Transit-timing variations in the AU Mic system observed with CHEOPS

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Volume 694 (February 2025)

A&A, 694 (2025) A137

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Table B.1

Posterior and derived parameters of the AU Mic system from the four dynamical analyses with TRADES.

configuration planets	(1)		(2)		(3)		(4)
	b, c		b, c, d		b, c, d		b, c, d
	i_c fixed, no RV		i_c fixed, no RV		i_c fitted, no RV		i_c fitted, with RV

Parameter (unit)	Prior	MAP (HDI)	Prior	MAP (HDI)	Prior	MAP (HDI)	Prior	MAP (HDI)
fitted parameters
M_b/M_⋆ (M_⊙/M_⋆) × 10⁻⁶	𝒰(0.057, 319.520)	$110_{- 4}^{+ 6}$ $110_{ - 4}^{ + 6}$	𝒰(0.057, 319.520)	$159_{- 10}^{+ 9}$ $159_{ - 10}^{ + 9}$	𝒰(0.057,319.520)	$28_{- 2}^{+ 3}$ $28_{ - 2}^{ + 3}$	𝒰(0.057, 319.520)	$187_{- 15}^{+ 16}$ $187_{ - 15}^{ + 16}$
P_b (days)	𝒰(8.3, 8.6)	${8.461705}_{- 0.000099}^{+ 0.000016}$ $8.461705_{ - 0.000099}^{ + 0.000016}$	𝒰(8.3, 8.6)	${8.463364}_{- 0.000031}^{+ 0.000017}$ $8.463364_{ - 0.000031}^{ + 0.000017}$	𝒰(8.3, 8.6)	${8.462697}_{- 0.000035}^{+ 0.000037}$ $8.462697_{ - 0.000035}^{ + 0.000037}$	𝒰(8.3, 8.6)	${8.463067}_{- 0.000084}^{+ 0.000053}$ $8.463067_{ - 0.000084}^{ + 0.000053}$
$\sqrt{e} \cos ω_{b}$ $\sqrt e \cos {\omega _{\rm{b}}}$	𝒰(-0.71,0 .71)	${0.5429}_{- 0.0206}^{+ 0.0098}$ $0.5429_{ - 0.0206}^{ + 0.0098}$	𝒰(−0.71, 0.71)	${0.120}_{- 0.028}^{+ 0.027}$ $0.120_{ - 0.028}^{ + 0.027}$	𝒰(−0.71,0.71)	${0.05568}_{- 0.05032}^{+ 0.00096}$ $0.05568_{ - 0.05032}^{ + 0.00096}$	𝒰(−0.71, 0.71)	$- {0.027}_{- 0.020}^{+ 0.011}$ $- 0.027_{ - 0.020}^{ + 0.011}$
$\sqrt{e} \sin ω_{b}$ $\sqrt e \sin {\omega _{\rm{b}}}$	𝒰(-0.71, 0.71)	${0.452}_{- 0.023}^{+ 0.014}$ $0.452_{ - 0.023}^{ + 0.014}$	𝒰(−0.71, 0.71)	${0.5369}_{- 0.0028}^{+ 0.0259}$ $0.5369_{ - 0.0028}^{ + 0.0259}$	𝒰(−0.71,0.71)	$- {0.30225}_{- 0.01469}^{+ 0.00069}$ $- 0.30225_{ - 0.01469}^{ + 0.00069}$	𝒰(−0.71, 0.71)	${0.060}_{- 0.025}^{+ 0.032}$ $0.060_{ - 0.025}^{ + 0.032}$
