Simulation of moving exoplanets detection using the VLT instrument SPHERE/IRDIS I. Smitha , M. Carbilleta , A. Ferraria , D. Mouilletb , A. Boccalettic , K. Dohlend a UMR
6525 H. Fizeau, Universit´e de Nice Sophia Antipolis/CNRS/Obesrvatoire de la Cˆote d’Azur, Campus Valrose, F-06108 Nice cedex 2
b UMR
5571 LAOG, Observatoire des Sciences de l’Univers de Grenoble, Universit´e Joseph Fourier/CNRS, 414 Rue de la Piscine, BP 53, F-38041 Grenoble cedex 9
c UMR
8109 LESIA, Observatoire de Meudon/CNRS, 5 place Jules Janssen, F-92195 Meudon
d UMR
6110 LAM, Observatoire Astronomique de Marseille Provence, Universit´e de Provence/CNRS, F-13388 Marseille cedex 13 ABSTRACT
Exoplanet direct imaging involves very low signal-to-noise ratio data that need to be carefully acquired and processed. This paper deals with data processing for the VLT planet finder SPHERE, that will include extreme adaptive optics and high-contrast coronagraphy, and where field rotation will occur. First, we propose estimators of the intensity, the intensity estimate uncertainty, and the initial position of a potential exoplanet. Because of the very large amount of data to process, they are derived from a simple gaussian data model relying on the time-stationarity of the background, where the so-called background is everything but the exoplanet. Analytical properties of the estimators are given, under the gaussian data model and under a more sophisticated data model. Then, in order to relate the detection procedure to a probabillity of false alarm, the detection consists simply in thresholding the intensity estimate at a given initial position. Finally, this detection-estimation algorithm is applied on a dataset simulated using the CAOS-based Software Package SPHERE, including time evolution of the atmospheric, pre-, and post-coronagraphic quasi-static aberrations. As a preliminary result, the detectionestimation algorithm proves to be totally satisfactory for a 8 × 10−5 intensity ratio for exoplanets located from 0. 2 to 2 . The stationarity assumption is discussed. Keywords: Exoplanets - Detection - Coronagraphy - Adaptive Optics - SPHERE - Simulation
1. INTRODUCTION The Very Large Telescope (VLT) planet finder SPHERE (Spectro-Polarimetric High Contrast Exoplanet REsearch), that will include extreme adaptive optics (AO) and high-contrast coronagraphy, is under development. The high contrast between the star and the exoplanet (typically 106 in near-infrared) and their small angular separation (∼ 0. 5) make the exoplanet detection very difficult. The detection of the exoplanet for a given Probability of False Alarm (PFA) and the estimation of the exoplanet’s intensity (and uncertainty) and initial position require dedicated techniques that have to be tested on realistic simulated data. This study focuses on the IRDIS (Infra-Red Dual-beam Imaging and Spectroscopy) facility and uses only one wavelengths band, but this framework can be quite easily extended to two bands under rough assumptions. This paper presents in Sec. 2 a first set of end-to-end realistic data containing quasi-static aberrations, simulated with the CAOS-based Software Package SPHERE 3.0. Sec. 3 is devoted to the detection-estimation technique, including the derivations of the relevant estimators and their analytical properties. Sec. 4 finally shows the current practical performance of this algorithm applied to the data presented in Sec. 2. The differences of the performances when the data are stationary (meaning that the star background intensity reaching the detector is constant with time) and non-stationary (real case) is stressed. Further author information: E-mail:
[email protected], Telephone: +33 (0) 492 076 390
Adaptive Optics Systems, edited by Norbert Hubin, Claire E. Max, Peter L. Wizinowich, Proc. of SPIE Vol. 7015, 70156F, (2008) 0277-786X/08/$18 · doi: 10.1117/12.789413 Proc. of SPIE Vol. 7015 70156F-1 2008 SPIE Digital Library -- Subscriber Archive Copy
