The spectroscopic exposure time package, SPECTIME, consists of a general calculation engine, SPTIME, and a collection of user or database defined IRAF scripts. The scripts are one type of user interface for SPTIME. Other user interfaces are Web-based forms and IRAF graphics/window applications. The user interfaces customize the general engine to specific spectrographs by hiding components and parameters not applicable to that spectrograph and guiding the user, through menus or other facilities, in the choice of filters, gratings, etc. However, SPTIME is a standard IRAF task that can be executed directly.
SPTIME takes an input source spectrum, either a reference blackbody, power law, or a user spectrum, a background "sky", observing parameters such as exposure time, central wavelength, seeing, airmass, and moon phase, instrument parameters such as aperture sizes and detector binning, a description of the spectrograph, and desired output. The output consists of a description of the observation, signal-to-noise statistics, and optional graphs and tables of various quantities as a function of wavelength over the spectrograph wavelength coverage.
SPTIME takes an input source spectrum, either a reference blackbody, power law, or a user spectrum, a background "sky", reddening to apply to the spectrum, observing parameters such as exposure time, central wavelength, seeing, airmass, and moon phase, instrument parameters such as aperture sizes and detector binning, a description of the spectrograph, and desired output. The interstellar extinction law is that of Cardelli, Clayton, and Mathis, ApJ 345:245, 1989. The output consists of a description of the observation, signal-to-noise statistics, and optional graphs and tables of various quantities as a function of wavelength over the spectrograph wavelength coverage.
SPTIME models a spectroscopic system as a flow of photons from a source to the detector through various optical components. It then computes signal-to-noise ratios from the detected photons of the source and background, and from the instrumental noise characteristics. The spectroscopic system components are defined at a moderate level of detail. It is not so detailed that every optical element has to be described and modeled and not so coarse that a single throughput function is used (though one is free to put all the thruput information into one component). Not all components modeled by the task occur in all spectroscopic systems. Therefore many of the components can be left out of the calculation.
The components included in SPTIME (additional components may be added later) are:
Each of these components specify a transmission function as the fraction of incident light transmitted as a function of some parameter or parameters. Except for the aperture, which is a function of the incident source profile (typically the seeing profile) relative to the aperture size, the transmissions of the components listed above are a function of wavelength.
The spectrograph component can be used in place of some or all of the other components. However, if the other components can be described separately, the spectrograph transmission is used for "fudge" terms. As described later, it is also used to specify default components in order to simplify the parameter description in SPTIME.
To make SPTIME easily configurable by observatories and users, all the component transmission functions are given in text files (called tables since they often contain an interpolation table). The data files may include parameters and an interpolation table. The files are searched for in the users working directory and then using a list of directories. Thus, users may place files in their work area to override system supplied files and observatories can organize the data files in a database directory tree. The role of other user interfaces is to map particular sets of data files for a spectrograph to parameters which users either do not see (when a component is always the same) or to a menu selection (such as for gratings).
Another simplifying feature is that many of the numerical parameters may be specified as INDEF to select reasonable defaults. For example if a central wavelength is not specified then the grating/grism blaze wavelength will be used. Other parameters, such as detector gain, will take default values defined in the calibration files.
Many spectrographs provide a wide variety of wavelength regions and dispersions. For gratings (and to some extent for grisms) this means use of different gratings, orders, tilts, and possibly camera angles in the spectrograph. The transmission as a function of wavelength (the grating efficiency) changes with these different setups. If the transmission function is given as an interpolation table this would require data files for each setup of each disperser. The structure of SPTIME allows for this.
However, it is also possible to specify the grating and spectrograph parameters and have the task predict the grating efficiency in any particular setup. In many cases it may be easier to use the calculated efficiencies rather than measure them. Depending on the level of accuracy desired this may be adequate or deviations from the analytic blaze function can be accounted for in another component.
The following sections discuss the task parameters , the contents of the various calibration data tables , the calculations performed by the task, and guidelines for configuring SPTIME for particular spectrographs.
