Error Budget

Error Budget ms cm micron nm km 1.00E-03 sec 1.00E-02 1.00E-06 1.00E-09 1000 m m m m Spreadsheet created and maintained by Donald Gavel, UCO/Lick Observatory, gavel@ucolick.org Example Adaptive Optics Error Budget This is intended to be generic, and does not represent a particular telescope Version 1 March 6, 2006 arcsec 4.84814E-06 radians plank 6.63E-34 Joule-second c 3.00E+08 m/sec mas 4.85E-09 radians / milliarcsec Table of Assumed Observing Parameters Telescope diameter Wavelength of parameter definitions Coherence cell size Mean wind speed Greenwood frequecy Controller frequency Outer scale Vertical Turbulence Profile Isoplanatic angle Science field Actuator spacing, DM 0 D l0 r0 v fg fc L0 Cn 2 8 500 0.2 20 42.6 84 >30 Cerro Pachon 3.27 10 0.50 Notes on using the spreadsheets: Error Budget Summary Green cells are "inputs" - change them for parametric studies Strehl Yellow cells are interesting or summary results nm 1m 2m All other cells are intermediate calculations - do not change these cells Higher Order WF 194.14 0.23 0.69 Fitting 93.53 0.71 0.92 Measurement 53.53 0.89 0.97 m 8 10 30 Anisoplanatism 113.34 0.60 0.88 nm m m / sec Hz Hz m Cerro Pachon Mauna Kea arcsec arcsec m diameter Bandwidth Cone Effect 45.19 105.77 0.92 0.64 1m 25.78 13.06 8.96 8.59 4.04 0.80 0.89 0.90 0.98 0.98 0.90 2m 51.57 0.94 0.97 0.97 0.99 t0 3.14 ms Diffraction Limit, mas Tip/Tilt Measurement Isokinetic Bandwidth mas q0 q d High Order Wavefront Errors Fitting Error Fitting Parameter Fitting Error 2 Tip / Tilt Errors Diffraction Limit l 0 /D s spot = l 0 /r 0 mV Fl 12.89 0.52 16 mas at l 0 = 500 arcsec magnitude nm m 0.3 5/3 Measurement Error rms Seeing size Star Brightness Guide Star Flux Bandwidth Sodium Watts m km 2 Photons / sec / cm Hz Rayleigh Sample Rate Sample Time Signal Read Noise Number of Pixels Photon Noise Total Noise Signal to Noise Ratio Closed loop averaging factor Centroiding Error s = m (d /r 0 ) 93.53355414 nm Measurement Error (Laser Guide Star) Hartmann Spot Size Beacon Laser Power Range Gate (Rayleigh only) Guidestar Range (Rayleigh only) Guidestar Brightness Frame Rate Sample Time Signal Read Noise Number of pixels Photon Noise Total Noise Signal to Noise Ratio Centroiding Error Closed loop averaging factor Reconstructor Degrees of Freedom Noise propagator 2 s spot P 2 Sodium 10 100 35 1000 840 arcsec Dl f s = 10 f c T = 1/f s N photons 0.0035749 Photons / sec / cm2 / nm 100 nm 140 Hz 0.0071429 sec 1283.53 3 4 35.83 36.33 35.33 0.61 8.96 mas rms Photons / sample electrons / pixel pixels (quad cell) Dh h B f s = 10 f c T = 1/f s N photons sr N pix s photon (s photon 2 +N pix s r 2 )1/2 SNR Photons / sample Photons / sample 1.19047619 ms 2976.190476 Photons / frame / subap 3 4 54.55 2 1/2 sr N pix electrons / pixel pixels (quad cell) Photons / frame / subap Photons / frame / subap arcsec actuators factor from Noll, JOSA 68, 1978 nm rms c s = c s spot / SNR s photon (s photon +N pix s r ) SNR s spot / SNR 54.88 54.23 0.04 0.61 201 0.97 53.53 Isokinetic Error Guide