Design reference mission construction for planet finders SPIE Astronomical Telescopes and Instrumentation Dmitry Savransky and N. Jeremy Kasdin Department of Mechanical and Aerospace Engineering Princeton University June 26, 2008 Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 1 / 15
Introduction to DRMs Multiple teams are currently developing direct detection planet-finding missions. Multiple different mission scenarios and observing strategies have been proposed. Several tools have been developed to assess the capabilities of a planet-finding instrument and the probability of detecting a planet in a given system. We require a framework to produce end-to-end mission simulations for proposed instruments and scenarios: a Design Reference Mission constructor. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 2 / 15
Direct Detection Platforms Coronographs - multiple methods exist for removing light from the star entering a telescope s aperture. Occulters - a starshade is flown along with the telescope to block out star-light. Hybrid designs - combinations of these two schemes. Figure: Schematic of a Lyot stop. Figure: Pupil mask for high contrast imaging. [Vanderbei et al. 2003] Figure: Proposed star-shade design. [Vanderbei et al. 2007] Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 3 / 15
Outline 1 Components of DRM 2 Constructing the DRM 3 Preliminary Results Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 4 / 15
Modeling a Planetary Detection - Apparent Separation [Brown, 2004a, Brown, 2004b] Select a semi-major axis (a) and eccentricity (e). Figure: System Model Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 5 / 15
Modeling a Planetary Detection - Apparent Separation [Brown, 2004a, Brown, 2004b] Select a semi-major axis (a) and eccentricity (e). ψ rotation about z Figure: System Model Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 5 / 15
Modeling a Planetary Detection - Apparent Separation [Brown, 2004a, Brown, 2004b] Select a semi-major axis (a) and eccentricity (e). ψ rotation about z θ rotation about x Figure: System Model Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 5 / 15
Modeling a Planetary Detection - Apparent Separation [Brown, 2004a, Brown, 2004b] Select a semi-major axis (a) and eccentricity (e). ψ rotation about z θ rotation about x φ rotation about z The three Euler angles correspond to the argument of pericenter, the orbital inclination, and the longitude of the ascending node. Figure: System Model Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 5 / 15
Modeling a Planetary Detection - Apparent Separation [Brown, 2004a, Brown, 2004b] Select current true anomaly (ν). Define: r p/s s Position of planet in XYZ frame. Apparent separation (projection of r p/s onto XY plane). z Component of r p/s along Z. r r p/s = a(1 e2 ) e cos(ν) + 1 1 0 0 s = 0 1 0 r p/s 0 0 0 Figure: System Model r(cos ν sin θ sin φ + sin ν(cos θ cos ψ sin φ + cos φ sin ψ)) r p/s = r(cos ν cos φ sin θ + sin ν(cos θ cos φ cos ψ sin φ sin ψ)) r(cos θ cos ν cos ψ sin θ sin ν) Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 6 / 15
Modeling a Planetary Detection - Mag Figure: System Model. Dotted lines represent lines of sight. since r d: β cos 1 ( z r This approximation produces errors of less than 0.001% for the nearest star (1.3 pc). ) The phase of a planet (Φ) is a function of the star-planet-observer angle (β). The difference in magnitude between a star and planet is a function of the ratio of their spectral fluxes: mag = 2.5 log F p F = 2.5 log for a planet of radius R and albedo p. ( p ( ) 2 R Φ(β)) r Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 7 / 15
Completeness [Brown, 2005] Figure: Candidate stars plotted over the single visit completeness for Earth-like planets. Planets are defined as: a [0.7, 1.5], e [0, 0.35], p = 0.33, R = R E, and the Lambert phase function is used. Instrument specifications of: IWA = 72 mas, mag 0 = 25. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 8 / 15
Integration Time Calculation [Kasdin and Braems, 2006] By treating S/N ratio as a random variable, we can calculate the integration time for a pre-selected False Alarm Probability (FAP) and missed detection rate: ( t = 1 K γ ) 2 1 + Qξ/Ψ B QT A Ψ where B is a function of planet intensity and detector efficiency and area, Q is the ratio of the total photon count from the planet to the number of background counts at a pixel, T A is the throughput times the summed normalized PSF, ξ is a function of the normalized PSF, and Ψ is the sharpness. K is a threshold value selected according to confidence level P (1 P = P FA ) γ is a threshold value selected according to the missed detection rate P MD. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 9 / 15
Re-visit Timing - Approximating the Orbital Period [Savransky and Kasdin 2007] For a star with gravitational parameter µ, the orbital period of a planet is given by: T = 2π a 3 /µ Figure: Mass Luminosity Relation. [Henry & McCarthy, 1993] The stellar mass can be calculated from the star s luminosity by one of the empirical mass-luminosity relations such as: ( ) M log = 0.002456V 2 0.09711V M +0.4365 This relation is accurate to within about 7% for mass ranges corresponding to visual magnitudes between 1.45 and 10.25. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 10 / 15
