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- 1. Discovering Exo-Earths: optimization of an external occulter. Christophe Bellisario June 29, 2010
- 2. Abstract Before looking for extra-terrestrial life, detecting exo-planets is one of the most promising quest in astronomy.However, the diﬃculties that science and technology encounter make tough the detection and observation of exo-planets, especially exo-Earths. In this paper, I itemize concretely what we are looking for and the various methodsused up to now in the search of exo-planets. I will be paying particular attention to the direct imaging, which enablesspectral characterization of a planet. Suppressing the light coming from the host star by orders of magnitude to reveal the faint light coming from theexo-planet is one of the most eﬃcient way for characterizing an exo-Earth and maybe, ﬁnding life. This can be doneby the use of internal or external occulters (coronagraphs or starshades) with numerous diﬀerent properties. I discusshere how to compute the physical expression of the light intensity when combining a starshade with a telescope. Ialso explain how to build the shape of an occulter through an apodization function coming from a numerical andanalytical optimization, which I implemented. Then, I investigate all the related parameters such as the diameter,the petal length, the inner working angle, etc..., to highlight all the various behaviors of the apodization through arange of data corresponding to the science we aim to do.
- 3. Contents1 Science of extra-solar planets 2 1.1 Exo-Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Gas Giants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Terrestrial exoplanets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Detection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Radial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.4 Puslar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.5 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.6 Direct Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Starshade: design, optimization and properties 11 2.1 Design and optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Free-Space propagation from starshade to telescope . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Binary apodization, shape of the occulter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Optimization of the occulter apodization function . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Global study of the parameter space and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 First comparison between analytical and numerical methods . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Contrast as a function of the diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.4 Distance of the occulter as a function of the diameter . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.5 Action of the petal length on the contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.6 Action of the Shadow Oversize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.7 Another way to deﬁne the HyperGaussian function . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1
- 4. Chapter 1Science of extra-solar planets1.1 Exo-PlanetsAre we alone in the Universe? This fundamental and philosophical question could ﬁnd a simple scientiﬁc yes-or-noanswer by ﬁnding life in other solar systems, other Earths around stars. Since we discovered the ﬁrst exoplanets, theresearch in this ﬁeld has grown tremendously and more than 450 planets have been catalogued to create a huge archiveof various planets. There are planets with masses from 2 M ([1]) to 13 MJup (limiting mass for thermonuclearfusion of deuterium [2]), the radius varies from 2 R ([3]) to 2 RJup ([4]) and the temperature from 50◦ K ([5]) to1100◦ K ([6]). The boundaries of the high values are quite diﬃcult to establish as the limit between gas giants andbrown dwarfs/aborted stars is as of now not really clear and can be deﬁned based on diﬀerent formation scenariosinstead of the mass ([7] & [8]). Gravity is also important as it depends on the density and radius; it can hardly mouldthe shape and consistency of the surface. Moreover, if we look for life, the age of the star will be another importantfactor as young stars won’t be good candidates since it took Earth at most one billion year to appear. The next andlast step in the search of life deals with the characterization of the atmosphere. Nowadays, we count few planetswhich show evidence of carbon dioxide, methane and sodium ([9], [10] & [11]).In order to ﬁnd candidate planets sheltering life, we have to diﬀerentiate the kinds of exoplanets: gas giant, ice giantsand terrestrial planets, etc. For a given terrestrial planet, the habitability is deﬁned by the presence of liquid water(habitable zone – def HZ). If life exists, then we can search for it by using biomarkers. Many more hot Jupiter-likeplanets have been found in comparison to terrestrial-like exoplanets, and most of them have been classiﬁed in diﬀerentkinds. In the next two parts we give a general description of gas giants and terrestrial planets, more precisely wherethe search of life starts.1.1.1 Gas GiantsBased on Jovian planets of our solar system, gas giants are also called hot/cold Jupiters. They mainly represent aclass of gaseous planets, that are almost always the mass of Jupiter or more, and are also above a vague boundaryof 10 M . Their types have been classiﬁed by David Sudarsky in regards of some characteristics like temperatureand composition ([12]). The gas giants like Jupiter and Saturn are mostly composed of hydrogen and helium, whichare the most abundant elements in the Sun whereas the ice giants like Uranus and Neptune are primarily made ofheavier components such as oxygen, carbon, nitrogen, and sulfur. They also diﬀer by the size of their respective coresaround which the gas orbits. The scientiﬁc community ﬁrst attempted to observe them while they were far awayfrom their host stars, like it is the case for Jupiter, Saturn, Uranus and Neptune. However, a signiﬁcant proportionof discoveries of gas giants that were very close to their host stars (which helped for their detection) and also morediscoveries of retrograde orbits made scientists rethink their ideas of star system formation.Hydrogen, helium, methane and ammonia are the main components detected in gas giants. Age and temperature areimportant properties and the latter is principally governed by the distance to the host star, e.g. closer planets meanhotter planets. Their temperature will lead to diﬀerent structures as the ﬂuid becomes either a gas or solid. Moreover,younger gas giants are signiﬁcantly more luminous, which makes them easier to observe with direct imaging in thenear infrared.The big question about whether a detection is a gas giant, a brown dwarf, or a binary star is still relevant. Thereare many observations of speculative close gas giants that need conﬁrmation before they can be recognized oﬃciallyas new exoplanets. 2
- 5. 1.1.2 Terrestrial exoplanetsAs previously stated, exo-Earths are the Holy Grail of exo-planet research. Up to now, the lightest exoplanetdiscovered is GJ 581e with a minimum mass of 1.9 M ([13]) and the smallest is CoRoT-Exo-7b with a radius of1.76 R ([3]). In our quest for life, we mainly look around Sun-like stars, even if it might look anthropocentric tosearch for similar biological markers. However, starting with the various forms that life can take on our planet, wemight include huge possibilities of other means of life.The key parameters to describe a possible Earth-twin are orbit, mass, radius, visible/infrared spectrum, and alsotheir variations during time. These parameters and their combinations provide us information about numerous otherproperties, like eﬀective temperature, density (→ surface gravity with the help of the radius) and albedo (→ surfacereﬂectance).The ﬁrst step is to deﬁne an habitable zone. In our galaxy, disruptive gravitational forces or strong emissions ofinfrared radiation and X-rays could cause the impossibility for life to grow close to the galactic center. In the outerlimit of the galaxy, the abundance of heavy elements decreases due to galactic chemical evolution. Next, as the lifeform we know could not inhabit planets like Neptune or Venus, the stellar-habitable