Transit timing variations in HIP 41378: CHEOPS and TESS confirm a non-transiting sixth planet in the system

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Table A.3

Posteriors and derived orbital parameters (MAP and HDI) for HIP 41378 b, c, and g obtained from the dynamical analysis with TRADES.

Parameter	Unit	Prior	MAP (HDI±1σ)
HIP 41378 b
FITTED PARAMETERS
M_p/M_⋆	$[(\frac{M_{⊙}}{M_{⋆}}) \times 10^{- 6}]$ $\[[\left(\frac{M_{\odot}}{M_{\star}}\right) \times 10^{-6}]\]$	𝒰(0.30, 954.7)	$22_{- 1}^{+ 3}$ $\[22_{-1}^{+3}\]$
Orbital Period (P)	[days]	𝒰(13.5, 17.5)	${15.571306}_{- 0.000323}^{+ 0.000011}$ $\[15.571306_{-0.000323}^{+0.000011}\]$
$\sqrt{e} \cos ω$ $\[\sqrt{e} ~\cos~ \omega\]$	−	$U (- \sqrt{0.5}, \sqrt{0.5})$ $\[\mathcal{U}(-\sqrt{0.5}, \sqrt{0.5})\]$	${0.142}_{- 0.037}^{+ 0.011}$ $\[0.142_{-0.037}^{+0.011}\]$
$\sqrt{e} \sin ω$ $\[\sqrt{e} ~\sin~ \omega\]$	−	$U (- \sqrt{0.5}, \sqrt{0.5})$ $\[\mathcal{U}(-\sqrt{0.5}, \sqrt{0.5})\]$	$- {0.033}_{- 0.038}^{+ 0.032}$ $\[-0.033_{-0.038}^{+0.032}\]$
Mean Longitude (λ)	[deg]	𝒰(0, 360)	${274.34}_{- 0.28}^{+ 1.09}$ $\[274.34_{-0.28}^{+1.09}\]$
DERIVED PARAMETERS
Mass (m_p)	[M_⊕]	−	${9.06}_{- 0.51}^{+ 1.41}$ $\[9.06_{-0.51}^{+1.41}\]$
Eccentricity (e)	−	𝒩⁺(0,0.083)	${0.0213}_{- 0.0099}^{+ 0.0022}$ $\[0.0213_{-0.0099}^{+0.0022}\]$
Argument of Periastron (ω)	[deg]	−	$- 13_{- 17}^{+ 14}$ $\[-13_{-17}^{+14}\]$
Mean Anomaly (M_A)	[deg]	−	$107_{- 15}^{+ 16}$ $\[107_{-15}^{+16}\]$
HIP 41378 c
FITTED PARAMETERS
M_p/M_⋆	$[(\frac{M_{⊙}}{M_{⋆}}) \times 10^{- 6}]$ $\[[\left(\frac{M_{\odot}}{M_{\star}}\right) \times 10^{-6}]\]$	𝒰(0.30, 954.7)	$16_{- 1}^{+ 3}$ $\[16_{-1}^{+3}\]$
Orbital Period (P)	[days]	𝒰(30, 34)	${31.71054}_{- 0.00098}^{+ 0.00172}$ $\[31.71054_{-0.00098}^{+0.00172}\]$
$\sqrt{e} \cos ω$ $\[\sqrt{e} ~\cos~ \omega\]$	−	$U (- \sqrt{0.5}, \sqrt{0.5})$ $\[\mathcal{U}(-\sqrt{0.5}, \sqrt{0.5})\]$	$- {0.242}_{- 0.010}^{+ 0.039}$ $\[-0.242_{-0.010}^{+0.039}\]$
$\sqrt{e} \sin ω$ $\[\sqrt{e} ~\sin~ \omega\]$	−	$U (- \sqrt{0.5}, \sqrt{0.5})$ $\[\mathcal{U}(-\sqrt{0.5}, \sqrt{0.5})\]$	${0.096}_{- 0.015}^{+ 0.054}$ $\[0.096_{-0.015}^{+0.054}\]$
