Download On Designing NASA's Terrestrial Planet Finder Space Telescope

On Designing NASA’s Terrestrial Planet Finder
Space Telescope
Robert J. Vanderbei
N. Jeremy Kasdin
David N. Spergel
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October 20, 2003
INFORMS, Atlanta
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Princeton University
http://www.princeton.edu/∼rvdb
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. The Big Question: Are We Alone?
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• Are there
planets?
Earth-like
• Are they common?
• Is there life on some of
them?
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. Exosolar Planets—Where We Are Now
There are more than 100 Exosolar planets known today.
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Most of them have been discovered by detecting a sinusoidal doppler shift in the parent star’s
spectrum due to gravitationally induced wobble.
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This method works best for large Jupiter-sized planets with close-in
orbits.
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One of these planets, HD209458b, also transits its parent star once
every 3.52 days. These transits have been detected photometrically as
the star’s light flux decreases by about 1.5% during a transit.
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. Some of the ExoPlanets
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. Future Exosolar Planet Missions
• 2006, Kepler a space-based telescope to
monitor 100,000 stars simultaneously looking for “transits”.
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• 2007, Eclipse a space-based telescope to directly image Jupiter-like planets.
• 2009, Space Interferometry Mission (SIM)
will look for astrometric wobble.
• 2014, Darwin is a space-based cluster of 6 telescopes used as an
interferometer.
• 2014, Terrestrial Planet Finder (TPF) space-based telescope to directly image Earth-like planets.
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. Terrestrial Planet Finder Telescope
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• DETECT: Search 150-500 nearby (5-15 pc distant) Sun-like stars
for Earth-like planets.
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• CHARACTERIZE: Determine basic physical properties and measure
“biomarkers”, indicators of life or conditions suitable to support it.
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. Why Is It Hard?
• If the star is Sun-like and the planet is Earth-like, then the reflected
visible light from the planet is 10−10 times as bright as the star.
This is a difference of 25 magnitudes!
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• If the star is 10 pc (33 ly) away and the planet is 1 AU from the
star, the angular separation is 0.1 arcseconds!
Originally, it was thought that this would
require a space-based multiple mirror
nulling interferometer.
However, a more recent idea is to use a
single large telescope with an elliptical mirror (4 m x 10 m) and a shaped pupil for
diffraction control.
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. Visible vs. Infrared
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HD209458 is the bright (mag. 7.6) star in the center of this image.
The dimmest stars visible in this image are magnitude 16.
An Earth-like planet 1 AU from HD209458 would be magnitude 33,
and would be located 0.2 pixels from the center of HD209458.
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. Shape Optimization (Telescope Design)
The problem is to design and build a space telescope that will be able
to “see” planets around nearby stars (other than the Sun).
Consider a telescope. Light enters
the front of the telescope. This is
called the pupil plane.
The telescope focuses all the light
focal
light
passing through the pupil plane from plane
cone
pupil
a given direction at a certain point on
plane
the focal plane, say (0, 0).
However, the wave nature of light makes it impossible to concentrate
all of the light at a point. Instead, a small disk, called the Airy disk,
with diffraction rings around it appears.
These diffraction rings are bright relative to any planet that might be
orbiting a nearby star and so would completely hide the planet. The
Sun, for example, would appear 1010 times brighter than the Earth to
a distant observer.
By placing a mask over the pupil, one can design the shape and strength
of the diffraction rings. The problem is to find an optimal shape so as
to put a very deep null very close to the Airy disk.
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. Shape Optimization (Telescope Design)
The problem is to design and build a space telescope that will be able
to “see” planets around nearby stars (other than the Sun).
Consider a telescope. Light enters
the front of the telescope. This is
called the pupil plane.
The telescope focuses all the light
focal
light
passing through the pupil plane from plane
cone
pupil
a given direction at a certain point on
plane
the focal plane, say (0, 0).
However, the wave nature of light makes it impossible to concentrate
all of the light at a point. Instead, a small disk, called the Airy disk,
with diffraction rings around it appears.
These diffraction rings are bright relative to any planet that might be
orbiting a nearby star and so would completely hide the planet. The
Sun, for example, would appear 1010 times brighter than the Earth to
a distant observer.
By placing a mask over the pupil, one can design the shape and strength
of the diffraction rings. The problem is to find an optimal shape so as
to put a very deep null very close to the Airy disk.
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. Airy Disk and Diffraction Rings
A conventional telescope has a circular openning as depicted by the left
side of the figure. Visually, a star then looks like a small disk with rings
around it, as depicted on the right.
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The rings grow progressively dimmer as this log-plot shows:
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. Airy Disk and Diffraction Rings—Log
Scaling
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Here’s the same Airy disk from the previous slide plotted using a logarithmic brightness scale with 10−11 = −110dB set to black:
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The problem is to find an aperture mask, i.e. a pupil plane mask,
that yields a −100 dB null somewhere near the first diffraction ring. A
hard problem! Such a null would appear almost black in this log-scaled
image.
