VOCET version 2.2 - User's Manual

This document intends to provide informations and help for the VST + OmegaCam Exposure Time calculator (VOCET). This is a versatile calculator allowing the observer to compute various quantities related to the VST + OmegaCam 16kx16k observing performance (source type, magnitude, filter, seeing, sky brightness, airmass, atmospheric transmission).

This release adopts different (from the previous version) transmission curves for the CCD and for the Johnson B and V filters. This fact implies that previous computations can be not correct! PLEASE, CHECK IT with this new version of VOCET. Moreover, a third option for the output of the program (computing the limit AB magnitude of the source, given the exposure time and the S/N), the characteristic curves for the various optical components (plot), and the resulting filter transmission (plot) have been implemented.
Finally, in this release there are two different available versions of VOCET:

1) VOCET running with LINUX operative system and PGPLOT Graphics Subroutine Library. It has been tested and runs on the server LINUX1, at INAF-OAC. It can also run with any other computer adopting LINUX operative system and PGPLOT Graphics Subroutine Library installed.
PGPLOT Graphics Subroutine Library is free at the web site http://www.astro.caltech.edu/~tjp/pgplot/.

2) VOCET running without PGPLOT Graphics Subroutine Library. This version has been tested to run with any computer adopting LINUX operative system.

To read the file READ.ME

To download the file vocet.tar

1.1 General Description

The adopted magnitude system for VOCET is the AB system of Oke e Gunn (1983, ApJ, 266, 713; hereafter referred to as OG). The great advantage of this magnitude system is that the magnitude is directly related to physical units, e.g. to physical fluxes.

For what concerns S/N or exposure time computation, there are three source options:

Point-like sources (seeing limited, e.g.: stars, QSOs). They are approximated with a moffat's profile having α = FWHM/2 (being FWHM = seeing) and β = 3.8 (see section 4.2). We assume that about 96% of the total flux from the source falls inside a circle (Point Spread Function) of radius R = 1.449*α (see section 4.2.1). The S/N is evaluated in the PSF area.

Other sources with a seeing dominated profile (e.g., distant galaxies, compact galaxies, etc.; hereafter named as integral photometry). For these sources the total magnitude is always intended over a given aperture area. The S/N ratio is evaluated over this selected aperture area.

Extended sources. Magnitude is always intended in mag/arcsec². The S/N is evaluated per pixel, and the surface brightness of the source is assumed to be uniform.

In this document the terms source or object are equivalent: they identify the astronomical target of the observation.

The document is organized as follows. The Vega photometric systems, the AB magnitude system and the conversion between them are described in section 2 . The general features and the formulae used to compute S/N or the exposure time are described in section 3. The description of input parameters and their meaning is given in section 4 , while the computations performed by the code are listed in section 5 . Finally, a description of the outputs and useful tables and plots are in section 6 and section 7 , respectively.

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2. VEGA and AB Photometric Systems

2.1 Vega System

Vega is the primary standard star used to establish the zero-point of photometric systems in the optical and near-infrared. By definition, Vega's magnitudes are equal to zero in all bands:

m_Vega (obj) = - 2.5 log (∫ (F_λ (obj) S_λ dλ ) / ∫ (F_λ(Vega) S_λ dλ ))

which can also expressed as:

m_Vega (obj) = - 2.5 log (∫ (F_ν (obj) S_ν dν ) / ∫ (F_ν(Vega) S_ν dν ))

where:

λ is the wavelenght;

ν is the frequency;

F_&lambda(obj) is the object flux in erg sec^-1 cm^-2 Å^-1;

F_&nu(obj) is the object flux in erg sec^-1 cm ^-2 Hz^-1;

F_&lambda (Vega) is the Vega's flux in erg sec^-1 cm^-2 Å^-1;

F_ν (Vega) is Vega's flux in erg sec^-1 cm^-2 Hz^-1;

S_λ, S_ν are the instrumental (filter+CCD+optics) responses;

2.2 AB System

The ABν magnitude is defined by OG so that the flux of Vega (α Lyrae) corresponds to V = 0.03 at 5480 Å. This led to the following definition of a monochromatic magnitude:

AB_ν = - 2.5 log f_ν - 48.6 [1 ]

where f_ν is the flux from an object per unit frequency in ergs sec^-1 cm^-2 Hz^-1.

