Bands at the 30m MRT, sizes of pixels and arrays for various FOVs, Power load, NEP, NET, and NEFD from the background
S.Leclercq - Nov 2009
The numbered variables are the free parameters for the optical and photometric calculations
The grey font is used for comments or optional calculations given for information but not used in the optical and photometric calculations
F# (30m+NIKA) = 1.6
Diffraction size on the 30m, number of beam and pixels per FOV NEFD & NET from planets Flux=BnW=noc*2hn^3/c^2*W noc = occupation number (Bose-Einstein in Planck formula)
AKID LEKID  Ko = Nb pix/HPBW (2D) = Wbm / Wp
Speed of light [m/s] c 3E+08 Pix pitch (mm) 1.6 2.3 x/zb = 2.6 zb/x = 0.4
Boltzman constant [J/K] k 1.38E-23 Pix Size (Fl) 0.5 0.72 zb 37%  => use to convert extended <-> point (= no diffract <-> diffract)
Planck constant [J*s] h 6.63E-34 Pix size (") 7.4 10.6  => W ('^2) 0.03 beam/pix (1D) theo 0.46 0.63
Beam width factor (Fl) 1.08 1.14 (convol(pix*diffr) see my Mathcad or Griffin) beam/pix (1D) mes 0.41 0.56
M1 Diameter [m] 30 HPBW theory (bt) (") 15.9 16.8  => Wbt ('^2) 0.06 Ko theo 3.6 2.0 Xavier Ko=1/0.63^2 2.5
1) Wavelength [mm] l 2 HPBW measure (bm) (") 18.0 19.0  => Wbm ('^2) 0.08 Ko mes 4.7 2.5   => Xavier uses square theo ?!
Frequency [GHz] n 150 Pix eff bm correct zpm 9% 17% <= the pixel samples Wbm/Wbt less bm than bt Wbm/Wbt 1.28 1.28
Diffraction Pattern FWHM [mrad] 69 T venus|mars (K) 232 205 Xavier Wb: 0.15 How ?! >HPBW but <Dark ring: 0.47
Diffraction Pattern FWHM ["] 14 Size venus|mars (") 10.7 7.5 Xavier dilution factor: 0.103 How does he get it ? (Mars/Beam)^2 = 0.156
Dilution factor in beam 0.135 0.060 Wplanet/Wb*z/x (=geom*part of diffr pattern in HPBW) *50% = 0.078 *zb = 0.057
Forward efficiency x (empiric fit) 96% T venus|mars (K) 31 12 beam Tbeam = Textended (no diffr) giving same P Xavier (K): 21.1 beam
Beam efficiency b (Ruze, Airy dark ring) 72% F venus|mars (Jy) 338 147 <- F(kT/hn)| F(noc) -> 145 noc 28.0  ~kT/hn 28.5
2) Fraction of unvigneted pupil diameter 93% 27.9 T/F venus|mars (K/Jy) 0.69 1.40 planet (l^2/2kW) Xavier square Mars F(l=2mm)(Jy): 184 F(l=2.1): 167
Telescope effective area [m^2] A 611 T/F beam (K/Jy) 0.24 0.22 <=> no diffr dilution <=> all F in beam Xavier T/F: 0.13 square/round = 1.3
Vigneted Diffraction Pattern FWHM ["] 15 F planet beam (Jy) 129 56  using Tbeam <=> F(z) on array = F(extended_source=without_z)
3) Pixel angular size u [Fl] 0.72 0.5 Tbeam/Fplanet (K/Jy) 0.09 0.08  = T on array / F in space = T equivalent extended / F point source
Pixel solid angle in the sky [sr] W 2.7E-09 1.3E-09 S/N pixel (Hz^0.5) 500 706 S/Nb=S/Np*beam/pix pour LEKIDs je prend S/N ~ Fmars et NEFDmap comme Xavier, pas correct !
