Calculations relevant to ELT MEIFU Study

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Randell Robinson
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1 Calulations relevant to ELT MEIFU Study Simon Morris 9.9. Angstrom := m µm := 6 m nm := 9 m Plank := Js λ Lyα := 5.67Angstrom m := se Photon := arse := deg 66 armin := deg 6. p := m kp := p 3 Mp := p 6 Gyr := 9 yr njy := 9 3 ergm s Hz F AB( F) :=.5 log ( ergm se Hz ) 48.6 Cosmology from Carroll, Press and Turner, 99, ARAA, 3, 499 H := h km semp H =.397 yr Ω M.3 Ω k := Ω M Ω Λ h.7 Ω Λ.7 Ω k = q := Ω M Ω Λ q =.55 Now alulate proper distane d M ( z) := z sinh ( Ω k ) ( + z) ( + Ω M z) z( + z) Ω Λ z H d if Ω k > ( Ω k ) z sin ( Ω k ) ( + z) ( + Ω M z) z( + z) Ω Λ z H d if Ω k < ( Ω k ) z H ( ) ( + z) + Ω M z z( + z) Ω Λ dz otherwise

2 d L ( z) := ( + z) d M ( z) Luminosity distane d A ( z) := d M ( z) Angular diameter distane + z Now do Volume out to some redshift V( d M ) := H H 3 4π Ω k 3 4π Ω k 4π 3 d M 3 d M H H + Ω k d M ( ) Ω k asinh ( ) Ω k H d M d M H H Ω k d + M H asin ( Ω k ) d M if Ω k < otherwise Volume per square degree out to given redshift ( ) Ω k if Ω k 5 4 log V( d M ( z p )) 4π armin Mp z p

3 Volume between z=5 and 6.5 per square armin ( ) V( d M ( 5) ) V d M ( 6.5) 4π armin = Mp 3 Chek on redshift ranges and resolutions for default 3 band MEIFU design R := 75 λ b := 6Angstrom λ r := 7Angstrom λ b λ Lyα z b := z b = λ Lyα λ b + λ r λ := λ = Angstrom λ := λ R λ =.873 Angstrom dispersion per pixel λ r λ b npix λ := npix λ = number of pixels in wavelength λ diretion λ b := 7Angstrom λ r := 843Angstrom λ b λ Lyα z b := λ Lyα z b = λ b + λ r λ := λ = Angstrom λ λ := R λ =. Angstrom dispersion per pixel λ r λ b npix λ := npix λ = number of pixels in wavelength λ diretion λ 3b := 843Angstrom λ 3r := Angstrom λ 3b + λ 3r λ 3 := λ 3 = Angstrom λ 3b λ Lyα z 3b := z 3b = λ 3r λ Lyα λ z Lyα 3r := λ Lyα z 3r = 7.6 λ 3 := λ 3 R λ 3 =.633 Angstrom dispersion per pixel λ 3r λ 3b npix λ3 := npix λ3 = number of pixels in wavelength λ 3 diretion

4 z := z b z b z := z 3b z b z 3 := z 3r z 3b z =.93 z =.86 z 3 =.9 Calulate some volumes in various surveys V survey ( arminx, arminy, z, z) := ( ) V( d M ( z) ) V d M ( z) 4π arminxarminy ( armin) ( ) = V survey.68, 3., z b, z 3r Mp 3 MEIFU V survey (.68,.55,.95, 4.44) =.77 3 Mp 3 SAURON.5e-8 in hours on 4m V survey ( 6, 6, 3.4, 3.46) = Mp 3 Fynbo typial table entry, e-7 V survey ( 36, 36, 4.37, 4.57) = Mp 3 Rhoads et al., e-7 in 6 hours on 4m Geometry Calulations for MEIFU onept Calulate number of slitlets pix := slit L := slit W :=.arse.arse.arse FOV tot :=.846arse FOV tot Nslit L := slit L FOV tot Nslit W := slit W Nslit L = 4 Nslit W = 85 Nslit tot := Nslit L Nslit W Nslit tot =. 5 onsider single 8k x 8k amera - alulate pixels used npix := 84 all alulations in pixels

5 Use Content numbers for 8m MEIFU geometry - i.e. stay with same slit size in pixels and same offsets for slits relative to eah other. dx := 75 dy := 33. offset between mioslits npix nx := ny := dx slit_len := npix dy nx = 9.7 ny = 6.5 nslit := nxny nslit = total number of miroslits on the detetor nspe := nxnyslit_len nspe = spe_len := 6 npix_used := nspespe_len npix_used = total number of pixels with data on them npix_used fra_used := fra_used =.7 npixnpix now onsider what that means in terms of sky overage onsider x pixel miroslits,. arse per pixel in both diretions dx_slit_arse := slit_len.arse dy_slit_arse :=.arse area := nslitdx_slit_arsedy_slit_arse area =.448 armin area on sky overed by spetrograph for 4 spetrographs get: equivalent to: tot_area := 4area tot_area =.748 armin tot_area = 3.78 armin npixnpix4 =.6 9 total number of detetor pixels in all spetrographs