λ_b (°)	𝒰(0, 360)	${176.66}_{- 0.70}^{+ 1.75}$ $176.66_{ - 0.70}^{ + 1.75}$	𝒰(0, 360)	$205_{- 2}^{+ 1}$ $205_{ - 2}^{ + 1}$	𝒰(0, 360)	${208.69}_{- 0.17}^{+ 1.81}$ $208.69_{ - 0.17}^{ + 1.81}$	𝒰(0, 360)	${211.094}_{- 0.084}^{+ 0.178}$ $211.094_{ - 0.084}^{ + 0.178}$
M_c/M_⋆ (M_⊙/M_⋆) × 10⁻⁶	𝒰(0.057, 319.520)	$127_{- 6}^{+ 2}$ $127_{ - 6}^{ + 2}$	𝒰(0.057, 319.520)	$26_{- 1}^{+ 7}$ $26_{ - 1}^{ + 7}$	𝒰(0.057, 319.520)	$86_{- 5}^{+ 8}$ $86_{ - 5}^{ + 8}$	𝒰(0.057, 319.520)	$86_{- 2}^{+ 8}$ $86_{ - 2}^{ + 8}$
P_c (days)	𝒰(18.6, 19.0)	${18.859052}_{- 0.000013}^{+ 0.000432}$ $18.859052_{ - 0.000013}^{ + 0.000432}$	𝒰(18.6, 19.0)	${18.86404}_{- 0.00014}^{+ 0.00135}$ $18.86404_{ - 0.00014}^{ + 0.00135}$	𝒰(18.6, 19.0)	${18.859167}_{- 0.000077}^{+ 0.000101}$ $18.859167_{ - 0.000077}^{ + 0.000101}$	𝒰(18.6, 19.0)	${18.85263}_{- 0.00057}^{+ 0.00063}$ $18.85263_{ - 0.00057}^{ + 0.00063}$
$\sqrt{e} \cos ω_{c}$ $\sqrt e \cos {\omega _{\rm{c}}}$	𝒰(-0.71, 0.71)	${0.1608}_{- 0.0074}^{+ 0.0043}$ $0.1608_{ - 0.0074}^{ + 0.0043}$	𝒰(−0.71, 0.71)	${0.012}_{- 0.016}^{+ 0.012}$ $0.012_{ - 0.016}^{ + 0.012}$	𝒰(−0.71, 0.71)	${0.0230}_{- 0.0028}^{+ 0.0250}$ $0.0230_{ - 0.0028}^{ + 0.0250}$	𝒰(−0.71, 0.71)	$- {0.049}_{- 0.018}^{+ 0.016}$ $- 0.049_{ - 0.018}^{ + 0.016}$
$\sqrt{e} \sin ω_{c}$ $\sqrt e \sin {\omega _{\rm{c}}}$	𝒰(-0.71, 0.71)	$- {0.4473}_{- 0.0129}^{+ 0.0039}$ $- 0.4473_{ - 0.0129}^{ + 0.0039}$	𝒰(−0.71, 0.71)	${0.0510}_{- 0.0587}^{+ 0.0035}$ $0.0510_{ - 0.0587}^{ + 0.0035}$	𝒰(−0.71, 0.71)	$- {0.0548}_{- 0.0246}^{+ 0.0076}$ $- 0.0548_{ - 0.0246}^{ + 0.0076}$	𝒰(−0.71, 0.71)	$- {0.142}_{- 0.017}^{+ 0.023}$ $- 0.142_{ - 0.017}^{ + 0.023}$
λ_c (°)	𝒰(0, 360)	${202.46}_{- 0.31}^{+ 1.17}$ $202.46_{ - 0.31}^{ + 1.17}$	𝒰(0, 360)	$189_{- 2}^{+ 4}$ $189_{ - 2}^{ + 4}$	𝒰(0, 360)	${155.51}_{- 2.29}^{+ 0.57}$ $155.51_{ - 2.29}^{ + 0.57}$	𝒰(0, 360)	$72_{- 1}^{+ 1}$ $72_{ - 1}^{ + 1}$
i_c (°)	fixed	-	fixed		𝒰(80, 100)	${90.89}_{- 0.46}^{+ 0.98}$ $90.89_{ - 0.46}^{ + 0.98}$	𝒰(80, 100)	${90.55}_{- 1.05}^{+ 0.37}$ $90.55_{ - 1.05}^{ + 0.37}$
Ω_c (°)	𝒰(0, 360)	${15.09}_{- 0.33}^{+ 1.36}$ $15.09_{ - 0.33}^{ + 1.36}$	𝒰(0, 360)	$352_{- 2}^{+ 4}$ $352_{ - 2}^{ + 4}$	𝒰(0, 360)	${318.09}_{- 2.26}^{+ 0.59}$ $318.09_{ - 2.26}^{ + 0.59}$	𝒰(0, 360)	$233_{- 1}^{+ 2}$ $233_{ - 1}^{ + 2}$
