Date de
mise à jour : 12 novembre 2011
Band |
l |
DL/l |
Flux at m=0 |
Reference |
|
µm |
µm |
Jy |
|
U |
0.36 |
0.15 |
1810 |
Bessel (1979) |
B |
0.44 |
0.22 |
4260 |
Bessel (1979) |
V |
0.55 |
0.16 |
3640 |
Bessel (1979) |
R |
0.64 |
0.23 |
3080 |
Bessel (1979) |
I |
0.79 |
0.19 |
2550 |
Bessel (1979) |
J |
1.26 |
0.16 |
1600 |
Campins, Reike, & Lebovsky (1985) |
H |
1.60 |
0.23 |
1080 |
Campins, Reike, & Lebovsky (1985) |
K |
2.22 |
0.23 |
670 |
Campins, Reike, & Lebovsky (1985) |
g |
0.52 |
0.14 |
3730 |
Schneider, Gunn, & Hoessel (1983) |
r |
0.67 |
0.14 |
4490 |
Schneider, Gunn, & Hoessel (1983) |
i |
0.79 |
0.16 |
4760 |
Schneider, Gunn, & Hoessel (1983) |
z |
0.91 |
0.13 |
4810 |
Schneider, Gunn, & Hoessel (1983) |
For C band (clear) in visible spectrum, we can take :
l = 0.6 µm, Dl : 0.3 µm, Fm0 = 3100 Jy
Lunar Age |
U |
B |
V |
R |
I |
(days) |
|
|
|
|
|
0 |
22.0 |
22.7 |
21.8 |
20.9 |
19.9 |
3 |
21.5 |
22.4 |
21.7 |
20.8 |
19.9 |
7 |
19.9 |
21.6 |
21.4 |
20.6 |
19.7 |
10 |
18.5 |
20.7 |
20.7 |
20.3 |
19.5 |
14 |
17.0 |
19.5 |
20.0 |
19.9 |
19.2 |
Foclen = (F/D) * D
Pixels as result of binning of
photocells on axis 1 (m) :
pixsize1 = cellsize1 * Bin1
Pixels
as result of binning of
photocells on axis 2 (m) :
pixsize2 = cellsize2 * Bin2
Pixel spatial sampling on axis1 (arcsec/pix) :
cdelt1 = 2 * atan ( pixsize1 / Foclen / 2) * 180 / pi * 3600
Pixel spatial sampling on axis2 (arcsec/pix) :
cdelt2 = 2 * atan ( pixsize2 / Foclen / 2) * 180 / pi * 3600
Pixel solid angle (arcsec2/pix) :
W = cdelt1 * cdelt2
Field of view of the axis1 CCD image (deg) :
FoV1 = 2 * atan ( naxis1 * cellsize1 / foclen / 2) * 180 / pi * 3600
Field of view of the axis2 CCD image (deg) :
FoV2 = 2 * atan ( naxis2 * cellsize2 / foclen / 2) * 180 / pi * 3600
Fwhm of the seeing in the image plane (m) :
Fwhm_psf_seeing = seeing / 3600 * pi / 180 * Foclen
Fwhm of the PSF in the image
plane (m) :
Fwhm_psf = sqrt ( Fwhm_psf_opt^2 + Fwhm_psf_seeing^2 )
Flux fraction in the brightest
pixel in
any case (numerical algorithm) for the best case (max flux at the
center of the pixel). For the worst case (max flux at the corner of
the pixel), take x1=0 and y1=0 :
oversampling = 20; (must be even >10)
sigma = Fwhm_psf / (2*sqrt(2*log(2)));
sigma2=sigma*sigma;
a1d=1 / sigma / sqrt(2*p);
a2d=a1d*a1d;
if (pixsize1>=pixsize2)
p=pixsize2;
P=pixsize1;
else
p=pixsize1;
P=pixsize2;
end
dp=p/oversampling;
som=0;
x1=-p/2;
y1=-P/2;
for x=x1:dp:x1+p
dx2=x*x;
for y=y1:dp:y1+P
dy2=y*y;
d2=dy2+dx2;
som=som+exp(-0.5*d2/sigma2);
end
end
fpix=som*a2d*dp*dp;
Apparent magnitude knowing absolute magnitude and distance :
