Asteroid orbital inversion using Asteroid orbital inversion using - PowerPoint PPT Presentation

Asteroid orbital inversion using Asteroid orbital inversion using Markov-chain Monte Carlo methods Markov-chain Monte Carlo methods Karri Muinonen 1,2 1,2 , , Dagmara Dagmara Oszkiewicz Oszkiewicz 3,4,1 3,4,1 , , Karri Muinonen Tuomo Pieniluoma Pieniluoma 1 1 , , Mikael Mikael Granvik Granvik 1 1 & & Jenni Jenni Virtanen Virtanen 2 2 Tuomo 1 Department of Physics, University of Helsinki, Finland 2 Finnish Geodetic Institute, Masala, Finland 3 Lowell Observatory, Flagstaff, Arizona, U.S.A. 4 Northern Arizona University, Flagstaff, Arizona, U.S.A. Solar System science before and after Gaia, Pisa, Italy, May 4-6, 2011

Introduction Introduction  Asteroid orbit determination Asteroid orbit determination is one of the oldest inverse is one of the oldest inverse  problems problems  Paradigm change from deterministic to probabilistic Paradigm change from deterministic to probabilistic  treatment near the turn of the millennium near the turn of the millennium treatment  Uncertainties in orbital elements, ephemeris Uncertainties in orbital elements, ephemeris  uncertainties, collision probabilities, classification uncertainties, collision probabilities, classification  Identification of asteroids, linkage of asteroid Identification of asteroids, linkage of asteroid  observations observations  Incorporation of statistical orbital inversion methods into Incorporation of statistical orbital inversion methods into  the Gaia/DPAC data processing pipeline the Gaia/DPAC data processing pipeline  Markov-chain Monte Carlo (MCMC, Markov-chain Monte Carlo (MCMC, Oszkiewicz Oszkiewicz et al. et al.  2009) 2009)  OpenOrb OpenOrb open source software ( open source software (Granvik Granvik et al. 2009) et al. 2009) 

Statistical inversion Statistical inversion  Observation equation Observation equation   A A posteriori posteriori  probability density probability density function (p.d.f.) function (p.d.f.)  A priori p.d.f., A priori p.d.f., Jeffreys Jeffreys  or uniform or uniform  Observational error p.d.f., Observational error p.d.f.,  multivariate normal multivariate normal

Statistical inversion Statistical inversion  A posteriori A posteriori p.d.f. for p.d.f. for  orbital elements orbital elements  Linearization Linearization   A posteriori A posteriori p.d.f. p.d.f. in the linear approximation in the linear approximation   Covariance matrix for Covariance matrix for  orbital elements orbital elements

MCMC ranging ranging MCMC  Initial orbital inversion Initial orbital inversion using exiguous using exiguous  astrometric data (short observational time data (short observational time astrometric interval and/or a small number of observations) interval and/or a small number of observations)  Ranging algorithm Ranging algorithm   Select two observation dates Select two observation dates   V Vary ary topocentric topocentric distances and values of R.A. and distances and values of R.A. and  Decl. . Decl  From two Cartesian positions, compute elements and From two Cartesian positions, compute elements and  2 against all the observations against all the observations χ 2 χ  In MC ranging, systematic In MC ranging, systematic sampling sampling and and  weighted sample elements weighted sample elements  How to sample using MCMC? How to sample using MCMC? 

MCMC ranging MCMC ranging  Gaussian proposal p.d.f. in Gaussian proposal p.d.f. in  the space of two Cartesian Cartesian the space of two positions positions  Complex proposal p.d.f. in Complex proposal p.d.f. in  the space of the orbital orbital the space of the elements (not needed!) elements (not needed!)  Jacobians Jacobians and cancellation and cancellation  of symmetric proposal of symmetric proposal p.d.f.s p.d.f.s

Examples Examples

Gaia/DPAC DU456 demonstration Gaia/DPAC DU456 demonstration  DU456, development unit entitled DU456, development unit entitled “ “Orbital Orbital  inversion” ” inversion  MCMC ranging as standalone Java MCMC ranging as standalone Java  software including GaiaTools GaiaTools software including  Potential for a future online computational Potential for a future online computational  tool with a friendly interface tool with a friendly interface

Conclusion Conclusion  MCMC ranging more efficient than MC MCMC ranging more efficient than MC  ranging ranging  Operational within the Gaia/DPAC pipeline Operational within the Gaia/DPAC pipeline  and as stand-alone software and as stand-alone software  MCMC-Gauss under MCMC-Gauss under development and to development and to  be incorporated into Gaia/DPAC be incorporated into Gaia/DPAC  MCMC-VoV MCMC-VoV on the drawing board on the drawing board 