zone has be deﬁned to set arange of distance around the star where all these conditions are met to encourage the development of life mainlydeﬁned by the presence of water ([14]). To help ﬁnd this range, we can use our own solar system as a base. Here thehabitable zone goes from 0.7 AU to 1.5 AU (Earth being, of course, at 1 AU). We scale this frame by the square rootof the stellar luminosity, which will lead to the following equation: L∗ Habitable Zone (AU) ∈ [0.7 − 1.5] × L(see ﬁgure 1.1). We can also translate the habitable zone in AU into an angular distance and through a(AU) L∗ /L θ(”) = = d(pc) d(pc)which will lead to a frame of 70 milli-arcseconds (mas) to 120 mas (by taking the ratio L∗ /L between 0.5 and 1.5).A useful parameter which expresses the angular separation is the Inner Working Angle (IWA). It expresses the anglebetween the host star and the planet seen from the telescope and takes the previous values in the section devoted tothe starshade.Figure 1.1: Habitable zone (HZ) in Earth radius as a function of the star masses. As we can see, for increasing mass,the luminosity will be higher and it will push away the HZ. For a Sun-like star, Earth is between the [0.7 - 1.5] AUboundaries whereas Mars is just at the limit of the HZ. Figure credit: GFDL.In spite of the increase of discoveries thanks to transit and radial velocity methods, characterization of the atmosphererequires a higher level of planet detection. Direct imaging is the key to getting spectroscopic data. Currently, transitspectroscopy provides many spectral characterizations ([15]) but remains ineﬃcient for terrestrial planets closer totheir host stars (except for some dwarfs), in the habitable zone. The wavelengths of interest are included in a range 3
- 6. of 1 to 12 µm which reveal information about water, methane and carbon monoxide signatures. They help us tounderstand about the surface type, clouds or atmospheric retention too. Finally, O2 and O3 , as biogenic traces willindicate evidences for life ([16]).Figure 1.2: Histograms of the number of exoplanets discovered as a function of radius and mass (ﬁgure credits:exoplanet.eu).1.2 Detection methods1.2.1 Radial VelocityRadial Velocity has been so far the more proliﬁc method in detecting exo-planets. Also called the Doppler spec-troscopy, it uses gravitational laws and the Doppler eﬀect. Historically, it has also been the ﬁrst method used byastronomers ([17]) to discover a Jupiter-class planet around 51 Pegasi.The exoplanets in a system have elliptical orbits and the host star moves in a small counter-orbit around a commonbarycenter due to the attraction of planets. This movement will change the radial velocity of the star seen from Earthand the spectral lines will show small blue shifts and red shifts.At ﬁrst, the errors of radial velocity measurements were too big to detect exo-planets. For example, the Sun getsan additional movement due to Jupiter of 13 m/s and the errors were of 1000 m/s. However, in 1988, the Canadianastronomers Bruce Campbell, G. A. H. Walker, and S. Yang suggested that a planet was orbiting the star GammaCephei using a method that allowed them to detect radial velocity movements to a precision of 15 m/s. Now theHigh Accuracy Radial velocity Planet Searcher (HARPS) at La Silla Observatory in Chile, can reach a precision ofalmost 0.97 m/s (in comparison, Earth causes a ﬂuctuation of 10 cm/s for the Sun).This method provides a lower mass limit to the planet since radial velocity measure M∗ · sin(i) due to the inclinationof the orbital plane, with i being the angle of inclination. Further astrometric observations tracking the movement ofthe star may change an exo-planet discovery to a brown dwarf detection. As of now, there are 425 planets discovered,with numerous ground missions already working: AFOE, Anglo-Autralian Planet Search Program, Automated PlanetFinder, California & Carnegie Planet Search, Coralie at Leonard Euler Telescope, Elodie, Sophie, Exoplanet Tracker,HARPS, Hobby-Eberly Telescope Magellan 6.5m Telescope, Mc Donald Observatory, NK2 Consortium, TNG HighResolution Spectrograph, UVES. In the next years, the Absolute Astronomical Accelerometry, Carmenes, HARPS-N,OWL, PRVS will complete the huge panel of missions using radial velocity ([18]). 4
- 7. Figure 1.3: Principle of the radial velocity method: when the star moves away, the Doppler shift will be red, andwhen the star gets closer, the shift moves to blue. Figure credit: NASA/JPL image.1.2.2 TransitFirst proposed by Otto Struve in 1951, the idea is to study the luminosity of a star with a reasonable sized-telescope.Periodical variations of the luminosity could come from a planet located between the star and the Earth. In thatcase, we need to see the system in the ecliptic plane. Otherwise, no planet can be detected. Some calculations withgeometric probabilities suggest an estimate that there are almost 5% of stars with a detectable exo-planet. Here, detection will occur with a fall of luminosity and characterization is possible with the spectral analysis of Figure 1.4: Principle of the transit method. Figure credit: NASA/JPL imagethe light received. It is the primary transit. The composition and scale height used with the help of the absorptionof starlight passing through the planet’s atmosphere is called transit spectroscopy. Next, when the planet is almostbehind the star, the secondary eclipse happens and provides a direct detection of the planet’s spectrum. We cantherefore get information about components, reﬂectivity and temperature. The radius, mass and orbits follow fromtime and depth of the fall. When the planet goes behind the star, we can get the light from the planet by subtractionof the star light; this semi-direct method also allows planet characterization. For example, the Spitzer Space Telescope(NASA) managed to produce a 7.5-14.7 µm spectrum for the transiting extrasolar giant planet HD 189733b ([19]).Currently 81 planets have been discovered with the ongoing ground missions: Alsubai’s Project, ASP, BEST,E.P.R.G., HATNetwork, LCOGT and UStAPS (also doing lensing), MEarth, MONET, OGLE III, PASS, PIRATE,PISCES, STARE, Super WASP, STEPSS, UNSWEPS Project, Tenessee Automatic Photoelectric Telescope, TrES,TRESCA, Vulcain South, WHAT, XO Project. In space, CoRoT, EPOCh, Fabra-ROA Camera, Gaia, KEPLER al-ready found many planets, and in project: Plato, GEST for space telescopes and GITPO, STELLA for ground-basedmission ([18]).1.2.3 Gravitational LensingIn order to use gravitational lensing for the detection of exoplanets, we need ﬁrst a background star. Next, whenanother star crosses the distance between Earth and the background star, the gravitational ﬁeld acts like a lens andthe light of the background star is modiﬁed. If the star owns a planet, its gravitational ﬁeld will contribute to thelense and be detected from the Earth (see ﬁgure 1.5). However, this event is very rare and only occurs once for eachplanet as the alignment cannot happen again, making a conﬁrmation impossible.This method provides some advantages: the mass can be measured, we can reach Earth-like masses, and the angularseparation is also known. We can also detect planets in other galaxies. But there are drawbacks: the observationtime required is huge as well as the number of stars which need to be observed for a small number of detections.Moreover, the mass and orbit size depend on the properties of the host star, which also need to be known.With gravitational lensing, 10 planets have been observed, especially with the use of OGLE (the Optical Gravitational 5
