COMPUTE EXPOSURE TIME
HELP

BASICS

• OPTIONAL PARAMETERS field

  The syntax to be used is :
  -parameter Value
  The value field may be empty for some boolean parameters. The only parameters which are of interest for the normal user are :
  
  
  • -wspec N, where N = 1 to 5. This sets the width of the spectra on the CCD, from one to five pixels. Each spectrum will be "summed" over N columns. The default value is 5, and spectra separation on the CCD is 7 pixels. If one sets the value above 5, it is reduced to 5 by the simulator program.
    
  
  
  • -dir pathname. This tells the simulator program to look into directory pathname/data/ to find the many data files used, instead of the default ../data/ directory. It forces also the use of directory pathname/tmp/ instead of the default ../tmp/ for storing some small temporary files. Using this switch supposes that the necessary data files have been copied to the new directory.
    
  
  
  • -resolved. This forces the S/N computations to be done for the resolved spectral element, taken as the reunion of two spectral samples.
    
  
  
  • -none is there as a default, to circumvent an unresolved issue which causes an abnormal exit on the CFHT server if the optional parameters field is left empty. Do not delete it, it has no meaning, and no effect...
    
  
  
  
  

• DATA used

  • Object :
    The brightness of the astrophysical target may be given as a B, V, R or I magnitude, or as a W m^-2 Å^-1 flux, either global for stellar objects, or per arcsec² for extended ones. Conversion between flux and magnitude uses :
    
    F_lambda = F_0,mag x 10^-0.4xM
    M = -2.5 x log₁₀(F_lambda / F_0,mag)
    where F_0,mag is the zero magnitude flux (taken from C.W. Allen, Astrophysical Quantities) : F_0,B = 6.6069 10^-12 for B mag
    F_0,V = 3.8019 10^-12 for V mag
    F_0,R = 1.7378 10^-12 for R mag
    F_0,I = 8.3176 10^-13 for I mag
    
  
  
  • Dark Sky background
    The typical Mauna Kea night sky brightness is interpolated from the table which can be found in the site characteristics chapter of the CFHT observer's manual@. This table gives the flux density (in photons cm^-2 s^-1 µm^-1 arcsec^-2) for eight wavelengths in the [ 0.36µm , 2.22µm ] range.
  
  
  
  • Moon Light
    The light diffused at the object sky location by the moon is evaluated using formulae from Krisciunas & Schaeffer, PASP 103:1033, 1991, taking into account the moon phase, the zenithal distance of the moon, the zenithal distance of the direction of observation, and the angular distance between the moon and the direction of observation. The sky brightness, given in nanoLamberts, is related to V magnitude through :
    
    B_nL = 34.08^{[20.7233-0.92104 V]}
  
  
  • Atmospheric extinction
    The typical Mauna Kea extinction coefficient for the wavelength of interest is interpolated from the curve which can be found in the site characteristics chapter of the CFHT observer's manual@. This table gives the extinction (in magnitudes per air mass) between 3100Å and 8000Å.
  
  
  
  • Telescope transmission
    The telescope spectral transmission, including central obscuration and two reflections, has been modelized by Pierre Ferruit; the effective value is interpolated from this data.
  
  
  
  • AOB transmission, excluding the beamsplitter
    The spectral AOB transmission, without the beamsplitter, has been modelized by Pierre Ferruit; the effective value is interpolated from this data.
  
  
  
  • CCD dark signal
    The hourly value (electrons/hour) is taken from the CFHT CCD data@.
  
  
  
  • CCD quantum efficiency
    The Qeff value for the mean wavelength of the scenario used is interpolated from the data which are given in the CFHT CCD data@ WEB page.
  
  
  
  • AOB beamsplitters
    The spectral transmissions of the AOB beamsplitters are interpolated from data obtained from the CFHT Optical Group.
  
  
  
  • OASIS transmission
    The value suitable for the current scenario is computed as the product of the transmissions of the optical elements : enlarger,filter, array, collimator, beamsteerer, grism, camera. The original data come either from CFH (grisms@), from manufacturer's data (filters, array, coatings), or/and from Pierre Ferruit's models.
    
  
  
  

• ALGORITHM

  • Common part
    
    
    • Scenario wavelength
      This is defined as the mean of the filter bounds if the object is continuum-dominated, and as the observed wavelength of the emission line if such a feature is prominent in the spectrum. The TIGER mode filter bounds are given in the TIGER mode section of the general OASIS overview.
      
    
    
    • Dark sky brightness
      The night sky brightness per arcsec² is interpolated from CFHT data to the "mean" (see above) wavelength of the scenario used, and integrated over the spectral sampling width; The spectral sampling value is given in the TIGER mode section of the general OASIS overview.
      
    
    
    • Moon light
      The flux per arcsec² diffused by the atmosphere in the direction of observation is computed. It is then integrated over the spectral sample width.
      
