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