I R A F
Image Reduction and Analysis Facility
PACKAGE = spectime
TASK = sptime
(time = INDEF) Total exposure time (sec)
(maxexp = 3600.) Maximum time per exposure (sec)
(sn = 25.) S/N per pixel if time is INDEF
# Source and sky parameters
(spectru=blackbody) Spectrum
(sky = ) Sky spectrum
(sensfun= ) Sensitivity function
(airmass= 1.) Airmass
(seeing = 1.) Seeing (arcsec)
(phase = 0.) Moon phase (0-14)
# Spectral distribution if no source spectrum
(tempera= 6000.) Blackbody temperature (K)
(index = 0.) Power law index
(refwave= INDEF) Reference wavelength (Angstroms)
(refflux= 10.) Source flux at reference wavelength
(funits = AB) Source flux units
(abjohns=sptimelib$abjohnson) Magnitude conversion table
# Setup parameters
(wave = INDEF) Central wavelength (Angstroms)
(order = INDEF) Order for grating
(xorder = INDEF) Order for cross disperser grating
(width = INDEF) Slit width (-=pixels +=arcsec)
(length = INDEF) Slit length (-=pixels +=arcsec)
(diamete= INDEF) Circular aperture diameter (-=pixels +=arcsec)
(inoutan= INDEF) Incident to diffracted grating angle (deg)
(xinouta= INDEF) Incident to diffracted cross disperser grating a
(xbin = 1) CCD Binning (Spatial)
(ybin = 1) CCD Binning (Dispersion)
# Spectrograph information
(search =spectimedb$) List of directories to search
(spectro= ) Spectrograph table
(filter = ) Filter table
(filter2= ) Filter table
(dispers= ) Disperser table
(xdisper= ) Cross disperser table
(fiber = ) Fiber table
(telesco= ) Telescope table
(adc = ) Atmospheric dispersion compensator table
(collima= ) Collimator table
(correct= ) Corrector table
(camera = ) Camera table
(detecto= ) Detector table
(apertur= ) Aperture table
(extinct= ) Extinction table
# Detector override parameters
(gain = INDEF) Detector gain (e-/ADU)
(rdnoise= INDEF) Detector read noise (e-)
(dark = INDEF) Detector dark count rate (e-/s)
# Sky subtraction parameters
(skysub = longslit) Type of sky subtaction
(nskyaps= 10) Number of sky apertures
# Output parameters
(output = object) List of output quantities
(list = ) Output list file
(graphic= stdgraph) Graphics device
(interac= yes) Interactive pause after graphs?
(nw = 101) Number of wavelength points
The spectrum file is a two column text table of wavelength in Angstroms and flux in ergs/s/cm^2/A. See the spectrum section for more details.
SPTIME uses text files to provide parameters and interpolation tables. These the source and sky/background input spectra are also given by interpolation tables. The files may contain comments, parameters, and tables.
Comment lines begin with '#' and may contain any text. They can occur anywhere in the file, though normally they are at the beginning of the file.
Parameters are comment lines of the form
# [parameter] = [value]
where whitespace is required between each field, [parameter] is a single
word parameter name, and [value] is a single word value. A quoted string
is a single word so if the value field contains whitespace, such as in
titles, it must be quoted. Any text following the value is ignored and may
be used for units (not read or used by the program) or comments.
Some parameters are common to all tables and others are specific to a particular type of file. The common parameters are
title - Title for the file used in documentation and labels.
description - Descriptive information that may be included in
documentation. There may be multiple parameters of
this type.
tablescale - Multiplicative factor to be applied to the table
values.
The optional table is a multicolumn list of numeric values. The list must be in increasing order in the independent columns. Only 1D (two columns) and 2D (three columns) tables are currently supported. 2D tables must form a regular grid. This means that any particular value from column one must occur for all values of column 2 and vice versa. The table is interpolated as needed. The interpolation is linear or bi-linear. Extrapolation outside of the table consists of the taking the nearest value; thus, a single line may be used to define a constant value for all values of the independent variable(s). If a "tablescale" parameter is specified then the table values in the last column are multiplied by this value before use by the program.
The types of tables used in SPTIME are listed below and described in the following subsections.
There are no special spectrum file parameters.
# title = "My spectrum"
3200.0 1.4072E-12
3250.0 1.3769E-12
3300.0 1.3111E-12
3350.0 1.2723E-12
3400.0 1.2015E-12
3450.0 1.1351E-12
3500.0 1.0928E-12
3571.0 1.0306E-12
3636.0 9.5812E-13
3704.0 8.9810E-13
3862.0 7.8894E-13
4036.0 7.4949E-13
4167.0 6.7146E-13
4255.0 6.2068E-13
4464.0 5.0959E-13
4566.0 4.7819E-13
4785.0 3.9346E-13
5000.0 3.2865E-13
5263.0 2.6804E-13
5556.0 2.1935E-13
5840.0 1.7776E-13
6056.0 1.5076E-13
6436.0 1.1520E-13
6790.0 9.7933E-14
7100.0 8.4752E-14
7550.0 6.5279E-14
7780.0 5.8710E-14
8090.0 4.9977E-14
units - Units of sky brightness (currently not used).