Star - Science Object Separation Mean Atmospheric Height Isokinetic Error D h1 30 3.97 8.5924 arcsec km mas rms c N DOF s isokinetic h Bandwidth Error Mean wind speed Tilt Tracking Frequency Control Loop Bandwidth Wavefront Error s =(2 p / l ) hc d s spot / SNR Anisoplanatic Error Isoplanatic angle Guide Star - Science Object separation Anisoplanatic Error s aniso q0 q = ( q / q 0 ) 5/3 3.27 5 113.34 arcsec arcsec nm (taken as 1/2 the science field) rms Bandwidth Error s BW v fT fc = (f T /f c )( l 0 /D) 20 4.39 14 4.04 m / sec Hz Hz mas rms Bandwidth Error Greenwood frequency Controller frequency Bandwidth error 2 Sky Coverage fg fc 42.6 84 5/3 Hz Hz nm rms Star Density Star Density Sky Coverage N (D ) P mV = 1 - e -N r 0.99 0.99 63% tip/tilt stars per square arcminute with magnitude m v and brighter such tip/tilt stars within radius D Probability of a tip/tilt star with magnitude m v or brighter within D of science object s BW = (f g /f c ) 45.19 Cone Effect (Focal Anisoplanatism) Height of Guide Star H gs Cone Effect Parameter d0 Cone Effect Error 90 5.69 105.77 km m nm rms s cone = (D/d 0 ) 2 5/3 Page 1 Profile r0 = l0= c0= q0= h1= Cerro Pachon 0.2 m 500 nm 1/3 2.189E-13 m 3.27 arcsec 3.9693836 km Fraction of Integral Cn2 3 5 Cerro Pachon Mauna Kea h (km) % Cn2 % int Cn2 h (km) 0 0.646 0.646 0.09 1.8 0.078 0.724 1.826 3.3 0.119 0.843 2.72 5.8 0.035 0.878 4.256 7.4 0.025 0.903 6.269 13.1 0.08 0.983 8.34 15.8 0.017 1 10.546 12.375 14.61 16.471 17.028 Cn2 Profiles given in Ellerbroek & Rigaut, SPIE, 2000 % Cn2 % int Cn2 0.003 0.003 0.136 0.139 0.163 0.302 0.161 0.463 0.167 0.63 0.235 0.865 0.068 0.933 0.032 0.965 0.023 0.988 0.006 0.994 0.007 1.001 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20775.37 87043.6 65531.19 70252.14 582383.2 169126.1 995111.6 3.9693836   5.422406896  2  0 2   37099.9676 r0   0.423    C n h dh    0   86391.12135 2 5 3 179954.1672 r0    2   355942.8342 c 0   C n h dh  0.423  2    806017.7638 3 5 2   344869.8205  2  2 5 3  0   2.905 dh    C n h h 211864.7269        200822.1116 35 2 63976.11481 h1  C n h h 5 3 dh c 0  78892.81318 2365836.864 6.674071834 2   Cn2 Profiles 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Cerro Pachon Mauna Kea 0 5 10 Altitude, km 15 20 Bahcall, Soneira, The Universe at Faint Magnitudes. I. Models for the Galaxy and the Predicted Star Counts, Astrophysical Journal Supplement, 44:73-110, Sept., 1981 N_V in equation B2 degrees C1 C2 alpha beta delta m_star kappa eta lambda m_dagger mu gamma sigma N_V 0.017453293 925 m_V 16 1050 gLat (b) 20 0.349066 -0.132 gLong (l) 0 0 0.035 3 L0 0 0 15.75 L90 90 1.570796 -0.18 L180 180 3.141593 0.087 sigma0 1.262061 2.5 sigma90 1.45 17.5 sigma180 1.637939 0.06 0.2 1.262061476 3558.978979 stars per square degree of magnitude m_V or brighter 0.988605272 stars per square arc minute 131.9354959 12.25649382 1617.067 56.10249421 34.61365516 1941.912 3558.979 Data from Table 6 in the paper Galactic Latitude (longitude=0) M_v 90 50 30 20 10 2.30E+00 3.19E+00 4.21E+00 6.23E+00 11 5.46E+00 7.77E+00 