- 8. Lensing Experiment) and the others projects: University of St. Andrews Planet Search (UStAPS), Las CumbresObservatory Global Telescope Network (LCOGT), MACHO, Microlensing Planet Search Project (MPS). In thefuture, GEST will join them ([18]).Figure 1.5: Principle of gravitational lensing: the lensing eﬀect of the star is disturbing by the presence of anexoplanet. Figure credit: Abe et al.1.2.4 Puslar TimingPulsars are neutron stars hosting a strong magnetic ﬁeld and are the fastest spinning objects discovered so far. Twobeams of radiation are ejected at the pole of the star, and we get on Earth a brief pulse each time the beam crossesthe path of the Earth. This need to have an alignment between Earth and a pulsar makes the proportion of pulsarswe cannot observe from Earth rather large. However, the existence of a planet around it makes them both orbitaround their center of mass. Similar to the radial velocity method, the act of measuring the periodic changes in thetime between each pulse will be a way to estimate the semi-major axis of the planet’s orbit, and a lower limit of theplanet’s mass ([20]).Actually, planets around pulsars are not very interesting in the search for life in our galaxy. Indeed, pulsars arecreated by stellar explosions like supernovae, which may prevent life to expand in such systems. At this moment,only 8 planets in 5 diﬀerent solar systems have been discovered with this method by the ongoing mission PulsarPlanet Detection.1.2.5 Astrometry Figure 1.6: Principle of the astrometry method. Figure credit: http://www.astro.wisc.edu/ Basically, astrometry is used for determining positions of stars. Using already well-known coordinates of nearstars, it determines the location of an unknown star in the same image by comparison. In the detection of extra-solarplanets, we measure the displacement of stars around a supposed center of mass of the system composed by a star andits planets, providing position and mass of the planet(see ﬁgure 1.6). Many white dwarfs have been discovered as theyinvolve high variations. However, the accuracy needed for exo-planets is very precise and ground-based astrometry isnot enough powerful, due to the distorting eﬀects of the Earth’s atmosphere for example. As of now, only one planet 6
- 9. has been discovered with astrometry (with HST Astrometry, [21]).There will be space missions using this method in the next few years, such as NASA’s SIM Lite (Space InterferometryMission) projected to launch in 2015, European Space Agency’s GAA, due to launch in 2012, Origins Billion StarSurvey (OBSS) in study, and on the ground: STEPS, Radio Interferometric Planet Search (RIPL), PRIMA-DDL(VLTI, under study), Keck Interferometer, ASPENS. All of them will detect terrestrial planets orbiting close to theirstars with astrometric techniques ([18]).1.2.6 Direct ImagingAbout direct imagingProviding a direct picture of exo-planets is the most diﬃcult challenge to take up. Their extremely faint luminosityrequires a high contrast for imaging. For young gas giants (1 MJup , 70 Myr), at a distance of 10 pc, a contrast of10−7 with its host star will be required ([22]). In comparison, for exo-Earths, still at 10 pc, 10−10 of contrast will benecessary. These values correspond to a diﬀerence of magnitude from 17.5 to 25 mag (as m = −2.5 ∗ log10 (F ) + C,with F the ﬂux and C a constant). There are some directly imaged exoplanets (12 as of now [18]).In detection of Earth-like planets, one issue for getting high contrast is dust. Primary disks of dust and asteroidsdating from the creation of stellar systems may still remain like our Jupiter and Neptune Trojans. We also add thedust coming from the numerous collisions between asteroids and comets. All of that represents the exozodiacal dust.It is a source of background noise contaminating the spectrum. For example, the zodiacal dust in our solar system isthe most luminous component after the Sun ([23]). It is called the local zodi. Earth signature might appear in thisdisk as a clump inside it so that even if the exozodiacal disk is not helpful for a detection, its structure might revealthe presence of an unseen planet or could be sign of the system’s orbital dynamic.The exozodiacal light is measured with the ratio between the infrared luminosity and stellar luminosity: (LIR /L∗ ) ≈10−6 − 10−5 in most of the cases. In comparison, the local zodi represents the zodiacal disk in our asteroid belt andthe ratio (LIR /L∗ ) is approximately equal to 10−7 which is the referee value, 1 zodi. Even if it represents most of thesource of noise, the exozodiacal light provides us information about the elliptic inclination of the planetary systemby supposing a circularly distribution around the star. Therefore, dust may provide a clue about the planets’ orbits,which is very important for science. The ﬁeld of exozodiacal disks is of great importance to understand the behaviorin planetary systems as well as planetary formation and systems evolution.One other issue lies in interferences between wavefronts. Optical aberrations cause speckles on the image that canbe confused with a planet signal. Speckles are also created by small thermal variations. These aberrations are timedependent which makes the subtraction by calibration diﬃcult. The use of deformable mirror and adaptive opticsreduces the atmospheric distortions for ground-based telescopes and the speckle intensity. However, 100% eﬃciencycannot be achieved and there will still be remnants.From visible light to infrared, there are several ways to detect exoplanets. In the case of infrared light, the main methodis the use of an interferometer. It consists of a few telescopes connected together to build a larger telescope withhigh resolution power. A nulling interferometer can also be built to reduce the intensity of the host star to show thefaint light of the planet, e.g. the canceled Darwin mission and the ongoing ground-based Large Binocular Telescope.Many projects are already in use or planned for the future ([18]): Keck Interferometer ([24]), Very Large TelescopeInterferometer VLTI (ESO, Paranal), both ground-based telescopes and others are in project: The Antarctic PlateauInterferometer (API) and CARLINA Hypertelescope Project on Earth, Space Infrared Interferometric Telescope(SPIRIT, NASA) in space.The coronagraph, invented by the French astronomer Bernard Lyot in the 1930’s, is a powerful instrument usedﬁrstly to observe the corona around the Sun. It simulates an artiﬁcial eclipse by blocking the light with an occultingspot or mask whereas the surrounding light stays undisturbed. In search for exoplanets, the coronagraph helps toget the faint light coming from a planet around its star. There are several types of coronagraphs, from the band-limited coronagraph (present on the JWST’s Near-Infrared Cam), phase-mask coronagraph, apodized pupil lyotcoronagraph to the optical vortex coronagraph. More information about coronagraphs are listed by Guyon ([25]) andclassiﬁed by Quirrenbach ([26]). The space missions are: Pupil mapping Exoplanet Coronagraphic Observer (PECO,under study), Super-Earth Explorer (SEE-COAST, under project [27]), and the ground-based missions: Spectro-Polarimetric Imaging (SPHERE, under construction [28]), Gemini Planet Imager (GPI, under construction [22]).And the last method, subject of our interest, is the external occulter. The idea of building an occulter in orderto block the light coming from a star was emitted by Lyman Spitzer in 1962 ([29], see ﬁgure 1.7). As one wouldthink, a circular-shaped screen could block the light of a star to help observe the faint light coming from a nearbyplanet. However, strong edges cause Fresnel diﬀraction eﬀects that will brighten the shadow created by the occulterand cause starlight to come through the telescope. An apodization has to remove the strong edges of the occulter.By advanced calculations, many shapes have been designed in the early 1980s ([30]) but ﬁnally, the petal-shape hasbeen adopted (a 20 point star-shaped mask was also another candidate, [31]). To ensure the best achievement, theexternal occulter has to remain accurately at its position during observation time. In our case, the telescope will be 7