With the use of Hayes (1985, IAU Symp. 111, p. 225) and Castelli & Kurucz (1994, AA, 281, 817) Vega's SED, the zero point in the AB_ν magnitude system is the flux of Vega at 5480 Å:

f_ν = 3.59 x 10^-20 erg sec^-1 cm^-2 Hz^-1

Strictly speaking, the AB magnitude system as defined above is a monochromatic system, defining a magnitude for a single frequency ν. It follows that an object with a flat energy distribution (constant flux) has the same monochromatic magnitude in all bands, and all colours are equal to zero.

In order to tie a broadband system to a monochromatic flux, we adopt an AB broadband magnitude, as defined by Fukugita et al. (1996, AJ, 111, 1748), in the following way:

AB_ν = - 2.5 log (∫ (F_ν S_ν dν ) / ∫ (S_ν dν )) - 48.6

where F_ν is the flux per unit frequency from an object on the atmosphere in ergs sec^-1 cm^-2 Hz^-1, and S_ν is the system response.

2.3 AB &harr VEGA System Conversion

From the definition of AB magnitude:

AB_ν(obj) = - 2.5 log (∫ (F_ν(obj) S_ν dν ) / ∫ (S_ν dν )) - 48.6

= - 2.5 log (∫ { (F_ν(obj) S_ν dν ) (F_ν(Vega) S_ν dν )} /∫ { (S_ν dν ) (F_ν(Vega) S_ν dν )} - 48.6

AB_ν (obj) = m_Vega(obj) + AB_ν (Vega)

that means that the conversion between systems is simply given by the AB Vega's magnitude:

conv. fact. = AB_ν(Vega) = - 2.5 log (∫ F _ν (Vega) S_ν dν / ∫ S_ν dν ) - 48.6

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3. General Features

VOCET simulates the four main components of an astronomical observation: the source, the atmosphere, the telescope and the camera.

At the moment, the source is characterized by a flat spectral energy distribution (in a next version, we will provide different spectral energy distribution and the redshift); the costant flux value is computed from the AB magnitude of the object, using [ 1 ] .

The atmosphere is characterized by the sky brightness at different days from the new moon, the extinction as a function of wavelength, the airmass, and the seeing.

The simulated telescope is characterized by its effective area and optical transmission, including the lens of the camera, as a function of wavelength.

The simulated camera is characterized by the filter + CCD.

The filters adopted by VST + OmegaCam are listed in Table 1 : they are described by a transmission curve as a function of the wavelength.

The CCD is characterized by its Quantum Efficiency (Q.E.) as a function of the wavelength, gain, readout noise, dark current, saturation level, and plate scale.

When the transmission of the filter T_λ is combined with the CCD's Q.E. (Q_λ ) and the optical transmission of the telescope, we obtain the Response Quantum Efficiency of the system S_λ . Then, we can define the effective wavelength &lambda_e as:

λ _e = ∫ λ S _λ dλ / ∫ S_λ dλ

and the corresponding effective width of the bandpass Δ λ_e. These data are in Table 1.

3.1 Basic Formulae

The formulae we used in the following sections are:

Given the AB limit magnitude and the exposure time, evaluate the S/N:

S/N computation [2]

Given the AB limit magnitude and the S/N, evaluate the exposure time:

exposure time
computation [3]

Given the exposure time t and the S/N, evaluate the AB limit magnitude:

AB limit magnitude
computation

where the symbols are for:

C_O           count rate from the source (e ^- s^-1) integrated over the proper moffat's profile (see section 4.2), where:
                 C_O = N_obj (point-like or integral photometry);
                 C_O = N_obj * arcsec² (extended);

C_S count rate from the sky (e ^- s^-1), where: C_S = N_sky * arcsec²;

F ff² (C_O + C_S)² with ff = flat field accuracy;

D dark current noise (e ^- s^-1);

RN readout noise (e ^-s^-1);

n_pix           a) n_pix = π *(R_PSF /arcsec)² for point-like sources;
                   b) n_pix = 1 for extended sources;
                   c) n_pix = π *(R_aperture/arcsec)²for integral photometry;

S/N signal to noise;

t exposure time (s).