    
    
    • Total sky flux
      The per arcsec² value is computed as the sum of the two above values; it is then integrated over the microlens aperture. Microlenses are hexagonal, and their area (arcsec²) is given by :
      
      S_lens = 0.8660 (Lens size)²
      The lens size parameter is the spatial sampling which is choosen in the exposure simulator form, and is part of the scenario definition.
      The transmission factors of the telescope, of OASIS, and of the AOB and beamsplitter if they are used, all calculated at the scenario wavelength, are applied to the total sky flux per lens.
      It is then converted to photons per second at the CCD level using :
      
      Nphot_sky = F_lambda x Lambda x K
      where K = 10^-10 / h x c = 5.03 10²² (mksa).
      Finally, the number of electrons per second per spatial element is computed as :
      
      Ne_sky = Nphot_sky x Qeff
      
      where Qeff is the quantum efficiency of the CCD at the mean wavelength of the scenario used.
      
      
    • Emission-line dominated objects
      If the flux from the object is dominated by an emission line of rest wavelength Lambda0_line and rest FWHM W_line, the two values are first corrected from the redshift effect. If the observed W_line is greater than the spectral sample width L_step, a sampling factor K_SampSpec is computed : assuming that the line has a gaussian shape, this factor gives the fraction of the energy which falls on a single pixel line (along the cross-dispersion axis); it is assumed that the line peak falls onto this pixel line. This dilution factor is evaluated by numerical interpolation of a standard gaussian distribution table.
      In the case of an unresolved emission line, the incident flux is divided by 2 to take into account the fact that the spectral PSF covers two pixels along the dispersion axis.
      In the case of a continuum-dominated object, no reduction of the flux is performed.
      
    
    
    • Influence of the image quality
      If the astrophysical target is star-like (for instance a star cluster), the image quality is taken into account : if the seeing (FWHM) is greater than the spatial sampling, then a spatial sampling factor K_SampSpat is computed. Assuming that the object is gaussian, it gives the fraction of the energy which enters a single micro-lens; it is assumed that the peak falls at the center of the lens. This dilution factor is evaluated by performing a numerical interpolation on a standard gaussian distribution table. See note in the "Approximations" section at the end of this document.
      
    
    
    • CCD dark signal
      The dark signal per second is evaluated as :
      
      Ne_dark = W_spectra x Dark / 3600
      where :
      • W_spectra is the width of the spectra on the CCD. This parameter is usually set to 5, although it may be changed in the Optional parameters field (not wise...). The signal from the W_spectra pixels situated on the line perpendicular to the dispersion are "added" to get the spectrum value at this point.
      • Dark is the dark electron count per hour, as given in the CCD specs.
    
    
    • CCD readout noise
      The signal variance (per second) due to the CCD reading is evaluated as :
      
      Var_read = W_spectra x N_read x N_read
      
      where N_read is the readout noise of the CCD, as given in the CCD specs.
  
  
  • Compute the signal-to-noise ratio from given flux and integration time
    
    
    • If the object brightness has been given as a magnitude, it is converted to flux density (W m^-2 Å^-1) at the wavelength "equivalent" to the type of magnitude given. The flux is then integrated over the spectral sample width.
      
    • If the object is an extended one, the W m^-2 arcsec^-2 L_step^-1 flux is converted to W m^-2 lens^-1 L_step^-1using :
      
      F_Obj/lens = F_Obj x S_lens
      The computations are the same as the ones for the total sky flux quantity (see above), and the number of electrons produced by the object flux in a single lens, Ne_Obj, is finally evaluated in the same way.
      
    • The respective noise (variance/second) contributions are then evaluated :
      
      Var_obj = Ne_obj
      Var_sky = Ne_sky
      Var_dark = Ne_dark
      
      The total noise in the given exposure time T_exp seconds is then taken as :
      
      N_total = sqrt [ T_exp x ( Var_obj + Var_sky + Var_dark ) + Var_read ]
      
    • Lastly, the signal-to-noise ratio equation :
      
      SN = [ T_exp x Ne_obj ] / N_total
      gives the signal-to-noise ratio achieved in the given exposure time.
    
    
  • Compute the integration time to reach SN with the given flux
    
    
    • If the object brightness has been given as a magnitude, it is converted to flux (W m^-2 Å^-1) at the wavelength "equivalent" to the type of magnitude given. With no information regarding the spectrum shape, nothing else can be done. The flux is then integrated over the spectral sample width.
      
    • If the AOB is used in the scenario currently defined, the beam splitter transmission for the mean scenario wavelength is applied to the object flux.
      
    • If the object is an extended one, the W m^-2 arcsec^-2 L_step^-1 flux is converted to W m^-2 lens^-1 L_step^-1 using :
      
      F_Obj/lens = F_Obj x S_lens
      The computations are the same as the ones for the total sky flux quantity (see above), and the number of electrons produced by the object flux in a single lens, Ne_Obj, is finally evaluated in the same way.
      