The sky brightness table has two or three columns. The first column is wavelength in Angstroms. The last column is the sky brightness in flux units of ergs/s/cm^s/A/arcsec^2 (currently this is the only units allowed but in future other units will be added). If there are only two columns then the sky brightness "spectrum" is used for all moon phases. If there are three columns then the second column is the moon phase given between new moon (phase of 0) and full moon (phase of 14). If no table is given then a sky brightness of 0 is assumed.
# title = "CTIO sky brightness as a function of moon phase"
# units = ergs/s/cm^2/A/arcsec^2
3600 0 6.63797E-18
4400 0 5.48524E-18
5500 0 6.87303E-18
6400 0 9.83901E-18
7900 0 1.34297E-17
24000 0 5.57523E-16
3600 3 1.05170E-17
4400 3 7.23239E-18
5500 3 7.53758E-18
6400 3 1.07888E-17
7900 3 1.34297E-17
24000 3 5.57523E-16
3600 7 4.59085E-17
4400 7 1.51104E-17
5500 7 9.93574E-18
6400 7 1.29707E-17
7900 7 1.61454E-17
24000 7 5.57523E-16
3600 10 1.66694E-16
4400 10 3.46180E-17
5500 10 1.89324E-17
6400 10 1.71019E-17
7900 10 1.94116E-17
24000 10 5.57523E-16
3600 14 6.63797E-16
4400 14 1.04558E-16
5500 14 3.60735E-17
6400 14 2.47186E-17
7900 14 2.55972E-17
24000 14 5.57523E-16
There are no special file parameters.
2.5 * (log (countrate) - log (f_lambda))
where countrate is the measured count rate corrected to remove extinction
in counts/s/A and f_lambda is the source flux in units of ergs/s/cm^2/A.
... 4672.9375 37.940693 4729.4702 38.038231 4786.0029 38.129082 4842.5356 38.212406 ...
The spectrograph file provides default values for all the SPTIME table parameters. This allows a convenient way to fix components that do not vary for a particular spectroscopic system and set defaults so that in SPTIME only the spectrograph file needs to be specified. The default parameters are optional and any value given directly as a task parameter overrides the value in the spectrograph file. The task parameters that can be specified in this file, where the table parameter name is the same as the task parameter name, are
sky - Sky brightness table/spectrum.
extinction - Extinction table.
telescope - Telescope table.
adc - ADC table.
aperture - Aperture table.
fiber - Fiber table.
collimator - Collimator table.
filter - Primary filter table.
filter2 - Secondary filter table.
disperser - Disperser table.
xdisperser - Cross-disperser table.
corrector - Corrector table.
camera - Camera table.
detector - Detector table.
Default values for the incident to diffraction angles may also be defined
in the spectrograph file with the parameters
inoutangle - Incident to diffraction angle for
primary grating.
xinoutangle - Incident to diffraction angle for
cross disperser grating.
The two common magnification terms in a spectrograph are from the camera and from the grating. The former is determined from focal length parameters in the collimator and camera descriptions and the later is determined from the grating description. In some cases there may be other sources of magnification. The following parameters provide for such magnification (remember do not include camera and grating magnification)
apmagdisp - Aperture magnification in dispersion direction. apmagxdisp - Aperture magnification in cross dispersion direction.
One source of magnification is when the aperture mechanism (such as the slit plate) is tilted for viewing and the user input for the aperture size is the unprojected slit size. For instance if the slit is tilted about the long axis the "apmagdisp" would be the cosine of the tilt to convert input slit widths to projected slit widths.
The spectrograph table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
In most cases all the transmission components are given in other tables; telescope, collimator, camera, detector, etc. The table in the spectrograph file provides a place for 1) any components not taken into account in the other tables, 2) a place to put final fudge corrections, and 3) if desired this could be the only table used apart from the filter, disperser, and detector tables.
# title = "4-m R-C Spectrograph"
# telescope = KPNO/Telescopes/mayall4m
# collimator = KPNO/Collimators/rcspec
# camera = KPNO/Cameras/uvfast
# detector = KPNO/Detectors/t2kb
# extinction = KPNO/kpextinct
# sky = KPNO/kpsky
# aperture = slit
# inoutangle = 46 deg
There are no special extinction file parameters.