6 Do some S/N alulations for fiduial redshift z := 6 λ Lyα ( + z) =.85 µm Consider and 3m telesope with a total throughput of 7% r := 5m ε :=.7 expose := 4836s expose =.5 5 s Night Sky values for Optial/IR Magnitude entral wavelengths and zero points from ESO web site λ :=.95 µm Z BAND := ESO Night Sky Brightnesses at new moon (magnitudes per square arse) (IR magnitudes are atually Ks, L and M-NB) S :=

7 Calulate flux per square arse S Z BAND.5 F S := W m µm arse F S = W m µm arse Calulate Photons per square arse.67.6 F S λ P S := Photon Plank P S = Photons m µm arse Reord Content 996 equivalent bakground redution fators to get effetive ontinuum between OH lines: J=5 H=38 K=4.4 - divide above J,H,K fluxes by this number for OH supressed bakground. fitting funtion Sky λ (,, λ) ( ) := linterp λ P S ( ) = Photon s Sky.9 µm m µm arse

8 Flux to photon rate onversion F =.6 9 ergm s tune hoie to get 5 sigma detetion in 4 nights L := 4π d L ( z) F L =.37 4 ergs orresponding luminosity P := Fλ Lyα ( + z) Plank P =.4 7 m s N tot := Pπr ε N tot =.3 s - S := N tot expose S = apfa :=.5 total Lyalpha photons deteted in an exposure assume sum over an aperture aperture whih only gets / of the photons (atually will use optimal weighting and depens on seeing and objet morphology) Figure out sky bakground assume objet flux (and hene sky flux) is summed over various different sized regions assume line flux is all within 4 pixels spetrally (=xfwhm, rit samp)

9 'Natural Seeing' N totsky Sky λ Lyα ( + z) λ Lyα ( + z) := (.6arse) πr ε N totsky = 5.65 s - R B := N totsky expose B = total sky photons SN := + B SN = 5.64 resulting S/N AO with fuzzy targets N totsky Sky λ Lyα ( + z) λ Lyα ( + z) := (.arse) πr ε N totsky = 5.68 s - R B := N totsky expose B = total sky photons SN := + B SN = 5.65 resulting S/N Diffration Limited,.79 mirons, 3m (i.e. point soure, perfet AO - fantasy) D EE5% := arse haven't bothered to opy over real alulation for this, and so is hardwired - obviously varies with wavelength in pratie N totsky Sky λ Lyα ( + z) λ Lyα ( + z) := D EE5% R ( ) πr ε N totsky = s - B := N totsky expose B = total sky photons SN := + B SN = 8.89 resulting S/N F.6 9 ergm s Ly A eline flux transribed here

10 Now try to make a ballpark omparison with an FTS (know in advane S/N will be a lot worse for strongly bakground limited, although are some subtleties about smoothing out the OH line noise) Consider a sanning strategy where one observes for 3 x 4 nights (i.e. the time alloated to the 3 bands above), overing the same total wavelength range and aiming for a similar spetral resolution in the middle of the band (FTS redued ube will be equally spaed in frequeny, not in wavelength) M spe := R Spetral bins expose3 expose FTS := n samp n samp := M spe number of FTS samples (two sided) time per FTS step n samp = 7 3 Have ignored stuff about powers of above and requirements on sampling strategy, just going to order of magnitude estimates expose FTS = s ε FTS := ε.6 allow the FTS to have higher throughput (no grism) - assuming port setup. Ignoring readout noise, as did for IFU. Consider 'broad band' S/N per step Nsky BB ε FTS expose FTS 4πr := λ 3r λ b ( ) Sky λ dλ Nsky BB = Photonarse total number of photons deteted from the sky for the entire wavelength range per step FTS broad band S/N for pure emission line soure is P := Fλ Lyα ( + z) Plank P =.4 7 m s N tot := Pπr ε FTS N tot =.354 s - S := N tot expose FTS 'Natural Seeing' B := Nsky BB (.6arse) B = total sky photons SN := + B SN = resulting S/N Calulate the SNR in the Wavelength domain - equation from Graham et al 998 (astro-ph/98363), makes some sense - sales by sqrt of number of exposures, inversely with resolution. Additional fator to aount for fat that is pure eline soure. Fator is signal in wavelength range ontaining the eline, ompared to the signal over the whole band (see Davis, Abrams and Brault 999, Draft of book, for justifiation)

11 SN FTS := n samp SN λ 3r λ b M spe λ Lyα( + z) R SN FTS =.34 As expeted the S/N suks - to make FTS ompetitive would need to tune down to muh narrower band. See main report for fuller disussion of when an FTS would be an attrative option. AO with fuzzy targets B := Nsky BB (.arse) B = total sky photons SN := + B SN =. resulting S/N SN FTS := n samp SN λ 3r λ b M spe λ Lyα( + z) R SN FTS =. Diffration limited B := Nsky BB ( D EE5% ) B = total sky photons SN := + B SN =.35 resulting S/N SN FTS := n samp SN λ 3r λ b M spe λ Lyα( + z) R SN FTS = 3.447

12 k > <

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