Direct Imaging of Exoplanets

Direct Imaging of Exoplanets

I. Techniques

a) Adaptive Optics

b) Coronographs

c) Differential Imaging

II. Results

Reflected light from Jupiter ≈ 10–9

Challenge 1: Large ratio between star and planet flux (Star/Planet)

Direct Detections need contrast ratios of 10–9 to 10–10

At separations of 0.01 to 1 arcseconds

Challenge 2: Close proximity of planet to host star

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. These are easier to detect.

A Little Background: Fourier Transforms

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.

F(s) = f(x) e−2ixs dx

f(x) = F(s) e2ixs ds

Fourier Transforms

Two important features of Fourier transforms:

1) The “spatial or time coordinate” x maps into a “frequency” coordinate 1/x (= s or )

Thus small changes in x map into large changes in s. A function that is narrow in x is wide in s

A Pictoral Catalog of Fourier Transforms

Time/Space Domain Fourier/Frequency Domain

Comb of Shah function (sampling function)

x 1/x

Time Frequency (1/time)

Period = 1/frequency

0


Cosine is an even function: cos(–x) = cos(x)

Positive frequencies

Negative frequencies


Sine is an odd function: sin(–x) = –sin(x)


The Fourier Transform of a Gausssian is another Gaussian. If the Gaussian is wide (narrow) in the temporal/spatial domain, it is narrow(wide) in the Fourier/frequency domain. In the limit of an infinitely narrow Gaussian (-function) the Fourier transform is infinitely wide (constant)

w 1/w

e–x2e–s2


Note: these are the diffraction patterns of a slit, triangular and circular apertures

All functions are interchangeable. If it is a sinc function in time, it is a slit function in frequency space

Convolution

Fourier Transforms : Convolution

f(u)(x–u)du = f *

f(x):

(x):

Cross Correlation

(x-u)

a1

a2

g(x)a3

a2

a3

a1

CCF

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

Background: Fourier Transforms

x

Suppose you wanted to smooth your data by n points.

You can either:

1. 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.

2. Compute FT of data, the FT of box function, multiply the two and inverse Fourier transform

2) In Fourier space the convolution is just the product of the two transforms:

Normal Space Fourier Space f*g F G

Fourier Transforms

The second important features of Fourier transforms:

f g F * G

sinc sinc2

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.

2“1“0.5“0.25“Seeing →

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

where: • P() is the light intensity in the focal plane, as a function of angular coordinates ; • 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 D of from the center.This is often taken as a measure of resolution (diffraction limit) in an ideal telescope.

The Ideal Telescope

D= 1.22 /D = 251643 /D (arcsecs)

This is the Fourier transform of the telescope aperture

Telescope

Diffraction Limit

5500 Å 2 m 10 m

TLS 2m

VLT 8m

Keck 10m

ELT 42m

0.06“ 0.2“ 1.0“

0.017“

0.014“

0.003“

0.06“

0.05“

0.01“

0.3“

0.25“

0.1“

Seeing

2“

0.2“

0.2“

0.2“

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)

• Turbulence causes temperature fluctuations

• Temperature fluctuations cause refractive index variations

- Turbulent eddies are like lenses

• Plane wavefronts are wrinkled and star images are blurred


Original wavefront

Distorted wavefront

ro: the coherence length or „Fried parameter“ is

r0 = 0.185 6/5 cos3/5(∫Cn² dh)–3/5

ro is the maximum diameter of a collector before atmospheric distortions limit performance (is in meters and is the zenith distance)

r0 is 10-20 cm at zero zenith distance at good sites

To compensate adequately the wavefront the AO should have at least D/r0 elements


Definitions

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 is the residual amplitude of phase variations then

= 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=0.3-0.4 is more typical.

Definitions

Isoplanetic Angle: Maximum angular separation (0) between two wavefronts that have the same wavefront errors. Two wavefronts separated by less than 0 should have good adaptive optics compensation

0 ≈ 0.6 r0/L

Where L is the propagation distance. 0 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

1. You need to have a mathematical model representation of the wavefront

2. You need to measure the incoming wavefront with a point source (real or artifical).

3. 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:


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 polynomial where each term describes a specific aberation and how much it is present.