# title = " KPNO Standard Extinction"
3200. 1.017
3250. 0.881
3300. 0.787
...
9832. 0.053
10256. 0.023
10400. 0.024
# Last four extinction coefficients from CTIO list - no data available
# for KPNO.
area - Effective collecting area in m^2 delivered to the instrument.
scale - Scale in arcsec/mm delivered to the instrument.
The area needs to take into account any obscurations. If the telescope
has different instrument focii and configurations then different telescope
tables would be used.
The telescope table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "KPNO Mayall 4-m Telescope"
# area = 9.186 m^2
# scale = 6.624 arcsec/mm (at R-C)
5000. 0.65
There are no special ADC file parameters.
The ADC table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "4m ADC"
5000 0.95
type - Aperture type (circular or rectangular).
diameter - Circular aperture diameter in mm or pixels.
width - Rectangular width (dispersion direction) in mm or pixels.
length - Rectangular length (spatial direction) in mm or pixels.
The aperture dimensions are optional and provide defaults if not given as program parameters. The dimensions are in mm if the values are positive and in projected pixels if negative. For fibers the diameter is normally given in mm and for slits a default of some number of pixels is defined.
The aperture table has two columns for circular apertures and three columns for rectangular apertures. The last column is the fraction of light transmitted though percentages may be given if the "tablescale" parameter is given as 0.01. The independent columns are the ratio of the aperture size divided by the seeing FWHM, that is, the aperture size in units of seeing FWHM. For rectangular apertures the first column is for the width (dispersion direction) and the second is the length (spatial direction). If no table is given then 100% transmission at all aperture sizes is assumed.
Aperture transmission tables can be made for various types of profiles. However, the most common one would be for Gaussian seeing profiles. The transmission values are then the fraction of the Gaussian profile passing through the aperture as a function of aperture size in units of profile FWHM.
1. Aperture description for a slit spectrograph.
# title = "Slit with Gaussian profile"
# width = -2 (-=pixels +=mm)
# length = -20 (-=pixels +=mm)
0.00 0.00 0.000
...
0.00 0.63 0.000
0.10 0.63 0.051
....
2.51 0.63 0.541
3.98 0.63 0.542
0.00 1.00 0.000
0.10 1.00 0.071
....
2.51 3.98 0.997
3.98 3.98 1.000
2. Aperture description for a fiber spectrograph.
# title = "Hydra red fiber aperture with Gaussian profile"
# diameter = 0.31 mm
0.00 0.000
0.10 0.007
0.13 0.011
0.16 0.017
0.20 0.027
0.25 0.043
0.32 0.067
0.40 0.104
0.50 0.160
0.63 0.241
0.79 0.354
1.00 0.500
1.26 0.667
1.58 0.825
2.00 0.937
2.51 0.987
3.16 0.999
3.98 1.000
aperture - Default aperture table.The default aperture table overrides the default value in the spectrograph file. The program aperture parameter overrides any value specified in this file.
The fiber table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "Hydra red fibers"
# aperture = "KPNO/Fibers/redfibap"
3400 0.0
4000 0.6
5000 0.81
6000 0.86
7000 0.88
8000 0.90
9000 0.92
10000 0.93
fl - Focal length in m.
The collimator table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "R-C Spectrograph Collimator"
# fl = 1.161 m
5000 0.85
There are no special filter file parameters.
The filter table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "RC Spectrograph GG455 Filter"
4370.00 0.000
4400.00 0.085
4450.00 0.032
4500.00 0.189
...
11800.00 0.963
12000.00 0.961
type - Type of disperser with a value of "grating".
gmm - Grooves per mm.
blaze - Blaze angle in degrees.
reflectance - Fractional reflectance of grating.
The grating table has three columns, the first order wavelength in Angstroms, the order, and the fractional efficiency at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given the efficiency (blaze function) is calculated. Normally the calculated function is used since the grating may be used in a wide range of configurations. However if the grating is used in a limited set of configurations (same spectrograph and a few orders) a table may be appropriate.
Note that the wavelengths are always at first order so that the table may be given in the required regular form. So for multiple orders the sample points are at the same first order values (wavelength * order) though the efficiency values will be for the particular order.
