TOI-1438: A rare system with two short-period sub-Neptunes and a tentative long-period Jupiter-like planet orbiting a K0V star

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Volume 702 (October 2025)

A&A, 702 (2025) A69

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Table 5

Best-fit transit and RV model of the TOI-1438 system, as described in Sect. 3.4.

		Planet b		Planet c
Parameter	Units	Priors ^a	Final value	Priors ^a	Final value
Fitted parameters
T₀	Transit epoch (BJD_TDB−2 457 000)	U[1683.621,1683.630]	1683.6256 ± 7e-4	U[1689.837,1689.984]	1689.9136 ± 1.4e-3
P_orb	Orbital period (d)	U[5.12,5.16]	5.139670 ± 3e-6	U[9.32,9.50]	9.428089 ± 1e-6
cos i	Cosine of inclination	U[0, 1]	${0.0660}_{- 0.0021}^{+ 0.0019}$ $0.0660^{+0.0019}_{-0.0021}$	U[0, 1]	${0.0417}_{- 0.018}^{+ 0.014}$ $0.0417^{+0.014}_{-0.018}$
a/R_⋆ ^b	Scaled semi-major axis	U[10, 20]	14.5 ± 0.4	U[15, 25]	${21.6}_{- 0.6}^{+ 0.7}$ $21.6^{+0.7}_{-0.6}$
R_p/R_⋆	Scaled planet radius	U[0.0, 0.2]	0.03410 ± 0.0013	U[0.0, 0.2]	0.0308 ± 0.0008
K	Doppler semi-amplitude (m s⁻¹)	U[0, 150]	3.8 ± 0.7	U[0, 150]	3.5 ± 0.7
e	Eccentricity	F	0	F	0
ω	Argument of periastron (deg)	F	90	F	90
Derived Parameters
M_p	Planet mass (M_⊕)	...	9.4 ± 1.8	...	10.6 ± 2.1
R_p	Planet radius (R_⊕)	...	3.04 ± 0.19	...	2.75 ± 0.14
i^c	Inclination (deg)	...	${86.21}_{- 0.11}^{+ 0.12}$ $86.21^{+0.12}_{-0.11}$	...	${87.61}_{- 0.08}^{+ 0.10}$ $87.61^{+0.10}_{-0.08}$
b	Impact parameter	...	0.956 ± 0.004	...	${0.902}_{- 0.008}^{+ 0.009}$ $0.902^{+0.009}_{-0.008}$
a	Semi-major axis (au)	...	0.0553 ± 0.0015	...	0.083 ± 0.003
F	Instellation (F_⊕)	...	145 ± 10	...	65 ± 4
ρ_p	Planet density (g cm⁻³)	...	1.8 ± 0.5	...	2.9 ± 0.7
g_p	Planet surface gravity (cm s⁻²)	...	997 ± 232	...	1425 ± 297
T_eq ^d	Equilibrium temperature (K)	...	971 ± 11	...	794 ± 9
Λ ^e	Jeans escape (cm² g erg⁻¹ s⁻²)	...	24 ± 5	...	37 ± 8
TSM ^f	Transmission spectroscopy metric	...	67 ± 18	...	36 ± 9
T₁₄	Total transit duration (h)	...	${1.04}_{- 0.03}^{+ 0.05}$ $1.04^{+0.05}_{-0.03}$	...	1.67 ± 0.04
T₂₃	Full transit duration (h)	...	0.65 ± 0.05	...	1.21 ± 0.05

		Signal d
Fitted parameters
T₀	Transit epoch (BJD_TDB- 2 457 000)	U[3000,3500]	3267-54	...	...
ln P_orb	Logarithm of orbital period (d)	U[0.8,10]	${7.9}_{- 0.3}^{+ 0.2}$ $7.9^{+0.2}_{-0.3}$	...	...
K	Doppler semi-amplitude (m s⁻¹)	U[0, 150]	$35_{- 5}^{+ 3}$ $35^{+3}_{-5}$	...	...
e	Eccentricity	B(0.867,3.03)	${0.25}_{- 0.11}^{+ 0.08}$ $0.25^{+0.08}_{-0.11}$	...	...
ω	Argument of periastron (deg)	U(−180,180)	$18_{- 10}^{+ 7}$ $18^{+7}_{-10}$	...	...
Derived Parameters (assuming planetary origin of signal d)
P_orb	Orbital period (yr)	...	${7.6}_{- 2.4}^{+ 1.6}$ $7.6^{+1.6}_{-2.4}$	...	...
M_p sin i ^g	Lower limit on planet mass (M_J)	...	2.1 ± 0.3	...	...
a sin i	Lower limit on semi-major axis (au)	...	3.6 ± 0.8	...	...

Additional Parameters			...	...
Y1	Systemic velocity HARPS-N (m s⁻¹)	U[−30000, −29 000]	$- 29476_{- 6}^{+ 10}$ $-29476^{+10}_{-6}$	...	...
σ₁	RV jitter HARPS-N (m s⁻¹)	U[0,50]	4.4 ± 0.4	...	...
γ₂	Systemic velocity HIRES (m s⁻¹)	U[−1000,1000]	$- 15_{- 7}^{+ 9}$ $-15^{+9}_{-7}$	...	...
σ₂	RV jitter HIRES (m s⁻¹)	U[0,50]	${5.0}_{- 1.1}^{+ 0.8}$ $5.0^{+0.8}_{-1.1}$	...	...
^q₁ + ^q₂	Limb-darkening coeff. sum	N[0.62, 0.10]	0.63 ± 0.09	...	...
q₁ − q₂	Limb-darkening coeff. difference	F	0.16	...	...

Adopted stellar parameters
M_⋆	Stellar mass (M_⊙)	F	0.876 ± 0.038	...	...
R_⋆	Stellar radius (R_⊙)	F	0.820 ± 0.017	...	...
T_eff	Effective temperature (K)	F	5230 ± 60	...	...

Notes. The given values are the median and the uncertainty is the highest posterior density at a confidence level of 0.68. ^(a)U[a,b] refers to uniform priors in the range a − b, F [a] to a fixed value a, N [a,b] to Gaussian priors with mean a and standard deviation b, and B(α,β) refers to a Beta prior. ^(b) A constraint on a/R_⋆ is applied through the stellar density. ^(c)Orbit inclination relative to the plane of the sky. ^(d)Dayside equilibrium temperature without heat redistribution and zero albedo. ^(e)Jeans escape parameter defined as Λ = GM_pm_H/(k_BT_eqR_p) (Fossati et al. 2017). ^(f)The transmission spectroscopy metric (TSM) is a proxy for the S/N of the James Webb Space Telescope (JWST) transmission spectroscopy and recommended to be >90 (Kempton et al. 2018). ^(g)Lower limit on the mass of planet d assuming that the long-period signal d has a planetary origin.

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