m = M + 5 * log10 (DL_Mpc * 1e6 / 10)
Total flux of the object outside
atmosphere (Jy) :
F_Jy = Fm0 * 10 ^ ( -0.4 * m )
Total flux of the object outside
atmosphere (photons / sec /m2) :
F_ph = F_Jy * 1.51e7 * Dl/l
Transmission of the atmosphere at elevation :
Tatm = Tatm0 * sin(E)
Total flux of the object after
passed
thru the optics (photons / object) :
Ftot_ph = F_ph * p * D^2 / 4 * Tatm * Topt * t
Total flux of the object after
passed
thru the optics (electrons / object) :
Ftot_el = Ftot_ph * h
Brightest pixel flux of the
object
after passed thru the optics (electrons / pixel) :
Fpix_el = Ftot_el * fpix
Brightness of the sky
(Jy/arsec2) :
Sky_Jy = Fm0 * 10 ^ ( -0.4 * msky )
Brightness of the sky (photons /
sec
/m2) :
Sky_ph = Sky_Jy * 1.51e7 * Dl/l
Brightness of the sky after
passed thru
the optics (photons / pixel) :
Skypix_ph = Sky_ph * p * D^2 / 4 * W * Topt * t
Brightness of the sky after
passed thru
the optics (electrons / pixel) :
Skypix_el = Skytot_ph * h
EMCCD excess noise factor (empirical formula derived from a figure of a paper) :
fex = 1 + ( 2/pi * atan( (Em-1)*3 ) )^3
All expressions are in (electrons/pixel). In order to convert into (ADUs/pixel) divide them by G.
Thermic signal (electrons/pixel)
:
S_th = C_th * Bin1 * Bin2 * t * Em
Sky signal (electrons/pixel) :
S_sk = Skypix_el * Em
Object signal (electrons/pixel) :
S_ph = Fpix_el * Em
Thermic noise (electrons/pixel) :
N_th = sqrt(C_th * Bin1 * Bin2 * t * fex) * Em
Sky noise (electrons/pixel) :
N_sk = sqrt(Skypix_el * fex) * Em
Object noise (electrons/pixel) =
shot
noise :
N_ph = sqrt(Fpix_el * fex) * Em
Total noise (electrons/pixels) :
N_tot = sqrt ( N_ro^2 + N_th^2 + N_sk^2 + N_ph^2 )
Object signal/noise at the
brightest
pixel :
SNR_obj = S_ph / N_tot
L'Exposure Time Calculator (ETC)
utilise
la théorie décrite dans le paragraphe 1 et
utilise une
série de fonctions qui sont décrites dans ce
paragraphe.
L'ETC doit d'abord être initialisé avec la fonction etc_init utilisée dans la Console de AudeLA :
etc_initL'initiatialisation donne des valeurs par défaut à la valeur des configurations instrumentales et atmosphériques. L'ETC de AudeLA définit deux types de configuration :
etc_params_set
# C_th Em G N_ro bin1 bin2 eta naxis1 naxis2 photocell1 photocell2 Dl Fm0 l Elev Tatm0 msky seeing band D FonD Fwhm_psf_opt Topt
On peut lire la valeur courante d'un paramètre :
etc_params_set C_thOn peut modifier la valeur d'un paramètre :
# 0.002 "Thermic coefficient (electrons/sec/photocell)"
etc_params_set C_th 0.4On peut afficher tous les paramètres d'un seul coup avec etc_disp (cf. paragraphe 2.4).
