N. Jeremy Kasdin

EXTRASOLAR PLANET FINDING VIA OPTIMAL APODIZED-PUPIL AND SHAPED-PUPIL CORONAGRAPHS

In this paper we examine several different apodization approaches to achieving high-contrast imaging of extrasolar planets and compare different designs on a selection of performance metrics. These approaches are characterized by their use of the pupils transmission function to focus the starlight rather than by masking the star in the image plane as in a classical coronagraph. There are two broad classes of pupil coronagraphs examined in this paper: apodized pupils with spatially varying transmission functions and shaped pupils, whose transmission values are either 0 or 1. The latter are much easier to manufacture to the needed tolerances. In addition to comparing existing approaches, numerical optimization is used to design new pupil shapes. These new designs can achieve nearly as high a throughput as the best apodized pupils and perform significantly better than the apodized square aperture design. The new shaped pupils enable searches of 50%-100% of the detectable region, suppress the stars light to below 10-10 of its peak value, and have inner working distances as small as 2.8λ/D. Pupils are shown for terrestrial planet discovery using square, rectangular, circular, and elliptical apertures. A mask targeted at Jovian planet discovery is also presented, in which contrast is given up to yield greater throughput.

Circularly Symmetric Apodization via Star-shaped Masks

In a recent paper, we introduced a class of shaped pupil masks, called spiderweb masks, that produce point-spread functions having annular dark zones. With such masks, a single image can be used to probe a star for extrasolar planets. In this paper, we introduce a new class of shaped pupil masks that also provide annular dark zones. We call these masks star-shaped masks. Given any circularly symmetric apodization function, we show how to construct a corresponding star-shaped mask that has the same point-spread function (out to any given outer working angle) as obtained by the apodization.

SPIDERWEB MASKS FOR HIGH-CONTRAST IMAGING

Motivated by the desire to image exosolar planets, recent work by us and others has shown that high-contrast imaging can be achieved using specially shaped pupil masks. To date, the masks we have designed have been symmetric with respect to a Cartesian coordinate system but were not rotationally invariant, thus requiring that one take multiple images at different angles of rotation about the central point in order to obtain high contrast in all directions. In this paper we present a new class of masks that have rotational symmetry and provide high contrast in all directions with just one image. These masks provide the required 10-10 level of contrast to within 4λ/D, and in some cases 3λ/D, of the central point, which is deemed necessary for exosolar planet finding/imaging. They are also well suited for use on ground-based telescopes, and perhaps the James Webb Space Telescope too, since they can accommodate central obstructions and associated support spiders.

Adaptive Neural Control of Deep-Space Formation Flying

A novel nonlinear adaptive neural control methodology is presented for the challenging problem of deep-space spacecraft formation flying. When the framework of the circular restricted three-body problem with the sun and Earth as the primary gravitational bodies is utilized, a nonlinear model is developed that describes the relative formation dynamics. This model is not confined to the vicinity of the Lagrangian libration points but rather constitutes the most general nonlinear formulation. Then, a relative position controller is designed that consists of an approximate dynamic model inversion, linear compensation of the ideal feedback linearized model, and an adaptive neural-network-based element designed to compensate for the model inversion errors. The nominal dynamic inversion includes the gravitational forces, whereas the model inversion errors are assumed to stem from disturbances such as fourth-body gravitational effects and solar radiation pressure. The approach is illustrated by simulations, which confirm that the suggested methodology yields excellent tracking and disturbance rejection, thus, permitting submillimeter formation keeping precision.

Nonlinear Modeling of Spacecraft Relative Motion in the Configuration Space

A reduced-input system problem is formed with the objective of preserving system input–output properties. For exogenous inputs, a reduced-order input system H∞ norm is maximized and an analytic solution is given. For control inputs, a reduced-order input system Hankel norm is maximized. Necessary conditions for the Hankel norm maximization are stated and an iterative solution is proposed. Although global convergence cannot be guaranteed, many practical applications have shown the achieved maxima to be very close to a known upper bound.

Optimal one-dimensional apodizations and shaped pupils for planet finding coronagraphy

The realization that direct imaging of extrasolar planets could be technologically feasible within the next decade or so has inspired a great deal of recent research into high-contrast imaging. We have contributed several design ideas, all of which can be described as shaped pupil coronagraphs. We offer a complete and unified survey of one-dimensional shaped pupil designs, some of which have been published in our previous papers. We also introduce a promising new design, which we call bar-code masks. With these masks we can achieve the required contrast with a fairly large discovery zone and throughput, but most importantly they are perhaps the easiest to manufacture and might therefore stand up best to refined analyses.