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4. Inputs for VOCET

In this section we report the input parameters the user is requested to introduce in the program for what concerns the source, the atmosphere and the telescope.

4.1 Source Input

For this version of VOCET it is not possible to select a SED for the observed source, so that the only available option is flat spectrum, i.e., constant flux at each wavelength.

4.2 Spatial Distribution

We assume that the source's profile can be fitted by a moffat's function:

I = I_c[ 1+(r/α )²]^-β [4]

where I_c is the peak value, r is the radius of the integration, and α and β are two parameters characterizing the moffat's function. For seeing dominated profiles (e.g., point-like sources and distant or compact galaxies) the parameter α is equal to half the FWHM of the source, with FWHM = seeing. The parameter β defines the profile type, that is how much energy is distributed in the wings of the profile.

We adopt the following mean values to fit the moffat's function:

a) β = 3.8 for point-like source profile;

b) β = 2.5 for galaxies with seeing dominated profile;

If we integrate the equation [4] over the profile, the flux within an aperture r = R is:

F(R) = I_cπ [ α ² / (1 - β)] [ (1+(r/α )²)^(1-β ) - 1] [5]

When R tends to infinite, the flux represents the total flux received from the source:

F_Total = I_cπ [ α ² / (β - 1 )] [6]

and the equation [5] becomes:

F(R) = F_Total [ 1 - [1+(r/α )²]^(1-β ) ] [7]

where F_Totalcan be simply derived from the equation [1] .

4.2.1 Optimal radius of flux integration

We consider thre possibilities:

point-like source. The user is requested to introduce the limit AB magnitude in one of the associated broad or narrow band filters: the program will provide the total limit flux of the source by using the [1] .

Maximizing the S/N (equation [2], with C_ot = F(R)) over R for a point-like source (seeing dominated profile), we obtain the relationship: R_OPT = 1.449*α. This result implies that the best S/N is obtained for R_PSF = R_OPT= 1.449*α , with F(R_PSF) = 0.9579 F_Total.
The S/N ratio and the exposure time are computed over the PSF area, in which falls about 96% of the total flux coming from the source.

integral photometry . The user is requested to provide the limit AB magnitude in an aperture in one of the associated broad or narrow band filters: the program provides the limit flux of the source in this aperture using the [1] . It is also requested the aperture in which the AB limit magnitude is given.

For galaxies with a seeing dominated profile (β = 2.5) the best S/N is obtained with a radius R_OPT = 2.80*α , which corresponds to a diameter of integration containing ~ 96% of the total flux: for a 0.8 arcsec seeing, this value corresponds to a 2.2 arcsec aperture. The program also computes the best aperture to use in order to optimize the S/N at the given seeing.

The S/N ratio and the exposure time are computed over the aperture area in which fall less than the total flux coming from the source, depending from the adopted aperture radius.

If the aperture area is smaller than the PSF area, then the S/N ratio and the exposure time are computed over the PSF area.

extended source. The user is requested to provide the limit AB mag/arcsec² in one of the associated broad or narrow band filters: the program provides the limit flux of the source per arcsec²by using the [ 1 ] .

The S/N ratio and the exposure time are computed over a single pixel, though the magnitude have to be provided per arcsec².

From the above assumptions, it is evident that using R = seeing = FWHM is slightly pessimistic for point-like sources, while it is optimist for galaxies with a seeing dominated profile.

4.2.2 The relation between the error on the magnitude and S/N.

Let us consider the expression of a magnitude as a function of the flux:

m = 2.5 log F

The error ε_m on the magnitude m can be written as:

ε_m = (dm/dF) ε_F

where ε_F is the error on the flux F. Since is:

(dm/dF) = (1/F) 2.5 [ 1/ln(10)]

we get:

ε_m = 0.92 [ 1/(F/ε_F) ] = 0.92 (ε_F/F)

If we put S/N = F/ε_F, the final relation is:

S/N = 0.92 / ε_m

For example, with S/N = 9.2 we will have ε_m = 0.1 = 10%.

