 # Direct Imaging of Exoplanets

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Direct Imaging of Exoplanets
Techniques Adaptive Optics Coronographs Differential Imaging Nulling Interferometers External Occulters Results

Challenge 1: Large ratio between star and planet flux (Star/Planet)
Reflected light from Jupiter ≈ 10–9

Direct Detections need contrast ratios of 10–9 to 10–10
Challenge 2: Close proximity of planet to host star Direct Detections need contrast ratios of 10–9 to 10–10 At separations of 0.01 to 1 arcseconds Earth : ~10–10 separation = 0.1 arcseconds for a star at 10 parsecs Jupiter: ~10–9 separation = 0.5 arcseconds for a star at 10 parsecs 1 AU = 1 arcsec separation at 1 parsec

Younger planets are hotter and they emit more radiated light
Younger planets are hotter and they emit more radiated light. These are easier to detect.

= A0 [cos(kx –wt) + i sin(kx –wt)]
Background: Electromagnetic Waves F(x,t) = A0 ei(kx – wt) = A0 [cos(kx –wt) + i sin(kx –wt)] where kx is the dot product = |k| |x| cos q where q is the angle in 3 dimenisions kx  kr r = (x,y,z)

Background: Electromagnetic Waves
|k| = 2p/l = wave number kr –wt is the phase kr is the spatial part wt is the time varying part

Background: Fourier Transforms
Cosines and sines represent a set of orthogonal functions (basis set). Meaning: Every continuous function can be represented by a sum of trigonometric terms

Background: Fourier Transforms
Eg. Consider the Step function: y(x) = kx over the interval x = (0, L) y x L y = k x y(x) =  Bn sin (npx/L) n=1 Expressed as an infinite sine series:

I. Background: Fourier Transforms
To determine Bn use the fact that sine functions are orthogonal Suppose we want the amplitude of the sine term associated with “frequency” n1 : L n1px L n1px npx  y(x) sin( ) dx =  Bn  sin( )sin( ) dx L L L n=1 Use: sin q sin f = 1/2[cos(q–f) – cos(q+f)]

I. Background: Fourier Transforms
If you do the above intergral and insert the limits you will find that all terms are zero except for n = n1 and we get: Bn = 2/L  y(x) sin( ) dx npx L y(x) = 2kL/p {sin (px/L –1/2 sin(2px/L) + 1/3 sin(3px/L) ...} Bn = (–1) n+1 (2kL/np)

I. Background: Fourier Transforms

Background: Fourier Transforms
The continous form of the Fourier transform: F(s) =  f(x) e–ixs dx f(x) = 1/2p  F(s) eixs ds eixs = cos(xs) + i sin (xs)

I. Background: Fourier Transforms
Compare this to the sine series f(x) = 1/2p  F(s) eixs ds y(x) =  Bn sin (npx/L) The Fourier transform of a function (frequency spectrum) tells you the amplitude (contribution) of each sin (cos) function at the frequency that is in the function under consideration. The square of the Fourier transform is the power spectra and is related to the intensity when dealing with light.

Background: Fourier Transforms
In interferometry and imaging it is useful to think of normal space (x,y) and Fourier space (u,v) where u,v are frequencies Two important features of Fourier transforms: The “spatial coordinate” x maps into a “frequency” coordinate 1/x (= s) Thus small changes in x map into large changes in s. A function that is narrow in x is wide in s

Background: Fourier Transforms
x n x

Background: Fourier Transforms
x n x n

Background: Fourier Transforms
sinc x n J1(2px) 2x x Diffraction patterns from the interference of electromagnetic waves are just Fourier transforms!

Background: Fourier Transforms
In Fourier space the convolution (smoothing of a function) is just the product of the two transforms: Normal Space Fourier Space f*g F  G x Suppose you wanted to smooth your data by n points. You can either: Move your box to a place in your data, average all the points in that box for value 1, then slide the box to point two, average all points in box and continue. Compute FT of data, the FT of box function, multiply the two and inverse Fourier transform

Adaptive Optics : An important component for any coronagraph instrument
Atmospheric turbulence distorts stellar images making them much larger than point sources. This seeing image makes it impossible to detect nearby faint companions.