Runge-Kutta Algorithm for the Numerical Integration of Stochastic Differential Equations

This paper presents a new Runge-Kutta (RK) algorithm for the numerical integration of stochastic differential equations. These equations occur frequently as a description of many mechanical, aerospace, and electrical systems. They also form the basis of modern control design using the linear quadratic regulator/Gaussian (LQR/LQG) method. It is convenient, and common practice, to numerically simulate these equations to generate sample random processes that approximate a solution of the system (often called Monte Carlo simulations). It is shown in the paper that the standard deterministic solution techniques are inaccurate and can result in sample sequences with covariances not representative of the correct solution of the original differential equation. A new set of coefficients is derived for a RK-type solution to these equations. Examples are presented to show the improvement in mean-square performance.

Shaped pupil Lyot coronagraphs: high-contrast solutions for restricted focal planes

Abstract. Coronagraphs of the apodized pupil and shaped pupil varieties use the Fraunhofer diffraction properties of amplitude masks to create regions of high contrast in the vicinity of a target star. Here we present a hybrid coronagraph architecture in which a binary, hard-edged shaped pupil mask replaces the gray, smooth apodizer of the apodized pupil Lyot coronagraph (APLC). For any contrast and bandwidth goal in this configuration, as long as the prescribed region of contrast is restricted to a finite area in the image, a shaped pupil is the apodizer with the highest transmission. We relate the starlight cancellation mechanism to that of the conventional APLC. We introduce a new class of solutions in which the amplitude profile of the Lyot stop, instead of being fixed as a padded replica of the telescope aperture, is jointly optimized with the apodizer. Finally, we describe shaped pupil Lyot coronagraph (SPLC) designs for the baseline architecture of the Wide-Field Infrared Survey Telescope–Astrophysics Focused Telescope Assets (WFIRST-AFTA) coronagraph. These SPLCs help to enable two scientific objectives of the WFIRST-AFTA mission: (1) broadband spectroscopy to characterize exoplanet atmospheres in reflected starlight and (2) debris disk imaging.

Stability and control of spacecraft formation flying in trajectories of the restricted three-body problem

Abstract This paper addresses the problem of relative position control of spacecraft formation flying (SFF) utilizing the framework of the circular restricted three-body problem (CR3BP) with the Sun and Earth as the primary gravitational bodies. Particularly, the results are not confined to the close vicinity of the collinear Lagrangian libration points. Rather, a linearization is performed relative to an arbitrary non-Keplerian reference trajectory, so that linear time-varying differential equations result. It is rigorously proved that the open-loop linearized SFF dynamics is unstable but controllable. Uncontrolled formations with bounded relative separation that constitute the stable subspace of the dynamics are found using the dual system method. This constitutes a powerful observation, since bounded formations in three-body trajectories have been found thus far only at the vicinity of the Lagrangian equilibrium points. Assuming ideal state measurements, a time-varying continuous linear-quadratic control law is subsequently developed. A complete internal disturbance model is used, rendering a robust disturbance rejection performance. An illustrative example is used to show that the propulsion for deep-space formationkeeping requires a considerable dynamic range. It is concluded that plasma electric propulsion having a micro-thrusting capability is a most suitable means for deep-space formationkeeping.

Coronagraph instrument for WFIRST-AFTA

Abstract. The Wide-Field Infrared Survey Telescope (WFIRST) is a NASA observatory concept, now in phase A study, which is designed to perform wide-field imaging and slitless spectroscopic surveys for dark energy research and other astrophysical studies. It will also perform microlensing surveys to look for distant exoplanets in our galaxy, and direct imaging studies of some of the very nearest exoplanets. The current astrophysics focused telescope assets (AFTA) design of the mission makes use of an existing 2.4-m telescope, which yields enhanced sensitivity and imaging performance in all these science programs. AFTA also enables the addition of a coronagraph instrument (CGI) for direct imaging and spectroscopy of nearby giant exoplanets (including some that were discovered by radial velocity and other methods), and also for observing debris disks around the candidate host stars. This paper outlines the context for the other papers in this special volume on the WFIRST-AFTA CGI, covering the science, design, engineering, and technology development of the observatory and its CGI.