TNO orbit computation: analysing the observed population

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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

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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
• Numerical techniques & examples
• Observed TNO population: statistical view
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: 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

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

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 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

LS

Numerical techniques: Volumes of variation

2001 QX322 (418 days)

• Strong nonlinearities
• Nonlinear correlations
• Non-gaussian features

CZ124 (1161 days)

Numerical techniques: Volumes of variation

velocity data: HD 28185

Muinonen et al. 2005

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 ranging
• 20 % of TNOs have 1-day arcs (in 2003, 17 %)
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
• Nonlinear collapse in uncertainties
• Sequence of numerical techniques across the transition region
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
• 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?