Adaptive Optics The scientific and engineering discipline whereby the performance of an optical signal is improved by using information about the environment through which it passes AO Deals with the control of light in a real time closed loop and is a subset of active optics. Adaptive Optics: Systems operating below 1/10 Hz Active Optics: Systems operating above 1/10 Hz

Example of an Adaptive Optics System: The Eye-Brain
The brain interprets an image, determines its correction, and applies the correction either voluntarily of involuntarily Lens compression: Focus corrected mode Tracking an Object: Tilt mode optics system Iris opening and closing to intensity levels: Intensity control mode Eyes squinting: An aperture stop, spatial filter, and phase controlling mechanism

The Ideal Telescope where:
This is the Fourier transform of the telescope aperture where:  P(a) is the light intensity in the focal plane, as a function of angular coordinates a   ; l is the wavelength of light; D is the diameter of the telescope aperture; J1 is the so-called Bessel function. The first dark ring is at an angular distance Dl of from the center. This is often taken as a measure of resolution (diffraction limit) in an ideal telescope. Dl = 1.22 l/D = l/D (arcsecs)

Diffraction Limit Telescope 5500 Å 2 mm 10 mm Seeing TLS 2m 0.06“ 0.2“ 1.0“ 2“ 0.017“ 0.06“ 0.3“ 0.2“ VLT 8m Keck 10m 0.014“ 0.05“ 0.25“ 0.2“ 0.1“ 0.2“ ELT 42m 0.003“ 0.01“ Even at the best sites AO is needed to improve image quality and reach the diffraction limit of the telescope. This is easier to do in the infrared

Atmospheric Turbulence
A Turbulent atmosphere is characterized by eddy (cells) that decay from larger to smaller elements. The largest elements define the upper scale turbulence Lu which is the scale at which the original turbulence is generated. The lower scale of turbulence Ll is the size below which viscous effects are important and the energy is dissipated into heat. Lu: 10–100 m Ll: mm–cm (can be ignored)

Atmospheric Turbulence
Original wavefront Turbulence causes temperature fluctuations Temperature fluctuations cause refractive index variations Turbulent eddies are like lenses Plane wavefronts are wrinkled and star images are blurred Distorted wavefront

Atmospheric Turbulence
ro: the coherence length or „Fried parameter“ is r0 = l6/5 cos3/5z(∫Cn² dh)–3/5 r0median = (l/5.5×10–7) cos3/5z(∫Cn² dh)–3/5 ro is the maximum diameter of a collector before atmospheric distortions limit performance (l is in meters and z is the zenith distance) r0 is cm at zero zenith distance at good sites To compensate adequately the wavefront the AO should have at least D/r0 elements

Definitions t0 ≈ r0/Vwind
to: the timescale over which changes in the atmospheric turbulence becomes important. This is approximately r0 divided by the wind velocity. t0 ≈ r0/Vwind For r0 = 10 cm and Vwind = 5 m/s, t0 = 20 milliseconds t0 tells you the time scale for AO corrections

Definitions Strehl ratio (SR): This is the ratio of the peak intensity observed at the detector of the telescope compared to the peak intensity of the telescope working at the diffraction limit. If D is the residual amplitude of phase variations then D = 1 – SR The Strehl ratio is a figure of merit as to how well your AO system is working. SR = 1 means you are at the diffraction limit. Good AO systems can get SR as high as 0.8. SR= is more typical.

Definitions Isoplanetic Angle: Maximum angular separation (q0) between two wavefronts that have the same wavefront errors. Two wavefronts separated by less than q0 should have good adaptive optics compensation q0 ≈ 0.6 r0/L Where L is the propagation distance. q0 is typically about 20 arcseconds.

If you are observing an object here
You do not want to correct using a reference star in this direction

Basic Components for an AO System
You need to have a mathematical model representation of the wavefront You need to measure the incoming wavefront with a point source (real or artifical). You need to correct the wavefront using a deformable mirror

Describing the Wavefronts
An ensemble of rays have a certain optical path length (OPL): OPL = length × refractive index A wavefront defines a surface of constant OPL. Light rays and wavefronts are orthogonal to each other. A wavefront is also called a phasefront since it is also a surface of constant phase. Optical imaging system:

Describing the Wavefronts
The aberrated wavefront is compared to an ideal spherical wavefront called a the reference wavefront. The optical path difference (OPD) is measured between the spherical reference surface (SRS) and aberated wavefront (AWF) The OPD function can be described by a polynomical where each term describes a specific aberation and how much it is present.

