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  = Observed position; e.g.,   )  (P) = Computed position (the model)  =  = O – C residuals Nonlinear model, I.e., relationship  (P) between the orbital parameters and observed parameters Stochastic (random) variables: p(P), p( 

4. Orbit computation: theory Bayes’ theorem: a posteriori probability density for the orbital parameters Likelihood function, p  P  typically given as observational error p.d.f., p  A priori p.d.f. for orbital elements p pr (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  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: 1. Starting point: global least-squares solution 2. Mapping the variation intervals for parameters (compare to line-of-variation techniques, Milani et al.) 3. MC sampling within the (scaled) variation intervals 4. 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: 1. Starting point: global least-squares solution 2. Mapping the variation intervals for parameters - one (or more) mapping element, P m - variation interval for P m from global covariance matrix - a set of local ls solutions a.f.o. P m - local variation intervals from local covariances 3. MC sampling within the (scaled) variation intervals 4. Orbital-element p.d.f.: MC sample orbits with weights

8. Numerical techniques: Volumes of variation 2001 QX322 (418 days) 2002CZ124 (1161 days) n Strong nonlinearities n Nonlinear correlations n Non-gaussian features LS

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

10. Numerical techniques: Statistical ranging Sampling of orbital-element p.d.f. in observation space 1. Two observation pairs       C sampling in topocentric spherical coordinates -  wo topocentric ranges are randomly generated     - Random noise is added to angular observations - Coordinates   and    define a sample orbit 3. 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?