Throughput AW [mm^2sr] 1.62 0.79 NET (mK/Hz^0.5) 62 17 pixel (~mean HPBW) Xavier NETbeam = NEFDmapbeam*Tb/Fp = 15
AW/l^2 0.41 0.20 NET (mK/Hz^0.5) 13 11 beam mes  (poisson noise & square pix vs round beam) NET beam theo: 29 11
Pixel efficiency on diffraction spot z 22% 12% NEFD (mJy/Hz^0.5) 145 132 beam pt source  = au NEFD map car utilise le meme S/N hybride (Fmars/NEFDbmap)
4.4 8.3 NEFDext (mJy/Hz^0.5) 55 50 beam extended  <=> lack the x/z ratio to get standard NEFD (flux in space not on pixel !) verif: 145 132
Number of detectors 30 42  => Tb/Fp = measured T / F in space, so not F on the pix W which is x/z less due to the part of beam not in HPBW,
Array length ['] 1.1 0.9 Calculations of MAP NEFD from 2 LEKIDs observations  so NEText deduced from NEFDpoint extracted with map same as planet;
Array solid angle Wa [sr] 8.0E-08 1.1E-07 Map size ("^2) 14400 8100  ~OK if same observ mode, but planet and map should be independent
["] 3388 4744 Mapping time (s) 1050 1050
FOV diameter ['] Number pixels in FOV discs NEFDmap(mJy*s^0.5) 275 201 (<= why flux of 3C345 is 4.348+/-0.013 and not +/-rms=0.006 ?)
1 30 50 rms (mJy) 8.5 6.2 Xavier (calcul): Xavier email (why not = calcul ?):
2 100 200 NEFDbeam 133 130 211 206 190
3 220 440 S/N beam (Hz^0.5) 1102 1130 my NEFDpix with NEFDmap & HPBW => mean pix in HPBW, not central pix !
4 380 790 S/N pix 696 715 mean on HPBW, not central pix ! 211 205 carré theo beam/mean vs beam/central:
6 850 1760 NEFDpix (meanHPBW) 211 206 134 130 120 187 182 rond theo Wbm/Wp 2.5
7 1160 2390 NEFDpix (centreHPBW) 194 189 238 232 carré mes zb/zp 1.7
8 1520 3130 scaling to 2 polar: NEFDbeam2pol 94 92 211 206 rond mes zb/zpm 2.1
10 2360 4880 Best det rms/2: b2p= 47 46 S/N calculé avec Fmars, devrait utiliser le S/N de la carte ! (pix sample less bm than bt !)
NEP: see below (Noise Equiv) Flux, T, and P are same for MAP & planets only if same observing mode !
My simple ATM model
Based on fits on ATM from Astro in Gildas, for the 50-400GHz range, with the error constraint |Dt/t|<4% at 100kHz resolution in the bands and 1MHz resolution in the atmospheric "walls",
built with Tatm = 275 K, Psea_level = 1015 mb, Altitude = Pico Veleta (Tau~P and Tau~T^3 => wv has the strongest effect in the range of possible P and T,
so my model ignore dependence in T and P for simplicity)
Continuum
Reference frequency nc [GHz] 250
225
Water Vapor slope ac [1/mmwv] 0.071 0.058
Dry continnum at nc bc 0.005 0.004
Kinetic lines
Central frequency no [GHz] 58.2 60.2 118.7 183.3 325.1 368.5 380.2
Width ns [GHz] 2.5 2 1 2.96 3.47 0.56 3.49
Central tau to 3.2 11.5 9.4 2.2 2 1 19
Water power pl (0 = O2 line, 1 = H2O line) 0 0 0 1 1 0 1
Gaussians fitting groups of close-packeted lines
Central frequency no [GHz] 58.1 62 65.3 440
Width ns [GHz] 2.5 2.1 3.1 80
Central tau to 17.6 20.2 0.2 0.13
Water power pl (0 = O2 line, 1 = H2O line) 0 0 0 1
Simple photometry calculations (reminder for the units of NEP: prefix "a" = atto = 10^-18)
Remarks 1: the RJ temperatures below are defined so that when used in the RJ approximation formula of the brightness, the result = the unapproximated Planck law
Remarks 2: the Power formula below assumes constant T (brightness) and h (overall efficiency) in the integration over the bandwidth => correct @ <2 % error in the mm atmospheric windows for T>2K
Remarks 3: the NET and NEFD are given for a choosen observing mode and for one pixel, their value for a standard size (beam of HPBW) is given at the end of the sheet
(Blockage M2 & quadurpod, leackage) 97%
4) Cabin optics transmission (M>2) 94%
5) Cryostat 300K transmission 90% 85% <= total warm parts
6) Cryostat 77K transmission 86%
7) Cryostat 4K transmission 86%
8) Band pass filter transmission 95% 70% <= total cold parts
9) Detector absorbtion efficiency 60% 36% <= total optical chain Deff such as h = 30% (Alessandro, and used by Xavier)
Global optical efficiency h 33% Discussion with Alessandro and Markus about Markus 05/2010 measures: LEKIDs used are Deff>~65% for a Dn~20GHz !!
