direct imaging of exoplanets
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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
Time/Space Domain Fourier/Frequency Domain
Cosine is an even function: cos(–x) = cos(x)
Positive frequencies
Negative frequencies
Time/Space Domain Fourier/Frequency Domain
Sine is an odd function: sin(–x) = –sin(x)
Time/Space Domain Fourier/Frequency Domain
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
Time/Space Domain Fourier/Frequency Domain
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
Atmospheric Turbulence
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
Atmospheric Turbulence
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:
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 polynomial where each term describes a specific aberation and how much it is present.
Describing the Wavefronts
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
Shack-Hartmann Wavefront Sensor
f
Image Pattern
reference
disturbed
f
Lenslet array
Focal Plane detector
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)
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
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/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)
Applications of Adaptive Optics
Applications of Adaptive Optics
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
Cs
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)
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