M_c/M_⋆ (M_⊙/M_⋆) × 10⁻⁶	-	-	𝒰(0.057, 319.520)	${1.22}_{- 0.12}^{+ 0.10}$ $1.22_{ - 0.12}^{ + 0.10}$	𝒰(0.057, 319.520)	${1.4562}_{- 0.0061}^{+ 0.1593}$ $1.4562_{ - 0.0061}^{ + 0.1593}$	𝒰(0.057, 319.520)	${5.91}_{- 0.40}^{+ 0.50}$ $5.91_{ - 0.40}^{ + 0.50}$
P_d (days)	-	-	𝒰(11,15)	${12.6173}_{- 0.0114}^{+ 0.0050}$ $12.6173_{ - 0.0114}^{ + 0.0050}$	𝒰(11, 15)	${13.6140}_{- 0.0016}^{+ 0.0026}$ $13.6140_{ - 0.0016}^{ + 0.0026}$	𝒰(11, 15)	${13.9276}_{- 0.0043}^{+ 0.0052}$ $13.9276_{ - 0.0043}^{ + 0.0052}$
$\sqrt{e} \cos ω_{d}$ $\sqrt e \cos {\omega _{\rm{d}}}$	-	-	𝒰(−0.71, 0.71)	$- {0.2041}_{- 0.0423}^{+ 0.0084}$ $- 0.2041_{ - 0.0423}^{ + 0.0084}$	𝒰(−0.71, 0.71)	${0.059}_{- 0.012}^{+ 0.038}$ $0.059_{ - 0.012}^{ + 0.038}$	𝒰(−0.71, 0.71)	${0.3067}_{- 0.0049}^{+ 0.0133}$ $0.3067_{ - 0.0049}^{ + 0.0133}$
$\sqrt{e} \sin ω_{d}$ $\sqrt e \sin {\omega _{\rm{d}}}$	-	-	𝒰(−0.71, 0.71)	${0.4993}_{- 0.0189}^{+ 0.0035}$ $0.4993_{ - 0.0189}^{ + 0.0035}$	𝒰(−0.71, 0.71)	${0.50045}_{- 0.01099}^{+ 0.00069}$ $0.50045_{ - 0.01099}^{ + 0.00069}$	𝒰(−0.71, 0.71)	$- {0.2443}_{- 0.0078}^{+ 0.0219}$ $- 0.2443_{ - 0.0078}^{ + 0.0219}$
λ_d (−)	-	-	𝒰(0, 360)	$18_{- 2}^{+ 8}$ $18_{ - 2}^{ + 8}$	𝒰(0, 360)	${348.65}_{- 2.99}^{+ 0.60}$ $348.65_{ - 2.99}^{ + 0.60}$	𝒰(0, 360)	$7_{- 3}^{+ 1}$ $7_{ - 3}^{ + 1}$
i_d(°)	-	-	𝒰(50, 130)	${127.08}_{- 0.52}^{+ 2.44}$ $127.08_{ - 0.52}^{ + 2.44}$	𝒰(50, 130)	${108.86}_{- 0.89}^{+ 1.54}$ $108.86_{ - 0.89}^{ + 1.54}$	𝒰(50, 130)	${101.07}_{- 2.73}^{+ 0.40}$ $101.07_{ - 2.73}^{ + 0.40}$
Ω_d (°)	-	-	𝒰(0, 360)	$120_{- 1}^{+ 5}$ ${120_{ - 1}^{ + 5}}$	𝒰(0, 360)	${184.39}_{- 2.17}^{+ 0.22}$ ${184.39_{ - 2.17}^{ + 0.22}}$	𝒰(0, 360)	${211.15}_{- 1.33}^{+ 0.86}$ ${211.15_{ - 1.33}^{ + 0.86}}$
γ_RV (m s⁻¹)	-	-	-	-	-	-	𝒰(−5412, 5243)	$- 100_{- 10}^{+ 3}$ $- 100_{ - 10}^{ + 3}$
log₂ σ_jitter.Rv	_-	_-	_-	_-	_-	_-	𝒰(−49.8, 6.6)	${6.6346}_{- 0.0041}^{+ 0.0091}$ ${6.6346_{ - 0.0041}^{ + 0.0091}}$
derived parameters
