Error Propagation in Asteroid Dynamics Models

Definitions and mathematical statement

Consider the source's uncertain asteroid model, dominated by the gravitational fields of the Sun and Jupiter. The questions are to quantify the error from omitting Saturn, compare it with initial-condition errors, and describe how initial error balls propagate.

To express the requested comparisons once a model is specified, let $\Phi_{t,0}$ be the evolution including Saturn and $\Psi_{t,0}$ the evolution omitting it, identified on a common asteroid state space with metric $d$. For a reference state $x_0$, a model-discrepancy quantity is \[ e_{\mathrm{Saturn}}(t;x_0)=d(\Phi_{t,0}(x_0),\Psi_{t,0}(x_0)). \] For an initial uncertainty ball $B_\delta(x_0)=\{x:d(x,x_0)\leq\delta\}$, the propagated uncertainty set and a corresponding initial-error radius are \[ \Phi_{t,0}(B_\delta(x_0)),\qquad e_{\mathrm{init}}(t;x_0,\delta)= \sup_{x\in B_\delta(x_0)}d(\Phi_{t,0}(x),\Phi_{t,0}(x_0)). \] These expressions illustrate the distinct errors the source asks to compare; they are not a supplied error bound or a claim that propagated balls remain balls. They are considered only where the chosen evolutions exist.

The actual planetary equations, masses and trajectories, reference frame, asteroid initial states, uncertainty model, state-space metric, time horizon, collision exclusions, and required accuracy are absent. Thus no particular restricted three-body model, perturbation parameter, exponential growth law, or quantitative criterion can be assigned from this record alone. A complete formulation must fix those choices and specify which bounds or geometric descriptions of the propagated sets are sought.

Short English statement

Quantify the effect of neglecting Saturn in a Sun--Jupiter asteroid model, compare it with uncertainty in the initial state, and describe the evolution of uncertainty balls. The dynamical and error models are unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30001004$, source code catalogue entry, "Error Propagation in Asteroid Dynamics Models." No individual proposer is supplied.
• Applied Dynamics and Geometric Mechanics (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/33, pp. 1851--1932: https://doi.org/10.4171/owr/2008/33. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It describes broad error-propagation questions involving Saturn, reports no resolved criterion in the report, and calls them open as stated. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.