2. GLOBAL INSTRUMENT SIMULATION 2.1 Test-Case Definition The physical model taken into account for the presented simulations, made within the framework of the “Data Reduction & Handling” (DRH) working group of the SPHERE consortium, aims at being as close as possible to the realistic behavior of the astrophysical and instrumental signal. In particular, and unlike the whole series of system simulations performed for the various sub-system studies up to now, here the goal is to have a reasonable test case for DRH purpose, but which anyway takes into account the temporal evolution of the atmospheric and instrumental effects. In order to benefit from the best possible AO correction possible with SAXO, a pupil-stabilized observation mode is chosen. Moreover, the test case concerns images obtained with IRDIS,1 in its DBI (dual-band imaging) observing mode. These two assumptions basically define the survey observing mode of SPHERE, which makes use of one of its near-infrared coronagraphs. In this paper we decided to consider the apodized Lyot coronagraph option, the alternative would be the four-quadrant coronagraph for this main observing mode. The test-case data described here include typical turbulent post-SAXO residuals, various independent realizations of these residuals for each long-exposure post-coronagraphic image of the whole set (in order to mimic a reasonable atmospheric speckle averaging), variability of the turbulence conditions (seeing, distribution and wind velocities and directions of the various turbulent layers) and subsequent AO correction quality from one long-exposure image to the other, instrument aberration variations directly associated with airmass (ADC (atmospheric dispersion corrector) residuals, optical defects), variations associated with long-timescale drifts (pointing stability, on-coronagraph focus stability, etc.), and detector effects and noises. The typical timescales involved in this physical modeling of the data are of different orders. For example, while typical residuals coherent times is ∼10 ms, the turbulence conditions stability is measured in minutes. For what concerns field-rotation rate, the time corresponding to a field rotation of λ/D (the expected full width at half-maximum of the SAXO-corrected point-spread function for a bright star, with λ the observing wavelength and D the diameter of the telescope) ranges from a few minutes to one hour, depending on the pointing direction and the separation ρ between the (central) star and its searched planet companion. While the typically detector integration time (DIT) is 10 s, the total observation time T is typically hours, distributed as symmetrically as possible around the meridian to maximize the detectability of a potential exoplanet. Concerning instrumental aberrations, the optical rotation rate associated with airmass is indeed function of the pointing direction and the correlation time related to the high-frequency part of the corresponding wavefront error is typically minutes. The chosen approach sticks with the numerical tool capabilities used for the various sub-system studies performed up to now, the Software Package SPHERE, in its appositely updated version 3.0.2 In fact, this approach separates in the one hand the simulation of a noiseless (i.e. without detector noises) post-coronagraphic stellar pattern, and in the other hand the detector noises, star-planet system properties, and field-rotation effect. This corresponds respectively to the “diffraction part” of the code, organized in modules which can be used within the CAOS Application Builder (the global interface of the CAOS “system”3 ), and the “photometric part” of the code, rather proposed as a classical library of IDL routines. The diffractive part of the code being very time-consuming, a one-shot (one long exposure) realization of it may be used with different noise, companion and field-rotation characteristics.
2.2 Test-Case Parameters The test-case parameters described in this subsection starts from the choice of coordinates for the typical starplanet system to be observed with SPHERE: a declination of -45 deg and an hour angle (HA) ranging from -2 hr to +2 hr. The time exposure chosen for each long-exposure post-coronagraphic image is 100 s (10 DITs), hence the total number of long-exposure images corresponding to T =4 hr of observation is N =144. Moreover, 100 independent realizations of SAXO-corrected wavefronts are considered for computing each long-exposure image. For each of these long-exposure image, optical parameters evolve – wavefront error, chromatic shifts associated with ADC, slow achromatic drifts, etc. The actual and main input values for the stellar halo simulation are reported in Tab. 1, while Fig. 1 shows the evolution of turbulence (in terms of seeing and wind speed) in the one hand, and the evolution of the overall pre-coronagraph achromatic wavefront error in the other hand.
