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 Karri Muinonen1,2

1,2,

, Dagmara Dagmara Oszkiewicz Oszkiewicz3,4,1

3,4,1,

, Tuomo Tuomo Pieniluoma Pieniluoma1

1,

, Mikael Mikael Granvik Granvik1

1 &

& Jenni Jenni Virtanen Virtanen2

2

Solar System science before and after Gaia, Pisa, Italy, May 4-6, 2011

1Department of Physics, University of Helsinki, Finland 2Finnish Geodetic Institute, Masala, Finland 3Lowell Observatory, Flagstaff, Arizona, U.S.A. 4Northern Arizona University, Flagstaff, Arizona, U.S.A.

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 treatment near the turn of the millennium near the turn of the millennium

• 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

• bservations
• bservations
• 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 (

• pen 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

• r uniform
• r 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

• rbital elements
• rbital 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

• rbital elements
• rbital elements

MCMC MCMC ranging ranging

• Initial orbital inversion

Initial orbital inversion using exiguous using exiguous astrometric astrometric data (short observational time data (short observational time 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 χ χ2

2

against all the observations against all the observations

• 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 the space of two Cartesian Cartesian positions positions

• Complex proposal p.d.f. in

Complex proposal p.d.f. in the space of the the space of the

• rbital
• rbital

elements (not needed!) elements (not needed!)

• Jacobians

Jacobians and cancellation and cancellation

• f symmetric proposal
• f 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 software including GaiaTools GaiaTools

• 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

• n the drawing board