- 10. in an orbit around the Sun-Earth L2 point and the occulter has to follow the orbit. To avoid noise coming from theSun reﬂectance on the occulter, the observations have to be reduced to the ones which match an angular position tothe Sun from 45 to 85 degrees (or slightly beyond if the occulter can be tilted, [32]). Currently, there is no missionin activity but a few missions are being studied e.g. the proposed THEIA (Telescope for Habitable Exoplanets andInterstellar/Intergalactic Astronomy) and NWO (New Worlds Observer). Now, the future JWST (James Webb SpaceTelescope) is the best candidate for an occulter mission on a short time scale.Figure 1.7: Principle of a starshade: the light coming from the star is stopped whereas the weak light from the planetis not stopped. Figure credit: Northrop Grumman CorporationManaging direct imaging is the most important march for the search of life. However, as indirect methods likeastrometry or radial velocities provide a direct measurement of the masses, orbital parameters and coordinates of theplanets, combining these methods would complete the characterization and supply all the information we want to getabout an exoplanet ([33]).Comparison between coronagraph and external occulterBoth occulter and coronagraph are acting in the same way: suppressing the light of the star to reveal the faintlight of the orbiting planet. However, if they do almost the same thing, we can wonder why one would send a hugespacecraft thousand of kilometers away from the telescope. An external occulter presents several advantages overinternal coronagraphs.But ﬁrst, we will discuss some of the drawbacks of an occulter. As the word ’external’ says, an occulter requiresanother spacecraft to maneuver it and thus the lifetime of the occulter depends on fuel consumption and on themission design. Since micro-engineering costs a lot for high performance coronagraphs, the cost of a occulter in orbitis not that much higher but it increases with its size. The time of travel between each target is also a burden forthe occulter. Almost two weeks of traveling is necessary to move to the next target (Design Reference Mission isbuilt to optimize as much as possible all these time constraints by reducing all the impacted parameters). Next,lower contrast caused by the deformations of the occulter, and speckles may arise from manufacturing, deploymentor micro-meteorite hits in ﬂight.The small size of a coronagraph may be source of material imperfections. One parameter only controlled by the ex-ternal occulter, as it can move backward and forward, is the inner working angle. It is not ﬁxed, and there is no limitfor outer working angle too (due to the absence of deformable mirror correction of the coronagraph speckles). Stilldealing with the inner working angle, in the case of coronagraph, it depends on the wavelength (IW A ∝ λ/D): thehigher the wavelength, the higher the inner working angle. Because exo-Earths will be found at low separations, thecoronagraphs are usually designed to provide a spectrum between 250 and 1000 nm ([34]). Starshades are typicallynot limited in the size of the bandpass. For internal coronagraphs, starlight suppression over a broad band is morechallenging and typically limited to 10-20%. The contrast obtained is also better for external occulters and 10−10 isachieved for most of the cases (see the result part) and this can be helped by the fact that the primary mirror andsupporting optics of the telescope have less constraints with an occulter. Finally, as a Lyot-type coronagraph is muchmore complex, a simple reasoning explains that with fewer optical systems, less signal is lost.The following table sums up the characteristics we compare between a coronagraph and the external equivalent: 8
- 11. Characteristics Coronagraph External Occulter Cost + - Signal - + Lifetime + - Deployment + - Scattered light - + Position control - + Higher suppression - + Inner Working Angle - + Usable for any telescope instruments - + Performance in spectral characterization - +Instead of wondering if an occulter is better than a coronagraph, the idea of doing both light suppression have beenexcogitated. Combining an occulter with a coronagraph makes the science really more complicated. The occulterhas to be designed for the coronagraph and moving the occulter will require the coronagraph to be changed. And foreach new conﬁguration, tests need to be done to verify the ability of the system. These are called hybrid occulters.They could reduce the size of the occulter and therefore the distance, economizing time and fuel. Occulters withapodized pupil Lyot coronagraph and achromatic interfero-coronagraph have been described by Cady ([35]) showingtheir advantages and drawbacks.JWSTJWST is a large space telescope, optimized for infrared observations, scheduled for launch in 2014. Its goals are toﬁnd ﬁrst galaxies, planetary systems formation, and evidence of the reionization ([36]) during a ﬁve-year mission butwill have enough fuel to run over 10 years. It is due to an international collaboration between NASA, the EuropeanSpace Agency (ESA), and the Canadian Space Agency (CSA). The James Webb Space Telescope was named after aformer NASA Administrator.The telescope will stand in the Sun-Earth Lagrangian 2 point, at 1.5 millions km from the Earth, after a trip of 30days. It is composed of a large mirror, 6.5 meters and oﬀers 4 scientiﬁc instruments covering infrared wavelengths:NIRCam (Near Infrared Camera), NIRSpec (Near Infrared Spectrograph), TFI (Tunable Filter Imager) and MIRI(Mid Infrared Instrument). Here we explain most of their abilities in the case of a starshade, at low resolution. • NIRCam: insuring observations between 0.6 and 5 microns (tow arms, one short from 0.6 µm to 2.5 µm and one longer from 2.5 µm to 5 µm), with a high sensitivity between 2.2 and 5 microns, the NIRCam is the ﬁrst camera of the JWST. It will be used for the search of planetary companions, mainly Jupiter-sized planets, for the search for protoplanetary disks, stellar populations in nearby galaxies, and also for the characterization of galaxies at very high redshift, mapping dark matter. NIRCam is composed of two modules for broad- and intermediate- band imaging where traditional focal plane coronagraphic mask plates will be used and two diﬀerent wavelength channel outputs. Lastly, NIRCam will be used as a wave front control module to check alignment and shape of the 18 hexagonal-shaped mirror segments ([37]). • NIRSpec: also covering from 0.6 µm to 5 µm with diﬀerent quantum eﬃciency above and below 1.0 µm (respectively ≥ 80% and ≥ 70%), the Near Infrared Spectrograph uses a 0.2 arcsec slit with resolution of 100, 1000 and 2700. Low resolution is advised with regards to sensitivity and exposure time whereas higher resolution could be used for giant planets. • TFI: the Tunable Filter Imager is, as it is called, an imager. The fact that the light is not dispersed reduces considerably the contrast sensitivity, making this instrument inappropriate with the use of a starshade. • MIRI: the Mid Infrared Instrument is composed of two spectrographs, one for low resolution and on for medium. Giant planets with an external occulter could be the goal of this instrument, however, similar eﬃciency as NIRCam and NIRSpec is reached for an habitable zone at 400 mas, and there is an impossibility to combine the low resolution spectrograph with ﬁlters. Theses particularities jeopardize a suﬃcient eﬃciency in the search of exo-Earths with the use of a starshade.A starshade for JWST is one of the New Worlds Observer mission ([38] & [39]), in supplementation of THEIA ([40]),CESO ( Celestial Exoplanet Survey Occulter [41]) and O3 (Occulting Ozone Observatory [42]) which all use apodizedof binary occulters, optimized for a wide variety of wavelengths. JWST by itself will require the help of an externalocculter to be capable of directly imaging planets in the habitable zone; the Hubble successor is one of the perfectcandidates for this mission. It would be the fastest and most aﬀordable path to the discovery of life as the resultingcost of this kind of mission have been estimated at about 1 billion dollars ([43]). 9
- 12. Figure 1.8: James Webb Space Telescope, with the on-axis primary mirror of 6.5 meters diameter composed of 18hexagonal mirrors. Figure credit: NASAThe starshade will be launched after the JWST (in the best case, 6 months after). The starshade will be in orbitaround the Sun-Earth L2 point. Due to the large distance between the occulter and the telescope, it has to covermany thousands of km for each star. In the 5 year planned mission, almost one week is required between each starobservation for the travel, 24 hours for imaging and 2 weeks for the spectroscopy science. There is a Design ReferenceMission (DRM, [34], [44] & [45]) built for optimizing the time travel and the number of discoveries based on thefrequency of Earth-like planets. Thus the DRM would maximize the number of planets discovered, their spectralcharacterizations, productions of orbital ﬁts. At the same time it would maximize the percentage of the targetlist observed. At the end, for an occurrence rate of planets of 0.3, there would be 5 habitable Earth-mass planetsdiscovered for a small fraction of JWST observing time (say 7%) and the probability of zero discoveries would be0.004 ([45]).Figure 1.9: Route followed by the starshade to join the Sun-Earth Lagrangian 2 point, for a case of a 50 meterocculter at 50000 km, launched 3 years after the space telescope. Figure credit: W.Cash et al. 10