[NE/Hz^0.5] = caracteristic of detector, [NE*s^0.5] = characteristic of observation => [NE*s^0.5] = (time_loss * 2{signal-bkg_ref} / 2{sample_freq_to_integr_time})^0.5 *[NE/Hz^0.5])
10) Bandwidht [GHz] Dn 40 <= NOT SURE: mesures Markus directly on pixels response 05/2010 = 20GHz, mesures in FTS (Martin Pupplet) before run 1 = 40 GHz limited by band-pass filter
Band pass min freq [GHz] 130 Dn/n 13% 27% 30% Verif Xavier mail (28/5/2010) P[pt]=exp(-t)[z/x](2AWh/p)kTDn/l^2=exp(-t)[z/x]AhFDn/p
Band pass max freq [GHz] 170 Dn (GHz) 20 40 45 A (tel) or Aeff (pupil) ?: 707 611
Band pass min wave [mm] 2.3 zb (HPBW) 37% 50% <= HPBW gaussian volume ? Besides Xavier counts h*z = 60%*50% = 30% not good !
Band pass max wave [mm] 1.8 Pmars(pW) beam 0.96 0.96 1.92 geom: 7.60 7.73 <= P calculated with F(noc) or (kT/hn) ?
Bandwidth [mm] 0.54 Power dilut diffr pattern: (zpm/zb) 0.47 Wp/Wbm 0.40 0.21 <= I guess Xavier uses geometry only Wp/Wb (with his 0.15'^2 ?!) and not the beam shape, so keeping his zHPBW
Pmars(pW) pix central 0.45 0.91 verif Pb/Ko: 1.59 1.62 <= some info missing in Xavier email => my reconstruction of his calcul is speculative
11) Degree of polarization 1 Pmars(pW) pix mean 0.38 0.77 0.38 0.93 1.7=3.6/5*2.3 I don't understand this central pix to mean pix in beam factor, =me by chance ?
Polarization parameter p 2 detector mean  NEP [aW/Hz^0.5] 770 1540 1505 <= 2^0.5*Pmpix*NEFDpix/Fmars, I disagree with 2^0.5 (see above) & remark NEFDmap vs NEFDplanet
770 770 <= verif NEPd = Pd/(S/N) = Pd*Ko^0.5*NEFDb/F = Pd*NEFDd/F (remark NEFDmap vs NEFDplanet)
  monomode or coherent or extended source: C = 1 ; multimode and incoherent point source: C ~= l^2/AW = 4/(pu^2) if u>3, C ~= exp(-(0.6*u^((u+2)/(u+1))) if 0<u<3 (empricial approx by me). C can't be >1.