Mean Longitude (λ)	[deg]	𝒰(0, 360)	$333_{- 2}^{+ 8}$ $\[333_{-2}^{+8}\]$
Longitude of Ascending Node (Ω)	[deg]	−	$173_{- 1}^{+ 9}$ $\[173_{-1}^{+9}\]$
DERIVED PARAMETERS
Mass (m_p)	[M_⊕]	−	${6.53}_{- 0.42}^{+ 1.33}$ $\[6.53_{-0.42}^{+1.33}\]$
Eccentricity (e)	−	𝒩⁺(0,0.083)	${0.0678}_{- 0.0097}^{+ 0.0078}$ $\[0.0678_{-0.0097}^{+0.0078}\]$
Argument of Periastron (ω)	[deg]	−	$158_{- 13}^{+ 4}$ $\[158_{-13}^{+4}\]$
Mean Anomaly (M_A)	[deg]	−	$2_{- 3}^{+ 13}$ $\[2_{-3}^{+13}\]$
HIP 41378 g
FITTED PARAMETERS
M_p/M_⋆	$[(\frac{M_{⊙}}{M_{⋆}}) \times 10^{- 6}]$ $\[[\left(\frac{M_{\odot}}{M_{\star}}\right) \times 10^{-6}]\]$	𝒰(0.30, 901)	$17_{- 2}^{+ 3}$ $\[17_{-2}^{+3}\]$
Orbital Period (P)	[days]	𝒰(32, 200)	${64.067}_{- 0.067}^{+ 0.026}$ $\[64.067_{-0.067}^{+0.026}\]$
$\sqrt{e} \cos ω$ $\[\sqrt{e} ~\cos~ \omega\]$	−	$U (- \sqrt{0.5}, \sqrt{0.5})$ $\[\mathcal{U}(-\sqrt{0.5}, \sqrt{0.5})\]$	$- {0.091}_{- 0.116}^{+ 0.073}$ $\[-0.091_{-0.116}^{+0.073}\]$
$\sqrt{e} \sin ω$ $\[\sqrt{e} ~\sin~ \omega\]$	−	$U (- \sqrt{0.5}, \sqrt{0.5})$ $\[\mathcal{U}(-\sqrt{0.5}, \sqrt{0.5})\]$	$- {0.043}_{- 0.011}^{+ 0.231}$ $\[-0.043_{-0.011}^{+0.231}\]$
Mean Longitude (λ)	[deg]	𝒰(0, 360)	$350_{- 22}^{+ 4}$ $\[350_{-22}^{+4}\]$
Inclination (i)	[deg]	−	$95_{- 10}^{+ 1}$ $\[95_{-10}^{+1}\]$
Longitude of Ascending Node (Ω)	[deg]	−	$184_{- 6}^{+ 6}$ $\[184_{-6}^{+6}\]$
DERIVED PARAMETERS
Mass (m_p)	[M_⊕]	−	${6.81}_{- 0.98}^{+ 1.14}$ $\[6.81_{-0.98}^{+1.14}\]$
Eccentricity (e)	−	𝒩⁺(0,0.083)	${0.010}_{- 0.010}^{+ 0.031}$ $\[0.010_{-0.010}^{+0.031}\]$
Argument of Periastron (ω)	[deg]	−	$205_{- 115}^{+ 5}$ $\[205_{-115}^{+5}\]$
Mean Anomaly (M_A)	[deg]	−	$- 39_{- 8}^{+ 117}$ $\[-39_{-8}^{+117}\]$
ADDITIONAL PARAMETERS
Radial Velocity Jitter (σ_jitter)	[m/s]	−	${2.36}_{- 0.10}^{+ 0.19}$ $\[2.36_{-0.10}^{+0.19}\]$
Radial Velocity offset (γ₁)	[m/s]	−	${50711.789}_{- 0.241}^{+ 0.095}$ $\[50711.789_{-0.241}^{+0.095}\]$

Notes. The symbols 𝒰, 𝒢, and 𝒩⁺ refer to uniform, Gaussian, and half-Gaussian distributions, respectively.

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