# 0.4 "Thermic coefficient (electrons/sec/photocell)"
etc_set_cameraPour Choisir une caméra :
# {Andor DW436} {Audine Kaf401ME}
etc_set_camera "Audine Kaf401ME"
etc_inputs_setOn peut lire la valeur courante d'une entrée. Par exemple, si l'on souhaite utiliser un calcul qui a besoin de connaître la magnitude apparente d'un astre :
# t snr DL_pc M m
etc_inputs_set mOn peut modifier la valeur d'un paramètre :
# 12.0102999566 "Apparent stellar magnitude in the V band"
etc_inputs_set m 9.5A noter que les entrées DL_pc (la distance en méga parsec), la magnitude absolue M et la magnitude apparente m sont reliées. Ainsi, la modification d'une valeur influera les autres.
# 9.5 "Apparent stellar magnitude in the V band"
etc_disp
#
===== input-ccd
# t = 30 : Exposure time (sec)
#
===== input-constraint
# snr = 5 : SNR constrained
#
===== input-object
# DL_pc = 40e6 : Distance luminosity (pc)
# M = -21 : Absolute stellar magnitude in the V band
# m = 12.0102999566 : Apparent stellar magnitude in the V band
#
# ===== param-ccd
# C_th = 0.4 : Thermic coefficient (electrons/sec/photocell)
# Em = 1 : Electron multiplier (>1 if EMCCD, else =1)
# G = 1.8 : CCD gain (electrons/ADU)
# N_ro = 8.5 : Readout noise (electrons/pixel)
# bin1 = 1 : Binning on axis1 (photocells/pixel)
# bin2 = 1 : Binning on axis2 (photocells/pixel)
# eta = 0.9 : CCD Quantum efficiency in the photometric band (electron/photon)
# naxis1 = 2048 : Number of pixels on an axis1
# naxis2 = 2048 : Number of pixels on an axis2
# photocell1 = 13.5e-6 : Photocell size on axis1 (m)
# photocell2 = 13.5e-6 : Photocell size on axis2 (m)
#
===== param-filter
# Dl = 0.088 : Wavelength bandpass of the filter (micrometers)
# Fm0 = 3640 : Flux for magnitude zero for the filter (Jy)
# l = 0.55 : Central wavelength of the filter (micrometers)
#
===== param-local
# Elev = 65 : Elevation above horizon (deg)
# Tatm0 = 0.5 : Zenith transmission of the atmosphere in the photometric band
# msky = 21.8 : Sky brightness in the V band at moon age 0 day (mag/arcsec2)
# seeing = 3.0 : Fwhm of the seeing (arcsec)
#
===== param-object
# band = V : Photometric system symbol
#
===== param-optic
# D = 0.3 : Optic diameter (m)
# FonD = 4 : Focal diameter ratio
# Fwhm_psf_opt = 15e-6 : Fwhm of the point spread function in the image plane (m)
# Topt = 0.5776 : Transmission of the optics in the photometric band (Reflec=0.8, Refrac=0.95)
etc_initOn trouve ainsi une magnitude 17.2. Par défaut, les paramètres de la caméra sont ceux d'une Andor DW436 avec un CCD aminci. On peut maintenant effectuer le même calcul en choisissant une caméra Audine :
etc_params_set D 0.20
etc_params_set FonD 10
etc_inputs_set snr 3
etc_inputs_set t 30
etc_snr2m_computations
# 17.285256851
etc_initOn voit nettement l'avantage d'utiliser un CCD aminci !