12) Spatial coherance factor C 1.0 0.7 2.5 836 1673 <= NEP central detector = Pd*NEFD/F
13) Observing mode useful time ratio 200% NEPc/m 1.09 Pc/m 1.18 NEFDc/m 0.92
Observing mode efficiency g 0.71  (zpm/b / Wp/bm)^0.5 1.09  (zpm/b / Wp/bm) 1.18  1/(zpm/b / Wp/bm)^0.5 0.92
NEP to NET = T/P(T) = Q/exp(-t) ; NEP to NEFD = F/P(F) = J/exp(-t) (Q and J are defined such that they don't depend on the atmospheric conditions)
   Q = pl^2/(2kAWhDn) [K/pW] 13 28 13 27 17
   J = px/(AhzDn) [Jy/pW] 108 204 275 Jm>J det samples less bm than bt 275 192
T/F: (Q/J)(x/z) = (l^2/2kW) = D^2/2ku^2 [K/Jy] 0.5 1.1 F/T 1.8 0.9 <= this T/F and F/T are without diffraction effects
 x/z 4.4 8.3 <= this factor must be used to count diffraction effect for point sources (generaly used with F)
Atmosphere
14) Atmosphere  temperature (Ta) [K] 275
Black body occupation number at n 38
Brightness for frequencies [fW/m^2/Hz/sr] 1.9
(RJ approx brightness [fW/m^2/Hz/sr]) 1.9
Black body RJ temperature T [K] 271
15) Elevation [deg] 72
Airmass 1.05
16) Precipitable water vapor (wv) [mm] 5
Opacity tau meter (225GHz) 0.29
Opacity tau meter (225GHz) 0.29
  Opacity components for each band:
Atm continuum only 0.130
Atm O2 kinetic lines 0.007
Atm H2O kinetic lines 0.020
Atm O2 gaussian bunch 0.000
Atm H2O gaussian bunch 0.000
Atm total t (including airmass) 0.16 0.85  <= transmission
Emissivity 15%
Spectral radiance of atmosphere [fW/m^2/Hz/sr] 0.28
Atmos emission T RJ [K] T 41
Power [pW] P=(2AWh/p)kTDn/l^2 3.1 1.5
NEPp = (2hnP)^0.5 [aW/Hz^0.5] 25 17
NEPb = P(pC/Dn)^0.5 [aW/Hz^0.5] 22 10
NEP 33 20
NETp = gNEPpQ/exp(-t) [mK*s^0.5] 0.28 0.40
NETb = gNEPbQ/exp(-t) [mK*s^0.5] 0.24 0.24
NET 0.37 0.47
NEFDp = gNEPpJ/exp(-t) [mJy*s^0.5] 2.2 2.9
NEFDb = gNEPbJ/exp(-t) [mJy*s^0.5] 1.9 1.8
NEFD 2.9 3.4
Spillover
Temperature of environment behind M1 [K] 275
Emissivity 4%
Spectral radiance of behind M1 [fW/m^2/Hz/sr] 0.08
T RJ [K] 11.3
fraction received 1.4%
Power [pW] P=(2AWh/p)kTDn/l^2 0.9 0.4
NEPp = (2hnP)^0.5 [aW/Hz^0.5] 13 9
NEPb = P(pC/Dn)^0.5 [aW/Hz^0.5] 6 3
NEP 15 10
NETp = gNEPpQ/exp(-t) [mK*s^0.5] 0.15 0.21
NETb = gNEPbQ/exp(-t) [mK*s^0.5] 0.07 0.07
NET 0.16 0.22
NEFDp = gNEPpJ/exp(-t) [mJy*s^0.5] 1.2 1.6
NEFDb = gNEPbJ/exp(-t) [mJy*s^0.5] 0.6 0.5
NEFD 1.3 1.6
300K optics
(17) Mean surface temperature [K] 280
Black body occupation number at n 38
Black body RJ temperature T [K] 276
Emissivity mirrors / cryostat 5.9% 9.8% 15% <= emissivity of system mirrors + cryostat