Describing the Wavefronts
Zernike Polynomials: Z= SKn,m,1rn cosmq + Kn,m,2rn sinm q

Measuring the Wavefront
A wavefront sensor is used to measure the aberration function W(x,y) Types of Wavefront Sensors: Foucault Knife Edge Sensor (Babcock 1953) Shearing Interferometer Shack-Hartmann Wavefront Sensor Curvature Wavefront Sensor

Shack-Hartmann Wavefront Sensor

Shack-Hartmann Wavefront Sensor
Lenslet array Image Pattern reference Focal Plane detector af disturbed a f

Shack-Hartmann Wavefront Sensor

Correcting the Wavefront Distortion
Adaptive Optical Components: Segmented mirrors Corrects the wavefront tilt by an array of mirrors. Currently up to 512 segements are available, but elements appear feasible. 2. Continuous faceplate mirrors Uses pistons or actuators to distort a thin mirror (liquid mirror)

Reference Stars You need a reference point source (star) for the wavefront measurement. The reference star must be within the isoplanatic angle, of about arcseconds If there is no bright (mag ~ 14-15) nearby star then you must use an artificial star or „laser guide star“. All laser guide AO systems use a sodium laser tuned to Na 5890 Å pointed to the 11.5 km thick layer of enhanced sodium at an altitude of 90 km. Much of this research was done by the U.S. Air Force and was declassified in the early 1990s.

Imaging Sun, planets, stellar envelopes and dusty disks, young stellar objects, etc. Can get 1/20 arcsecond resolution in the K band, 1/100 in the visible (eventually)

2. Resolution of complex configurations Globular clusters, the galactic center, stars in the spiral arms of other galaxies

3. Detection of faint point sources Going from seeing to diffraction limited observations improves the contrast of sources by SR D2/r02. One will see many more Quasars and other unknown objects

4. Faint companions The seeing disk will normally destroy the image of faint companion. Is needed to detect substellar companions (e.g. GQ Lupi)

5. Coronography With a smaller image you can better block the light. Needed for planet detection

Coronagraphs

Basic Coronagraph

Dl = D/l = number of wavelengths across the telescope aperture

b) The telescope optics then forms the incoming wave into an image. The electric field in the image plane is the Fourier transform of the electric field in the aperture plane – a sinc function (in 2 dimensions this is of course the Bessel function) Eb ∝ sinc(Dl, q) Normally this is where we place the detector

c) d) In the image plane the star is occulted by an image stop. This stop has a shape function w(Dlq/s). It has unity where the stop is opaque and zero where the stop is absent. If w(q) has width of order unity, the stop will be of order s resolution elements. The transfer function in the image planet is 1 – w(Dlq/s). W(q) = exp(–q2/2)

e) The occulted image is then relayed to a detector through a second pupil plane e) This is the convolution of the step function of the original pupil with a Gaussian

e) f) g) One then places a Lyot stop in the pupil plane

At h) the detector observes the Fourier transform of the second pupil

Increasing the Strehl ratio makes a coronograph more efficient
SR=0.30 0.53 0.69 0.82 0.9

The Solar Corona with a Coronagraph

Types of Masks: Simple opaque disk 4 Quadrant Phase mask : Shifts the phase in 4 quadrants to create destructive interference to block the light Vortex Phase Coronagraph: rotates the angle of polarization which has the same effect as ramping up the phase shift

A 4-quadrant phase mask The Airy Disk Phase shifted p phase shift
The exit pupil (FT of c) The exit pupil through the Lyot stop The image (FT of e) )

External Occulter 50000 km At a distance of km the starshade subtends the same angle as the star q

A complex starshape is needed to suppress diffraction

A coronagraph An external occulter!

Subtracting the Point Spread Function (PSF)
To detect close companions one has to subtract the PSF of the central star (even with coronagraphs) which is complicated by atmospheric speckles. One solution: Differential Imaging

Spectral Differential Imaging (SDI)
1.58 mm 1.68 mm 1.625 mm Split the image with a beam splitter. In one beam place a filter where the planet is faint (Methane) and in the other beam a filter where it is bright (continuum). The atmospheric speckles and PSF of the star (with no methane) should be the same in both images. By taking the difference one gets a very good subtraction of the PSF

Planet Bright Since the star has no methane, the PSF in all filters will look (almost) the same. Planet Faint

Nulling Interferometry
Principles of Long Baseline Stellar Interferometry, ed. Peter Lawson

A Basic Interferometer
s • B = B cos q q A2 x2 A1 x1 Beam Combiner Delay Line 2 d2 Delay Line 1 d1 1) Idealized 2-telescope interferometer

Interferometry Basics
Consider two wavefronts hitting our telescopes with apertures A1 and A2 separated by a baseline B. The waves are: f1 ~ eikx1 eiwt = eik ŝ ·x1 eiwt f2 ~ eikx2 eiwt = eik ŝ ·x2 eiwt ŝ = S/|S|

Interferometry Basics
But if we interpret kx as a phase then k ŝx2 = k ŝ x1 + k ŝ ·B in other words, the difference in phase is just caused by the difference in path length introduced by the baseline.

