EMPEROR - I. Exoplanet MCMC parallel tempering for RV orbit retrieval

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Volume 704 (December 2025)

A&A, 704 (2025) A323

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Table C.3

HD 55693 model parameter estimations.

Parameter	Prior	WN (K₁)	WN (K₂)	MA (K₁)	MA (K₂)	SA (K₀)	SAMA (K₂)	GProt(K₁)

Acceleration, offsets and jitter
γ_T1 (m s^-1)	𝒰 (-8.5, 8.5)	$- {1.17}_{- 0.15}^{+ 0.5}$ $- 1.17_{ - 0.15}^{ + 0.5}$	$- {1.95}_{- 0.68}^{+ 1.36}$ $- 1.95_{ - 0.68}^{ + 1.36}$	$- {1.46}_{- 0.7}^{+ 0.79}$ $- 1.46_{ - 0.7}^{ + 0.79}$	$- {1.32}_{- 0.47}^{+ 0.47}$ $- 1.32_{ - 0.47}^{ + 0.47}$	${1.2}_{- 0.16}^{+ 0.29}$ $1.2_{ - 0.16}^{ + 0.29}$	$- {1.38}_{- 0.42}^{+ 0.19}$ $- 1.38_{ - 0.42}^{ + 0.19}$	$- {1.55}_{- 0.63}^{+ 0.63}$ $- 1.55_{ - 0.63}^{ + 0.63}$
γPFS (m s^-1)	𝒰 (-6.7, 6.7)	${0.71}_{- 0.14}^{+ 0.37}$ $0.71_{ - 0.14}^{ + 0.37}$	${1.21}_{- 0.66}^{+ 0.56}$ $1.21_{ - 0.66}^{ + 0.56}$	${0.4}_{- 0.61}^{+ 0.79}$ $0.4_{ - 0.61}^{ + 0.79}$	${1.39}_{- 0.70}^{+ 0.99}$ $1.39_{ - 0.70}^{ + 0.99}$	$- {0.12}_{- 0.27}^{+ 0.28}$ $- 0.12_{ - 0.27}^{ + 0.28}$	${1.26}_{- 0.72}^{+ 0.58}$ $1.26_{ - 0.72}^{ + 0.58}$	${0.75}_{- 0.3}^{+ 0.31}$ $0.75_{ - 0.3}^{ + 0.31}$
γ_T2 (m s^-1)	𝒰 (-5.7, 5.7)	${0.11}_{- 1.44}^{+ 1.11}$ $0.11_{ - 1.44}^{ + 1.11}$	${0.14}_{- 1.64}^{+ 1.17}$ $0.14_{ - 1.64}^{ + 1.17}$	${0.64}_{- 2.06}^{+ 1.12}$ $0.64_{ - 2.06}^{ + 1.12}$	${0.98}_{- 2.52}^{+ 1.31}$ $0.98_{ - 2.52}^{ + 1.31}$	${0.3}_{- 0.27}^{+ 0.16}$ $0.3_{ - 0.27}^{ + 0.16}$	$- {0.55}_{- 0.62}^{+ 0.22}$ $- 0.55_{ - 0.62}^{ + 0.22}$	${0.23}_{- 1.34}^{+ 1.19}$ $0.23_{ - 1.34}^{ + 1.19}$
σ_T1 (m s^-1)	𝒩 (1, 5)	${2.01}_{- 0.41}^{+ 0.37}$ $2.01_{ - 0.41}^{ + 0.37}$	${2.42}_{- 0.38}^{+ 0.42}$ $2.42_{ - 0.38}^{ + 0.42}$	${1.79}_{- 0.41}^{+ 0.72}$ $1.79_{ - 0.41}^{ + 0.72}$	${2.15}_{- 0.02}^{+ 0.35}$ $2.15_{ - 0.02}^{ + 0.35}$	${2.37}_{- 0.1}^{+ 0.23}$ $2.37_{ - 0.1}^{ + 0.23}$	${1.63}_{- 0.42}^{+ 0.52}$ $1.63_{ - 0.42}^{ + 0.52}$	${0.16}_{- 0.16}^{+ 0.04}$ $0.16_{ - 0.16}^{ + 0.04}$
σPFS (m s^-1 )	𝒩 (1, 5)	${2.67}_{- 0.34}^{+ 0.08}$ $2.67_{ - 0.34}^{ + 0.08}$	${1.56}_{- 0.49}^{+ 0.96}$ $1.56_{ - 0.49}^{ + 0.96}$	${2.51}_{- 0.27}^{+ 0.09}$ $2.51_{ - 0.27}^{ + 0.09}$	${1.27}_{- 0.48}^{+ 1.06}$ $1.27_{ - 0.48}^{ + 1.06}$	${3.05}_{- 0.18}^{+ 0.22}$ $3.05_{ - 0.18}^{ + 0.22}$	${1.53}_{- 0.47}^{+ 0.79}$ $1.53_{ - 0.47}^{ + 0.79}$	${1.02}_{- 0.43}^{+ 0.14}$ $1.02_{ - 0.43}^{ + 0.14}$
σ_T2 (m s^-1)	𝒩 (1, 5)	${2.32}_{- 0.52}^{+ 0.52}$ $2.32_{ - 0.52}^{ + 0.52}$	${2.41}_{- 0.1}^{+ 0.37}$ $2.41_{ - 0.1}^{ + 0.37}$	${2.02}_{- 0.47}^{+ 0.52}$ $2.02_{ - 0.47}^{ + 0.52}$	${1.92}_{- 0.5}^{+ 0.62}$ $1.92_{ - 0.5}^{ + 0.62}$	${1.77}_{- 0.39}^{+ 0.33}$ $1.77_{ - 0.39}^{ + 0.33}$	${1.99}_{- 0.11}^{+ 0.27}$ $1.99_{ - 0.11}^{ + 0.27}$	${1.04}_{- 0.02}^{+ 0.73}$ $1.04_{ - 0.02}^{ + 0.73}$