# title = "BL 181: 316 l/mm
# type = grating
# gmm = 316 groves/mm
# blaze = 7.396 deg
# reflectance = 0.78
The following grism parameters are used.
type - Type of disperser with a value of "grism".
gmm - Grooves per mm.
prism - Prism angle (= blaze angle) in degrees.
index1 - Index of refraction of grism in first order
(also default).
indexN - Index of refraction of grism in Nth order;
index2, etc.
transmission - Fractional transmission of grism.
The grism table has three columns, the first order wavelength in Angstroms, the order, and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given the transmission and blaze function is calculated. Normally the calculated function is used.
Note that the wavelengths are always at first order so that the table may be given in the required regular form. So for multiple orders the sample points are at the same first order values (wavelength * order) though the transmission values will be for the particular order.
# title = "Grism 810: 150 l/mm
# type = grism
# gmm = 150 groves/mm
# prism = 10.6 deg
# index = 1.52
# transmission = 1.
There are no special corrector file parameters.
The corrector table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "Coude Spectrograph Blue Corrector"
5000 0.8
fl - Focal length in m. resolution - Resolution at detector in mm. unvignetted - Unvignetted dispersion region in mm at the detector.
The detector specifies the number of pixels along the dispersion. If the full detector is vignetted the unvignetted parameter is used.
The camera table has two columns, the wavelength in Angstroms and the fractional transmission at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% transmission at all wavelengths is assumed.
# title = "UV Fast Camera"
# fl = 0.265 m
# resolution = 0.06 mm
5000 0.85
type - Type of detector. Currently only "ccd".
ndisp - Number of pixels along the dispersion.
pixsize - Pixel size in mm.
gain - Detector gain in electrons/ADU.
rdnoise - Detector readout noise in electrons RMS.
dark - Detector dark count rate in electrons/s/pixel.
saturation - Saturation in electrons.
dnmax - Data number maximum.
The gain, readout noise, and dark current are task parameters that can be used to override the default values in the detector file. The saturation and data number maximum values are for an extracted object and so is profile dependent. It is intended to provide a warning of possible error to users.
The detector table has two columns, the wavelength in Angstroms and the fractional detected quantum efficiency (DQE) at that wavelength. For values in percent use a "tablescale" parameter of 0.01. If no table is given then 100% DQE at all wavelengths is assumed.
# title = "T2KB - SITe 2048x2048 CCD"
# type = ccd
# ndisp = 2048
# pixsize = 0.024 mm
# gain = 2.0 e-/ADU
# rdnoise = 4.0 e-
# dark = 0 e-/s
# saturation = 70000 e-
# dnmax = 65535 counts
2500 0.00
3200 0.160
3650 0.498
4050 0.649
5000 0.765
6000 0.818
7000 0.799
8000 0.603
9000 0.331
10000 0.10
10700 0.00
There are no special parameters.
The table has two columns, the wavelength in Angstroms and the AB magnitude zero point. The relation is
AB = mag + abzero (lambda)
where "AB" is the AB magnitude corresponding to the broadband magnitude "mag"
at wavelength "lambda" and "abzero" is the interpolated table value.
# AB zeropoints.
# Values are from Astrophysical Quantities converted from
# log f (in W/cm^2/micron) to AB = -2.5 * log (F_lambda) - 5 log (lambda) - 2.4
# where lambda is in Angstroms and F_lambda is in ergs/s/cm^2/A. The
# values are for U, B, V, R, I, J.
3650 0.714
4400 -0.167
5500 -0.052
7000 0.275
9000 0.529
12500 0.815
This section describes the calculations, and assumptions behind the calculations, performed by SPTIME. These include the dispersion and efficiencies of gratings and grisms , the dispersion resolution , the spatial resolution and how it applies to the number of object and sky pixels in the apertures, the object and sky detected photons/counts, the signal-to-noise ratio , and the exposure time for a given S/N .
Gratings are assumed to tilted only around the axis parallel to the groves and with the incident angle greater than the blaze angle. The grating equation is then
g * m * w = sin(tilt+phi/2) + sin(beta)
where g is the number of groves per wavelength unit, m is the order, w is
the wavelength, tilt is the grating tilt measured from the grating normal,
phi is the angle between the incident and diffracted rays, and beta is the
diffracted angle. Phi is a spectrograph parameter and g is a grating
parameter. At the desired central wavelength beta is tilt-phi/2 and at the
blaze peak it is 2*blaze-tilt-phi/2 where blaze is the blaze angle.
