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# Direct Imaging of Exoplanets - PowerPoint PPT Presentation

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.

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### Direct Imaging of Exoplanets

• Techniques

• Coronographs

• Differential Imaging

• Nulling Interferometers

• External Occulters

• Results

Reflected light from Jupiter ≈ 10–9

Challenge 2: Close proximity of planet to host star (Star/Planet)

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. These are easier to detect.

Background: Electromagnetic Waves light. These are easier to detect.

F(x,t) = A0ei(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 light. These are easier to detect.

|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 light. These are easier to detect.

Cosines and sines represent a set of orthogonal functions (basis set).

Meaning: Every continuous function can be represented by a sum of trigonometric terms

Expressed as an infinite sine series: light. These are easier to detect.

y = k x

y(x) =  Bn sin (npx/L)

y

n=1

x

L

Background: Fourier Transforms

Eg. Consider the Step function:

y(x) = kx over the interval x = (0, L)

I. Background: Fourier Transforms light. These are easier to detect.

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

L

n1px

n1px

npx

 y(x) sin( ) dx =  Bn  sin( )sin( )

dx

L

L

L

0

n=1

0

Use: sin q sin f = 1/2[cos(q–f) – cos(q+f)]

L light. These are easier to detect.

npx

Bn = 2/L  y(x) sin( ) dx

L

0

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

If you do the above intergral and insert the limits you will find that all terms are zero except for n = n1and we get:

I. Background: Fourier Transforms light. These are easier to detect.

Background: Fourier Transforms light. These are easier to detect.

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 light. These are easier to detect.

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 light. These are easier to detect.

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

x light. These are easier to detect.

x

Background: Fourier Transforms

n

x light. These are easier to detect.

x

n

n

Background: Fourier Transforms

sinc light. These are easier to detect.

x

x

n

n

J1(2px)

2x

Background: Fourier Transforms

Diffraction patterns from the interference of electromagnetic waves are just Fourier transforms!

x light. These are easier to detect.

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

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

Atmospheric turbulence distorts stellar images making them much larger than point sources. This seeing image makes it impossible to detect nearby faint companions.

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

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 instrument

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 = 251643 l/D (arcsecs)

Diffraction Limit instrument

Telescope

5500 Å

2 mm

10 mm

Seeing

TLS 2m

0.06“

0.2“

1.0“

2“

0.06“

0.3“

0.017“

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 instrument

A Turbulent atmosphere is characterized by eddy (cells) that decay from larger to smaller elements.

The largest elements define the upper scale turbulenceLuwhich 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 instrument

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 instrument

ro: the coherence length or „Fried parameter“ is

r0 = 0.185 l6/5 cos3/5z(∫Cn² dh)–3/5

r0median = 0.114 (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 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 instrument

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 instrument

• 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=0.3-0.4 is more typical.

Definitions instrument

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 instrument

You do not want to correct using a reference star in this direction

Basic Components for an AO System instrument

• 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 instrument

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 instrument

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 instrument

Zernike Polynomials:

SKn,m,1rn cosmq + Kn,m,2rn sinmq

Z=

Measuring the Wavefront instrument

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 instrument

Lenslet array

Image Pattern

reference

Focal Plane detector

af

disturbed

a

f

Correcting the Wavefront Distortion instrument

• 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)

Reference Stars instrument

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.

• 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 instrument

Basic Coronagraph instrument

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

b) instrument

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

d) instrument

c)

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) instrument

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) instrument

f)

g)

One then places a Lyot stop in the pupil plane

SR=0.30

0.53

0.69

0.82

0.9

• 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

The Airy Disk Phase shifted

p phase shift

0 phase shift

The Airy Disk

p phase shift

0 phase shift

The exit pupil (FT of c)

The exit pupil through the Lyot stop

The image (FT of e) )

External Occulter efficient

50000 km

At a distance of 50.000 km the starshade subtends the same angle as the star

q

50 m

A coronagraph efficient

An external occulter!

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) efficient

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 efficient

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

Planet Faint

### Nulling Interferometry efficient

Principles of Long Baseline Stellar Interferometry, ed. Peter Lawson

A Basic Interferometer efficient

S

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 efficient

Consider two wavefronts hitting our telescopes with apertures A1 and A2 separated by a baseline B. The waves are:

f1 ~ eikx1eiwt = eikŝ ·x1eiwt

f2 ~ eikx2eiwt = eikŝ ·x2eiwt

ŝ = S/|S|

Interferometry Basics efficient

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 efficient

We can absorb the k ŝ ·x1term in normalization. We can also introduce the effects of delay lines:

f1 ~ eikd1eiwt

f2 ~ eikd2eikŝ Beiwt

fTotal = f1 + f2 ~ eiwt (eikd1 + eikd2e–ik ŝ B )

Interferometry Basics efficient

The power, P, is:

P fTotalf*Total =

eiwte+iwt(eikd1 + eikd2e–ik ŝ B ) (eikd1 + eikd2e-ik ŝ B )

= 1 + 1 + cross terms

Cross terms are what the important part

Interferometry Basics efficient

= 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 efficient

Adjacent fringe crests projected on the sky are separated by an angle given by:

Ds = l/B

I efficientmax –Imin

V =

Imax +Imin

Interferometry Basics: The Visibility Function

Michelson Visibility:

Visibility is measured by changing the path length and recording minimum and maximum values

efficient

Ds ŝo

Interferometry Basics: Cittert-Zernike theorem

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

V( efficientk, B) =  da db A(a,b) F(a,b) e 2pi(au+bv)

Interferometry Basics: Cittert-Zernike theorem

The visibility :

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.

• 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)

N efficient

B

E

Interferometric Basics: Aperture Synthesis

Aperture Space

Fourier Space

V

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 efficient

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! Interferometry

Coronography of Debris Disks Interferometry

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

Cs Interferometry

The Planet Candidate around GQ Lupi system on the VLT

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

a ~ 115 AU system on the VLT

P ~ 870 years

Mass < 3 MJup, any more and the gravitation of the planet would disrupt the dust ring

Photometry of Fomalhaut b system on the VLT

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.

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

The Planet around „Vortex Phase“ coronagraph at the 5m Palomar Telescopeb Pic

Mass ~ 8 MJup

2003 „Vortex Phase“ coronagraph at the 5m Palomar Telescope

2009

Imaging Planets „Vortex Phase“ coronagraph at the 5m Palomar Telescope

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