Re-visit Timing - Approximating the Semi-Major Axis Define parameter Ξ equal to the ratio s/a: Ξ 1 e2 (cos ν sin ν + cos θ cos ψ sin ν) 2 + (sin(ν) sin(ψ)) e cos ν + 1 2 Figure: Probability density functions of Ξ. The PDF of Ξ always has a maximum at 1 (s = a). The observed apparent separation is the best estimate for the semi-major axis. The obscurational and photometric limitations actually make Ξ = 1 an even better estimate. Using these two estimates produces a mean 16% error in orbital period in simulation. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 11 / 15
Visits as a Graph Figure: Visit graph for 3 target pool. Each set of possible transitions on the visit graph can be represented as a weighted adjacency matrix. The weights of the matrix entries represent the cost of choosing the next star. The cost of transitioning from target i to target j is calculated as: A ij = cos 1 (u i u j ) B inst 2π Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 12 / 15
Visits as a Graph Figure: Visit graph for 3 target pool. Each set of possible transitions on the visit graph can be represented as a weighted adjacency matrix. The weights of the matrix entries represent the cost of choosing the next star. The cost of transitioning from target i to target j is calculated as: A ij = cos 1 (u i u j ) B inst + comp j 2π Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 12 / 15
Visits as a Graph Figure: Visit graph for 3 target pool. Each set of possible transitions on the visit graph can be represented as a weighted adjacency matrix. The weights of the matrix entries represent the cost of choosing the next star. The cost of transitioning from target i to target j is calculated as: A ij = cos 1 (u i u j ) B inst + comp j e t t f B unvisited 2π Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 12 / 15
Visits as a Graph Figure: Visit graph for 3 target pool. Each set of possible transitions on the visit graph can be represented as a weighted adjacency matrix. The weights of the matrix entries represent the cost of choosing the next star. The cost of transitioning from target i to target j is calculated as: A ij = cos 1 (u i u j ) B inst + comp j e t t f B unvisited + pb visited (1 B revisit ) 2π Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 12 / 15
Visits as a Graph Figure: Visit graph for 3 target pool. Each set of possible transitions on the visit graph can be represented as a weighted adjacency matrix. The weights of the matrix entries represent the cost of choosing the next star. The cost of transitioning from target i to target j is calculated as: [ cos 1 ] (u i u j ) A ij = B inst + comp j e t t f B unvisited + pb visited (1 B revisit ) (1 B ko ) 2π Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 12 / 15
Simulation Algorithm Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 13 / 15
Preliminary Results Comparing Occulter, Coronograph and Hybrid Designs 4m internal coronograph with IWA of 3 λ/d. 4m telescope with a 50m occulter at a separation of 72,000 km. 4m hybrid system with a 28m occulter at a separation of 40,000 km. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 14 / 15
References Vanderbei, R. J., Spergel, D. N., Kasdin, N. J., Circularly symmetric apodization via star-shaped masks, Aph. J, 599:686-694, 2003. Vanderbei, R. J., Cady, E., Kasdin, N. J., Optimal Occulter Design for Finding Extrasolar Planets, Aph. J, 655:794-798, 2007. Brown, R. A. Obscurational completeness, Astrophysical Journal, Vol. 607, p. 1003-1017, 2004. Brown, R. A. New information from radial velocity data sets, Astrophysical Journal, Vol. 610, p. 1079-1092, 2004. Brown, R. A. Single-visit photometric and obscurational completeness, Astrophysical Journal, Vol. 624, p. 1010-1024, 2005. Kasdin, N. J., and Brames, I. Linear and bayesian planet detection algorithms for the Terrestrial Planet Finder, ApJ 646, 1260-1274, 2006. Henry, T. J. and McCarthy, D. W. Jr. The Mass-Luminosity Relation for stars of mass 1.0 to 0.08M, Astronomical Journal, Vol. 106, No. 2, p. 773-789, 1993. Savransky, D., Kasdin, N. J., Optimal return visit timing for planet finding missions, in AAS Bulletin, 39(4), 134, 2007. Butler, R. P., et al. Catalog of Nearby Exoplanets, Astrophysical Journal, Vol. 646, Pg. 505, 2006. De Vaucouleurs, G. Geometric and Photometric Parameters of the Terrestrial Planets, Icarus, 3, 187-235, 1964. Kane, T. R., Likins, P. W., and Levnison, D. A. Spacecraft Dynamics, New York: McGraw-Hill Book Co., 1983. Kasting, J. F., Whitmire, D. P., Reynolds, R. T., Habitable zones around main sequence star, ICARUS 101, 1993. Sudarsky, D., Burrows, A., Hubney, I., and Li, A., Phase functions and light curves of wid-separation extrasolar giant planets, Aph. J, 627:520-533, 2005. Sobolev, V. V., Light Scattering in Planetary Atmospheres, New York: Pergamon Press, 1975. Savransky and Kasdin (Princeton University) DRM Construction for Planet Finders SPIE 2008: 7010-64 15 / 15