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. Electric Field
Consider an aperture mask consisting of an openning given by
1
1
(x, y) : − ≤ x ≤ , −A(x) ≤ y ≤ A(x) .
2
2
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We only consider masks that are symmetric with respect to both the x
and y axes. Hence, the function A() is a nonnegative even function.
In such a situation, the electric field E(ξ, ζ) is real and also symmetric
about both the x and y axes. It is given by
Z
1
2
Z
A(x)
e
− 21
=4
j
0
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dydx
−A(x)
XZ
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i(xξ+yζ)
E(ξ, ζ) =
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1
2
cos(xξ)
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sin(A(x)ζ)
dx
ζ
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The intensity of the light at (ξ, ζ) is given by the square of the electric
field.
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. Maximizing Throughput
Because of the symmetry, we only need to optimize in the first quadrant:
Z 12
maximize 4
A(x)dx
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0
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subject to − 10−5E(0, 0) ≤ E(ξ, ζ) ≤ 10−5E(0, 0), for (ξ, ζ) ∈ O
0 ≤ A(x) ≤ 1/2,
for 0 ≤ x ≤ 1/2
The objective function is the total open area of the mask. The first
constraint guarantees 10−10 light intensity throughout a desired region
of the focal plane, and the remaining constraint ensures that the mask
is really a mask.
If the set O is a subset of the x-axis, then the problem is entirely linear
(a linear programming problem).
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. One Pupil w/ On-Axis Constraints
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PSF for Single Prolate Spheroidal Pupil
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. Best Mask: 8-Pupil Mask
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50
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. Circularly Symmetric Masks
• My original question was “Why not work with circularly symmetric
optics?” In this case, one could think of making a variable filter.
That is, at point (x, y) have the filter transmit a fraction A(x, y)
of the light.
• Such a filter is called an apodization.
• The answer is that apodizations are hard to make accurately.
• For small working bands, the square-aperture masks are essentially
bang-bang all-or-nothing masks.
• It suggests looking for similar circularly symmetric masks.
• They can be thought of as apodizations in which the apodizing
function A(r) is zero-one valued.
• On the next few slides we derive the formulas for circularly symmetric apodization and then restrict attention to the zero-one valued
case.
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. Circularly Symmetric Apodization
Instead of a square mask, we consider now a circularly symmetric
apodized aperture:
Z 1/2 Z π
E(ξ, ζ) =
A(r)e−2πi(xξ+yζ)rdθdr
0
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−π
where, of course, x = r cos θ and y = r sin θ.
WLOGWMAT, ζ = 0 and hence we look at
Z π
Z 1/2
E(ξ) =
rA(r)
e−2πiξr cos θ dθ dr
−π
0
Z 1/2
=
2πrA(r)J0 (2πrξ) dr
0
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. Circularly Symmetric Masks
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Let
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A(r) =
1 r2j ≤ r ≤ r2j+1,
0 otherwise,
j = 0, 1, . . . , m − 1
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where
0 ≤ r0 ≤ r1 ≤ · · · ≤ r2m−1 ≤ 1/2.
The integral on the previous slide can now be written as a sum of
integrals and each of these integrals can be explicitly integrated to get:
E(ξ) =
m−1
X
1
j=0
ξ
r2j+1J1 2πξr2j+1 − r2j J1 2πξr2j
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. Circularly Symmetric Masks Optimization Problem
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maximize
m−1
X
2
2
π(r2j+1
− r2j
)
j=0
subject to: − 10−5E(0) ≤ E(ξ) ≤ 10−5E(0), for ξ0 ≤ ξ ≤ ξ1
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where E(ξ) is the function of the rj ’s given on the previous slide.
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. ξ0 = 4 and ξ1 = 40 and m = 18
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. Starshaped Masks
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. Apodization—Tinting Glass
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Contents
1
The Big Question: Are We Alone?
2
2
Exosolar Planets—Where We Are Now
3
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Some of the ExoPlanets
4
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Future Exosolar Planet Missions
5
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Terrestrial Planet Finder Telescope
6
6
Why Is It Hard?
7
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Visible vs. Infrared
8
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Shape Optimization (Telescope Design)
10
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Shape Optimization (Telescope Design)
11
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10 Airy Disk and Diffraction Rings
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11 Airy Disk and Diffraction Rings—Log Scaling
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12 Electric Field
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13 Maximizing Throughput
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14 One Pupil w/ On-Axis Constraints
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15 Best Mask: 8-Pupil Mask
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16 Circularly Symmetric Masks
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17 Circularly Symmetric Apodization
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18 Circularly Symmetric Masks
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19 Circularly Symmetric Masks Optimization Problem
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20 ξ0 = 4 and ξ1 = 40 and m = 18
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21 Starshaped Masks
23
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