M_b(M_⊕)	ℬ(0.01, 50)	$18_{- 1}^{+ 2}$ $18_{ - 1}^{ + 2}$	ℬ(0.01, 50)	$26_{- 2}^{+ 2}$ ${26_{ - 2}^{ + 2}}$	ℬ(0.01, 50)	${4.70}_{- 0.40}^{+ 0.55}$ ${4.70_{ - 0.40}^{ + 0.55}}$	ℬ(0.01, 50)	$31_{- 3}^{+ 3}$ ${31_{ - 3}^{ + 3}}$
e_b	ℬ(0.0, 0.5)	${0.4988}_{- 0.0148}^{+ 0.0012}$ ${0.4988_{ - 0.0148}^{ + 0.0012}}$	ℬ(0.0, 0.5)	${0.303}_{- 0.013}^{+ 0.040} (2 σ)$ ${0.303_{ - 0.013}^{ + 0.040}(2\sigma )}$	ℬ(0.0, 0.5)	${0.0945}_{- 0.0029}^{+ 0.0063}$ ${0.0945_{ - 0.0029}^{ + 0.0063}}$	ℬ(0.0, 0.5)	${0.0044}_{- 0.0023}^{+ 0.0033}$ ${0.0044_{ - 0.0023}^{ + 0.0033}}$
ω_b (°)	𝒰(0, 360)	$40_{- 2}^{+ 2}$ ${40_{ - 2}^{ + 2}}$	𝒰(0, 360)	$77_{- 3}^{+ 3}$ ${77_{ - 3}^{ + 3}}$	𝒰(0, 360)	$- 80_{- 15}^{+ 3} (2 σ)$ ${ - 80_{ - 15}^{ + 3}(2\sigma )}$	𝒰(0, 360)	$114_{- 22}^{+ 17}$ $114_{ - 22}^{ + 17}$
ℳ_b (°)	𝒰(0, 360)	${316.89}_{- 0.40}^{+ 1.35}$ ${316.89_{ - 0.40}^{ + 1.35}}$	𝒰(0, 360)	${307.30}_{- 2.04}^{+ 0.99}$ ${307.30_{ - 2.04}^{ + 0.99}}$	𝒰(0, 360)	${108.249}_{- 0.030}^{+ 11.401}$ ${108.249_{ - 0.030}^{ + 11.401}}$	𝒰(0, 360)	$- 83_{- 17}^{+ 22}$ ${ - 83_{ - 17}^{ + 22}}$
M_c (M_æ)	ℬ(0.01, 50)	$21_{- 2}^{+ 1}$ ${21_{ - 2}^{ + 1}}$	ℬ(0.01, 50)	${4.40}_{- 0.23}^{+ 1.28}$ ${4.40_{ - 0.23}^{ + 1.28}}$	ℬ(0.01, 50)	${14.24}_{- 1.00}^{+ 1.71}$ ${14.24_{ - 1.00}^{ + 1.71}}$	ℬ(0.01, 50)	${14.31}_{- 0.73}^{+ 1.76}$ ${14.31_{ - 0.73}^{ + 1.76}}$
e_c	ℬ(0.0, 0.5)	${0.2260}_{- 0.0037}^{+ 0.0105}$ ${0.2260_{ - 0.0037}^{ + 0.0105}}$	ℬ(0.0, 0.5)	${0.0027}_{- 0.0027}^{+ 0.0057} (2 σ)$ ${0.0027_{ - 0.0027}^{ + 0.0057}(2\sigma )}$	ℬ(0.0, 0.5)	${0.00353}_{- 0.00023}^{+ 0.00287}$ ${0.00353_{ - 0.00023}^{ + 0.00287}}$	ℬ(0.0, 0.5)	${0.0226}_{- 0.0042}^{+ 0.0045}$ ${0.0226_{ - 0.0042}^{ + 0.0045}}$
ω_c (°)	𝒰(0, 360)	${289.77}_{- 1.17}^{+ 0.52}$ ${289.77_{ - 1.17}^{ + 0.52}}$	𝒰(0, 360)	$77_{- 58}^{+ 40}$ ${77_{ - 58}^{ + 40}}$	𝒰(0, 360)	$- 67_{- 9}^{+ 19}$ ${ - 67_{ - 9}^{ + 19}}$	𝒰(0, 360)	$- 109_{- 7}^{+ 8}$ ${ - 109_{ - 7}^{ + 8}}$
ℳ_c (°)	𝒰(0, 360)	${257.60}_{- 0.80}^{+ 1.45}$ ${257.60_{ - 0.80}^{ + 1.45}}$	𝒰(0, 360)	$121_{- 40}^{+ 58}$ ${121_{ - 40}^{ + 58}}$	𝒰(0, 360)	$265_{- 19}^{+ 9}$ ${265_{ - 19}^{ + 9}}$	𝒰(0, 360)	$307_{- 8}^{+ 7}$ ${307_{ - 8}^{ + 7}}$
