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TNO orbit computation: analysing the observed population

TNO orbit computation: analysing the observed population. Jenni Virtanen Observatory, University of Helsinki. Workshop on Transneptunian objects - Dynamical and physical properties Catania, Hotel Nettuno, July 3-7, 2006. TNO orbit computation. Summary of theory: statistical inverse problem

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TNO orbit computation: analysing the observed population

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  1. TNO orbit computation: analysing the observed population Jenni Virtanen Observatory, University of Helsinki Workshop on Transneptunian objects - Dynamical and physical properties Catania, Hotel Nettuno, July 3-7, 2006

  2. TNO orbit computation • Summary of theory: statistical inverse problem • Numerical techniques & examples • Observed TNO population: statistical view

  3. Orbit computation: theory • Observation equations • P = Orbital elements • y = Observed position; e.g., (a,d) • Y = Y(P)= Computed position (the model) • e = Dy = O – C residuals • Nonlinear model, I.e., relationship Y(P) between the orbital parameters and observed parameters • Stochastic (random) variables: p(P), p(e)

  4. Orbit computation: theory • Bayes’ theorem: a posteriori probability density for the orbital parameters • Likelihood function, p(y | P), typically given as observational error p.d.f., p(e), • A priori p.d.f. for orbital elements ppr (P), e.g., regularizing a priori by Jeffreys

  5. Orbit computation: numerical techniques How to solve for the orbital-element p.d.f.? • Single (point) estimates • Maximum likelihood estimates: Least squares, Bernstein & Khushalani (2000) • Monte Carlo sampling of p.d.f. • Sampling in observation space (r, a, d): Statistical ranging (Virtanen et al. 2001) (Goldader & Alcock 2003) • Sampling in orbital-element space P: Volumes of variation (Muinonen et al. 2006) In order of increasing degree of nonlinearity: 1) Least squares, 2) Volumes of variation, 3) Statistical ranging

  6. Numerical techniques: Volumes of variation Sampling of orbital-element p.d.f. in phase space: • Starting point: global least-squares solution • Mapping the variation intervals for parameters (compare to line-of-variation techniques, Milani et al.) • MCsampling within the (scaled) variation intervals • Orbital-element p.d.f.: MC sample orbits with weights

  7. Numerical techniques: Volumes of variation Sampling of orbital-element p.d.f. in phase space: • Starting point: global least-squares solution • Mapping the variation intervals for parameters • one (or more) mapping element, Pm • variation interval for Pm from global covariance matrix • a set of local ls solutions a.f.o. Pm • local variation intervals from local covariances • MCsampling within the (scaled) variation intervals • Orbital-element p.d.f.: MC sample orbits with weights

  8. LS Numerical techniques: Volumes of variation 2001 QX322 (418 days) • Strong nonlinearities • Nonlinear correlations • Non-gaussian features CZ124 (1161 days)

  9. Numerical techniques: Volumes of variation Exoplanet orbits from radial velocity data: HD 28185 Muinonen et al. 2005

  10. Numerical techniques: Statistical ranging Sampling of orbital-element p.d.f. in observation space • Two observation pairs (a, d)1, (a, d)2 • MC sampling in topocentric spherical coordinates • Two topocentric ranges are randomly generated r1, r2 • Random noise is added to angular observations • Coordinates(r, a, d)1 and (r, a, d)2define a sample orbit • Orbital-element p.d.f.: MC sample orbits with weights

  11. Numerical techniques: Statistical ranging • 20 % of TNOs have 1-day arcs (in 2003, 17 %)

  12. Numerical techniques: Statistical ranging

  13. Statistical view of the observed population • Joint orbital-element p.d.f. for the observed population • Ranging and VoV -solution for 975 (725+250) objects • Phase transition in orbital uncertainties • Ephemeris prediction • Dynamical classification

  14. TNO orbital distribution: joint p.d.f. All objects (975)

  15. TNO orbital distribution: joint p.d.f. Objects > 180 d (471)

  16. Phase transition in orbital uncertainties • Nonlinear collapse in uncertainties • Sequence of numerical techniques across the transition region

  17. Phase transition in orbital uncertainties

  18. Ephemeris prediction: current uncertainties July 4, 2006 • Large fraction of the population lost • Bayesian approach for recovery attempts (e.g., Virtanen et al. 2003)

  19. Conclusions • Sequence of orbit computation techniques applicable over the phase transition region • Orbit computation for ESA’s Gaia mission • Orbital element database at Helsinki Observatory • ~15 years of observations: a poorly observed population • > 50 % of objects have a > 1 AU • Nonlinear techniques needed • 3-10 year survey needed to improve the situation • Effect of improving observational accuracy?

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