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Table 1. Actual and main input values for the stellar halo simulation. Star and exoplanet system star intensity (in space), intensity ratio in “H2” separation angles declination, init. hour angle, zenith angle (range) Atmosphere+VLT wind velocity of the 3 turbulent layers (range) seeing at 500 nm (range) wave-front outer-scale L0 diameter D obscuration ratio zero point SAXO system sensor type guide star magnitude sensing central wavelength sensing integration time loop delay, loop gain RON, dark current spatial filter efficiency range Near-IR coronagraph wavelength band coronagraph type mask and Lyot stop diameters Aberrations instrumental jitter VLT M1, M2, M3 global pre-coronagraph achromatic wavefront error (but tilt) range pre-coronagraph chromatic defocus ADC residuals range on-coronagraph achromatic tilt range post-coronagraph chromatic defocus post-coronagraph chromatic wavefront error IRDIS imaging device RON, flat-field noise sampling rate DIT long-exposure time ∆t total observation time T field rotation velocity in pupil-stabilized mode (range) Global (VLT + SPHERE/IRDIS) transmission
2.80×106 γ/m2 /s, 8.3×10−5 0. 2, 0. 5, 1 and 2 -45 deg , -2 hr, 20–32 deg
2.1×1010
7–25 m/s 0. 5–1. 1 25 m 8m 0.14 e− /m2 /s
40×40 spatially filtered Shack-Hartmann mV =8 0.65 µm 0.83 ms 1 ms, ∼0.47 0.5 e− , 2 e− /s 0.3–0.9 “H2”: (1.56645 - 1.63922) µm apodized Lyot 4λ/D and 0.96 D 3 mas 11.9 nm, 11.9 nm, 16.6 nm 29.2–30.1 nm 0.8 nm 11.5–12.0 nm 0.07–0.28 nm 1.8 nm 7.5 nm
rms rms rms rms rms rms rms rms
10 e− rms, 10−3 rms Shannon at 0.95 µm 10 s 100 s 4 hr 0.005–0.010 deg/s 0.09
//
Figure 1. Left: Seeing evolution with observation time. Center: time evolution of the wind speed corresponding to the turbulent atmospheric layers. Right: time evolution of the global pre-coronagraph achromatic wavefront error (but tilt).
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Figure 2. Typical dataset made of long exposure images simulated using the CAOS-based Software Package SPHERE with parameters given in section 2.2. From left to right, exposures k = 1, k = 50, k = 100 and k = 144 (at the power of 0.2). Pixel size = 0. 0122, field-of-view shown ≈ 4. 5× 4. 5.
3. DETECTION-ESTIMATION ALGORITHM The inputs of the algorithm are the set of data (N images, as illustrated in Fig. 2), some estimates of the instantaneous off-axis post-coronagraphic point-spread function (PSF) and an expected PFA. The output is first the answer to the question: “Given a PFA, do we detect an exoplanet in the dataset?” and if the answer is yes, other outputs are some estimates of the initial position of the exoplanet, its intensity and the uncertainty of this intensity estimate. The first goal of the detection-estimation algorithm consists in defining a detection procedure, related as precisely as possible to a PFA and with a satisfactory probability of detection. For the detected exoplanets, the point is then to estimate their intensity and position. Actually, due to the large amount of data, both the detector and the estimators have to be quite simple functions of the data. This also helps reducing the analytical complexity of deriving an accurate PFA associated to the detector but may not be enough. At the end, the detection step consists here merely in thresholding the intensity estimate at a given initial position. The choice of the estimator –among estimators adapted to a so large amount of data!– has to be made according to its statistical properties: small bias (expected error), small uncertainty, fast decreasing uncertainty with increasing time integration, etc. Because of the complexity and the time-length of the simulations, Monte Carlo simulations cannot be used to estimate its statistical properties. An explicit modeling of the dataset is needed for such a purpose and will also give systematic methods to derive the formulation of the estimators themselves.