On this basis, we can define the following relations between S/N and photometry quality for VST observations:

S/N = 3 : detection (ε_m ~31%)

S/N = 5 : fair detection (ε_m ~18%)

S/N = 10 : good detection (ε_m ~9%)

S/N = 25 : quality photometry (ε_m ~4%)

S/N = 100 : high quality photometry (ε_m ~1%)

4.3 Atmosphere

In this section, the user is requested to introduce the following parameters:

Days from new moon. This parameter is used to to define the sky brightness as a function of the days from the new moon. These values are reported in Table 2. We adopted the corresponding values from ESO-Paranal for B and V filters. For Hα and Hα off-band filters we adopted the R values from ESO-Paranal values. For u', g', r', i', z' filters we transformed the Johnson-Cousins values from ESO-Paranal to SDSS by using the formulae in Fukugita et al. (1996, AJ, 111, 1748). The program provides the flux of the sky per arcsec²by using the [ 1] .

Airmass. The user is requested to introduce a value for the airmass at the moment of the observation. We adopt the extinction curve of ESO-Paranal.

Seeing. The user is requested to introduce a value for the seeing, in arcsec.

4.4 Output

The user is requested to choose the output between:

given the AB limit magnitude and the S/N (over the PSF area, per pixel, or over the aperture area for the point-like, extended and integral photometry options, respectively), to compute the requested Exposure Time;

given the AB limit magnitude and the Exposure Time, to compute the requested S/N (over the PSF area, per pixel, or over the aperture area for the point-like, extended and integral photometry options, respectively);

given the Exposure Time and the S/N (over the PSF area, per pixel, or over the aperture area for the point-like, extended and integral photometry options, respectively), to compute the AB limit magnitude.

Note that the flat-field accuracy is fixed at 1%, and it can not be modified by the user.

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5. VOCET Computations

In this section we describe the steps and the computations made by VOCET to arrive to the final output.

A total efficiency, both for the source and the sky is created. For the source, it is the combination between the system efficiency S_e(λ) (filter+CCD+optics), the effective band-width Δλ_e, the atmospheric extinction and the effective area of the telescope. For the sky, it is the combination between the system efficiency S_e(λ) (filter+CCD+optics), the effective band-width Δλ_e, and the effective area of the telescope. The atmospheric extinction correction is computed (only for the source) in the following way:

ext_λ = 10^{-0.4[Xe(λ)airmass]}

being:

X_e(λ) = ∫ X_λ S_λ dλ / ∫ S_λ dλ

the coefficient of atmospheric extinction at λ_e.

The effective band-width and the system efficiency as a function of λ are reported in Table 1 .

The effective area of the telescope is defined as:

Area_tel = π * (R²_mirror - R²_{central_hole})

The AB limit magnitude of the source is scaled to the limit AB flux at the effective wavelength:

F_λ = (c/λ_e²) 10^{-0.4(AB+48.6)}

The sky brightness is first converted in AB magnitude ( section 2), and then transformed in flux as in 2).

Both the source and sky fluxes are scaled for the total efficiency:

total_efficiency_source = S_e(λ) ext_λ Δλ_e Area_tel

total_efficiency_sky = S_e(λ) Δλ_e Area_tel

Both the fluxes (source and sky) are transformed in electrons by dividing for the single photon energy at the given wavelength: E_λ = (hc)/λ_e, where:

c = 2.9979 * 10¹⁸ Å s^-1

h = 6.6262 * 10^-27 erg s^-1

The final count-rates for the source and the sky are:

point-like sources:

counts_source = C_O = (F_λ/ E_λ) * total_efficiency_source

counts_sky = C_S = (F_λ/ E_λ) * total_efficiency_sky * arcsec² * n_PSF

where n_PSF = π *(R²_PSF/arcsec²) are the pixels in the PSF area.

extended sources:

counts_source = C_O = (F_λ/ E_λ) * total_efficiency_source* arcsec²

counts_sky = C_S = (F_λ/ E_λ) * total_efficiency_sky * arcsec².

integral photometry:

counts_source = C_O = (F_λ/ E_λ) * total_efficiency_source

counts_sky = C_S = (F_λ/ E_λ) * total_efficiency_sky * arcsec² * n_aper

where n_aper = π *(R²_aper/arcsec²) are the pixels in the aperture area.