- 13. Chapter 2Starshade: design, optimization andproperties2.1 Design and optimization2.1.1 Free-Space propagation from starshade to telescopeAs the name tells us, an external occulter, also called a starshade, obscures the light coming from a star. We placea large occulter in the path of the light between the star and our telescope. The latter has to remain in the shadowproduced by the occulter. We manipulate the design and position of the occulter to have a control over the size ofthe shadow, and also over the contrast between both stellar and planet luminosity.Our purpose is to get the expression of the light intensity, traveling in ﬁnite distances. One of the unwanted eﬀectsdue to a circular disk mask is the Poisson’s spot, a bright spot in the center of the shadow which irradiance is nearlythe same as without any occulter ([46]). It results in a diﬀraction pattern dependent on size, shape and distance ofthe starshade relative to the telescope.On paper, we work with the electric ﬁeld in the telescope’s pupil plane to express the light observed since its intensityis given by the squared modulus of the electric ﬁeld. We start with the Babinet’s theorem ([47]). The light propagatingfrom an unobstructed star (Eu ) is the same as the light coming from an on-axis hole (Eh ) plus the light coming fromthe complement of that hole (Eo ): Eu = Eh + Eo .Next we can write the plane wave equation in terms of the polar coordinates (ρ, φ) of the telescope pupil plane, (ρ=0being the center of the plane and ρmax the top) 2πiz Eu (ρ, φ) = E0 e λ ,with z being the distance between the occulter and the telescope. As previously said, a circular-shaped screen cannotstop the light of a star without Fresnel diﬀraction eﬀects. In this way, Spitzer ([29]) built up a way to suppressthe Poisson’s spot with the help of an apodization function A(r, θ), meaning that we consider the occulter beingpartially attenuated, with r and θ, the polar coordinates of the occulter. As we assume circular symmetry, we haveA(r, θ) = A(r) (same result for φ). This function is equal to 1 for total obscurity and 0 when all the light propagates,so that it describes the whole occulter. All these tools combined, we have ([46]): Eo = Eu − Eh , Eu (ρ) = E0 e2πiz/λ , R E (ρ) = E 2π e2πiz/λ eπiρ2 /λz × 2πrρ 2 )A(r)e(πi/λz)r rdr, h 0 J0 ( iλz 0 λzwith J0 , Bessel function of the ﬁrst kind, order 0, and the Fresnel integral being Eh (ρ), which gives us the total ﬁeldfor the occulter-telescope system: R 2π 2πrρ πi (r2 +ρ2 ) Eo (ρ) = E0 e2πiz/λ 1− × A(r)J0 ( )e λz rdr . iλz 0 λz 11
- 14. 2.1.2 Binary apodization, shape of the occulterA continuous graded occulter cannot be built. The idea has been proposed by Spitzer ([29]) to build a binary occulterwith a workable shape. Thus we approximate our continuous apodization by a binary occulter composed of N evenidentical petals arranged around a circular central part ([31] then [38]). We use a similar new expression of theimage-plane electric ﬁeld E(ρ, φ) = e−2πiρ cos(θ−φ) rdrdθ Swith S being the mathematical description of the petal-shape mask. Using among others the Jacobi-Anger expression([48]), we compute the following result for a propagated ﬁeld with an occulter ([49] & [50]): Eo (ρ, φ) = ∞ R 2π(−1)j 2 +ρ2 )/λz 2πrρ sin(jπA(r)) Eo (ρ) − E0 e2πiz/λ eπi(r JjN rdr ∗ (2 cos(jN (φ − π/2))) j=1 iλz 0 λz jπwith Eo (ρ) being the previous electric ﬁeld for a graded apodization, JjN the jN th order Bessel function whose eﬀectexponentially decreases for high N and j > 0 so that Eo (ρ) becomes predominant. Indeed, terms in the sum over jconverge to 0 quickly enough so that our optimization codes only require writing Eo (ρ) ([31]).Next, the translation between the apodization function and the ﬁnal shape of the occulter is easy: the angular width∆θ(r) will be expressed as: 2π ∆θ(r) = A(r) Nwhere N , the number of petals. Hence, the width of the petals, R∆θ, is directly mapping the apodization functionwith R being the radius: 2π ∆ = R ∗ ∆θ = R A(r) NIf N → ∞ we result in a graded occulter. However to keep a range for a feasible occulter, calculations of the averagesuppression over the shadow proﬁle on the telescope for diﬀerent wavelengths have shown that 16 petals were a goodcompromise ([51] & [38]).2.1.3 Optimization of the occulter apodization functionAmong the numerous variables, we can ﬁt the best occulter based on the physical constraints. As previously said,we need to get a contrast (or a suppression) of about 10−10 in the focal plane. A starshade can be optimized fordiameter, suppression level, wavelength range, shadow size, petal length and inner working angle (IWA). Other linkedvariables, like the telescope aperture size itself, can also be changed in the research ﬁeld of adding external occultersto general space telescopes.ParametersTo shape an occulter, there are several parameters which come into play. Here we describe the main characteristicsof the occulter and we will see in the result section how they behave together.The diameter is one of the most important features. As the suppression increases with the size, bigger occulterswould be more favorable. However, if the contrast increases with the size for a constant geometric IWA, so does thedistance, the cost and the time for moving it. A good frame size for an occulter is 60 - 100 meters in diameters for alarge telescope of the size of JWST since the size of the occulter depends on the size of the shadow. As of now, anocculter larger than 80 meters would encounter technological issues.Then, with the addition of a binary apodization function, we establish a diﬀerence between a circular central partand the petals. Their lengths modify the suppression achieved and they are also subject to deployment concerns, i.e.longer petals would be harder to bloom and control. Decreasing the size of the petals will reduce the suppression fora reasonable number of petals, as we come closer to the circular occulter without apodization.The size of the shadow is important too. If we need to suppress the light coming from the star, the shadow provided bythe occulter needs to be large enough to cover the telescope aperture over the wavelengths studied. The shadow needsto have at least a 1 meter margin to make sure we achieve the required contrast ([52] & [43] and see section 2.2.6).Even if we ﬁrst consider the contrast as a goal, it can also be considered as a parameter we get after optimization.How much contrast we get as a function of the shape will be explained in depth in the section 2.2.3.The Fresnel number is deﬁned by D2 F = λz 12
- 15. with z, the distance of the occulter. For a given Fresnel number, the form of the shadow, or the contrast, created bythe occulter will remain the same. Thus, for a deﬁned wavelength, we are able to establish a proportionality betweenthe distance z and the diameter D. Moreover, since the inner working angle equals IW A = D/2z, we have: 2 ∗ IW A ∗ D F = λThereby, the science we are looking for will set the range of parameters describing the starshade, and by taking aFresnel number, we can ﬁgure how IWA, diameter and wavelength are correlated. To keep the same contrast, thefresnel number needs to be identical. For a constant diameter, λ ∗ z remains constant. Therefore the starshade canbe used for observations at longer wavelengths by moving it closer to the telescope (this in turn increases the IWA).The range of wavelengths we look at matters in our case of a starshade for JWST. Spectral characterization of the life-signatures like O2 , O3 , H2 O, CO2 , or CH4 are optimally found between 0.7 µm and 2 µm. In the case of the JWST,NIRSpec and NIRCam detectors work best between 0.6 and 5 µm, with diﬀerent quantum eﬃciency depending onthe wavelength. However outside the optimal band pass, especially close to the red above 2 µm, the starshade startsto leak starlight, reducing considerably the contrast. This light takes the form of speckles in the focal plane and thusthe use of ﬁlters with good