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HAWK-I Exposure Time Calculator

Important notes and bug reports

Note: These tools are only provided for the technical assessment of feasibility of the observations. Variations of the atmospheric conditions can strongly affect the required observation time. Calculated exposure times do not take into account instrument and telescope overheads. Users are advised to exert caution in the interpretation of the results and kindly requested to report any result which may appear inconsistent.

General description

The HTML/Java based interface allows to set the simulation parameters and examine interactively the model generated graphs. The ETC programs allow easy comparison of the different options relevant to an observing program, including target information, instrument configuration, variable atmospheric conditions and observing parameters. Being maintained on the ESO Web servers, the ETCs are regularly updated to reflect the known performance of ESO instruments.

The exposure time calculator consists of two pages.
Input page: The observation parameters page presents the entry fields and widgets for the target information, expected atmospheric conditions, instrument configuration, observation parameters such as exposure time or signal-to-noise, and results selection. An "Apply" button submits the parameters to the model executed on the ESO Web server.
Output page: The results page presents the computed results, including number of counts for the object and the sky, signal-to-noise ratios, instrument efficiencies, PSF size etc.. The optional graphs are displayed in several formats. Finally a summary of the input parameters is appended to the result page.

Note: These tools are only provided for technical assessment of observation feasibility. Variations of the atmospheric conditions can strongly affect the required observation time. Calculated exposure time do not take into account instrument and telescope overheads. Users are advised to exert caution in the interpretation of the results and to report any result which may be suspected to be inconsistent.

The exposure time calculator models the observation chain which includes the target spectral distribution, atmosphere parameters, instrument configuration, and detector setup. An instrument description for HAWK-I is available on the instrument page.

Input Spectrum

The following options are available to describe the input spectrum of the target.


The flux density is constant at all wavelengths (F(λ) = const.) The flux density level is determined from the specified object magnitude.


The target model is a blackbody defined by its temperature, expressed in Kelvin. The intensity distribution is scaled to the object magnitude.

Spectral Type

The target model can be defined by a template spectrum which is scaled to the provided magnitude and filter. The spectrum can be red-shifted.
References: Pickles (1998, PASP 110, 863); Coleman et al.: 1980ApJS; Kinney at al.: 1996ApJ.


The target spectrum can also be selected from a subset of MARCS stellar model spectra, kindly provided by Bengt Edvardsson at the Uppsala Astronomical Observatory. The parameter space of the MARCS subsets are listed the following tables. Note that not all models (referring to all possible combinations of parameters) actually exist.

MARCS subset: Spherical Geometry
Parameter Number of
unique values
Unique Values
model 1 "st"
[Fe/H] 4 -4.00,-2.00,-1.00,0.00
Teff/K 9 4000,4500,5000,5500,6000,6500,7000,7500,8000
log(g) 5 -0.50,0.00,1.00,2.00,3.50
geometry 1 "s"
microturbulence 1 2
mass 2 1,5
total (product) 360 (this is the number of possible combinations, but only 87 models exist)

MARCS subset: Plane Parallel Geometry
Parameter Number of
unique values
Unique Values
model 1 "st"
[Fe/H] 6 -1.00,-2.00,-4.00,0.00,0.50,1.00
Teff/K 9 4000,4500,5000,5500,6000,6500,7000,7500,8000
log(g) 1 4.00
geometry 1 "p"
microturbulence 1 2
total (product) 54 (this is the number of possible combinations, but only 50 models exist)


Users can upload a file with the spectral flux distibution to the ETC server.

Supported formats:

In both cases the values in the first column should be the wavelength in nm units and ascending order; the second column is the the flux density in a unit proportional to erg/cm2/s/A. The absolute flux scale is not significant since the spectrum will be scaled to the given magnitude in the given band. The maximum file size is 2 MB.

NOTE! The wavelength range of the uploaded spectrum must cover the spectral range of the selected instrument mode as well as the wavelength range of the photometric band in which the magnitude is given.

Object Magnitude

Enter the V (650 nm), Y (1000 nm), J (1250 nm) , H (1650 nm), or K (2160 nm) magnitude, ideally closest in wavelength to the selected filter. The reference for the zero points used in conversion into photon fluxes:
Vega-system: (B,V,R,I): Bessel, 1979, PASP, 91, 589. (J,H,K): Bessel and Brett, 1998, PASP, 100, 1134.
AB-system:Oke, 1974, ApJS.

Emission Line

The input spectrum is a single emission line. It is an analytic Gaussian, centered on the Wavelength parameter, defined by its total Flux and full-width at half-maximum FWHM. Line flux is given in 10-16 erg.cm-2.s-1.

NB: When requesting a single line as input spectrum, the magnitude parameter is not taken into account. Only the line flux will be used to determine the signal magnitude.