3.1 Statistical data models From a qualitative description of the expected data, this section aims at expliciting some underlying assumptions defining a mathematical modeling of the data. The assumptions have to be strong enough to reduce the number of free parameters, but relevant enough to describe precisely the reality or at least make precise assessments about the searched signal (the exoplanet). Note that the present models allow the presence of only one exoplanet. The main underlying assumptions made in these models are the field-rotation effect that discriminates the exoplanets from the background4 and the stationarity of the (unknown) background. In a general framework, the dataset is mathematically modeled by a set of N successive images. The k th image is reordered column by column into a vector xk of size M , where M is the total number of pixels. The th ( = 1, .., M ) component of this vector is xk (). 3.1.1 Simple data model, used to derive the estimators and detector To derive practical estimators of the intensity and the position of the planet, a very simple modeling of xk is: xk = d0 + α0 pk (r0 ) + k
(1)
where d0 is the unknown background, α0 is the unknown exoplanet intensity, pk (r0 ) is the profile of the exoplanet for the exposure k when its position on the first image is r0 , and k is a gaussian noise, independent for k = k ,
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Figure 3. Comparison of an instantaneous exoplanet profile and a 100 s-exposure field-rotated image, for different initial positions and for the first exposure (where HA=-2 hr so that the field rotation has the lowest speed within the total observation time T ). Left: normalized “peas” (instantaneous off-axis post-coronagraphic PSF). Middle: normalized “beans” (rotated peas integrated over a 100 s exposure). Right: difference between normalized peas and beans. Note that the ratio of the maxima of the difference image and of the absolute profile(s) images is close to 0.9 ≈ 25% (see the 3.5 colormap). This appears at a large separation (around 2 ).
with k ∼ N (0, σ02 IM ), where IM is the identity matrix of size M . It is important to note that the background d0 is assumed to be stationary but not necessarily spatially homogeneous. Therefore, the model could quite accurately describe the data even if there are strong intensity variations in a given exposure. What matters is that the spatial intensity distribution remains static with time. Concerning the source profile, due to the field rotation effect that appears with alt-azimuthal mounts, pk (r0 ) is just the axi-symmetric instantaneous profile, nicknamed “pea” in the following, spread out deterministically in a circular arc, or “bean”, as displayed in Fig. 3. Note that the total flux only varies here by a factor of 3/1000 for peas located at 0. 2 and 2 from the center. The detection of the exoplanet consists in choosing among the two hypotheses H0 : α0 = 0 H1 : α0 > 0
(there is no exoplanet) (there is an exoplanet)
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3.1.2 More precise data model, used to derive statistical properties of the estimators As studied by J. Goodman5 and extended in 6 a more realistic mathematical model relies on the description of the complex amplitude on the focal plane: in a “speckle regime” the complex amplitude physically adds up and is therefore subject to the Central Limit Theorem and the complex amplitude of the background at exposure k is complex gaussian with mean µ0 and covariance matrix Σ0 . Then, the intensity at pixel , that is just the square of the complex amplitude ψk (), is subject to Poisson noise and other additive noises. The mathematical expression of this data model ends in:
complex amplitude of the background: intensity of the background: total intensity: Poisson noise: final data:
ψ k ∼ Nc (µ0 , Σ0 ) uk () = |ψk ()|2 ik = uk + α0 pk (r0 ) nk ()|ik () ∼ P(ik ()) xk = nk + k
∀ = 1, .., M ∀ = 1, .., M k ∼ N (0, ρ20 IM )
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Using spatial correlation (Σ0 not diagonal) and a non-zero mean µ0 , the distribution of the corresponding intensity cannot be expressed with its usual probability density function form, only with its moment generating function7 . More generally, this model is too complex to derive practical estimators and detectors. However, some sophisticated computation can be done in order to derive reliable statistical properties of any linear estimator8 , properties that are summarized thereafter in a simple form, as well as the following calculation.
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Figure 4. Estimation of the planet intensity for a given initial position r. The inputs are: r, the post-coronagraphic PSFs (possibly depending on r), the declination, the initial HA and the data. The output is the estimated intensity α ˆ located at position r.
3.2 Exoplanet intensity estimation for an assumed initial position A common method to derive estimators consists in choosing the parameters that make the data the most likely (probable), in the frame of the statistical model chosen. Here, under the first model (1) the probability of the dataset {xk }k=1,..,N is simply gaussian with unknown parameters d0 , α0 , r0 and σ0 , where α0 and r0 are of special interest. First, the likelihood p({xk }k=1,..,N |d, α, r, σ) = L(d, α, r, σ) is maximized for any tested initial position r and any σ. The data are linear with respect to the parameters d0 and α. The Maximum Likelihood of d0 , both as functions of the variable r, are: (ML) estimates α ˆ of α0 and d N
pk (r)t xk − k pk (r)t k xk α ˆ (r) = cN (r) 2 2 pk (r) − pk (r) with cN (r) = N k k = pk (r) − pl (r)2 ≥ 0 N
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l