etc_set_camera "Audine Kaf401ME"
etc_params_set D 0.20
etc_params_set FonD 10
etc_inputs_set snr 3
etc_inputs_set t 30
etc_snr2m_computations
# 15.721193582
etc_disp
#
===== comp1
# F_Jy = 0.0571454163189 : Total flux of the object outside atmosphere (Jy)
# F_ph = 138063.325826 : Total flux of the object outside atmosphere (photons / sec /m2)
# FoV1 = 0.198014016908 : Field of view of the CCD image on axis1 (deg)
# FoV2 = 0.132009417601 : Field of view of the CCD image on axis2 (deg)
# Foclen = 2.0 : Focal length (m)
# Fpix_el = 1280.91200087 : Brightest pixel flux of the object after passed thru the optics (electrons / pixel)
# Ftot_el = 17029.1261651 : Total flux of the object after passed thru the optics (electrons / object)
# Ftot_ph = 34058.2523303 : Total flux of the object after passed thru the optics (photons / object)
# Fwhm_psf = 3.27285731343e-05 : Fwhm of the PSF in the image plane (m)
# Fwhm_psf_seeing = 2.90888208666e-05 : Fwhm of the seeing in the image plane (m)
# Sky_Jy = 6.93587701339e-06 : Brightness of the sky (Jy/arsec2)
# Sky_ph = 16.7570788644 : Brightness of the sky (photons / sec /m2)
# Skypix_el = 3.92954189887 : Brightness of the sky after passed thru the optics (electrons / pixel)
# Skypix_ph = 7.85908379773 : Brightness of the sky after passed thru the optics (photons / pixel)
# Tatm = 0.453153893518 : Transmission of the atmosphere at elevation
# W = 0.861539698493 : Pixel solid angle (arcsec2/pix)
# cdelt1 = 0.92819162811 : Pixel spatial sampling on axis1 (arcsec/pix)
# cdelt2 = 0.92819162811 : Pixel spatial sampling on axis2 (arcsec/pix)
# fex = 1.0 : EMCCD excess noise factor (empirical formula derived from a figure of a paper)
# fpix1 = 0.0752188919416 : Flux fraction in the brightest pixel in the favorable case (max flux at the center of the pixel)
# fpix2 = 0.0715280179357 : Flux fraction in the brightest pixel in the intermediate case
# fpix3 = 0.0680182493763 : Flux fraction in the brightest pixel in the worst case (max flux at the corner of the pixel)
# pixsize1 = 9e-06 : Pixel length on axis1 (m)
# pixsize2 = 9e-06 : Pixel length on axis2 (m)
#
===== compsnr
# m = 15.721193582 : Apparent magnitude computed from a SNR and exposure value constrained
#
===== input-ccd
# t = 30 : Exposure time (sec)
#
===== input-constraint
# snr = 3 : SNR constrained
#
===== input-object
# DL_pc = 40e6 : Distance luminosity (pc)
# M = -21 : Absolute stellar magnitude in the V band
# m = 12.0102999566 : Apparent stellar magnitude in the V band
#
===== param-ccd
# C_th = 0.2 : Thermic coefficient (electrons/sec/photocell)
# Em = 1 : Electron multiplier (>1 if EMCCD, else =1)
# G = 2.1 : CCD gain (electrons/ADU)
# N_ro = 12 : Readout noise (electrons/pixel)
# bin1 = 1 : Binning on axis1 (photocells/pixel)
# bin2 = 1 : Binning on axis2 (photocells/pixel)
# eta = 0.5 : CCD Quantum efficiency in the photometric band (electron/photon)
# naxis1 = 768 : Number of pixels on an axis1
# naxis2 = 512 : Number of pixels on an axis2
# photocell1 = 9e-6 : Photocell size on axis1 (m)
# photocell2 = 9e-6 : Photocell size on axis2 (m)
#
===== param-filter
# Dl = 0.088 : Wavelength bandpass of the filter (micrometers)
# Fm0 = 3640 : Flux for magnitude zero for the filter (Jy)
# l = 0.55 : Central wavelength of the filter (micrometers)
#
===== param-local
# Elev = 65 : Elevation above horizon (deg)
# Tatm0 = 0.5 : Zenith transmission of the atmosphere in the photometric band
# msky = 21.8 : Sky brightness in the V band at moon age 0 day (mag/arcsec2)
# seeing = 3.0 : Fwhm of the seeing (arcsec)
#
===== param-object
# band = V : Photometric system symbol
#
===== param-optic
# D = 0.20 : Optic diameter (m)
# FonD = 10 : Focal diameter ratio
# Fwhm_psf_opt = 15e-6 : Fwhm of the point spread function in the image plane (m)
# Topt = 0.5776 : Transmission of the optics in the photometric band (Reflec=0.8, Refrac=0.95)
etc_initIl faut donc idéalement poser 28 secondes.