NB: The FWHM of a single line is limited by the sampling. If the requested FWHM is too narrow, it will be replaced by the minimum supported value, and a warning will be issued in the beginning of the result page.

Source Geometry

Sky Conditions

Turbulence Category (T category)

With the advent of instruments using new adaptive optics (AO) modes, new turbulence parameters need to be taken into account in order to properly schedule observations and ensure that their science goals are achieved. These parameters include the coherence time and the fraction of turbulence taking place in the atmospheric ground layer, in addition to the seeing. Starting from Period 105, the turbulence constraints are standardised to the turbulence conditions required by all instruments and modes, whether they are seeing-limited or AO-assisted.

The handling of atmospheric constraints thus changes for both Phase 1 (proposal preparation) and Phase 2 (OB preparation). In Phase 1, the seven current seeing categories are replaced by seven turbulence categories for all instruments. Each category can be defined by other parameters than a pure seeing threshold, depending on the instrument. For all instruments, all categories share the same statistical probability of realisation, which is key for an accurate time allocation process. In Phase 2, the image quality will still be the only applicable constraint for seeing-limited modes, whereas the same turbulence category as for Phase 1 will be used for diffraction-limited modes.

Users are encouraged to read the general description of these changes for Phase 1 and Phase 2 on the Observing Conditions webpage, as well as instrument User Manuals for specifics per instrument.

Seeing and Image Quality

The definitions of seeing and image quality used in the ETC follow the ones given in Martinez, Kolb, Sarazin, Tokovinin (2010, The Messenger 141, 5) originally provided by Tokovinin (2002, PASP 114, 1156) but corrected by Kolb (ESO Technical Report #12):

Seeing is an inherent property of the atmospheric turbulence, which is independent of the telescope that is observing through the atmosphere; Image Quality (IQ), defined as the full width at half maximum (FWHM) of long-exposure stellar images, is a property of the images obtained in the focal plane of an instrument mounted on a telescope observing through the atmosphere.

The IQ defines the S/N reference area for non-AO point sources in the ETC.

With the seeing consistently defined as the atmospheric PSF FWHM outside the telescope at zenith at 500 nm, the ETC models the IQ PSF as a gaussian, considering the gauss-approximated transfer functions of the atmosphere, telescope and instrument, with s=seeing, λ=wavelength, x=airmass and D=telescope diameter:

Image Quality
\( { \begin{equation} \mathit{FWHM}_{\text{IQ}} = \sqrt{\mathit{FWHM}_{\text{atm}}^2(\mathit{s},x,\lambda)+\mathit{FWHM}_{\text{tel}}^2(\mathit{D},\lambda)+\mathit{FWHM}_{\text{ins}}^2(\lambda)} \end{equation} } \)

For fibre-fed instruments, the instrument transfer function is not applied.

The diffraction limited PSF FWHM for the telescope with diameter D at observing wavelength λ is modeled as:
\( \begin{equation} \begin{aligned} \mathit{FWHM}_{\text{tel}} & = 1.028 \frac{\lambda}{D} \text{, } & \text{ with } \lambda \text{ and D in the same unit}\\ & = 0.000212 \frac{\lambda}{D} \text{arcsec, } & \text{ with } \lambda \text{ in nm and D in m}. \end{aligned} \end{equation} \)
For point sources and non-AO instrument modes, the atmospheric PSF FWHM with the given seeing \(s\) (arcsec), airmass \(x\) and wavelength \({ \lambda }\) (nm) is modeled as a gaussian profile with:
$${\mathit{FWHM}}_{\text{atm}}(\mathit{s},\mathit{x},\lambda) = \mathit{s} \cdot x^{0.6} \cdot (\frac {\lambda} {500})^{-0.2} \cdot \sqrt{[1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356})]}$$
Note: The model sets \({ \mathit{FWHM}}_{\text{atm}}\)=0 if the argument of the square root becomes negative \({ [1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356}] < 0 }\) , which happens when the Fried parameter \({ {r_0} } \) reaches its threshold of \({ r_{\text{t}} = L_{0} \cdot [1/(2.183 \cdot F_{\text{Kolb}})]^{1/0.356}}\). For the VLT and \({ L_{0} = 46m}\) , this corresponds to \({ r_{\text{t}} = 5.4m} \).
\({ L_{0} }\) is the wave-front outer-scale. We have adopted a value of \({ L_{0} }\)=46m (van den Ancker et al. 2016, Proceedings of the SPIE, Volume 9910, 111).

