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Direct Imaging of Exoplanets
I. Techniques
a) Adaptive Optics
b) Coronographs
c) Differential Imaging
d) Nulling Interferometers
e) External Occulters
II. Results
Challenge 1: Large ratio between star and planet flux (Star/Planet)
Reflected light from Jupiter ≈ 10–9
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. These are
easier to detect.
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
y=kx
x
L
Expressed as an infinite sine series:
∞
y(x) =  Bn sin (npx/L)
n=1
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
 y(x) sin(
0
n1px
L
L
∞
) dx =  Bn  sin(
n=1
0
n1px
L
Use: sin q sin f = 1/2[cos(q–f) – cos(q+f)]
npx
)sin(
) dx
L
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:
L
Bn = 2/L  y(x) sin(
0
npx
L
) dx
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:
a) 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
x
n
Background: Fourier Transforms
x
n
x
n
Background: Fourier Transforms
sinc
x
n
J1(2px)
2x
x
n
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
f*g
Fourier Space
F G
Suppose you wanted to smooth
your data by n points.
x
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
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
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
Telescope
5500 Å
2 mm
10 mm
Seeing
TLS 2m
0.06“
0.2“
1.0“
2“
VLT 8m
0.017“
0.06“
0.3“
0.2“
Keck 10m
0.014“
0.05“
0.25“
0.2“
ELT 42m
0.003“
0.01“
0.1“
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)
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 = 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
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=0.3-0.4
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
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:
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:
1. Foucault Knife Edge Sensor (Babcock 1953)
2. Shearing Interferometer
3. Shack-Hartmann Wavefront Sensor
4. Curvature Wavefront Sensor
Shack-Hartmann Wavefront Sensor
Shack-Hartmann Wavefront Sensor
Lenslet array
Image Pattern
Focal Plane
detector
reference
af
disturbed
a
f
Shack-Hartmann Wavefront Sensor
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)
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.
Applications of Adaptive Optics
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
Applications of Adaptive Optics
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
Applications of Adaptive Optics
4. Faint companions
The seeing disk will normally destroy the image of faint
companion. Is needed to detect substellar companions
(e.g. GQ Lupi)
Applications of Adaptive Optics
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:
1. Simple opaque disk
2. 4 Quadrant Phase mask : Shifts the phase in 4
quadrants to create destructive interference to
block the light
3. Vortex Phase Coronagraph: rotates the angle
of polarization which has the same effect as
ramping up the phase shift
A 4-quadrant phase mask
p phase
shift
0 phase shift
0 phase shift
The Airy
Disk
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
q
At a distance of
50.000 km the
starshade subtends
the same angle as
the star
A complex starshape is needed to suppress diffraction
50 m
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
Planet Faint
Since the star has no methane, the PSF in all
filters will look (almost) the same.
Nulling Interferometry
Principles of Long Baseline Stellar Interferometry,
ed. Peter Lawson
A Basic Interferometer
S
s • B = B cos q
q
A2
A1
x2
x1
Delay Line 1
Beam
Combiner
d1
1) Idealized 2-telescope interferometer
Delay Line 2
d2
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 )
= 1 + 1 + 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:
Imax –Imin
V=
Imax +Imin
Visibility is measured by changing the path length and
recording minimum and maximum values
Interferometry Basics: Cittert-Zernike theorem




b
a
Ds  ŝo
a, b are angles in „x-y“
directions of the source.
Ds = (a,b,0)
in the coordinate system
where ŝo =(0,0,1)
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:
1. Collect as many visibility curves as possible
2. 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
V
N
B
E
Spatial Resolution : l/B
Can resolve all angular scales up
to q > l/D, i.e. the diffraction limit
Fourier Space
U
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.
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
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
10
190
38
´-
HR 8799 d2
7
10
24
-
Fomalhaut b
<3
88
115
-
A3 V
0.025
2.06
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
1SIMBAD
2A
fourth planet around HR 8799 was reported at the 2011 meeting of the American
Astronomical Society