out-of-band rejection is required to suppress enough this red leak ([43]).In summary the basic parameters for the starshade are • occulter diameter • petal length • inner working angle • distance (related to occulter & IWA) • wavelength range • shadow size.We next use them as the ’x’ values of the apodization function and write the physical equation of the electric ﬁeldso that we can easily manipulate these parameters.Analytical optimizationA useful tool to shape the petals through the apodization function is the hypergaussian function. It has been usedand set up to be the best mathematical expression for the apodization. Developed by Cash ([38]) the function isbased on the following expression: 1 ∀ r ≤ a, n A(r) = exp − r − a ∀ r ≥ a, b where a is the radius of the central part of the occulter and therefore b is the complementary distance which gives usthe radius of the occulter, i.e. the petal length. As the Hypergaussian is a mathematical function, the exponentialis endless and the deﬁnition of the inner working angle has been provided by Cash at AIWA = 1 , corresponding to a etransmission of almost 63%. a and b are given intrinsically by the value of the occulter central part (OCP ) and theocculter diameter (D): a = OCP ∗ D and b = (1 − OCP ) ∗ D. n is a parameter for the petal shape set to the value6 ([52]). The hypergaussian function presents the advantage of being an analytical application reducing, the timeof calculation. This is in contrary to the following optimization which requires minimization of calculation paths.Moreover, the hypergaussian is independent of the number of wavelengths since it is monochromatic. For each steprequiring the wavelength, the average value λ = 1.7 µm is taken, corresponding to the maximum of the broadbandof interest. For smaller wavelengths, the contrast will be better (see ﬁgure 2.5).Another function given by Copi & Starkman ([53]), is based on the transmission function τ (r) which equals 1 − A(r).This transmission function expresses the expansion of the diﬀraction pattern through Chebyshev polynomials. Theywrote the transmission function as N τN (y) = cn y n n=0where N the order of the occulter and with (r/R)2 − y= 1− 13
- 16. where is the fractional radius of the center of the occulter, which gives us, for a four-order occulter: A4 (y) = 1 − τ4 (y) = 1 − (35y 4 − 84y 5 + 70y 6 − 20y 7 ).They took =0.15 in their consideration. It turns out that the hypergaussian function is similar to the ﬁrst order ofthe Copi & Starkman development, which makes it a harder but analytical way of optimization.I wrote a code to generate starshade proﬁles with the hypergaussian function which calculates the contrast for arange of parameters. I integrated the calculation with the existing functions for numerical optimization described inthe next section, in order to compare both approaches.Numerical optimizationThe main numerical way to get an optimized apodization function has been developed by Robert J. Vanderbei [51]using linear programming. This mathematical operation, also used in economics, management or engineering, appliesthis following operation: Minimize: (c · A) ≥ ≥ Subject to: (m · A) = b & A = d. ≤ ≤with c and A vectors, m a matrix, and b and d vectors corresponding to the constraints.In our case, the goal is to get the best occulter shape by using Fourier optics for the system telescope + starshade.To do so, we reproduce the approach used by Vanderbei et al. ([31]) and constrain the intensity ratio of the light overthe pupil plane to be less than 10−10 . As the intensity is not linear, we settle the matter by expressing the constrainton the related electric ﬁeld so that: |Eo (ρ)|2 ≤ 10−10 |Eo |2with R 2π 2πrρ πi (r2 +ρ2 ) Eo (ρ) = E0 e2πiz/λ 1− × A(r)J0 ( )e λz rdr . iλz 0 λzWe can already see that the ﬁrst exponential e2πiz/λ disappears in the calculation of the modulus. As written above,the electric ﬁeld is described by the apodization function A(r). We use linear programming to minimize the sumof the apodization function, which is described in matrix format by the scalar product c · A(r), with c, a simpleunit vector, so that c · A(r) = A(r). This objective function has little impact on the results in this case, becausethe contrast goal will be placed on the constraints (together with other constraints described below). Since Eo (ρ) iscomplex, the assumption is taken that the constraint Re(Eo )2 + Im(Eo )2 ≤ 10−10will correspond to −5 −5 −10 ≤Re(E ) ≤ 10 √ o √ 2 2 −5 −5 −10 ≤Im(E ) ≤ 10 , √ √ o 2 2 √with the amplitude E0 removed and the 2 coming from the constraint expression scaled to a circle around 10−5 .Thus, we get a system of four inequalities. In order to make it linear for optimal solutions ([54]), we write in the code πi 2 2the integral inside Eo (ρ) as the sum of all the area elements under the curve i J0 ( 2πri ρ )e λz (ri +ρ ) ri ∗ A(ri ), like a λzRiemann sum approximation of the integral. For convenience, we write 2πri ρ πi (ri +ρ2 ) 2 J0 ( )e λz ri → φi , χi λzso that Re(Eo ) ∝ φi ∗ A(ri ) & Im(Eo ) ∝ χi ∗ A(ri ).Moreover, we can add constraints on the apodizer itself. Firstly, by deﬁnition, A(r) will be bound by 0 and 1.Secondly, we impose A(r) to be equal to one for the central part of the occulter (the fully opaque central disk).Thirdly, in the monochromatic case, the natural solution of such an optimization is a ”bang-bang” solution which is adiscontinuous function (like a bar-code). In order to avoid this problem we add a smoothness constraint σ bounding 14
- 17. the second derivative of A(r). We also add a constraint on the ﬁrst derivative so that the petal width decreasesmonotonically (the width of the petal is directly proportional to the apodization function). We ﬁnally have for A(r): A(r) = 1 ∀ 0 ≤ r ≤ a, 0 ≤ A(r) ≤ 1 ∀ a ≤ r ≤ R, A (r) ≤ 0 ∀ 0 ≤ r ≤ R, |A (r)| ≤ σ ∀ 0 ≤ r ≤ R. The ﬁrst and second derivatives are expressed using respectively the ﬁnite diﬀerence expressions: A(r + 1) − A(r) A(r + 1) − 2A(r) + A(r − 1) lim and lim . h→0 h h→0 h2Here, h will correspond to the number of points n taken for the calculations, so that limh→0 will be limn→∞ . Thelinear programming formalism allows to combine diﬀerent constraints. For that, we simply combine all the constraintsin one large matrix: Minimize: c · A(r) = ( .. 1 .. ) · A(r), m·A b √ .. Re(Eo ) .. ≥ and ≤ ±10−5 / 2 √ ≥ and ≤ ±10−5 / 2 .. Im(E ) .. A(rr<a ) =1 o Suject to: & . A(rr≥a ) ∈ [0, 1] .. A (r) .. ≤0 .. |A (r)| .. ≤σand it returns a table of the function A(r) in the case where the data set can converge to a solution. If not, thecalculation stops and returns an error message.As Eo depends on the wavelength, the calculations have to be done over a suﬃcient range of λ covering the bandpasswe are looking at, meaning that all the constraints are also repeated for each wavelength.The contrast as a parameterI used the ﬁrst optimization code described above to generate starshade designs for a large range of parameters. Thisapproach is appropriate to design a starshade for a speciﬁc goal, but for a systematic study of the parameter spacethe optimizer may or may not deliver a result; e.g. there may not be a possible starshade proﬁle achieving 10−10suppression for a given set of constraints (diameter, IWA, shadow size, petal length). Given an existing code for thebasic optimization of starshade described above, I wrote a new optimization code that includes the contrast in theobjection function of the optimizer. This is described below following a method described by Cady ([35]). Next, Ichanged the code to add the contrast as the result of the optimization, no longer constraint.Starting from: −5 Re(E ) ≤ 10 −10−5 o √ & Re(Eo ) ≥ √ 2 2 10−5 Im(E ) ≤ √ & Im(E ) ≥ √ , −10−5 o o 2 2 −10the square root of the contrast 10 becomes k, a parameter, such as: k k Re(E ) ≤ √ & Re(E ) ≥ √ o o 2 2 k Im(E ) ≤ √ & Im(E ) ≥ √ , k o o 2 2The trick is to remove the contrast of the constraints. To do so, we write: k k Re(E ) − √ ≤ 0 & Re(E ) − √ ≥ 0 o o 2 2 k Im(E ) − √ ≤ 0 & Im(E ) − √ ≥ 0, k o o 2 2Then, the vector expressing the apodization requires the addition of one term, the contrast k. We rewrite the matrix 1to express Re(Eo ) − √2 so that: .. −1 A(ri ) k .. φi , χi .. √ · .. = φi , χi ∗ A(ri ) − √ 2 i 2 k 15