M_d(M_⊕)	-	-	ℬ(0.01, 50)	${0.203}_{- 0.024}^{+ 0.022}$ ${0.203_{ - 0.024}^{ + 0.022}}$	ℬ(0.01, 50)	${0.2424}_{- 0.0084}^{+ 0.0333}$ ${0.2424_{ - 0.0084}^{ + 0.0333}}$	ℬ(0.01, 50)	${0.983}_{- 0.087}^{+ 0.106}$ ${0.983_{ - 0.087}^{ + 0.106}}$
e_d	_-	_-	ℬ(0.0, 0.5)	${0.2910}_{- 0.0057}^{+ 0.0.028}$ ${0.2910_{ - 0.0057}^{ + 0.0.028}}$	ℬ(0.0, 0.5)	${0.2539}_{- 0.0086}^{+ 0.0017} (2 σ)$ ${0.2539_{ - 0.0086}^{ + 0.0017}(2\sigma )}$	ℬ(0.0, 0.5)	${0.1537}_{- 0.0059}^{+ 0.0033}$ ${0.1537_{ - 0.0059}^{ + 0.0033}}$
ω_d (°)	_-	_-	𝒰(0, 360)	${112.23}_{- 0.87}^{+ 5.02}$ ${112.23_{ - 0.87}^{ + 5.02}}$	𝒰(0, 360)	$83_{- 5}^{+ 1}$ ${83_{ - 5}^{ + 1}}$	𝒰(0, 360)	$321_{- 2}^{+ 3}$ ${321_{ - 2}^{ + 3}}$
ℳ_d (°)	_-	_-	𝒰(0, 360)	${145.02}_{- 1.89}^{+ 0.87}$ ${145.02_{ - 1.89}^{ + 0.87}}$	𝒰(0, 360)	${80.97}_{- 0.82}^{+ 3.67}$ ${80.97_{ - 0.82}^{ + 3.67}}$	𝒰(0, 360)	${194.56}_{- 4.14}^{+ 0.96}$ ${194.56_{ - 4.14}^{ + 0.96}}$
σ_{jitter, RV} (m s ⁻¹)	-	-	-	-	-	-	ℬ(10−¹⁵, 100)	${99.36}_{- 0.28}^{+ 0.63}$ $99.36_{ - 0.28}^{ + 0.63}$
BIC = −2 ln 𝒫 + n_fit ln n_data	208.1		−536.6		−520.3		1115.2

Notes. All the parameters have been defined at the reference time BJD_TDB − 2450000 = 8329. Priors indicated as 𝒩(µ, σ) means Gaussian-Normal distribution, as 𝒰(lower, upper) means Uniform distribution, and as ℬ (lower, upper) means the values have been used to force the boundaries, e.g we sampled uniformly in $\sqrt{e} \cos (ω)$ $\sqrt e \cos \left( \omega \right)$ and $\sqrt{e} \sin (ω)$ $\sqrt e \sin \left( \omega \right)$ , but we checked that e was within the boundaries. λ is the mean longitude, that is λ = ω + Ω + ℳ, where ω is the argument of the pericentre (or periastron), Ω is the longitude of the ascending node, and ℳ is mean anomaly. In some cases, the physical parameter converted from the MAP fitted parameters was not within the HDI at 68.27%, so we reported also the value of the 2σ uncertainty from the HDI at 95.5%.

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