1.04E+01 1.55E+01 12 1.31E+01 1.96E+01 2.65E+01 3.94E+01 13 3.03E+01 4.91E+01 6.97E+01 1.05E+02 14 6.67E+01 1.19E+02 1.85E+02 2.92E+02 15 1.35E+02 2.74E+02 4.83E+02 8.12E+02 16 2.53E+02 5.86E+02 1.19E+03 2.15E+03 17 4.37E+02 1.15E+03 2.68E+03 5.25E+03 18 7.09E+02 2.09E+03 5.43E+03 1.15E+04 19 1.10E+03 3.52E+03 9.89E+03 2.24E+04 20 1.64E+03 5.56E+03 1.64E+04 3.89E+04 22 3.26E+03 1.18E+04 3.62E+04 8.97E+04 24 5.66E+03 2.10E+04 6.53E+04 1.64E+05 26 8.85E+03 3.31E+04 1.04E+05 2.60E+05 28 1.27E+04 4.79E+04 1.51E+05 3.79E+05 30 1.70E+04 6.51E+04 2.06E+05 5.19E+05 Galactic Latitude (longitude=90) 90 50 30 20 2.30E+00 3.01E+00 3.84E+00 5.54E+00 Same table using the formula 3558.979 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 90 1.99E+00 4.68E+00 1.06E+01 2.33E+01 4.89E+01 9.82E+01 1.88E+02 3.42E+02 5.88E+02 9.49E+02 1.44E+03 2.81E+03 4.70E+03 7.21E+03 1.06E+04 1.55E+04 50 3.93E+00 9.42E+00 2.20E+01 4.97E+01 1.09E+02 2.30E+02 4.68E+02 9.03E+02 1.64E+03 2.77E+03 4.35E+03 8.94E+03 1.55E+04 2.44E+04 3.68E+04 5.47E+04 M_v 10 1183.971 90 50 10 1.99E+00 3.33E+00 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 5.46E+00 1.31E+01 3.03E+01 6.67E+01 1.35E+02 2.53E+02 4.37E+02 7.09E+02 1.10E+03 1.64E+03 3.26E+03 5.66E+03 8.85E+03 1.27E+04 1.70E+04 7.13E+00 1.72E+01 4.08E+01 9.20E+01 1.93E+02 3.72E+02 6.54E+02 1.06E+03 1.61E+03 2.33E+03 4.39E+03 7.35E+03 1.12E+04 1.57E+04 2.06E+04 9.09E+00 2.16E+01 5.13E+01 1.20E+02 2.69E+02 5.64E+02 1.08E+03 1.88E+03 2.97E+03 4.34E+03 8.03E+03 1.31E+04 1.95E+04 2.70E+04 3.47E+04 1.31E+01 3.04E+01 7.11E+01 1.67E+02 3.85E+02 8.48E+02 1.74E+03 3.26E+03 5.50E+03 8.42E+03 1.61E+04 2.64E+04 3.92E+04 5.44E+04 7.02E+04 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 4.68E+00 1.06E+01 2.33E+01 4.89E+01 9.82E+01 1.88E+02 3.42E+02 5.88E+02 9.49E+02 1.44E+03 2.81E+03 4.70E+03 7.21E+03 1.06E+04 1.55E+04 7.77E+00 1.75E+01 3.77E+01 7.76E+01 1.52E+02 2.81E+02 4.93E+02 8.17E+02 1.28E+03 1.88E+03 3.52E+03 5.70E+03 8.52E+03 1.23E+04 1.74E+04 M_v 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 90 2.30E+00 5.46E+00 1.31E+01 3.03E+01 6.67E+01 1.35E+02 2.53E+02 4.37E+02 7.09E+02 1.10E+03 1.64E+03 3.26E+03 5.66E+03 8.85E+03 1.27E+04 1.70E+04 Galactic Latitude (longitude=180) 50 30 20 2.90E+00 3.62E+00 5.13E+00 6.78E+00 8.48E+00 1.20E+01 1.16E+01 1.98E+01 2.74E+01 3.73E+01 4.60E+01 6.27E+01 8.17E+01 1.04E+02 1.43E+02 1.65E+02 2.25E+02 3.16E+02 3.07E+02 4.51E+02 6.62E+02 5.18E+02 8.26E+02 1.29E+03 8.02E+02 1.37E+03 2.28E+03 1.17E+03 2.07E+03 3.65E+03 1.63E+03 2.93E+03 5.38E+03 2.88E+03 5.15E+03 9.81E+03 4.62E+03 8.08E+03 1.56E+04 6.83E+03 1.17E+04 2.27E+04 9.35E+03 1.58E+04 3.09E+04 1.19E+04 1.98E+04 3.91E+04 Galactic Latitude (longitude-averaged) 850.18 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 90 1.99E+00 4.68E+00 1.06E+01 2.33E+01 4.89E+01 9.82E+01 1.88E+02 3.42E+02 5.88E+02 9.49E+02 1.44E+03 2.81E+03 4.70E+03 7.21E+03 1.06E+04 1.55E+04 50 2.97E+00 6.92E+00 1.55E+01 3.29E+01 6.66E+01 1.27E+02 2.29E+02 3.89E+02 6.23E+02 9.39E+02 1.34E+03 2.38E+03 3.69E+03 