- 18. and the last modiﬁcation will be in the scalar product minimized: c · A(r). As we need to get the best contrast, wewrite: c1 A(r1 ) 0 .. .. .. c such as ci · A(ri ) minimized, give us the minimum value of k, i.e. c = 0 . .. .. .. cn k 1Constraints dealing with the ﬁrst and second derivatives of the apodization function just need to be resized to thedimension of A(r) and we burke the suppression by putting a 0 inside the new vectors. With this scheme, the sum ofthe apodization proﬁle is no longer minimized. This does not impact the result since the optimizer is entirely drivenby the constraints in this case. With this new version of the code, the optimizer delivers a result whatever the set ofconstraints may be. The output of the optimizer is both the apodization proﬁle and contrast, concatenated in a longvector. This code is more adapted for the parameter space study we describe in the next section.Creation of widths at tips and gapsAnother improvement I made is creating a concrete expression of the apodization function. The numerical solutionshave limited size petals, corresponding to the size of the array used for A(r), however the width of the tip of thepetal is unconstrained and may reach non-realistic values. Analytical optimization has endless petals that need tobe truncated at some radius, which is studied in section 2.2.2. The problem is the same for the ”valleys” betweentwo petals. The purpose is to create a width characterizing both tip and gap size at the same time (they can takediﬀerent values). In order to do so, we add two lines of constraints ([35]): one will ask the translation of apodizationFigure 2.1: Positions of the tip and gap on an external occulter. We want to build width of the order of the millimeter,and as the petal size is more than 10 meters, gap and tip widths will not be observable.function in width at the bottom of the petal (r1 ) to be 1 minus the gap, ∆gap . The second one will ask it to be πequal to a thickness ∆tip at the edge (r2 ). We use ∆ = R∆θ(r) = R N A(r) (here, the factor of 2 disappears as wedeal with half of the petal) to write the following new constraints: π R ∗ (1 − A(r1 )) ≥ ∆gap N π R ∗ A(r2 ) ≥ ∆tip N π ∆ − ∗ A(r1 ) ≥ gap − π N R N π ∆tip ∗ A(r2 ) ≥ N r1 R (.. 0 .. − π .. 0 ..) · A(r) ≥ ∆gap − π i.e., in our matrix: N R N r2 ∆ (.. 0 .. 0 .. .. π ) · A(r) ≥ tip , N R 16
- 19. As we ask the function to make a leap in two points, the smoothness constraints might be relaxed to allow the op-timization convergence. Here, I wrote it by nullifying the expression of the second derivative on three points aroundthe gap so that the discrepancy is not taken into account.Of course, we can combine both improvements in order to get the best suppression for these new petals. A realisticnumber for manufacturability would be a minimum size of 2 mm for these features. This lower limit is suﬃcient forthe tip and gap widths, without changing the eﬃciency of the starshade as we can observe in ﬁgure 2.2 (this is easyto imagine when we think about the size of a petal, between 15 and 20 meters). More calculations have to be made in 10 4. 10 10 3. 10 Contraste 10 2. 10 10 1. 10 0 1 2 3 4 Width mmFigure 2.2: Contrast as a function of the width at tips and gaps (both the same) for an occulter of 75 meters indiameter, at 100 mas and with an occulter central part of 50%. The contrast remains under the 10−10 requirementat 3 mm and starts going up exponentially after, exceeding 10−10 . Here the changes in the program made the startat 0 mm diﬀerent of 1 · 10−11 between the original one, creating a small discrepancy in the results.order to see the diﬀerence in behavior between the tip and the gap over all ranges of diameter, petal length and innerworking angle. Following similar logic, the addition of tensioning elements and some changes in the petal structurehave been used to see how the contrast varied between them ([49]). However, in general, the more constraints weadd to the problem, the larger the starshade becomes. 17
- 20. 2.2 Global study of the parameter space and results2.2.1 GoalWe now have enough tools to describe the starshade properties giving the suppression we are looking for. We runmany calculations to explore the abilities of the starshade over the range of parameters describing the stellar system.The inner working angle will take values from 80 to 120 mas, corresponding to most of the range of the habitablezone. We start with starshade diameters of 60 up to 100 meters. 100 meters would certainly be unrealistic, but weinclude a large range of diameter to understand the behavior of the starshade according to its design parameters.The distance between the occulter and the telescope is between 50,000 km and 100,000 km. Next, the petal lengthvaries between 30% and 70% of the total size. More than 70% would be unfeasible and less than 30% would makethe occulter come closer to a one without apodization. The shadow size will take values from -1 to 8 meters (for themargin in comparison to the telescope size). At ﬁrst, the contrast, as a constraint, will be set at 10−10 as planned.Then, we want to optimize this value in regards to all the others.About time of calculationEach creation of an apodizer proﬁle for a given set of data, in the case of numerical optimization runs between 10and 20 minutes depending on the precision (number of points, constraints). Typically we use 4000 points alongthe apodizer proﬁle, 11 wavelengths and 11 points along the shadow proﬁle at the telescope aperture. For ﬁgureslike ﬁgure 2.11, ﬁgure 2.9, or ﬁgure 2.6, the diameter takes almost 8 values, the inner working angle 5 values, thecentral part 9 values, which gives us almost 360 diﬀerent panels. Thus we have 360*20*60/3600 = 120 hours ofcalculation. This long time can be reduced by decreasing the number of wavelengths, or the number of points acrossthe apodization proﬁle A(r), and the number of points across the shadow proﬁle at the telescope aperture. Howeverthis will signiﬁcate less accurate values (it gets a useful practice only for testing the programs). Moreover, trying toget an higher accuracy or changing the values of the shadow oversize, gap and tip sizes, or any other parameter (likechanging the step of the diameter to 1 or 2 meters) will increase considerably the time of calculation.On the contrary, since the hypergaussian function proceeds to take an analytical optimization so that the apodizationis already known, we only calculate which contrast it returns. Because the hypergaussian is always better at shorterwavelengths, we calculate the monochromatic contrast at the longest wavelength of the band. This monochromaticcharacter also reduces the time and makes the hypergaussian easier to manipulate. Each calculation takes less than aminute mainly limited by the Fresnel propagation from occulter to the aperture. However, for the second method ofcalculating an analytical optimization, seen in 2.2.7, we need to compute a complete numerical apodization so thatit takes as much time as the numerical solution.2.2.2 First comparison between analytical and numerical methodsFor similar properties, we have a look at the behavior of the two optimization methods for the occulter’s shape.Figure 2.3 shows the diﬀerences between linear programming and hypergaussian in the apodization function. We 1.0 0.8 Occulter profile 0.6 Numerical 0.4 Analytical 0.2 0.0 0 1000 2000 3000 4000 Radial positionFigure 2.3: Apodization function for numerical approximation with linear programming and analytical optimizationwith the hypergaussian function for an occulter diameter of 75m, an IWA of 100 mas, and a central part of 50% inthe case of a JWST-like telescope.can see that there are some steps on the numerical apodization. We also note the hypergaussian function is ﬁtted 18
