Delta Chi2

delta-chi2

Astrometric exoplanet detection with Gaia — Perryman et al. (2014) (arXiv:1411.1173, 2014)

What this evaluates

Evaluates the detectability and orbital parameter recovery of long-period exoplanets using simulated Gaia astrometric observations. It quantifies detection significance by comparing the goodness-of-fit between a standard astrometric model and a full orbital model.

Datasets

• Simulated Gaia exoplanet systems (AGISLab/TRILEGAL) — total ?; splits: (unstated)

Metrics

• Δχ² (primary) — range: other
  • Δχ² = χ²_min(5-parameter) - χ²_min(12-parameter), where the 5-parameter fit models only astrometry (position, parallax, proper motion) and the 12-parameter fit adds orbital elements. It serves as a likelihood ratio test statistic for orbital motion significance.

Input / output format

Input: Simulated along-scan astrometric observations per field crossing (time, position angle, parallax factor, standard error) for a target star, given its sky coordinates, magnitude, and true orbital elements.

Output: Fitted 12 orbital/astrometric parameters and the resulting Δχ² value comparing the 5-parameter and 12-parameter least-squares fits.

Scoring recipe

# 1. Fit 5-parameter model (astrometry only) to observations -> chi2_5
# 2. Fit 12-parameter model (astrometry + orbit) to observations -> chi2_12
# 3. Compute delta_chi2 = chi2_5 - chi2_12
# 4. Apply threshold: delta_chi2 > 30 (marginal), > 50 (reliable), > 100 (precise)
# 5. For large samples, use noncentrality parameter lambda: lambda + 7 > threshold

Common pitfalls

• Relying solely on S/N per field crossing (α/σ_fov) without Δχ² confirmation, as high S/N can yield low Δχ² if observations cover only a small orbital arc.
• Using formal least-squares uncertainties for precision estimates, which are misleading due to strong non-linearity; requires Monte Carlo noise realizations instead.

Evidence (verbatim from paper)

Let χ²_min(12 parameter) be the minimum χ² obtained when adjusting all 12 parameters... Omitting the orbit parameters... results in a fit with χ²(5 parameter)≥χ²_min(12 parameter). The increase in χ² when omitting the orbit parameters is Δχ²=χ²_min(5 parameter)-χ²_min(12 parameter), which can therefore be used as a test statistic for the significance of the orbit.

Citation

@misc{perryman2014astrometric,
  title={Astrometric exoplanet detection with Gaia},
  author={Perryman et al. (2014)},
  year={2014},
  note={arXiv:1411.1173}
}

• arXiv: 1411.1173