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6. VOCET output

The VOCET output consists of: 1) an output text displayed on the screen, containing a summary of input parameters, general informations, the requested computation (S/N or exposure time); 2) an interactive graphic showing the log(S/N) vs. the log(exposure time) in the selected filter; 3) a gif file (vocet_plot.gif) containing the graphic output; 4) a ps files (vocet_plot.ps) containing the graphic output; 5) a text file (graphic.dat) containing the data used to create the graphic; 6) (optional) a file (vocet_out.dat) containing the output text.

6.1 General

Pixel scale (arcsec pix^-1);

readout noise (e- pix^-1);

dark current (e- pix^-1 s^-1);

flat field accuracy;

saturation level (e-).

6.2 Summary

Source geometry;

AB limit magnitude;

filter;

airmass;

seeing (arcsec);

days from new moon;

aperture (only for integral photometry computation);

S/N in the PSF area, in a single pixel, in the aperture area (depending from the source geometry) or exposure time (seconds).

6.3 Computation

Best aperture to optimize the S/N at the given seeing (only for integral photometry);

Flux level at AB limit magnitude (erg s^-1 Å^-1 cm^-2).

Sky brightness (mag arcsec^-2) in the AB system.

Sky flux level (erg s^-1 Å^-1 cm^-2).

S/N in the PSF area, or in a single pixel, or in the aperture area (depending from the source geometry) or exposure time (seconds).

Number of pixels in the PSF area, per arcsec², or in the aperture area (depending from the source geometry). For point-like sources and integral photometry this is the area over which the S/N is estimated. It is a circular area of radius given by R_PSF or R_aper, divided by the pixel scale.

Only for the source: electrons in the PSF area, in a single pixel, or in the aperture area (depending from the source geometry). This is the total flux contribution from the source, integrated over the PSF area, over a single pixel, or over the aperture area, expressed in electrons. This value corresponds to C_O in [2] e [3] .

Sky background value. This is the total flux contribution from the sky for a single pixel of the detector: it corresponds to C_S in [2] e [3].

Peak pixel value. This value is the sum of the sky background level (C_S in [2] e [3] ) and the fraction of the source signal falling on: i) one pixel at the center of the PSF area; ii) one pixel; iii) one pixel at the center of the aperture area.

S/N at central pixel. Only this S/N is computed for extended sources.

The user can decide to save these outputs in a text file (vocet_out.dat).

6.4 Graphic

An interactive plot showing log(S/N) vs. log(exposure time) for the selected source geometry and filter, at the given seeing, airmass, days from new moon and aperture (only for integral photometry).

The user can type single-character commands, after positioning the cursor. The following are accepted:

1) a (Adds) or clicking the LEFT button of the mouse: select a point on the plot;

2) d (Delete) or clicking the CENTER button of the mouse: delete the selected point on the plot;

3) x (eXit) or clicking the RIGHT button of the mouse: leave the program.

On leaving the program, the values corresponding to the selected point(s) (exposure time and S/N), will be displayed on the screen, in order of increasing exposure time, not in order they were entered.

The graphic output will be saved in gif format in the file vocet_plot.gif.

The graphic output will be saved in ps format in the file vocet_plot.ps.

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7. Useful Tables and Plots

7.1 Tables for VST:

7.1.1 General

Table 1.

Filter	*λ_e* (Å)	*Δλ_e* (Å)	*X_e* (mag/airmass)	*S_e* (%)
u'	3540.5	541.5	0.6582	32.84
g'	4779.2	1423.8	0.2080	49.14
r'	6197.4	1423.5	0.1033	42.48
i'	7558.3	1513.3.	0.0508	32.41
z'	9177.7	1566.4	0.0200	11.83
B	4386.6	931.6	0.2366	33.55
V	5490.1	1052.5	0.1371	41.66
H_α	6579.9	112.2	0.0747	41.73
H_α off	6440.5	110.1	0.0747	42.34

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7.1.2 Sky Brightness

Table 2.