\(F_{\text{Kolb}} \) is the Kolb factor (ESO Technical Report #12):
$$F_{\text{Kolb}} = \frac {1}{1+300 {\text{ }} D/L_{0}}-1$$
For the VLT and \({ L_{0} }\)=46m, this corresponds to \(F_{\text{Kolb}} = -\)0.981644.
\( {r_0} \) is the Fried parameter at the requested seeing \(s\), wavelength \({ \lambda }\) and airmass \(x\):
$$r_0 = 0.100 \cdot s^{-1} \cdot (\frac{\lambda}{500})^{1.2} \cdot x^{-0.6} \text{ m, } \text{ } \text{ } \text{ } \text{ with } s \text{ in arcsec } \text{and } \lambda \text{ in nm.} $$

For AO-modes, a model of the AO-corrected PSF is used instead.

Sky Model

The sky background model is based on the Cerro Paranal Advanced Sky Model, also for instruments at la Silla, except for the different altitude above sea level. The observatory coordinates are automatically assigned for a given instrument.


By default, the airmass and moon phase parameters are entered manually. The sky model will use fixed typical values for all remaining relevant parameters (which can be seen in the output page by enabling the check box "show skymodel details").

Alternatively, a dynamic almanac widget can be enabled to facilitate assignment of accurate sky model parameters for a given target position and time of observation. The sky radiation model includes the following components: scattered moonlight, scattered starlight, zodiacal light, thermal emission by telescope and instrument, molecular emission of the lower atmosphere, emission lines of the upper atmosphere and airglow continuum.

The almanac is updated dynamically by a service on the ETC web server, without the need to manually update the web application.

Notes about the algorithms, resources and references for the almanac are available here. A more advanced version of the almanac is included in our SkyCalc web application, which provides more input and output options.

Hovering the mouse over an input element in the almanac normally displays a pop-up "tooltip" with a short description.


The upper left part of the almanac box refers to the date and time of observation.
This can be done with a UT time or a MJD. A date/time picker widget will appear when the UT input field is clicked, but the UT can also be assigned manually. In any case, the UT and MJD fields are dynamically coupled to be mutually consistent.

The two +/- buttons can be used to step forward or backward in time by the indicated step and unit per click. The buttons can be held down to step continuously until released.

The third of night corresponding to the currently selected time is indicated. This is an input parameter to the airglow component in the sky model. Twilight levels (civil, nautical and astronomical) referring to the sun altitude ranges are also indicated in the dynamic text. These levels refer to the sun altitude:


The target equatorial coordinates RA and dec can be assigned manually in the two input fields or automatically using the SIMBAD resolver to retrieve the coordinates.
If the lookup is successful, an "info" link will open a window in which the raw SIMBAD response can be inspected.
The units can be toggled between decimal degrees and hh:mm:ss [00:00:00 - 23:59:59.999] for RA and dd:mm:ss (or dd mm ss) for dec. A whitespace can be used as separator instead of a colon.

Output Table

The table dynamically displays the output from the server back-end service, including temporal and spatial coordinates for the target, Moon and Sun. The bold-faced numbers indicate the parameters normally relevant in the phase 1 proposal for optical instruments. The numbers appear in red color if they are out of the range supported by the sky model.

Visiblity Plot

The chart dynamically shows the altitude and equivalent airmass as function of time for the moon and target, centered on midnight for the currently selected date.
The green line, which refers to the currently selected time, can be dragged left and right to change the time, dynamically coupled with the sections in the Time section.

A more advanced version of the almanac is included in our SkyCalc web application, which provides more input and output options.

Instrument Setup


Choosing the instrument filter determines for which band the exposure time will be computed. For information about the filters (incl. transmission curves), please refer to the Instrument Description.


ModeThe pixel scale is fixed to 0.106 arcsec/pixel. The read-out mode is fixed to Non-Destructive Read-out (NDR) - the detector is continuously read-out in a non-destructive mode. Providing a read-out noise of about 5 electrons.


The DIT (Detector on-chip integration time in seconds) is needed as input for both of the following cases:

In both cases, the total exposure time will be given a DIT*NDIT, with the specified DIT. The output form will give the estimates for SNR or Exposure Time, together with the selected output graphs.

Do not confuse exposure time and total observation time, the latter being a sum of exposure time and overheads in the telescope and instrument. Please consult the user manuals for guidance on the choice of the integration parameters.

Graphical Outputs

Resultant spectrum including sky

The sum of object signal and sky background spectrum for the central pixel, in e-/pixel/DIT.

Object spectrum only

The total integrated counts contribution from the object per pixel as a function of wavelength, in e-/pixel/DIT.

Sky Emission Spectrum

The sky contribution to each pixel as a function of wavelength, in e-/pixel/DIT.

Sky Transmission Spectrum

The sky transmission in percent as a function of wavelength.

S/N as a function of Exposure Time

The S/N as a function of Exposure Time

Total efficiency and Wavelength range

This option will display a curve showing the total efficiency in percent of the system.

Input spectrum in physical units

The input flux distribution is displayed in units of photons/cm^2/s/A

Version Information

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