Dynamics of an Occulter Based
Planet Finding Telescope
Operating
range
Egemen Kolemen, N. Jeremy Kasdin
Due to reflection of sunlight from the occulter Including the constraint of
the formation not being able to look towards or away from the Sun
direction, we get a time dependent ordering problem as shown on the
right.
Dept. of Mechanical and Aerospace Eng., Princeton University
Finding Quasi-Halos:
A fast, robust, fully numeric method employing multiple Poincare
sections to find the quasi-periodic orbits around libration points is
developed along with a numerical method to transport the R3BP
results to the full ephemeris model.
1. Mission Basics
Relative motion between a Quasi-Halo and a Halo
in the inertial frame.
Further Optimization with best TSP outputs
Once done choose the best ~1000 solutions and go
to SQP for fine tuning (we averaged and didn’t use
the full ephemeris) and more importantly optimizing
the time between each imaging session
3. Trajectory Optimization and Control
Occulter
Telescope
Formation flying of an occulter is proposed to enhance the optical performance of an exo-planet imaging telescope.
Need for control:
Fuel-free Quasi-Halo trajectories are very suitable to place the occulter but they are too slow to image the required
number of stars in the approximate 5 year life time without control. Two approaches:
1. Multiple occulters employing the fuel-free trajectories.
2. Single occulter with minimal-fuel consumption trajectory.
a. Single Occulter Optimization
)
Further Optimization with best TSP outputs
Hh
J
Optimal Control of the occulter between two imaging sessions can be converted into a two-point boundary value
problem (TPBVP) with algebraic constraints via Euler-Lagrange Formulation. Under simplifying assumptions Hu can be
solved for and the problem is converted to a convex TPBVP. A very fast implementation of this algorithm enables finding
the optimal trajectories as a function of the significant parameters.
j
b. Multiple Occulter Optimization
Configuration
Space of the
Occulter
The contrast between the planet and the target star is reduced by suppressing most of the light from the target star
before it enters the optical system. The occulter shape is optimized such that the intensity of light from the star is
minimal in the possible planet locations on the image plane.
C
E
A
D
t1
Telescope’s
Orbit
F
B
t0
Multiple occulters using the fuel free Quasi-Halo Trajectories

(degrees)
Possible Occulter Trajectories from t0 to t1
The formation is proposed to be around a Halo orbit near the L2 point of the Earth-Sun system. Reasons:
1. Far away from Earth to avoid interference.
2. Good telecommunication properties.
3. Minimal fuel is needed to get to and station keep the orbits.
Left: The sphere of possible occulter locations about the telescope at two times and example optimal trajectories
connecting them.
Right: Surface of optimal Delta-V's as a function of distance from the telescope and angle between the LOS vectors of
consecutively imaged star (Averaged over millions of simulations).
2. Dynamics around the Libration Point
Quasi-Halo Orbit
2 multiple s/c
Figure – multiple s/c in the skyplot
3 parameters per s/c that define the unique quasi-halo and position on it x number of s/c = total variables
Using the fast quasi-halo generation; Find the best configuration for the most sky coverage by optimizing the
above parameters
Fine tune to find the orbits that come close to the given star set (Assume no control); This gives you the ball
part of the solution;
Combine both – find the minimum fuel maximum imaging space of variables
Optimization in stages:
1. Maximize total sky coverage, obtain the region of parameters
2. Maximize target star coverage within Stage 1 solutions
Time Dependent Dynamical Traveling Salesman Problem
References
Earth
Moon Orbit
1. E. Kolemen & N. J. Kasdin, “Optimal Trajectory Control of an Occulter Based Planet Finding Telescope”, To be
presented at the AAS Guidance & Control Conference, Breckenridge, Colorado, AAS 07-036, Feb. 2007.
2. E. Kolemen & N. J. Kasdin, “Optimal Configuration of a Planet-Finding Mission Consisting of a Telescope and
a Constellation of Occulters”, To be presented at the AAS/AIAA Space Flight Mechanics Meeting, Sedona,
Arizona, AAS 07-202, Jan. 2007.
3. E. Kolemen, N. J. Kasdin & P. Gurfil, “Quasi-Periodic Orbits of the Restricted Three Body Problem Made
Easy”, Proceedings of the New Trends in Astrodynamics and Applications, Aug. 2006
4. W. Cash, “Detection of Earth-like planets around nearby stars using a petal shaped occulter”, Nature, 442, 5153, 6 July 2006.
5. J. Arenberg, A. Lo, C. Lillie, R. Malmstrom, R. Polidan, C. Noecker & W. Cash, “Occulter Systems Terrestrial
Planet Finding”, Terrestrial Planet Finder Coronagraph Workshop, Pasadena, CA, Sep. 28-29, 2006.
L2 Point
Periodic phase-space around L2 shown in 3D and on a Poincare section.
An example sequence of star imaging on the skyplot and the Branch-And-Cut Algorithm for solving TSP.
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What the mission is
Explain the optical system
L2 and mission formation
Dynamics around L2
Talk about new method to find the quasi-periodic orbits and how to move them to JPL-406
Quasi-Halos – why they are good
2 options since quasi-halos are slow
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1 s/c find the minimum fuel consumption
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Optimization between 2 star imaging session.
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TSMP
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Once done choose the best ~1000 solutions and go to SQP for fine tuning (we averaged and didn’t use the full ephemeris)
and more importantly optimizing the time between each imaging session
SQP equation
Figure – SQP with time optimization
2 multiple s/c
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Figure – Tree
Figure – Constraint and the matrix (traveling salesman)
Talk about time dependent – dynamics TSMP
Fine tune and optimize time between imaging sessions
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Figure – example trajectories
Problem statement
Simplified E-L equations
Figure - After 20 million simulations and averaging
Figure – multiple s/c in the skyplot
3 parameters per s/c that define the unique quasi-halo and position on it x number of s/c = total variables
Using the fast quasi-halo generation; Find the best configuration for the most sky coverage
Fine tune to find the orbits that come close to the given star set (Assume no control); This gives you the ball part
of the solution;
Combine both – find the minimum fuel maximum imaging space of variables