Filter	Sky brightness at 0 days from new moon (mag/arcsec²)	Sky brightness at 3 days from new moon (mag/arcsec²)	Sky brightness at 7 days from new moon (mag/arcsec²)	Sky brightness at 10 days from new moon (mag/arcsec²)	Sky brightness at 14 days from new moon (mag/arcsec²)
u'	22.7	22.2	20.5	19.0	17.6
g'	22.1	21.9	21.3	20.5	19.5
r'	21.2	21.2	21.2	20.6	20.1
i'	20.4	20.4	20.4	19.9	19.5
z'	19.9	19.9	19.9	19.4	19.1
B_AB	22.5	22.2	21.4	20.5	19.3
V_AB	21.6	21.5	21.2	20.5	19.8
Hα_AB	21.3	21.2	21.0	20.7	20.3
Hα_AB off	21.1	21.0	20.8	20.5	20.1

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7.1.3 Photometric Sensitivity

Table 3.
Counts and signal-to-noise ratios for for stellar objects with 1 arcsecond FWHM (seeing limited source), an assumed sky at 3 days from the new moon, and 1.2 airmass. The assumed exposure time is 60 seconds.

Filter	u'		g'		r'		i'		z'		B		V		Hα		Hα off

AB_λ	Counts	S/N	Counts	S/N	Counts	S/N	Counts	S/N	Counts	S/N	Counts	S/N	Counts	S/N	Counts	S/N	Counts	S/N

17.0	58432	225.6	280120	429.6	209602	383.9	147724	328.1	47555	196.1	132094	324.5	165275	352.7	15777	118.3	15838	118.1
17.5	36868	178.0	176744	360.2	132250	316.6	93208	264.4	30005	152.6	83345	264.8	104281	289.0	9955	92.0	9993	91.7
18.0	23262	141.8	111518	295.2	83444	255.3	58810	208.2	18932	116.3	52587	212.8	65797	232.4	6281	70.6	6305	70.1
18.5	14667	110.2	70363	237.0	52650	201.7	37107	160.0	11945	86.6	33180	168.6	41515	183.6	3963	53.2	3978	52.6
19.0	9261	84.3	44396	186.6	33220	155.9	23413	119.9	7537	62.8	20935	131.7	26194	142.5	2500	39.2	2510	38.6
19.5	5843	63.2	28012	143.9	20960	117.7	14772	87.3	4756	44.3	13209	101.1	16527	108.3	1578	28.1	1584	27.6
20.0	3687	46.3	17674	108.5	13225	86.6	9321	61.8	3001	30.5	8335	76.2	10428	80.4	995	19.7	999	19.2
20.5	2326	33.0	11152	79.8	8344	62.0	5881	42.6	1893	20.5	5259	56.1	6580	58.2	628	13.4	651	13.0
21.0	1468	22.9	7036	57.1	5265	43.2	3711	28.6	1195	13.5	3318	40.2	4152	40.9	396	9.0	398	8.7
21.5	926	15.5	4440	39.7	3322	29.3	2341	18.9	754	8.8	2094	28.1	2619	28.1	250	5.9	251	5.7
22.0	584	10.3	2801	27.0	2096	19.5	1477	12.3	476	5.6	1321	19.1	1653	18.8	158	3.8	158	3.7
22.5	369	6.7	1767	17.9	1323	12.7	932	7.9	300	3.6	833	12.7	1043	12.4	100	2.5	100	2.4
23.0	233	4.4	1115	11.7	834	8.2	588	5.1	189	2.3	526	8.3	658	8.0	63	1.6	63	1.5
23.5	147	2.8	704	7.6	527	5.3	371	3.2	119	1.5	332	5.4	415	5.2	40	1.0	40	1.0
24.0	93	1.8	444	4.9	332	3.4	234	2.0	75	0.9	209	3.5	262	3.3	25	0.6	25	0.6
24.5	58	1.1	280	3.1	210	2.1	148	1.3	48	0.6	132	2.2	165	2.1	16	0.4	16	0.4
25.0	37	0.7	177	2.0	132	1.4	93	0.8	30	0.4	83	1.4	104	1.3	10	0.2	10	0.2