Photo-dynamical characterisation of the TOI-178 resonant chain - Exploring the robustness of transit-timing variations and radial velocity mass characterisations

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Volume 688 (August 2024)

A&A, 688 (2024) A211

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Table 3

Fitted and derived properties of the planets of TOI-178.

Parameter	Prior	Photo-dynamical	Photo-dynamical + RV
TOI-178e

λ (deg)	𝒰(−360.00, 360.00)	${74.265}_{- 0.017}^{+ 0.017}$ ${74.265_{ - 0.017}^{ + 0.017}}$	${74.23}_{- 0.11}^{+ 0.11}$ ${74.23_{ - 0.11}^{ + 0.11}}$
P (day)	𝒰(9.66, 10.26)	${9.96336}_{- 1.5 e - 04}^{+ 1.6 e - 04}$ ${9.96336_{ - 1.5e - 04}^{ + 1.6e - 04}}$	${9.96318}_{- 1.2 e - 04}^{+ 1.5 e - 04}$ ${9.96318_{ - 1.2e - 04}^{ + 1.5e - 04}}$
e cos ϖ	(*)	$- 9.8 e - 05_{- 5.1 e - 04}^{+ 1.7 e - 04}$ ${ - 9.8e - 05_{ - 5.1e - 04}^{ + 1.7e - 04}}$	$- 2.0 e - 05_{- 6.7 e - 04}^{+ 4.3 e - 04}$ ${ - 2.0e - 05_{ - 6 - 7e - 04}^{ + 4.3e - 04}}$
e sin ϖ	(*)	$- 2.7 e - 05_{- 5.2 e - 04}^{+ 3.5 e - 04}$ ${ - 2.7e - 05_{ - 5.2e - 04}^{ + 3.5e - 04}}$	$1.0 e - 07_{- 5.7 e - 04}^{+ 5.7 e - 04}$ ${1.0e - 07_{ - 5.7e - 04}^{ + 5.7e - 04}}$
M/M_⊙	(*)	$1.0 e - 05_{- 1.0 e - 06}^{+ 1.1 e - 06}$ ${1.0e - 05_{ - 1.0e - 06}^{ + 1.1e - 06}}$	$1.0 e - 05_{- 8.6 e - 07}^{+ 8.6 e - 07}$ ${1.0e - 05_{ - 8.6e - 07}^{ + 8.6e - 07}}$
R/R_★	𝒰(0.0e+00, 2.0e-01)	${0.03205}_{- 4.7 e - 04}^{+ 4.5 e - 04}$ ${0.03205_{ - 4.7e - 04}^{ + 4.5e - 04}}$	${0.03191}_{- 3.7 e - 04}^{+ 3.8 e - 04}$ ${0.03191_{ - 3.7e - 04}^{ + 3.8e - 04}}$
b	𝒰(0.0e+00, 1.5e+00)	${0.5842}_{- 0.0073}^{+ 0.0099}$ ${0.5842_{ - 0.0073}^{ + 0.009}}$	${0.574}_{- 0.027}^{+ 0.015}$ ${0.574_{ - 0.027}^{ + 0.015}}$
dF (ppm)	derived	${1026.9}_{- 30.0}^{+ 28.9}$ ${1026.9_{ - 30.}^{ + 28.9}}$	${1018.1}_{- 23.5}^{+ 24.7}$ ${1018.1_{ - 23.5}^{ + 24.7}}$
e	derived	$4.3 e - 04_{- 3.6 e - 04}^{+ 7.5 e - 04}$ ${4.3e - 04_{ - 3.6e - 04}^{ + 7.5e - 04}}$	$3.8 e - 04_{- 2.5 e - 04}^{+ 6.7 e - 04}$ ${3.8e - 04_{ - 2.5e - 04}^{ + 6.7e - 04}}$
ϖ (deg)	derived	$- 49_{- 95}^{+ 175}$ ${ - 49_{ - 95}^{ + 175}}$	$2_{- 134}^{+ 120}$ ${2_{ - 134}^{ + 120}}$
M(M_Earth)	derived	${3.34}_{- 0.33}^{+ 0.37}$ ${3.34_{ - 0.33}^{ + 0.37}}$	${3.48}_{- 0.29}^{+ 0.29}$ ${3.48_{ - 0.29}^{ + 0.29}}$
R (R_Earth)	derived	${2.325}_{- 0.038}^{+ 0.034}$ ${2.325_{ - 0.038}^{ + 0.034}}$	${2.301}_{- 0.039}^{+ 0.038}$ ${2.301_{ - 0.039}^{ + 0.038}}$
ρ(ρ_Earth)	derived	${0.264}_{- 0.023}^{+ 0.029}$ ${0.264_{ - 0.023}^{ + 0.029}}$	${0.287}_{- 0.026}^{+ 0.028}$ ${0.287_{ - 0.026}^{ + 0.028}}$
∆_M		0.54	0.49