Spectral radiance of mirrors [fW/m^2/Hz/sr] 0.11
Spectral radiance of warm optics [fW/m^2/Hz/sr] 0.19
T RJ for mirrors [K] 16.2
T RJ for cryostat warm optics [K] 27.0 41.5 <= system mirrors + cryostat
    For P: apply x to mirrors (correct or not ?) but not to cryostat 300K
Power [pW] P=(2AWh/p)kTDn/l^2 3.8 1.9
details of P: mirror / cryostat for u(col_1) only 1.3 2.5 3.9 3.7 <= for u(col_1) only: system mirrors + cryostat with/without x
NEPp = (2hnP)^0.5 [aW/Hz^0.5] 28 19
NEPb = P(pC/Dn)^0.5 [aW/Hz^0.5] 27 13
NEP 39 23
NETp = gNEPpQ/exp(-t) [mK*s^0.5] 0.31 0.45
NETb = gNEPbQ/exp(-t) [mK*s^0.5] 0.31 0.31
NET 0.44 0.54
NEFDp = gNEPpJ/exp(-t) [mJy*s^0.5] 2.5 3.3
NEFDb = gNEPbJ/exp(-t) [mJy*s^0.5] 2.4 2.2
NEFD 3.5 4.0
77K stage
Temperature cryostat optics on N2 stage [K] 77
Black body occupation number at n 10
Black body RJ temperature T [K] 73
Emissivity 14%
Spectral radiance of N2 stage [fW/m^2/Hz/sr] 0.07
T RJ for [K] 10
Power [pW] P=(2AWh/p)kTDn/l^2 1.1 0.6
NEPp = (2hnP)^0.5 [aW/Hz^0.5] 15 11
NEPb = P(pC/Dn)^0.5 [aW/Hz^0.5] 8 4
NEP 17 11
NETp = gNEPpQ/exp(-t) [mK*s^0.5] 0.17 0.24
NETb = gNEPbQ/exp(-t) [mK*s^0.5] 0.09 0.09
NET 0.19 0.26
NEFDp = gNEPpJ/exp(-t) [mJy*s^0.5] 1.4 1.8
NEFDb = gNEPbJ/exp(-t) [mJy*s^0.5] 0.7 0.7
NEFD 1.5 1.9
TOTAL BACKGROUD
Power [pW] 9 4
NEPp = (2hnP)^0.5 [aW/Hz^0.5] 42 29
NEPb = P(pC/Dn)^0.5 [aW/Hz^0.5] 63 31
NEP [aW/Hz^0.5] 76 42 NEPdet_c/NEPbkg 11.0
NETp = gNEPpQ/exp(-t) [mK*s^0.5] 0.47 0.68
NETb = gNEPbQ/exp(-t) [mK*s^0.5] 0.71 0.71 NETdet/NETbkg 20.3  *((zpm/zb)*(Wbm/Wp))^0.5
NET [mK*s^0.5]     (extended) 0.85 0.98 (point) 3.7 8.1 22.0
x/z = 4.36
NEFDp = gNEPpJ/exp(-t) [mJy*s^0.5] 3.8 5.0
NEFDb = gNEPbJ/exp(-t) [mJy*s^0.5] 5.7 5.2 NEFDdet_c/NEFDbkg 28.2  * zpm/zp 22.1
NEFD [mJy*s^0.5]     (point) 6.81 7.2 6.81 (extended) 1.6 0.9
TOTAL BACKGROUD for a standard elementary size
(18) Standard elementary size us [Fl] 1.14
(19) spatial coherence factor Cs 1 (<= extended source) 0.5 1.0 (<= incoherent point source approxs : u<3 and u>3)
Diffractive gaussian efficiency zs 43%
zs/z 1.97
P ~ (us/u)^2 [pW] 22.4 22.4
NEPp ~ (us/u) 67 67
NEPb ~ (us/u)^2*(Cs/C)^0.5 159 159
NEP [aW/Hz^0.5] 172
NETp ~ 1/(us/u) 0.30 0.30
NETb ~ (Cs/C)^0.5 0.71 0.71
NET [mK*s^0.5]     (extended) 0.77 (point) 1.7
T/F = 0.22 x/zs = 2.21
NEFDp ~ (us/u)/(zs/z) 3.0 3.0
NEFDb ~ (us/u)^2*(Cs/C)^0.5/(zs/z) 7.2 7.2
NEFD [mJy*s^0.5]     (point) 7.8 (extended) 3.5