - 7, the optimum circular aperture size is ~1"
and ~1.3" for 0.4" and 0.8" seeing respectively.
The optimum slit width is ~0.75" for 0.4" and ~1"
for 0.8" seeing.
In order to get a more refined determination of the
optimum aperture diameter/slit width, we decided to construct
histograms showing the distribution of the position of the peak
S/N ratio value versus aperture size. The plots are shown in Figures
7 - 10. The results are quite revealing. The effect of seeing
making all galaxies "look the same" is quite apparent
especially for slits. The width of the distributions clearly narrows
as one compares the results obtained with no seeing, 0.4"
seeing, and 0.8" seeing. Figures 7a, 8a, 9a, and 10a illustrates
very well the shift toward smaller sizes as galaxies get fainter.
The effect of seeing on the circular aperture peak S/N ratio positions
is revealed in Figures 7 - 8. At 0.4" seeing, the histogram
is effectively truncated below 0.8" aperture diameter. There
are very few galaxies for which the optimum aperture diameter
is below 1.4" with 0.8" seeing. For slits, the distribution
becomes surprisingly peaked. Part of the reason may lie in the
sky correction applied when processing the curves of growth. But
the outcome of this correction is likely to have shifted some
of the objects that would appear in the 1.8" and 1.96"
slit width bins into smaller bins (1.48" and 1.64").
The limitations in detector dark current affecting the slit in
the case of variable pixel scale is obvious in the difference
between the histograms shown in Figures 9a-c and Figures 10a-c.
The variable pixel scale configuration gives optimum slit widths
of 1" and 1.3" for seeings of 0.4" and 0.8"
respectively. In the case of a fixed scale of 0.4"/pixel,
the optimum slit width is 0.8" and 1.1" for seeings
of 0.4" and 0.8" respectively.
7. Summary and Conclusions
We have carried out detailed S/N calculations for a near-IR OH-suppressed spectrograph looking at faint galaxies. The HDF images were used to calculate the shape of galaxies under no seeing, 0.4", and 0.8" seeing conditions. Both options for MOS instruments were explored: fiber-fed spectrograph with circular apertures, and multi-slit instrument. Furthermore, we simulated two different instrument configurations. The first option is to match the aperture scale to 2 detector pixels. This allows adequate sampling of the spectrum and makes maximum use of the detector real-estate, but has the disadvantage of requiring impossibly fast cameras for aperture sizes larger than 0.8". The second option is to set the pixel scale at 0.4"/pixel, corresponding to an f/1.2 camera (about as fast as is achievable). This has the disadvantage that narrow slits and small apertures have undersampled spectra which might lead to some complication for software OH-suppression. Wide slits and large circular apertures require a larger number of pixels, thus making less efficient use of the detector pixels if maximum multiplex gain is the goal. And, if spread over to many pixels, the spectrum becomes detector dark current limited.
The optimum circular aperture diameters are 0.8"
- 1.0" and 1.3" - 1.5", for seeings of 0.4"
and 0.8" respectively. The corresponding values for slit
widths are 0.8" - 1.0" and 1.1" - 1.3" for
seeings of 0.4" and 0.8" respectively. To these values
must be factored in the uncertainty in relative astrometry and
fiber/slit positioning. Our results are entirely compatible with
the AUSTRALIS design study. Without knowing the full details of
the VIRMOS simulations, it is not possible to make a judgement
call as to the origin of the discrepancy between our and their
Cowie et al. (1996) "HAWAII IR observed galaxies" at http:///www.ifa.hawaii.edu/~cowie/hdf.html.
Le Fèvre et al. (1996) "VIRMOS Phase A Study".
Hodapp et al. (1996) New Astronomy, 1, 177.
Maihara et al. (1993) P.A.S.P., 105, 940.
Taylor et al. (1996) "The AUSTRALIS Concept Study".
Williams et al. (1996) A. J., 112, 1335
Sample: IR observed HDF galaxies from the HAWAII group.
Image: HDF version 2 drazzled data.
Seeing appplied to the HDF data: None, 0.4",
Telescope Diameter: 8 m
Instrument Throughput: 30%
Spectral Resolution: 3000
Detector Dark Current: 0.1 e-/s/pixel
Detector ReadNoise: 10 e-/pixel/read
Detector pixel size: 18.5 mm
Pixel scale: 0.4"/pixel (fixed pixel simulation cases only)
OH-Suppressed Sky Level: 500 g/m2 s arcsec2 mm
Total Integration Time: 4 hours - 14400 s
Number of Individual Exposures: 4
Wavelength used in all calculations: 1.25 mm
Slit length: 2"
Number of extraneous pixels added to account for optical aberrations:
1 in spatial direction + 1 in spectral direction.
Aperture diameters: 0.2", 0.4", 0.6", 0.8", 1.0", 1.2", 1.4", 1.6", 1.8", 2.0"
Circular Aperture Photometry Calibration: Done with 0.3" diameter aperture.
Sky Circular Aperture: Done on an annulus with 0.4" inner diameter and 0.2" width
Slit widths: 0.2", 0.36", 0.52", 0.68", 0.84", 1.0", 1.16", 1.32", 1.48", 1.64", 1.8", 1.96"
Slit Photometry Calibration: Done at 1.96" slit width.
Slit Sky Aperture: Square 8" on a side minus
inner square 4" on a side.