Making Dark Shadows with Linear Programming

Transcription

1 Making Dark Shadows with Linear Programming Robert J. Vanderbei 28 October 13 INFORMS 28 Washington DC rvdb

2 Are We Alone?

3 Indirect Detection Methods Over 3 planets found more all the time

4 Wobble Methods Radial Velocity. For edge-on systems. Measure periodic doppler shift. Astrometry. Best for face-on systems. Measure circular wobble against background stars.

5 Transit Method HD29458b confirmed both via RV and transit. Period: 3.5 days Separation:.45 AU (.1 arcsecs) Radius: 1.3R J Intensity Dip: 1.7% Venus Dip =.1%, Jupiter Dip: 1% Venus Transit (R.J. Vanderbei)

6 Direct Detection

7 Why It s Hard Bright Star/Faint Planet: In visible light, our Sun is 1 1 times brighter than Earth. That s 25 mags. Close to Each Other: A planet at 1 AU from a star at 1 parsecs can appear at most.1 arcseconds in separation. Far from Us: There are less than 1 Sunlike stars within 1 parsecs.

8 Telescope w/ Unobstructed Aperture Doesn t Work! Requires an aperture measured in kilometers to mitigate diffraction effects. -.5 Pupil Mask -2 Image (PSF) linear Image (PSF) Cross Section -2 Image (PSF) log

9 Space-based Occulter (TPF-O) Telescope Aperture: 4m, Occulter Diameter: 5m, Occulter Distance: 72, km

10 Plain External Occulter (Doesn t Work!) Shadow Circular Occulter Note bright spot at center (Poisson s spot) y in meters x in meters y in meters 1 2 Telescope Image x in meters Shadow (Log Stretch)

11 Apodized Occulters Apodized Occulter Radial Attenuation A(r) The problem is diffraction. Abrupt edges create unwanted diffraction. Solution: Soften the edges with a partially transmitting material an apodizer. Let A(r, θ) denote attenuation at location (r, θ) on the occulter. The intensity of the downstream light is given by the square of the magnitude of the electric field E(ρ, φ). Babinet s principle plus Fresnel propagation gives a formula for the downstream electric field: E(ρ, φ) = 1 1 iλz where 2π e iπ λz (r2 +ρ 2 2rρ cos(θ φ)) A(r, θ)rdθdr. z is distance downstream and λ is wavelength of light.

12 Attenuation Profile Optimization Specific choice: minimize γ subject to γ R(E(ρ)) γ for ρ R, λ L γ I(E(ρ)) γ for ρ R, λ L A (r) for r R d A (r) d for r R R = 25, d =.4, R = [, 3], L = [.4, 1.1] 1 6 where all metric quantities are in meters. An infinite dimensional linear programming problem. Discretize: [, R] into 5 evenly space points. R into 15 evenly spaced points. L into increments of

13 Petal-Shaped Occulters From Jacobi-Anger expansion we get: E(ρ, φ) = 1 2π R ( ) e iπ λz (r2 +ρ 2) 2πrρ J A(r)rdr iλz λz ( 2π( 1) k R ( ) ) e iπ λz (r2 +ρ 2) 2πrρ sin(πka(r)) J kn rdr iλz k=1 λz πk (2 cos(kn(φ π2 ) )) 16-Petal Occulter A(r, θ) where N is the number of petals. For small ρ, truncated summation wellapproximates full sum. Truncated after 1 terms. λ [.4, 1.1] microns. z = 72, km, R = 25 m. In angular terms, R/z =.73 arcseconds Radial Attenuation A(r)

14 Shaped Occulter Shadow Shaped Occulter Bright spot is gone y in meters x in meters y in meters 1 2 Telescope image shows planet x in meters Shadow is dark (Log Stretch)

15 Sub-Optimal Hypergaussian Hypergaussian profile: A(r) = 1 r < 9.5 e (r/9.5 1)6 9.5 r < r. 2 λ=.4 λ=.7 λ=1. -2 Log Intensity Radius in meters Need to increase size and distance. New distance: 158, 4 km.

16 AMPL Model Data/Constants function J; param pi := 4*atan(1); param pi2 := pi/2; param N := 4; param M := 15; param c := 25.; param z := 72e+3; param lambda {3..11}; param rho1 := 25; param rho_end := 3; # discretization parameter at occulter plane # discretization parameter at telescope plane # overall radius of occulter # distance from telescope to apodized occulter # set of wavelengths # max radius investigated at telescope s pupil plane # radius below which high contrast is required # a few convenient shorthands param lz {j in by 1.} := lambda[j]*z; param pi2lz {j in by 1.} := 2*pi/lz[j]; param pilz {j in by 1.} := pi/lz[j]; param dr := c/n; set Rs ordered; let Rs := setof {j in 1..N by 1} c*(j-.5)/n; set Rhos ordered; let Rhos := setof {j in..m} (j/m)*rho1;

17 AMPL Model Model function J; var A {r in Rs} >=, <= 1; var contrast >= ; var Ereal {j in by 1., rho in Rhos} = 1-pi2lz[j]* sum {r in Rs} sin(pilz[j]*(r^2+rho^2))*a[r]*j(-pi2lz[j]*r*rho)*r*dr; var Eimag {j in by 1., rho in Rhos} = pi2lz[j]* sum {r in Rs} cos(pilz[j]*(r^2+rho^2))*a[r]*j(-pi2lz[j]*r*rho)*r*dr; minimize cont: contrast; subject to main_real_neg {j in by 1., rho in Rhos: rho < rho_end}: -contrast <= Ereal[j,rho]; subject to main_real_pos {j in by 1., rho in Rhos: rho < rho_end}: Ereal[j,rho] <= contrast; subject to main_imag_neg {j in by 1., rho in Rhos: rho < rho_end}: -contrast <= Eimag[j,rho]; subject to main_imag_pos {j in by 1., rho in Rhos: rho < rho_end}: Eimag[j,rho] <= contrast; subject to monotone {r in Rs: r>first(rs)}: A[prev(r)] >= A[r]; subject to smooth {r in Rs: r>first(rs) && r<last(rs)}: -.44 <= (A[next(r)]-2*A[r]+A[prev(r)])/dr^2 <=.44; let {j in by 1.} lambda[j] := (j-.5)*1e-7; solve;