Past exam of the astrophysics course of the University of Oxford / 2021 / 3 / e / Solution

Apply the same symmetric-ejection and causality argument to the launch proper motions, when the proposed ejecta are fastest. One obtainsandThe fitted angular decelerations describe subsequent evolution; they do not relax the bound already imposed at launch. They are observer-time phenomenological coefficients, not directly identical physical decelerations in the source frame.

The stellar parallax distance $2.96\pm 0.33\mathrm{kpc}$ is compatible with the constant-velocity bound $3.435\mathrm{kpc}$, but its central value exceeds the decelerating-launch bound by $0.717\mathrm{kpc}$, or about $2.17$ quoted standard deviations if the uncertainty is treated as symmetric Gaussian. Even its one-standard-deviation lower endpoint, $2.63\mathrm{kpc}$, exceeds $2.243\mathrm{kpc}$. At $D=2.96\mathrm{kpc}$ the respective inferred intrinsic speeds are $\beta =0.885$ and $\beta =1.300$. The latter is a concrete violation of causality under the symmetric model assumptions.

Thus the supplied distance favours the constant-velocity fit. This is a consistency comparison, not a full statistical model selection: uncertainties in the fitted proper motions, their correlations, intrinsic asymmetry and the distance likelihood would be required for that.