Astronomical Seeing

Innovations Foresight

Astronomical Seeing

Northeast Astro-Imaging Conference

April 7 & 8, 2016

Dr. Gaston Baudat Innovations Foresight, LLC

Innovations Foresight

• Astronomical seeing is the blurring of astronomical objects caused by Earth's atmosphere turbulence and related optical refractive index variations (air density fluctuations).

• It impacts the intensity (scintillation) and the shape (phase) of the incoming wave front.

• This presentation will ignore scintillation.

Innovations Foresight An incoming plane wave is perturbed by the Earth atmosphere turbulent structure leading to phase errors.

Δ휙(푟) 푟

Innovations Foresight The Reynolds’s (O. Reynolds 1883) number Re predicts flow patterns.

Density Velocity Characteristic length Viscosity

For air flow with velocity of few km/h and for length scales of 0.1km to 1km Re > 106. Therefore the atmosphere is almost always turbulent.

Innovations Foresight Energy is transferred by a turbulence cascade form the outer scale L0 to the inner scale l0 until heat dissipation (friction).

Eddies exhibit different air density -> different refraction index

Innovations Foresight

The Kolmogorov’s turbulence model (A. Kolmogorov 1941) assumes an infinite outer scale turbulence (OST) value L0.

The related wave front phase error variance, known as the phase structure function, is given by:

(25l) ퟓ ퟑ ퟐ 풓 푫흓 풓 = 휟흓(풓) = ퟔ. ퟖퟖ 풓ퟎ Some r

푟0 is known as the Fried’s parameter, or coherence length. (David Fried 1960) (-15l)

Innovations Foresight

The Fried’s parameter 푟0 is the diameter for which the rms wave front phase error is about 1 radian, or ~l/6. It is the “average” turbulence cell size.

−ퟑ ퟓ ퟐ ∞ ퟐ흅 ퟐ 풓ퟎ = ퟎ. ퟒퟐퟑ 퐬퐞퐜(풛) 푪푵 풉 풅풉 흀 ퟎ z zenith angle, 휆 the wavelength and 2 퐶푁 ℎ is the atmospheric turbulence strength at the altitude h.

6 5 푟0 is a function of the wavelength and increases as 휆 .

Innovations Foresight

2 퐶푁 ℎ is the profile of the atmospheric strength as function of the altitude h.

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푟0 is the equivalent diameter of a seeing limited scope of aperture D>>푟0 푫 Diffraction limited < ퟏ 풓ퟎ 흀 퐅퐖퐇퐌 ≅ [퐫퐚퐝퐢퐚퐧] 푫

푫 Seeing limited > ퟏ 풓ퟎ 흀 FWHM ≅ 퐫퐚퐝퐢퐚퐧 풓ퟎ 풍풐풏품 풆풙풑풐풔풖풓풆

" ≅ radian × ퟐ ∙ ퟏퟎퟓ

FWHM [“] 1 1.5 2 2.5 3

푟0 [mm/inch] 110 / 4.3 74 / 2.9 56 / 2.2 44 / 1.7 37 / 1.5

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푟0 histogram and seasonal variations for Lick Observatory (CA USA).

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흉ퟎ corresponds to the time for which the changes in the turbulence becomes significant (1 radian rms, ~휆 6). One assumes that the turbulence pattern is moved by the wind faster than it evolves: 풓 흉 ≅ ퟎ. ퟑퟏ ퟎ ퟎ 풗 where 풗 is the average wind speed Typical 풗 ~10m/s (~20mph)

Credit Tokovinin 풓ퟎ = ퟓퟎmm → 흉ퟎ = ퟏ. ퟓms

6 5 휏0 is a function of the wavelength and increases as 휆 .

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흉ퟎ histogram & cumulative distribution for Xinglong observatory (China).

풓ퟎ [mm] 흉ퟎ [ms] @ 550nm V = 10m/s (~20mph) 30 0.93

50 1.5

70 2.2

100 3.1

300 9.3

Innovations Foresight For exposure times t<<흉ퟎ the seeing is “frozen”.

