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On Designing NASA’s Terrestrial Planet Finder Space Telescope Robert J. Vanderbei N. Jeremy Kasdin David N. Spergel Home Page Title Page Contents JJ II J I Page 1 of 25 October 20, 2003 INFORMS, Atlanta Go Back Full Screen Close Princeton University http://www.princeton.edu/∼rvdb Quit . The Big Question: Are We Alone? Home Page Title Page Contents • Are there planets? Earth-like • Are they common? • Is there life on some of them? JJ II J I Page 2 of 25 Go Back Full Screen Close Quit . Exosolar Planets—Where We Are Now There are more than 100 Exosolar planets known today. Home Page Title Page Most of them have been discovered by detecting a sinusoidal doppler shift in the parent star’s spectrum due to gravitationally induced wobble. Contents JJ II J I Page 3 of 25 This method works best for large Jupiter-sized planets with close-in orbits. Go Back Full Screen 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. Close Quit . Some of the ExoPlanets Home Page Title Page Contents JJ II J I Page 4 of 25 Go Back Full Screen Close Quit . Future Exosolar Planet Missions • 2006, Kepler a space-based telescope to monitor 100,000 stars simultaneously looking for “transits”. Home Page Title Page Contents • 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. JJ II J I Page 5 of 25 Go Back Full Screen Close Quit . Terrestrial Planet Finder Telescope Home Page Title Page • DETECT: Search 150-500 nearby (5-15 pc distant) Sun-like stars for Earth-like planets. Contents JJ II J I Page 6 of 25 • CHARACTERIZE: Determine basic physical properties and measure “biomarkers”, indicators of life or conditions suitable to support it. Go Back Full Screen Close Quit . 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! Home Page Title Page Contents • 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. JJ II J I Page 7 of 25 Go Back Full Screen Close Quit . Visible vs. Infrared Home Page Title Page Contents JJ II J I Page 8 of 25 Go Back Full Screen Close Quit 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. Home Page Title Page Contents JJ II J I Page 9 of 25 Go Back Full Screen Close Quit . 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. Home Page Title Page Contents JJ II J I Page 10 of 25 Go Back Full Screen Close Quit . 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. Home Page Title Page Contents JJ II J I Page 11 of 25 Go Back Full Screen Close Quit . 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. Home Page Title Page Contents The rings grow progressively dimmer as this log-plot shows: JJ II J I Page 12 of 25 Go Back Full Screen Close Quit . Airy Disk and Diffraction Rings—Log Scaling Home Page Here’s the same Airy disk from the previous slide plotted using a logarithmic brightness scale with 10−11 = −110dB set to black: Title Page Contents JJ II J I Page 13 of 25 Go Back 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. Full Screen Close Quit . Electric Field Consider an aperture mask consisting of an openning given by 1 1 (x, y) : − ≤ x ≤ , −A(x) ≤ y ≤ A(x) . 2 2 Home Page Title Page 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 II J I dydx −A(x) XZ JJ Page 14 of 25 i(xξ+yζ) E(ξ, ζ) = Contents 1 2 cos(xξ) Go Back sin(A(x)ζ) dx ζ Full Screen Close The intensity of the light at (ξ, ζ) is given by the square of the electric field. Quit . Maximizing Throughput Because of the symmetry, we only need to optimize in the first quadrant: Z 12 maximize 4 A(x)dx Home Page Title Page 0 Contents 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). JJ II J I Page 15 of 25 Go Back Full Screen Close Quit . One Pupil w/ On-Axis Constraints Home Page Title Page PSF for Single Prolate Spheroidal Pupil Contents JJ II J I Page 16 of 25 Go Back Full Screen Close Quit . Best Mask: 8-Pupil Mask Home Page Title Page Contents 50 100 JJ II J I 150 200 250 300 Page 17 of 25 350 400 Go Back 450 500 50 100 150 200 250 300 350 400 450 500 Full Screen Close Quit . 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. Home Page Title Page Contents JJ II J I Page 18 of 25 Go Back Full Screen Close Quit . 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 Title Page Contents −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 Home Page JJ II J I Page 19 of 25 Go Back Full Screen Close Quit . Circularly Symmetric Masks Home Page Let Title Page A(r) = 1 r2j ≤ r ≤ r2j+1, 0 otherwise, j = 0, 1, . . . , m − 1 Contents 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 JJ II J I Page 20 of 25 Go Back . Full Screen Close Quit . Circularly Symmetric Masks Optimization Problem Home Page Title Page 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 Contents JJ II J I Page 21 of 25 Go Back where E(ξ) is the function of the rj ’s given on the previous slide. Full Screen Close Quit . ξ0 = 4 and ξ1 = 40 and m = 18 Clipboard Home Page Title Page Contents JJ II J I Page 22 of 25 0 10 -2 10 Go Back -4 10 -6 10 Full Screen -8 10 -10 10 Close -12 10 -14 10 0 5 10 15 20 25 30 35 40 45 50 Quit . Starshaped Masks Home Page Title Page Contents JJ II J I Page 23 of 25 Go Back Full Screen Close Quit . Apodization—Tinting Glass 1 Home Page 0.9 0.8 0.7 Title Page 0.6 0.5 Contents 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 JJ II J I 0.5 Page 24 of 25 0 -20 Go Back -40 -60 Full Screen -80 -100 Close -120 -140 -160 Quit -180 -60 -40 -20 0 20 40 60 Contents 1 The Big Question: Are We Alone? 2 2 Exosolar Planets—Where We Are Now 3 3 Some of the ExoPlanets 4 4 Future Exosolar Planet Missions 5 5 Terrestrial Planet Finder Telescope 6 6 Why Is It Hard? 7 7 Visible vs. Infrared 8 8 Shape Optimization (Telescope Design) 10 9 Shape Optimization (Telescope Design) 11 Home Page Title Page Contents 10 Airy Disk and Diffraction Rings 12 11 Airy Disk and Diffraction Rings—Log Scaling 13 12 Electric Field 14 13 Maximizing Throughput 15 14 One Pupil w/ On-Axis Constraints 16 15 Best Mask: 8-Pupil Mask 17 16 Circularly Symmetric Masks 18 17 Circularly Symmetric Apodization 19 18 Circularly Symmetric Masks 20 19 Circularly Symmetric Masks Optimization Problem 21 20 ξ0 = 4 and ξ1 = 40 and m = 18 22 21 Starshaped Masks 23 JJ II J I Page 25 of 25 Go Back Full Screen Close Quit