Long-term monitoring of WASP-19 b: Signs of apsidal precession and molecular signatures

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Volume 708 (April 2026)

A&A, 708 (2026) A313

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Table A.3

Statistical comparison of linear, quadratic, and cubic ephemeris models for WASP-19 b under different data-quality selections.

Probability	Without TESS				With TESS
	Quantity	Linear	Quadratic	Cubic	Quantity	Linear	Quadratic	Cubic
All data	χ²	532.46	493.65	433.72	χ²	547.02	516.29	468.47
	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	4.98	4.66	4.13	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	4.84	4.61	4.22
	AIC	536.46	499.65	441.72	AIC	551.02	522.29	476.47
	BIC	541.84	507.73	452.48	BIC	556.51	530.53	487.45
	rms (s)	48.40	47.85	45.72	rms (s)	47.40	46.50	45.45
	X_MAD	3.25×10⁻⁴	2.91×10⁻⁴	2.51×10⁻⁴	X_MAD	3.14×10⁻⁴	2.90×10⁻⁴	2.49×10⁻⁴

Best 20%	χ²	146.50	134.37	111.78	χ²	151.42	148.51	114.08
	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	8.14	7.90	6.99	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	8.41	8.74	7.13
	AIC	150.50	140.37	119.78	AIC	155.42	154.51	122.08
	BIC	152.49	143.35	123.76	BIC	157.41	157.50	126.06
	rms (s)	28.73	26.67	25.45	rms (s)	29.12	28.42	26.11
	X_MAD	3.05×10⁻⁴	2.80×10⁻⁴	2.51×10⁻⁴	X_MAD	3.54×10⁻⁴	3.41×10⁻⁴	2.59×10⁻⁴

Best 30%	χ²	163.82	148.72	133.44	χ²	170.07	165.55	138.04
	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	5.85	5.51	5.13	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	6.07	6.13	5.31
	AIC	167.82	154.72	141.44	AIC	174.07	171.55	146.04
	BIC	170.62	158.93	147.04	BIC	176.87	175.75	151.65
	rms (s)	26.19	24.21	23.89	rms (s)	26.65	25.88	24.62
	X_MAD	2.42×10⁻⁴	2.48×10⁻⁴	2.26×10⁻⁴	X_MAD	2.76×10⁻⁴	2.62×10⁻⁴	2.50×10⁻⁴

Best 40%	χ²	209.07	195.98	181.31	χ²	207.89	203.88	184.86
	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	5.50	5.30	5.04	$χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$	5.47	5.51	5.14
	AIC	213.07	201.98	189.31	AIC	211.89	209.88	192.86
	BIC	216.45	207.05	196.06	BIC	215.27	214.95	199.61
	rms (s)	31.96	30.61	30.36	rms (s)	30.44	29.98	29.12
	X_MAD	2.88×10⁻⁴	2.47×10⁻⁴	2.16×10⁻⁴	X_MAD	2.80×10⁻⁴	2.69×10⁻⁴	2.53×10⁻⁴

Notes. Results are shown with and without the inclusion of TESS sectors and are derived from bootstrap-resampled MCMC fits with N_boot = 2000. The reported χ², reduced $χ_{ν}^{2}$ $Mathematical equation: $\chi^{2}_{\nu}$$ , AIC, and BIC values are computed directly from the timing uncertainties listed in Table A.2, without additional uncertainty rescaling (see Sect. 5.1). The cross-validation metric X_MAD is obtained from random hold-out tests removing 20% of the data and refitting the remaining 80% over 10000 iterations. Although none of the models achieves $χ_{ν}^{2} \approx 1$ $Mathematical equation: $\chi^{2}_{\nu}\approx1$$ , the cubic ephemeris consistently yields lower information-criterion values and stable predictive performance across different subsets. The overall timing scatter limits the statistical significance of any single ephemeris model.

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