퐷 푟0 = 1 퐷 푟0 = 3 퐷 푟0 = 10

2 Nb. of speckle ~ 퐷 푟0 Nb. of AO degree of freedom ~ 퐷 푟0 6 5 퐷 푟0 decreases as 휆 Diffraction limited operation is easier at longer wavelengths

Innovations Foresight

Atmospheric energy transfer goes from the largest (L0) down to smaller scales (l0), the strongest distortions are on the lowest-order Zernike’s modes (~tilt/tip, piston ignored). ퟓ ퟑ ∞ 2 1 2 ퟐ 푫 (Noll, 1976) 휙 decreases as 휆 흓 = 휟풋 풓ퟎ 풋=ퟐ Type of aberration Residual rms phase removed (inclusive) error 휟 [ %] Tilt/tip 36% Defocus 33% Astigmatism 25% Coma (3rd order) 23% Spherical (4th order) 21% Trefoild (3rd order) 19% Zernike’s polynomials (c) Innovations Foresight 2016 - Dr. Gaston Baudat 14 Long term seeing t >> 흉ퟎ

Innovations Foresight

For exposure times t >> 흉ퟎ the seeing is averaged.

휆 FWHM ≅ 푟0

Tilt/tip removed 휆 FWHM ≅ 0.3 푟0

Innovations Foresight The angle for which the wavefront error remains almost the same (~l/6) is known as the isoplanatic angle: 풓 휽 ≅ ퟎ. ퟑퟏ ퟎ ퟎ 풉

풉 ~ 5km, 휽ퟎ is usually few arc-second across (@550nm): 풓ퟎ = 50mm → ~ 0.6” 풓ퟎ = 200mm → ~ 2.6”

6 5 휽ퟎ increases as 휆

Innovations Foresight AO operation is usually only effective in a very narrow FOV.

Guide IR bands: 1200nm, 1600nm, and 2200nm star FWHM offset [“] [“]

0 0.2

5.5 0.3

0.45 13 - 0.59 0.51 Credit R. Dekany, Caltec Palomar AO system 23 - 0.68

Innovations Foresight Temporal phase fluctuations derive from spatial fluctuations under the “frozen turbulence” model. Turbulence pattern moves “untouched” at the average wind speed across the aperture. Small cells contribute to the high frequency turbulences and are eventually averaged during image exposition (풕풆풙풑) at the expense of the resolution (larger star FWHM). Large cells are related to low frequencies. Long exposure times increase the apparent isoplanatic angle:

풕풆풙풑 풗 휽풆풙풑 ≅ 휽ퟎ = 풕풆풙풑 D>풓ퟎ, 풕풆풙풑 > 흉ퟎ 흉ퟎ 풉

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풗 ~ 10m/s (~ 20mph), 풉 ~ 5km, D = 279mm (11”) @550nm Diffraction limit = 0.4” Tilt/tip corrections v.s. seeing function of exposure time

풕풆풙풑 0 50ms 0.5s 1s 5s 10s

휽풆풙풑 0.6” 20” 200” 400” 2000” 4000” 0.01’ 0.3’ 3’ 7’ 33’ 67’ FWHM tilt/tip removed 0.7” 1.5” 2.2” 2.2” 2.3” 2.3” 풓ퟎ=50mm FWHM tilt/tip removed 0.5” 1.0” 1.5” 1.5” 1.5” 1.5” 풓ퟎ=75mm

Innovations Foresight Some time a short exposure image can be close to the diffraction limit. The probability P for a rms phase error at, or below, ~l/6 (one radian) is (Fried 1977): 푫 ퟐ −ퟎ.ퟏퟓퟓퟕ 푷 ≅ ퟓ. ퟔ 풆 풓ퟎ 푫 >3.5, t<<흉ퟎ, inside isoplanatic patch 풓ퟎ Example: D=279mm (11”) 풓ퟎ=50mm @550nm 푷 ≅0.04 (~1/25 frame) 풓ퟎ=84mm @850nm 푷>0.7 (~18/25 frame) (c) Innovations Foresight 2016 - Dr. Gaston Baudat 20 Lucky imaging and PSF shape

Innovations Foresight Set of 10 short exposure frames (tilt/tip removed):

Innovations Foresight The seeing is wavelength dependent. It is worse at shorter wavelengths and better at longer wavelengths.

ퟔ ퟓ ퟏ.ퟐ . 풓ퟎ 흀 ∝ 흀 = 흀 coherence length (Fried) ퟔ ퟓ ퟏ.ퟐ . 흉ퟎ 흀 ∝ 흀 = 흀 coherence time ퟔ ퟓ ퟏ.ퟐ . 휽ퟎ 흀 ∝ 흀 = 흀 isoplanatic angle (patch) . 흓 흀 ퟐ ∝ 흀−ퟓ ퟑ~ 흀−ퟏ.ퟕ wave front phase variance . FWHM ∝ 흀−ퟏ ퟓ = 흀−ퟎ.ퟐ Full Width at Half Maximum −ퟔ ퟓ −ퟏ.ퟐ . 푫 풓ퟎ 흀 ∝ 흀 = 흀 DL performance threshold . 푷 흀 ∝ 흀ퟏퟐ ퟓ = 흀ퟐ.ퟒ Lucky imaging rate of success

(c) Innovations Foresight 2016 - Dr. Gaston Baudat 22 Limitations of the standard Kolmgorov’s model Innovations Foresight

. The Kolmogorov’s model is only valid for 퐿0 → ∞ and 푙0 → 0.