etc_params_set D 0.20
etc_params_set FonD 10
etc_inputs_set m 11
etc_inputs_set snr 100
etc_snr2t_computations
# 28.0398673386
mc_cosmology_calculator 0.1On notera que DL_Mpc = 455 méga parsecs (Mpc). Le calcul nous signale aussi que la lumière aura voyagé pendant DTT_Gyr = 1.3 milliard d'années.
# {H0 71.000000 {current Hubble constant km/s/Mpc}} {Omega_M 0.270000 {Omega_matter}} {Omega_vac 0.730000 {Omega_vac = 1-Omega_M gives a flat Universe}} {z 0.100000 redshift} {cz 29979.245800 {non relativistic velocity km/s}} {DTT_Gyr 1.287066 {time from z to now in Gyr}} {age_Gyr 13.680177 {age of Universe in Gyr}} {zage_Gyr 12.393111 {age of Universe at redshift z in Gyr}} {DCMR_Mpc 413.871021 {comoving radial distance in Mpc}} {V_Gpc 0.296950 {comoving volume within redshift z in Gpc3}} {DA_Mpc 376.246333 {angular size distance in Mpc}} {kpc_DA 1.824094 {angular scale in kpc/arcsec}} {DL_Mpc 455.258063 {luminosity distance in Mpc}}
etc_initAinsi, pour détecter une supernova à z=0.1 avec un SNR=5 il faut poser 1 minute. La magnitude apparente correspondante peut être aussi affichée :
etc_params_set D 0.20
etc_params_set FonD 10
etc_inputs_set M -21
etc_inputs_set DL_pc 455e6
etc_inputs_set snr 5
etc_snr2t_computations
# 59.2449949385
etc_inputs_set m
# 17.2900569833 "Apparent stellar magnitude in the V band"
loadima m57
simulimage * * * * * USNO c:/d/usno 30 2.3 0.20 C 20.5 0.4 3 15 1 300 0.4 0.6 0.7 1 0
simulimage * * * * * USNO c:/d/usno 30 2.3 0.20 C 20.5 0.4 3 15 1 300 0.4 0.6 0.7 1 0 ADD 18h53m20s +33d00m51s 7
simulimage2 test [mc_date2listdates 2011-11-11T00:00:00 0.021 100] {FOURIER 164.630566 67.529504 2011-11-08T00:00:00 0.28 12.5 -0.86 -0.53 -0.45 0.32 0.039 0.021} 200 200 164.589733 67.515479 13.5 13.5 0.84587 USNO c:/d/usno/ 90 2.5 0.25 R 20.0 0.07 1.8 8.5 1 1000 0.5 0.6 0.85 1 0Au départ, apparaît la courbe de lumière théorique pour les dates calculées :
photrel_wcs2cat test 100 newA la fin on a un nouveau graphique :
photrel_cat2var test
photrel_cat2per test test 164.630566 67.529504 C 0
simulimage2 test [mc_date2listdates 2011-11-11T00:00:00 1 60] {SN Ia 164.630566 67.529504 2011-11-30T00:00:00 15} 200 200 164.589733 67.515479 13.5 13.5 0.84587 USNO c:/d/usno/ 90 2.5 0.25 R 20.0 0.07 1.8 8.5 1 1000 0.5 0.6 0.85 1 0Au départ, apparaît la courbe de lumière théorique pour les dates calculées :
photrel_wcs2cat test 60 newA la fin on a un nouveau graphique :
photrel_cat2var test
photrel_cat2per test test 164.630566 67.529504 C 0