TOI-178f

λ(deg)	𝒰(-360.00, 360.00)	${157.735}_{- 0.005}^{+ 0.021}$ ${157.735_{ - 0.005}^{ + 0.021}}$	${157.67}_{- 0.09}^{+ 0.11}$ ${157.67_{ - 0.09}^{ + 0.11}}$
P (day)	𝒰(14.78, 15.69)	${15.23351}_{- 1.6 e - 04}^{+ 1.4 e - 04}$ ${15.23351_{ - 1.6e - 04}^{ + 1.4e - 04}}$	${15.23335}_{- 2.5 e - 04}^{+ 2.4 e - 04}$ ${15.23335_{ - 2.5e - 04}^{ + 2.4e - 04}}$
e cos ϖ	(*)	$- 2.1 e - 04_{- 4.6 e - 04}^{+ 2.9 e - 04}$ ${ - 2.1e - 04_{ - 4.6e - 04}^{ + 2.9e - 04}}$	$- 8.4 e - 06_{- 6.7 e - 04}^{+ 8.1 e - 04}$ ${ - 8.4e - 06_{ - 6.7e - 04}^{ + 8.1e - 04}}$
e sin ϖ		$8.8 e - 04_{- 0.0009}^{+ 0.0015}$ ${8.8e - 04_{ - 0.0009}^{ + 0.0015}}$	${0.0017}_{- 0.0016}^{+ 0.0016}$ ${0.0017_{ - 0.0016}^{ + 0.0016}}$
M/M_⊙	(*)	$1.9 e - 05_{- 1.5 e - 06}^{+ 1.2 e - 06}$ ${1.9e - 05_{ - 1.5e - 06}^{ + 1.2e - 06}}$	$1.7 e - 05_{- 1.2 e - 06}^{+ 1.4 e - 06}$ ${1.7e - 05_{ - 1.2e - 06}^{ + 1.4e - 06}}$
R/R_★	𝒰(0.0e+00, 2.0e-01)	${0.03370}_{- 5.9 e - 04}^{+ 5.7 e - 04}$ ${0.03370_{ - 5.9e - 04}^{ + 5.7e - 04}}$	${0.03351}_{- 6.0 e - 04}^{+ 5.6 e - 04}$ ${0.03351_{ - 6.0e - 04}^{ + 5.6e - 04}}$
b	𝒰(0.0e+00, 1.5e+00)	${0.7399}_{- 0.0042}^{+ 0.0098}$ ${0.7399_{ - 0.0042}^{ + 0.0098}}$	${0.734}_{- 0.014}^{+ 0.015}$ ${0.734_{ - 0.014}^{ + 0.015}}$
dF (ppm)	derived	${1135.6}_{- 39.3}^{+ 38.6}$ ${1135.6_{ - 39.3}^{ + 38.6}}$	${1123.1}_{- 40.0}^{+ 38.0}$ ${1123.1_{ - 40.0}^{ + 38.0}}$
e	derived	${0.0011}_{- 0.0009}^{+ 0.0014}$ ${0.0011_{ - 0.0009}^{ + 0.0014}}$	$4.5 e - 04_{- 2.6 e - 04}^{+ 7.0 e - 04}$ ${4.5e - 04_{ - 2.6e - 04}^{ + 7.0e - 04}}$
τ (deg)	derived	${98.2}_{- 73.8}^{+ 28.8}$ ${98.2_{ - 73.8}^{ + 28.8}}$	${87.3}_{- 40.1}^{+ 28.5}$ ${87.3_{ - 40.1}^{ + 28.5}}$
M(M_Earth)	derived	${6.24}_{- 0.49}^{+ 0.40}$ ${6.24_{ - 0.49}^{ + 0.40}}$	${5.63}_{- 0.41}^{+ 0.45}$ ${5.63_{ - 0.41}^{ + 0.45}}$
R (R_Earth)	derived	${2.443}_{- 0.053}^{+ 0.047}$ ${2.443_{ - 0.053}^{ + 0.047}}$	${2.417}_{- 0.048}^{+ 0.041}$ ${2.417_{ - 0.048}^{ + 0.041}}$
ρ(ρ_Earth)	derived	${0.426}_{- 0.037}^{+ 0.040}$ ${0.426_{ - 0.037}^{ + 0.040}}$	${0.402}_{- 0.036}^{+ 0.033}$ ${0.402_{ - 0.036}^{ + 0.033}}$
∆_M		0.93	1.15