The tilt is computed from the desired central wavelength. It is also used to compute the grating magnification
magnification = cos(tilt-phi/2) / cos(tilt+phi/2)
which is used in calculating the projected slit size at the detector.
This number is less than zero so the aperture is actually demagnified.
The dispersion, treated as constant over the spectrum for the sake of simplicity, is given by the derivative of the grating equation at the blaze peak,
dispersion = cos(blaze-phi/2) / (g * m * f)
where f is the camera focal length.
The grating efficiency or blaze function is computed as described by Schroeder and Hilliard (Applied Optics, vol 19, 1980, p. 2833). The requirements on the grating noted previously correspond to their case A. As they show, use of incident angles less than the blaze angle, their case B, significantly degrades the efficiency due to back reflection which is why this case is not included. The efficiency formulation includes variation in the peak efficiency due light diffracted into other orders, shadowing of the groves, and a reflectance parameter. The reflectance parameter is basically the blaze peak normalization and does not currently include a wavelength dependence. Thus the peak efficiency is near the reflectance value but somewhat lower and is order dependent due to the other effects.
Grisms are assumed to have a prism angle equal to the blaze angle of the inscribed grating. The index of refraction is treated as constant over the wavelength range of an order, though different index of refraction values can be specified for each order.
The grism formula used is a variation on the grating equation.
g * m * w = n * sin (theta+prism) - sin (beta+prism)
where n is the index of refraction, prism is the prism or blaze angle,
theta is the incident angle relative to the prism face, and beta is the
refracted angle relative to the prism face. Theta and beta are defined so
that at the undeviated wavelength they are zero. In other words at the
undeviated wavelength the light path is a straight through transmission.
The efficiency is also computed in an analogous manner to the reflection grating except that shadowing is not included (a consequence of the blaze face being parallel to the prism face and theta being near zero). Instead of a reflectance value normalizing the blaze function a transmission value is used.
The scale between arc seconds on the sky and millimeters at the aperture(s) of the spectrograph is a parameter of the telescope data file. This parameter is used to convert aperture sizes between arc seconds and millimeters.
The aperture sizes are magnified or demagnified by three possible factors. The basic magnification is given by the ratio of the collimator focal length to the camera focal length. These are parameters of the collimator and camera descriptions. This magnification applies both along and across the dispersion.
For grating dispersers there is a demagnification along the dispersion due to the tilt of the grating(s). The demagnification is computed (as given previously) from the grating parameters and the spectrograph parameter giving the angle between the incident and diffracted rays at the detector center.
The last magnification factor is given by the spectrograph parameters "apmagdisp" and apxmagdisp". These define magnifications of the aperture along and across the dispersion apart from the other two magnifications. These parameters are often missing which means no additional magnifications.
One use for the last magnification parameters is to correct aperture sizes given as millimeters or arc seconds on a plane tilted with respect to the focal plane. Such tilted apertures occur with aperture mechanisms (usually slits) that reflect light for acquisition and guiding. Note that one only needs to use these terms if users are expected to define the apertures sizes on the tilted plane. If instead the projection factors are handled by the spectrograph system and users specify aperture size as millimeters or arc seconds on the sky then these terms are not needed.
The above scale factors map arc seconds on the sky and aperture sizes in millimeter to arc seconds and millimeters projected on the detector. To convert to pixels on the detector requires the pixel size as given in the detector description. One option in SPTIME is to specify aperture sizes as projected pixels on the detector (either in the user parameters or in the aperture description file). Using the detector pixel size and the scale factors allows conversion of aperture sizes specified in this way back to the actual aperture size.
A camera resolution parameter may be set in the camera description. If a resolution value is not given it is taken to be 2 pixels. This parameter is used to define the dispersion resolution element and the number of pixels across the dispersion imaged by the detector for the aperture and the object. The latter usage is discussed in the next section.
The dispersion resolution element, in pixels, is given by
| 2 pixels
disp resolution = maximum of | camera resolution
| 1 + min (seeing, apsize)
where seeing is the FWHM seeing diameter in pixels and apsize is the
aperture size in pixels. For circular apertures the aperture size is
the diameter and for rectangular apertures it is the width. The first term
comes from sampling considerations, the second from the camera resolution,
and the third from the finite resolution of a pixel (the factor of 1) and
the spread of wavelengths across the aperture or seeing disk. The
dispersion resolution is printed for information and the S/N per dispersion
resolution element is given in addition to the per pixel value.