Interferometry Basics
We can absorb the k ŝ ·x1 term in normalization. We can also introduce the effects of delay lines: f1 ~ eikd1 eiwt f2 ~ eikd2 eikŝ B eiwt fTotal = f1 + f2 ~ eiwt (eikd1 + eikd2 e–ik ŝ B )

Interferometry Basics
The power, P, is: P  fTotal f*Total = eiwt e+iwt(eikd1 + eikd2 e–ik ŝ B ) (eikd1 + eikd2 e-ik ŝ B ) = cross terms Cross terms are what the important part

Interferometry Basics
= eik(d2–d1) eik ŝ B + eik(d1–d2) eik ŝ B = eik(d2–d1 + ŝ B) + eik(d2–d1 + ŝ B) = 2 cos k (sB+d1–d2) P = 2(1 + cos k (sB+d1–d2)) = 2A(1+cos k ·D) A = telescope aperture D  sB+d1–d2 s can be interpreted as an angle on the sky with dimensions of radians

Interferometry Basics
Adjacent fringe crests projected on the sky are separated by an angle given by: Ds = l/B

Interferometry Basics: The Visibility Function
Michelson Visibility: V = Imax –Imin Imax +Imin Visibility is measured by changing the path length and recording minimum and maximum values

Interferometry Basics: Cittert-Zernike theorem
Ds  ŝo b a, b are angles in „x-y“ directions of the source. Ds = (a,b,0) in the coordinate system where ŝo =(0,0,1) a

Interferometry Basics: Cittert-Zernike theorem
The visibility : V(k, B) =  da db A(a,b) F(a,b) e 2pi(au+bv) Cittert-Zernike theorem: The interferometer response is related to the Fourier transform of the brightness distribution under certain assumtions (source incoherence, small-field approximation). In other words an interferometer is a device that measures the Fourier transform of the brightness distribution.

Interferometry Basics: Cittert-Zernike theorem
Procedure: Collect as many visibility curves as possible Compute the inverse Fourier transform F(a,b) = ∫ (du dv V(u,v) e 2pi(au + bv))/A(a,b)

Interferometric Basics: Aperture Synthesis
Aperture Space Fourier Space V N E B U Spatial Resolution : l/B Can resolve all angular scales up to q > l/D, i.e. the diffraction limit Frequency Resolution : B/l Can sample all frequencies out to D/l One baseline measurement maps into a single location in the (u,v)-plane (i.e. it is only one frequency measurement of the Fourier transform)

Nulling Interferometers
Adjusts the optical path length so that the wavefronts from both telescope destructively intefere at the position of the star Technological challenges have prevented nulling interferometry from being a viable imaging method…for now

Darwin/Terrestrial Path Finder would have used Nulling Interferometry
Earth Venus Mars Ground-based European Nulling Interferometer Experiment will test nulling interferometry on the VLTI

Results!

Coronography of Debris Disks
Structure in the disks give hints to the presence of sub-stellar companions

Coronographic Detection of a Brown Dwarf

Cs

Spectral Features show Methane and Water

Another brown dwarf detected with the NACO adaptive optics system on the VLT

The Planet Candidate around GQ Lupi
But there is large uncertainty in the surface gravity and mass can be as low as 4 and as high as 155 MJup.

Estimated mass from evolutionary tracks: 13-14 MJup

Coronographic observations with HST

a ~ 115 AU P ~ 870 years Mass < 3 MJup, any more and the gravitation of the planet would disrupt the dust ring

Photometry of Fomalhaut b
Planet model with T = 400 K and R = 1.2 RJup. Reflected light from circumplanetary disk with R = 20 RJup Detection of the planet in the optical may be due to a disk around the planet. Possible since the star is only 30 Million years old.

Imaged using Angular Differential Imaging (i. e
Imaged using Angular Differential Imaging (i.e. Spectral Differential Imaging)

The Planets of HR 8799 on Evolutionary Tracks

Image of the planetary system around HR 8799 taken with a „Vortex Phase“ coronagraph at the 5m Palomar Telescope

The Planet around b Pic Mass ~ 8 MJup

2003 2009

Imaging Planets Planet Mass (MJ) Period (yrs) a (AU) e Sp.T. Mass Star
2M1207b 4 - 46 M8 V 0.025  AB Pic 13.5 275 K2 V GQ Lupi 4-21 103 K7 V 0.7 b Pic 8 12 ~5 A6 V 1.8 HR 8799 b 10 465 68 F2 V1 HR 8799 c 190 38 ´- HR 8799 d2 7 24 Fomalhaut b < 3 88 115 A3 V 2.06 1SIMBAD lists this as an A5 V star, but it is a g Dor variable which have spectral types F0-F2. Tautenburg spectra confirm that it is F-type 2A fourth planet around HR 8799 was reported at the 2011 meeting of the American Astronomical Society