- 21. to the constraints so that it goes down faster than the numerical. Moreover, because the hypergaussian function isexponential, it never reaches the 0 value, but reaches very small values quickly as the typical value for the hypergaus-sian exponent (which is characteristic of the fall) is n=6. It is diﬃcult to deﬁne where to end the written apodizer.Indeed, in our plot, the true diameter of the analytical apodizer is not known. In order to calculate the IWA, Cash([38]) deﬁned the diameter for a transmission to be up to 1 − 1/e. In this way, we understand why the diﬀerenceis well marked. A 50 meter diameter deﬁned in this way would actually be a 60 meter diameter, tip to tip ([38]).However, the propagation takes into account all the apodizer proﬁles up until a zero value. The next plot 2.4 showsus how the contrast varies when we expand the diameter to its true value. To do so, we make a run over a coeﬃcientwhich extends the radius for the propagation calculation. We start from 1 up to 1.6. As we can see, the radius needs to be multiplied at least by 1.3 to satisfy the constraints and provide a good 1 0.01 Contrast 4 10 6 10 60 65 70 75 80 85 90 95 True Diameter mFigure 2.4: Behavior of the contrast as a function of the true diameter. We multiply the value of the radius by acoeﬃcient which will express how much longer the true diameter is (in the case of a 60 meter diameter). Beyond acertain diameter the truncation of the hypergaussian has virtually no eﬀect (here beyond 75m). Therefore, we use acoeﬃcient from the 1/e diameter to include the entire tip of the hypergaussian. Since the propagation is calculatedfor the entire array, the result does not depend on the coeﬃcient value.suppression (here, the 60 meters occulter made with the analytical optimization provides 10−8 suppression). Theocculter diameter will proceed from 60 to 78 meters (and so on for bigger starshade).As previously said, the hypergaussian function is monochromatic. Figure 2.5 shows us how the contrast changes whenwe change the maximum wavelength. Clearly, increasing the range of the spectrum, and therefore the wavelengthof interest, will damage the contrast quickly and thus a bigger starshade will be required. Later in section 2.2.7, wewill create the hypergaussian in another way by ﬁtting a numerical optimization to it in order to ﬁnd a consistentdeﬁnition for both approaches. 10 5 10 10 1 10 11 5 10 Contrast 11 1 10 12 5 10 0.8 1.0 1.2 1.4 1.6 Maximum Wavelength ΜmFigure 2.5: Contrast of the hypergaussian apodization as a function of the wavelength. Smaller wavelengths willprovide a better suppression, and increasing the wavelength will deteriorate the contrast in a logarithmic way. 19
- 22. 2.2.3 Contrast as a function of the diameterHere, we have a look at the general behavior of the contrast subject to the variations of diameter at diﬀerent innerworking angles for a constant central part size of the occulter. We make an imposing run over the whole range ofdiameters and inner working angles. In ﬁgure 2.6, we can see a logarithmic behavior of the contrast. As the inner working angle gets smaller, the 7 7 10 10 8 8 10 80 mas 10 80 mas 90 mas 90 mas 9 9 10 10 Contrast Contrast 100 mas 100 mas 10 10 10 10 110 mas 110 mas 11 11 10 120 mas 10 120 mas 12 12 10 10 60 70 80 90 100 60 70 80 90 100 Occulter Diameter m Occulter Diameter mFigure 2.6: Values of the contrast as a function of the occulter diameter for diﬀerent inner working angles. Left:obtained with the numerical optimization. Right: obtained with the help of the hypergaussian function. For thenumerical optimization, we notice that up to 85 meters would be suﬃcient to get all the range of inner working angleswanted. 7 10 1.0 Numerical 8 10 0.8 AnalyticalOcculter profile 9 10 0.6 Contrast Numerical 10 10 0.4 Analytical 11 0.2 10 12 0.0 10 0 1000 2000 3000 4000 60 70 80 90 100 Radial position Occulter Diameter mFigure 2.7: Left: Shape of the apodization for a similar contrast between numerical and analytical optimization at62m. Right: their comparison in suppression at various diameter, overlaying at 62m. Very early, the numericaloptimization becomes more eﬃcient than the other one, with a peak at around 85 meters. At 75m, the numericalsolution is nearly 9 times better (Contrast for the numerical: 4.383·10−11 and contrast for the analytical: 4.301·10−10 ).Similarly, the 75 m numerical solution has a contrast of 4.383·10−11 ; this contrast is obtained for a diameter of almost93 meters with an analytical solution.suppression requirement of 10−10 is achieved for high values of diameter. On the contrary, high inner working angleswill easily reach the suppression for smaller diameters. This is consistent with the normal thought: bigger occulter= higher suppression = smaller inner working angle.In the second plot of ﬁgure 2.7, we have the two methods for a given inner working angle (here, 100 mas). We can seethat for deeper contrast, the diﬀerence between the hypergaussian and linear programming apodization gets higher,and for the 10−10 requirements, we have a diﬀerence of almost 12 meters in the telescope diameter. The following isthe evidence of the advantage of a numerical optimized apodization-built occulter in comparison to the hypergaussianfunction. If we increase the value of the inner working angle, the separation between the two methods will go down,for as much as 8 meters for 120 mas. However, at 100 mas, an occulter smaller than 62 meters would be preferablybuilt with the help of the hypergaussian function. The ﬁrst plot is similar to ﬁgure 2.15 and expresses how the shapeof both apodizations will look like for a similar contrast (our 62 meter occulter in that case). It is interesting to seethat even for a similar contrast, the shape is actually quite diﬀerent. But we have to remember that the size of thisanalytical apodization is deﬁned up until the 1/e transmission point and will in fact correspond to a diameter of atleast 10 meters more (considering the supposed-inﬁnite length of the function). 20
- 23. 2.2.4 Distance of the occulter as a function of the diameterIn this part, we change the variables. Using the fact that the inner working angle, diameter and distance betweenthe occulter and telescope are linked together through IW A = (Diameter D)/(2 ∗ Distance z), we get the distancewith: D(m) 1 z(km) = ∗ 10−3 2 IW A(mas) ∗ 10−3 ∗ 180 3600 πFor a given ratio between the central part and petal length (here it is 49%), we select the 10−10 contrast. For eachdiameter, we select the smaller distance between the occulter and the telescope. In ﬁgure 2.8, the plot describes 130 000 130 000 120 000 120 000 Distance Distance 110 000 110 000 80 mas 80 mas Distance km Distance km 100 000 90 mas 100 000 90 mas 90 000 100 mas 90 000 100 mas 80 000 110 mas 80 000 110 mas 70 000 120 mas 120 mas 70 000 60 000 60 000 65 70 75 80 85 90 95 100 65 70 75 80 85 90 95 100 Occulter Diameter m Occulter Diameter mFigure 2.8: Distance of the occulter for less than 10−10 as a function of its diameter. Left: by numerical optimization.Right: by analytical optimization. In the case of the analytical apodization, each time the diameter is not plotted, itmeans that a design was not generated for this set of constraints (too small of a diameter).the distance from the occulter to the telescope required to reach 10−10 suppression. There are two similar behaviorsbetween the numerical and analytical optimizations but with diﬀerent conﬁgurations of data. We can see for example,an angular resolution of 100 mas in the extra-solar system will require an occulter of about 89 m at a distance of90,000 km for the hypergaussian apodization. In comparison, the numerical apodization requires an occulter of about75 m at 75,000 km. Calculations of the distance are made in regards to the formula IW A = D/2z. By translatingthe curve, we clearly see how much we gain in the occulter size and distance: almost 15 meters in diameter and 15000 km in distance in average.2.2.5 Action of the petal length on the contrastUsing the basic optimization scheme with the contrast as a constraint, the program stops each time the constraintscannot be satisﬁed. This method presents the advantage to directly show the limit of the starshade. However, wedon’t have any information about what would be the best contrast for this set of constraints. Here, we make a runover diﬀerent sizes of petal and occulter in order to see how behave these two parameters together. In ﬁgure 2.9, 1.0 24 0.8 22Petal length ratio Petal length m 0.6 20 0.4 18 0.2 16 0.0 65 70 75 80 85 90 95 65 70 75 80 85 90 95 Starshade diameter m Starshade diameter mFigure 2.9: Minimum of the petal length ratio (ﬁrst plot) and petal length (second plot) as a function of the diameterin order to reach the suppression requirement of 10−10 . Increasing the size of the diameter means decreasing thepetal length, which is more practical for the starshade’s engineering. 21

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