The number of pixels across the dispersion in the object and the sky are required to compute the S/N statistics. The number of pixels in the projected aperture image is taken to be
| diameter + resolution (circular apertures)
aperture pixels = |
| length + resolution (rectangular apertures)
where resolution is the camera resolution discussed previously. The value
is rounded up to an integer.
Objects are assumed to fill circular (fiber) apertures. Therefore the number of object pixels is the same as the number of pixels in the aperture. In rectangular (slit) apertures the number of object pixels is taken to be
| 3*seeing + resolution
object pixels = minimum of |
| number of aperture pixels
where seeing is the FWHM seeing diameter converted to pixels. The values
are rounded up to an integer.
The number of sky pixels depends on the type of sky subtraction. For "longslit" sky subtraction the number of sky pixels is given by the difference of the number of aperture pixels and the number of object pixels. For circular apertures this always comes out to zero so it does not make sense to use longslit sky subtraction. For rectangular apertures the number of sky pixels in the aperture depends on the aperture size and the seeing. If the number of sky pixels comes out to zero a warning is printed.
For "multiap" sky subtraction the number of sky pixels is the number of sky apertures times the number of pixels per aperture.
The source spectrum flux at each wavelength is converted to units of photons per second per Angstrom per square centimeter. This is multiplied by the telescope effective area, the exposure time, and the pixel size in Angstroms to give the source photons per dispersion pixel per exposure. This is then multiplied by any of the following terms that apply to arrive at the number of source photons detected over all spatial pixels. The spatial integration is implicit in the aperture function.
The sky spectrum, if one is given, defines a photon flux per square arc second. When it is given as a function of the moon phase it is interpolated to the specified moon phase. The sky brightness is multiplied by the area of the aperture occupied by the object (in arc seconds) and divided by the aperture transmission of the source. This is the quantity reported in the output for the sky photon flux. It is comparable to the source photon flux.
Next this flux is multiplied by the telescope effective area, the exposure time, and the pixel size in Angstroms. Finally it is multiplied by the same transmission terms as the object except for the extinction. Note that this removes the aperture transmission term included earlier giving the sky photons as the number passing through the aperture per object profile. The final value is the number of sky photons from the object. To get the sky photons per spatial pixel the value is divided by the number of spatial pixels occupied by the source.
If no sky subtraction is specified then the sky counts are added to the source counts to define the final source counts and the sky counts are set to zero.
The noise attributed to the source and sky is based on Poisson statistics; namely the noise is the square root of the number of photons. The detector noise is given by a dark count component and a readout noise component. The noise from the dark counts is obtain by multiplying the dark count rate by the exposure time and the number of spatial pixels used in extracting the source and taking the square root. The readout noise is the detector readout noise parameter multiplied by the square root of the number of spatial source pixels.
If sky subtraction is selected and the number of available sky pixels is greater than zero then the uncertainty in the sky estimation is computed. The uncertainty in a single pixel is the square root of the sky photons per pixel, the dark counts per pixel, and the readout noise per pixel. This is divided by the square root of the number of sky pixels to get the uncertainty in the sky estimation for subtraction from the source.
The total noise is the combination of the source, sky, dark count, and readout noise values and the sky subtraction uncertainty added in quadrature.
The signal-to-noise ratio per pixel per exposure is the source counts divided by total noise. This value is multiplied by the square root of number of pixels per resolution element to get the S/N per resolution element. If multiple exposures are used to make up the total exposure time then the single exposure S/N is multiplied by the square root of the number of exposures.
If no exposure time is specified, a value of INDEF, then the exposure time required to reach a desired signal-to-noise ratio per pixel is determined. The computation is done at the specified central wavelength. The task iterates, starting with the specified maximum time per exposure, by computing the S/N and adjusting the exposure time (possibly breaking the total exposure up into subexposures) until the computed S/N matches the desired S/N to 0.1%.
This section gives suggestions for how to set up the calibration files for a spectrograph. There are many other different ways to do this.
Calibration files are specified by filenames. They are first sought in the current directory and then in a list of directories given by the search parameter. For developing the calibration files create the files in your working directory. As a starting point copy files from the demonstration directories under spectimedb$KPNO. Rather than copying some files they can be specified in SPTIME by setting the search parameter to spectimedb$KPNO and then specifying the files as [directory]/[filename]. The calibration files in the demonstration directory should be checked. Many were created with ad hoc values or by rough measurement from graphs for the purpose of testing.