    • The respective noise (variance/second) contributions are then evaluated :
      
      Var_obj = Ne_obj
      Var_sky = Ne_sky
      Var_dark = Ne_dark
      
      The total noise in T_exp seconds is then taken as :
      
      N_total = sqrt [ T_exp x ( Var_obj + Var_sky + Var_dark ) + Var_read ]
      
    • The signal-to-noise ratio equation :
      
      SN = [ T_exp x Ne_obj ] / N_total
      is then solved for T_exp.
  
  
  • Compute the Minimum flux to reach SN in the given integration time
    
    
    • The total number of electrons due to the dark signal during the given integration time is computed as :
      
      Ne_dark,time = Ne_dark x IntTime
      
    • The total number of electrons due to the sky during the given integration time is computed as :
      
      Ne_sky,time = Ne_sky x IntTime
      
    • The signal-to-noise ratio equation :
      
      SN = [ T_exp x Ne_obj ] / N_total
      where :
      
      N_total = sqrt [ (T_exp x Ne_obj) + Ne_sky,time + Ne_dark,time + Ne_read ]
      is then solved for Ne_obj :
      
      N_obj = SNx[SN+sqrt( SN²+x4 (Var_sky+Var_dark+Var_read)] / [2xIntTime]
      
    • The value obtained is converted to photons per second using
      
      Nphot_obj = Ne_{obj / Qeff}
      and then to W m^-2 Å^-1 using :
      
      F_lambda = K x Nphot_obj / ( Transmission_OASIS x Lambda x L_step )
      where K = 10¹⁰ h x c = 19.8648 10^-16 (mksa).
      
    • If the object is star-like, the spatial dilution factor related to the seeing is taken into account :
      
      F_{lambda,global} = F_lambda / K_SampSpat
      
    • If the flux is dominated by an emission line of width FWHM_line, the spectral dilution factor is taken into account :
      
      F_obj = F_{lambda,global} / K_SampSpec
      
    • The flux is corrected from OASIS+Telescope (+PUEO if applicable) absorption.
      
    • Finally, the flux is divided by the telescope area, and corrected from the atmospheric extinction.
      

INFAMOUS TRICKS
&
DIRTY APPROXIMATIONS USED

• The Mauna Kea extinction curve has been visually extrapolated from 8000Å to 12000Å.
  
• The sky brightness at Mauna Kea is taken as constant over the spectral sample width, that is over 1 to 4.8Å, according to the spectral configuration used.
  
• In the same manner, the moon spectral flux is taken as constant over the spectral sample width.
  
• The V band and R band AOB beamsplitter spectral transmission curves, not yet available as of January 1998, have been set to a constant 90% over their respective spectral coverage.
  
• If the Atmospheric Dispersion Compensator is not used, the overall transmission of PUEO is enhanced by a factor K⁸, to take into account the missing eight refractions through the four uncoated prism surfaces. As of February 1998, K has been estimated to 0.98 (in lack of any known measurement).
  
• In the continuum light case, the various OASIS+PUEO+Telescope optical elements reflection/transmission factors are evaluated at the central wavelength of the filter used, and taken as constant over the filter bandpass.
  
• The AOB PSF is taken as gaussian, which may lead to a substantial error in some cases. This PSF may be more realistically described by a sum of two gaussians : the core one, and the extended one. The energy sharing between the two is variable, depending on the quality of the correction, (magnitude and distance of the guide star, atmosphere status, zenithal distance, wavelength, number of modes corrected), and the spatial sampling effects on one and the other are different. In this first release of the TIGER simulator, when planning AOB observations, use the CFHT's AOB performance estimator to evaluate the single gaussian which represents the "best eye fit" to the AOB PSF you expect. This single gaussian is the one to be used in this simulator.
  
• The spectral PSF of OASIS is taken as a Gaussian, which is not the case, as it shows a central dip (it is an image of the primary mirror).
  
• The spectral PSF "FWHM" of OASIS is taken as a constant 30µm (although it varies from 27 to 35µm according to the spatial sampling used), that is 2 CCD pixels; as the most frequent values are lower, the number of object electrons s^-1 pixel^-1 should be higher than it is in the simulator.
  
• If the brightness of the target is given as a, say, B magnitude, it is converted to flux density at the equivalent wavelength, 4400Å in this case. The object flux at the mean wavelength of the scenario is then supposed to be the same, although this last wavelength maybe far from the first one; with no information regarding the spectrum shape, nothing else can be done.
  
• The approximate magnitude which is computed from the minimum flux is evaluated in the band (B, V, R or I) which is the nearest to the scenario wavelength, assuming that the flux at the band equivalent wavelength is the same as the flux at the sceanrio wavelength.
  

DATA FILES

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