. When the outer scale turbulence (OST) 퐿0 is finite the seeing may exhibit a significant departure from the model.

. When the low level turbulence dominates 퐿0 is small, around 10m, possibly down to about 1m.

. The resulting effect is the greatest for the first two (tilt/tip) Zernike terms.

. The seeing usually exhibits a stronger dependency with l compared to the standard Kolmogorov’s model.

Innovations Foresight . The Von Karman’s (VK) model explicitly accounts for a finite OST 퐿0 value (Theodore Van Karman, 1948). . Worldwide extensive measurements correlate with VK model.

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The VK model depends on a new parameter 퐿0 푟0

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• Finite OST values improve the guide star PSF (less tilt/tip). • Longer wavelengths benefice the most.

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• The goal of auto-guiding is to correct for tracking errors.

• Those errors are independent of the seeing effect and are fully correlated across the FOV (unlike seeing).

• Long term errors: PE, King’ rate, flexure, polar alignment, …, require a slow correction rate (10s to minutes).

• Short term errors: Mount mechanics, wind burst, vibrations, …, require a fast correction rate (few seconds).

• Short term corrections are more prone to seeing effects.

Innovations Foresight • The acceptable tracking error is a function of the seeing.

Rule of thumb: RMS tracking error < 1/4 FWHMseeing

Seeing Excellent Good Average Bad 0.5” 1.0” 2.0 ” 3.0”

흈풕풓풂풄풌풊풏품 rms error 0.13” 0.25” 0.50” 0.75”

• The rms auto-guiding error 흈(푡푐)품풖풊풅풊풏품, before any correction, for a given exposure time 푡푐 is given by:

ퟐ ퟐ 흈(푡푐)품풖풊풅풊풏품 = 흈(푡푐)풕풓풂풄풌풊풏품 + 흈(푡푐)풔풆풆풊풏품

Innovations Foresight

• Let’s assume that 푡푐 has been chosen to cancel the tracking errors (mount, setup).

• Question: How does the seeing impact auto-guiding errors?

• Answer: It depends whether we are inside or outside the isoplanatic patch.

• Total error after correction inside the isoplanatic patch:

흈(푡푐)품풖풊풅풊풏품 ≅ ퟏ − 푲 흈풔풆풆풊풏품 ퟎ ≤ 푲 ≤ ퟏ

푲 → ퟎ seeing limited image: 풕풄 ≫ 흉ퟎ 푲 → ퟏ ~ diffraction limited image: 풕풄 ≪ 흉ퟎ (tilt/tip correction only)

Innovations Foresight • Total error after correction outside the isoplanatic patch:

ퟐ 흈(푡푐)품풖풊풅풊풏품 ≅ ퟏ + 푲 흈풔풆풆풊풏품 ퟎ ≤ 푲 ≤ ퟏ

푲 → ퟎ seeing limited image: 풕풄 ≫ 흉ퟎ 푲 → ퟏ ~1.4 x seeing limited image: 풕풄 ≪ 흉ퟎ

• K should be kept as close as possible to zero.

풕풄 ≫ 흉ퟎ, guiding at longer wavelengths, better mount

Innovations Foresight Mario Motta’s relay telescope 32” (0.8m) @ f/6 M86 images with STL11000 + AO-L

M83 with full spectrum guiding (OAG) M83 with NIR guiding (ONAG)

Innovations Foresight

 Do not rush auto-guiding, unless you have too (>5~10s).  Use a long guiding exposure time whenever possible.  Minimize errors with a good polar alignment.  Use a model for minimizing the drifts and the guiding rate.  Consider guiding at longer wavelengths (such as NIR).  Do not minimize the guiding error at any cost.  Use small aggressiveness at fast guiding rates.  Consider using red filters for image detail retrieval.  Image high above the horizon to minimize seeing.  Know your local seeing and conditions.  Watch forecasts (cleardarksky.com) to mange expectations.

Innovations Foresight, LLC

Clear skies!