Begin by setting up the files with measured or known transmission curves such as as CCD DQE and filter curves. The calibration files are defined so that it is possible to import digital filter curves with minimal editing. The editing is to remove or comment most of the header data, add a title parameter, and add a tablescale parameter with value of 0.01 to scale from percent to the fractional values expected by SPTIME. See the example in the Filters section.
Next calibrate the telescope transmission if not known. This might be done with direct imaging of standard stars using a CCD with known DQE and filters with measured transmission curves. This is discussed further in the following sections. If it is not possible to measure the telescope transmission then use the number of mirrors and reflectance values (as a function of wavelength if possible) for the coatings to estimate the transmission. The same applies for any lenses using transmission values for the materials.
Try and measure or estimate the transmission of the ADC, collimator, corrector, and camera. Again this might be done using physical values for reflectance or transmission based on the composition of the elements.
Next setup up grating/grism files. As discussed elsewhere, it is best to calculate the efficiencies from the grating parameters than to use an efficiency table because the gratings can be used in different configurations. However, if a grating is always used in one spectrograph with fixed camera-collimator angle then it may be better to use a laboratory measured efficiency. Note that one can use SPTIME to look at the calculated efficiency curves by setting the output parameter to "disperser" and run SPTIME with appropriate calibration files.
The calculated efficiencies use the physical blaze angle rather than a configuration dependent blaze wavelength as is often tabulated in the documentation. One can compute the blaze angle from the wavelength and the camera-collimator angle using the equation
blaze angle = arcsin (1e-7*(g*w*m)/(2*cos(phi/2))
where g is the groves/mm, w is the blaze wavelength in Angstroms, m
is the order, and phi is the camera-collimator angle.
For the grisms, the prism angle (which is the same as the blaze angle) and the index of refraction can be computed from the blaze wavelength and dispersion using the equations
prism angle = arccos (1e-7 * (g*d*m) / f)
index of refraction = 1e-7 * (g*w*m) / sin(prism angle) + 1
where d is the dispersion in Angstroms per mm and f is the focal length
of the camera. It is best to use known values rather than those computed
from the blaze wavelength and dispersion if available.
The grism files in the demonstration directory had known prism angles
and the index of refraction was computed as above.
Finally standard star observations, as described below, are used to get the calibration files for one last element or as a fudge term in the spectrograph calibration file.
It is desirable to determine the telescope transmission independently of the spectrograph if it is not already known. This could be done with direct standard star photometry using filters and detectors with measured transmission and DQE. If it is possible to do photometry with the same ADC that is used with the spectrograph then the observations would be done with and without the ADC to separate the ADC transmission function.
Obtain standard star observations in photometric conditions. For slit spectrographs use a wide slit and, if possible, use an ADC or align the slit to avoid atmospheric dispersion effects. Assuming the disperser efficiency can be modeled or determined separately so that the calibration is for the rest of the spectrograph components, the observations need only be done with the lowest dispersion. As many setups as necessary to cover the operating range of the spectrograph need to be obtained.
The telescope and ADC transmission functions are determined first. Create the ADC data files. For development you probably would put it in your working directory and set filenames in SPTIME.
Reduce the spectroscopic standard star data for each spectrograph configuration using the standard IRAF reduction path. Use Angstroms for the wavelength calibration and flux units of ergs/s/cm^2/A for the flux calibrations (these are the defaults). The goal is to produce sensitivity function spectra. Convert the sensitivity function image files produced to sensitivity function tables with the command
cl> listpix [sens] wcs=world > sensfunc.dat
where [sens] is the sensitivity function spectrum produced by
SENSFUNC.
Set all the SPTIME parameters to match the observations. The input spectrum used does not matter though you might want to use a final flux calibrated spectrum. Set the output parameter to "correction".
The correction values produced by SPTIME can now be used to fix up any of the calibration tables. The easiest way to apply the correction is to assign the correction to one component (the spectrograph can be used as a fudge term on top of the other components or it could be attributed to the camera). For the target component create the file with no table so that it will use a transmission of 100%. The output correction values would then be those needed in that component to produce the observed count rates. Note that by setting the list parameter to a file name you can get the numbers in the plot as a text file suitable to used in the calibration file.
If several spectrograph settings are required to cover the spectral range of the spectrograph then there will have to be some intelligent merging of the corrections.