Zernike Polynomials:

Z= Kn,m,1n cosm + Kn,m,2n sinm

Measuring the Wavefront

A wavefront sensor is used to measure the aberration function W(x,y)

Types of Wavefront Sensors:

1. Foucault Knife Edge Sensor (Babcock 1953)

2. Shearing Interferometer

3. Shack-Hartmann Wavefront Sensor

4. Curvature Wavefront Sensor

Shack-Hartmann Wavefront Sensor


f

Image Pattern

reference

disturbed

f

Lenslet array

Focal Plane detector

Correcting the Wavefront Distortion

Adaptive Optical Components:

1. Segmented mirrors

Corrects the wavefront tilt by an array of mirrors. Currently up to 512 segements are available, but 10000 elements appear feasible.

2. Continuous faceplate mirrors

Uses pistons or actuators to distort a thin mirror (liquid mirror)

Unperturbed wavefront

Wavefront at telescope

corrected wavefront to camera

wavefront sensor

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 10-30 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.

1. 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)

Applications of Adaptive Optics


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/r0

2. 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

= D/= 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(D, )

Normally this is where we place the detector

In the image plane the star is occulted by an image stop. This stop has

a shape function w(D/s). It has unity where the stop is opaque and

zero where the stop is absent. If w() has width of order unity, the stop will be of order s resolution elements. The transfer function in the

image planet is 1 – w(D/s).

c)

W() = exp(–2/2)

d)

The occulted image is then relayed to a detector through a second pupil plane e)

e)

This is the convolution of the step function of the original pupil with a Gaussian

e)

One then places a Lyot stop in the pupil plane

f) g)

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

Difference Imaging : 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

Planet Bright

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

1.58 m 1.68 m

1.625 m

Spectral Differential Imaging (SDI)

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

Results!

Coronography of Debris Disks

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

Coronographic Detection of a Brown Dwarf

Spectral Features show Methane and Water

But there is large uncertainty in the surface gravity and mass can be as low as 4 and as high as 155 MJup.

The Planet Candidate around GQ Lupi

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

M = 4 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.

SPITZER Observations of Fomalhaut at 4.5 m

Marengo et al. 2009

Not detected in the Infrared. Limits of 3 MJup

and age of 200 Million years

2010 2012

Kalas et al. 2012

Recent observations by Kalas using HST confirm presence of planet.

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

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

A fourth planet has also been detected around HR 8799

The 2009-2010 orbital motions of the four planets are shown in the larger plot. A square symbol denotes the first 2009 epoch. The upper-right small panel shows a zoomed version of e's astrometry including the expected motion (curved line) if it is an unrelated background object. Planet e is confirmed as bound to HR 8799 and it is moving 46 ± 10 mas/year counter-clockwise. The orbits of the solar system's giant planets (Jupiter, Saturn, Uranus and Neptune) are drawn to scale (light gray circles). With a period of ~50 years, the orbit of HR 8799e will be rapidly constrained by future observations; at our current measurement accuracy it will be possible to measure orbital curvature after only 2 years.

asteroid belt

HR 8799 Compared to Our Solar System

The Planet around Pic

Mass ~ 8 MJup

20032009

UScoCTIO 108

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

Pic 8 12 ~5 - A6 V 1.8

HR 8799 b 7 465 68 - F2 V1

HR 8799 c 10 190 38 ´-

HR 8799 d 10 10 24 -

HR 8799 e 9 50 14.5

Fomalhaut b < 3 88 115 - A3 V 2.06

Some Imaging Planets

1SIMBAD lists this as an A5 V star, but it is a Dor variable which have spectral types F0-F2. Tautenburg spectra confirm that it is F-type

SummaryAdvantages:

1.Finds planets at large orbital radii. This fills an important region of the parameter space inaccessible with other methods.2.Can get spectroscopy of the planet directly3.Can Planets around hot stars as well4.Seeing is believing!

Disadvantages:

1.Only works for nearby stars 2.Planet mass relies on evolutionary tracks that are model dependent – mass uncertain!3.Orbital parameters poorly known (wait a long time!)4.Only massive and young planets detected so far5.Only planets far from the star have been detected

Smaller, close in planets will require space missions or extemely large telescopes (30m)

Direct Imaging of Exoplanets

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