TOI-178g

λ(deg)	𝒰(−360.00, 360.00)	${55.329}_{- 0.019}^{+ 0.007}$ ${55.329_{ - 0.019}^{ + 0.007}}$	${55.27}_{- 0.10}^{+ 0.06}$ ${55.27_{ - 0.10}^{ + 0.06}}$
P (day)	𝒰(20.09, 21.34)	${20.71700}_{- 3.4 e - 04}^{+ 4.1 e - 04}$ ${20.71700_{ - 3.4e - 04}^{ + 4.1e - 04}}$	${20.71663}_{- 3.0 e - 04}^{+ 3.1 e - 04}$ ${20.71663_{ - 3.0e - 04}^{ + 3.1e - 04}}$
e cos ϖ	(*)	$- 2.0 e - 04_{- 2.9 e - 04}^{+ 2.1 e - 04}$ ${ - 2.0e - 04_{ - 2.9e - 04}^{ + 2.1e - 04}}$	$1.2 e - 04_{- 4.2 e - 04}^{+ 7.9 e - 04}$ ${1.2e - 04_{ - 4.2e - 04}^{ + 7.9e - 04}}$
e sin ϖ	(*)	$- 9.3 e - 05_{- 6.9 e - 04}^{+ 2.6 e - 04}$ ${ - 9.3e - 05_{ - 6.9e - 04}^{ + 2.6e - 04}}$	$- 1.7 e - 04_{- 0.0012}^{+ 0.0005}$ ${ - 1.7e - 04_{ - 0.0012}^{ + 0.0005}}$
M/M_⊙	(*)	$1.4 e - 05_{- 7.9 e - 07}^{+ 8.8 e - 07}$ ${1.4e - 05_{ - 7.9e - 07}^{ + 8.8e - 07}}$	$1.3 e - 05_{- 1.1 e - 06}^{+ 1.2 e - 06}$ $1.3e - 05_{ - 1.1e - 06}^{ + 1.2e - 06}$
R/R_★	𝒰(0.0e+00, 2.0e-01)	${0.04086}_{- 5.4 e - 04}^{+ 4.4 e - 04}$ ${0.04086_{ - 5.4e - 04}^{ + 4.4e - 04}}$	${0.04075}_{- 4.4 e - 04}^{+ 5.1 e - 04}$ ${0.04075_{ - 4.4e - 04}^{ + 5.1e - 04}}$
b	𝒰(0.0e+00, 1.5e+00)	${0.8591}_{- 0.0029}^{+ 0.0027}$ ${0.8591_{ - 0.0029}^{ + 0.0027}}$	${0.8549}_{- 0.0054}^{+ 0.0055}$ ${0.8549_{ - 0.0054}^{ + 0.0055}}$
dF (ppm)	derived	${1669.2}_{- 43.7}^{+ 36.0}$ ${1669.2_{ - 43.7}^{ + 36.0}}$	${1660.5}_{- 35.5}^{+ 41.8}$ ${1660.5_{ - 35.5}^{ + 41.8}}$
e	derived	$4.5 e - 04_{- 3.5 e - 04}^{+ 5.4 e - 04}$ ${4.5e - 04_{ - 3.5e - 04}^{ + 5.4e - 04}}$	$4.3 e - 04_{- 2.8 e - 04}^{+ 6.5 e - 04}$ ${4.3e - 04_{ - 2.8e - 04}^{ + 6.5e - 04}}$
τ (deg)	derived	$- 98_{- 44}^{+ 227}$ ${ - 98_{ - 44}^{ + 227}}$	$- 44_{- 63}^{+ 120}$ ${ - 44_{ - 63}^{ + 120}}$
M(M_Earth)	derived	${4.58}_{- 0.26}^{+ 0.29}$ ${4.58_{ - 0.26}^{ + 0.29}}$	${4.40}_{- 0.37}^{+ 0.39}$ ${4.40_{ - 0.37}^{ + 0.39}}$
R (R_Earth)	derived	${2.964}_{- 0.048}^{+ 0.042}$ ${2.964_{ - 0.048}^{ + 0.042}}$	${2.939}_{- 0.055}^{+ 0.057}$ ${2.939_{ - 0.055}^{ + 0.057}}$
ρ(ρ_Earth)	derived	${0.177}_{- 0.012}^{+ 0.013}$ ${0.177_{ - 0.012}^{ + 0.013}}$	${0.174}_{- 0.015}^{+ 0.015}$ ${0.174_{ - 0.015}^{ + 0.015}}$
∆_M		1.34	0.79

Notes. Results of the photo-dynamical and photo-dynamical+RV fits for the final set of priors. The orbital elements are given at the date 2458352.55018382 BJD. λ is the mean longitude of the planet, σ its longitude of periastron. ∆_M is the robustness criterion defined in Eq. (1). b, e, M and R are the planet’s impact parameter, eccentricity, mass and radius respectively. M_⊙ is the mass of the sun and R_★ is the radius of the star.^(*) The mass and eccentricity priors depend on the case, see Sect. 3.2.

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