5.31E+03 7.38E+03 1.01E+04 M_v 10 11 12 13 14 15 16 17 18 19 20 22 24 90 2.30E+00 5.46E+00 1.31E+01 3.03E+01 6.67E+01 1.35E+02 2.53E+02 4.37E+02 7.09E+02 1.10E+03 1.64E+03 3.26E+03 5.66E+03 50 3.03E+00 7.23E+00 1.61E+01 4.24E+01 9.76E+01 2.11E+02 4.22E+02 7.74E+02 1.32E+03 2.10E+03 3.17E+03 6.36E+03 1.10E+04 30 3.89E+00 9.32E+00 2.26E+01 5.57E+01 1.36E+02 3.26E+02 7.35E+02 1.53E+03 2.89E+03 4.98E+03 7.89E+03 1.65E+04 2.88E+04 20 5.63E+00 1.35E+01 3.24E+01 7.96E+01 2.01E+02 5.04E+02 1.22E+03 2.76E+03 5.68E+03 1.05E+04 1.76E+04 3.85E+04 6.87E+04 10 11 12 13 14 15 16 17 18 19 20 22 24 90 1.99E+00 4.68E+00 1.06E+01 2.33E+01 4.89E+01 9.82E+01 1.88E+02 3.42E+02 5.88E+02 9.49E+02 1.44E+03 2.81E+03 4.70E+03 50 3.41E+00 8.04E+00 1.83E+01 4.01E+01 8.43E+01 1.70E+02 3.26E+02 5.95E+02 1.03E+03 1.66E+03 2.52E+03 4.95E+03 8.29E+03 26 28 30 8.85E+03 1.70E+04 1.27E+04 2.43E+04 1.70E+04 3.25E+04 4.51E+04 1.07E+05 6.46E+04 1.55E+05 8.68E+04 2.09E+05 26 7.21E+03 1.27E+04 28 1.06E+04 1.88E+04 30 1.55E+04 2.74E+04 Galactic latitude Integrated Star Density, gLong=0 deg 1.00E+06 30 1.06E+01 2.56E+01 6.04E+01 1.39E+02 3.09E+02 6.69E+02 1.39E+03 2.73E+03 5.04E+03 8.62E+03 1.37E+04 2.85E+04 4.98E+04 7.90E+04 1.20E+05 1.79E+05 20 2.63E+01 6.38E+01 1.51E+02 3.48E+02 7.82E+02 1.70E+03 3.56E+03 7.05E+03 1.31E+04 2.25E+04 3.58E+04 7.48E+04 1.31E+05 2.08E+05 3.17E+05 4.73E+05 1.00E+05 Stars per square degree 1.00E+04 1.00E+03 1.00E+02 1.00E+01 1.00E+00 4 8 12 16 Mv 20 24 28 32 30 20 7.69E+00 1.63E+01 Integrated Star Density, gLong=90 1.00E+05 1.79E+01 3.99E+01 8.48E+01 1.71E+02 3.26E+02 5.84E+02 9.85E+02 1.56E+03 2.34E+03 3.33E+03 5.83E+03 8.97E+03 1.28E+04 1.76E+04 2.39E+04 3.78E+01 8.41E+01 1.78E+02 3.56E+02 6.71E+02 1.18E+03 1.96E+03 3.05E+03 4.46E+03 6.19E+03 1.04E+04 1.54E+04 2.12E+04 2.81E+04 3.66E+04 1.00E+05 1.00E+04 Stars per square degree 1.00E+03 1.00E+02 1.00E+01 1.00E+00 4 8 12 16 Mv 20 24 28 30 6.27E+00 1.45E+01 3.23E+01 6.83E+01 1.37E+02 2.57E+02 4.53E+02 7.50E+02 1.16E+03 1.70E+03 2.36E+03 3.96E+03 5.86E+03 8.04E+03 1.06E+04 1.38E+04 20 1.20E+01 2.78E+01 6.16E+01 1.30E+02 2.59E+02 4.85E+02 8.50E+02 1.39E+03 2.14E+03 3.09E+03 4.24E+03 6.96E+03 1.01E+04 1.35E+04 1.74E+04 2.19E+04 Integrated Star Density, gLong=180 1.00E+05 1.00E+04 Stars per square degree 1.00E+03 1.00E+02 1.00E+01 1.00E+00 Stars per square degree 30 8.19E+00 1.93E+01 4.42E+01 9.73E+01 2.06E+02 4.17E+02 8.08E+02 1.49E+03 2.59E+03 4.22E+03 6.46E+03 1.28E+04 2.15E+04 20 1.82E+01 4.31E+01 9.89E+01 2.19E+02 4.66E+02 9.53E+02 1.86E+03 3.47E+03 6.08E+03 1.00E+04 1.54E+04 3.07E+04 5.21E+04 4 8 12 16 Mv 20 24 28 Integrated Star Density, longitude-averaged 1.00E+06 1.00E+05 1.00E+04 1.00E+03 3.33E+04 8.10E+04 4.94E+04 1.21E+05 7.22E+04 1.77E+05 Stars per square degree 1.00E+02 1.00E+01 1.00E+00 4 8 12 16 Mv 20 24 28 Lidar equation Exponential fit, Lidar Equation  d2   P  R 2 np    zTo T L T A   B n r  z  2   h   4 z  Backscatter coefficien Hardy, Table 9.4 Gardner, 1986 meas. Backscatter coefficient  B n R  z   3.6  1 0 R  31 Pz T z   4 .0 1 1 7