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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

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
TNO orbit computation
  • Summary of theory: statistical inverse problem
  • Numerical techniques & examples
  • Observed TNO population: statistical view
orbit computation theory
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)
orbit computation theory1
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
orbit computation numerical techniques
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

numerical techniques volumes of variation
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
numerical techniques volumes of variation1
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
numerical techniques volumes of variation2

LS

Numerical techniques: Volumes of variation

2001 QX322 (418 days)

  • Strong nonlinearities
    • Nonlinear correlations
    • Non-gaussian features

CZ124 (1161 days)

numerical techniques volumes of variation3
Numerical techniques: Volumes of variation

Exoplanet orbits from radial

velocity data: HD 28185

Muinonen et al. 2005

numerical techniques statistical ranging
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
numerical techniques statistical ranging1
Numerical techniques: Statistical ranging
  • 20 % of TNOs have 1-day arcs (in 2003, 17 %)
statistical view of the observed population
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
phase transition in orbital uncertainties
Phase transition in orbital uncertainties
  • Nonlinear collapse in uncertainties
    • Sequence of numerical techniques across the transition region
ephemeris prediction current uncertainties
Ephemeris prediction: current uncertainties

July 4, 2006

  • Large fraction of the population lost
  • Bayesian approach for recovery attempts (e.g., Virtanen et al. 2003)
conclusions
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?