Magnetic cycle

P₁ (days)	𝒰 (1, 6712)	${2463.4}_{- 96.3}^{+ 143.8}$ $2463.4_{ - 96.3}^{ + 143.8}$	${2523.6}_{- 81.3}^{+ 86.7}$ $2523.6_{ - 81.3}^{ + 86.7}$	${2395.5}_{- 903.5}^{+ 188.5}$ $2395.5_{ - 903.5}^{ + 188.5}$	${2540.1}_{- 44.9}^{+ 43.6}$ $2540.1_{ - 44.9}^{ + 43.6}$	-	${2557.0}_{- 36.7}^{+ 70.1}$ $2557.0_{ - 36.7}^{ + 70.1}$	${2457.7}_{- 87.0}^{+ 95.3}$ $2457.7_{ - 87.0}^{ + 95.3}$
A₁ (m s^-1 )	𝒰 (1e-6, 7.1)	${4.4}_{- 0.38}^{+ 0.15}$ $4.4_{ - 0.38}^{ + 0.15}$	${4.43}_{- 0.56}^{+ 0.52}$ $4.43_{ - 0.56}^{ + 0.52}$	${3.97}_{- 0.19}^{+ 0.03}$ $3.97_{ - 0.19}^{ + 0.03}$	${3.98}_{- 0.24}^{+ 0.02}$ $3.98_{ - 0.24}^{ + 0.02}$	-	${2.82}_{- 0.15}^{+ 0.16}$ $2.82_{ - 0.15}^{ + 0.16}$	${4.33}_{- 0.55}^{+ 0.49}$ $4.33_{ - 0.55}^{ + 0.49}$
M₀₁(rad)	𝒰 (0, 2π)	${0.79}_{- 0.79}^{+ 0.39}$ $0.79_{ - 0.79}^{ + 0.39}$	${2.85}_{- 2.04}^{+ 1.76}$ $2.85_{ - 2.04}^{ + 1.76}$	${0.4}_{- 0.4}^{+ 1.4}$ $0.4_{ - 0.4}^{ + 1.4}$	${3.93}_{- 1.84}^{+ 2.36}$ $3.93_{ - 1.84}^{ + 2.36}$	-	${2.05}_{- 0.09}^{+ 1.54}$ $2.05_{ - 0.09}^{ + 1.54}$	${0.47}_{- 0.1}^{+ 1.48}$ $0.47_{ - 0.1}^{ + 1.48}$
e₁	𝒩 (0, 0.1)	${0.2}_{- 0.2}^{+ 0.14}$ $0.2_{ - 0.2}^{ + 0.14}$	${0.14}_{- 0.14}^{+ 0.06}$ $0.14_{ - 0.14}^{ + 0.06}$	${0.17}_{- 0.17}^{+ 0.03}$ $0.17_{ - 0.17}^{ + 0.03}$	${0.11}_{- 0.11}^{+ 0.06}$ $0.11_{ - 0.11}^{ + 0.06}$		${0.17}_{- 0.12}^{+ 0.03}$ $0.17_{ - 0.12}^{ + 0.03}$	${0.2}_{- 0.2}^{+ 0.0}$ $0.2_{ - 0.2}^{ + 0.0}$
ω₁ (rad)	𝒰 (0, 2π)	${5.73}_{- 0.53}^{+ 0.54}$ $5.73_{ - 0.53}^{ + 0.54}$	${3.77}_{- 1.71}^{+ 2.46}$ $3.77_{ - 1.71}^{ + 2.46}$	${5.9}_{- 1.41}^{+ 0.38}$ $5.9_{ - 1.41}^{ + 0.38}$	${2.86}_{- 1.79}^{+ 3.38}$ $2.86_{ - 1.79}^{ + 3.38}$	-	${4.75}_{- 1.35}^{+ 0.11}$ $4.75_{ - 1.35}^{ + 0.11}$	${6.05}_{- 1.1}^{+ 0.23}$ $6.05_{ - 1.1}^{ + 0.23}$

Stellar rotation

P₂ (days)	𝒰 (1, 300)	-	${167.23}_{- 162.33}^{+ 83.81}$ $167.23_{ - 162.33}^{ + 83.81}$	-	${167.13}_{- 55.25}^{+ 13.64}$ $167.13_{ - 55.25}^{ + 13.64}$	-	${29.72}_{- 0.02}^{+ 0.01}$ $29.72_{ - 0.02}^{ + 0.01}$	-
A₂ (m s^-1)	𝒰 (0, 7.1)	-	${2.95}_{- 1.63}^{+ 1.07}$ $2.95_{ - 1.63}^{ + 1.07}$	-	${4.31}_{- 3.0}^{+ 1.39}$ $4.31_{ - 3.0}^{ + 1.39}$	-	${3.03}_{- 1.57}^{+ 1.52}$ $3.03_{ - 1.57}^{ + 1.52}$	-
M₀₂(rad)	𝒰 (0, 2π)	-	${0.99}_{- 0.99}^{+ 5.01}$ $0.99_{ - 0.99}^{ + 5.01}$	-	${0.76}_{- 0.66}^{+ 1.41}$ $0.76_{ - 0.66}^{ + 1.41}$	-	${5.16}_{- 0.39}^{+ 0.37}$ $5.16_{ - 0.39}^{ + 0.37}$	-
e₂	𝒩 (0, 0.1)	-	${0.74}_{- 0.13}^{+ 0.15}$ $0.74_{ - 0.13}^{ + 0.15}$	-	${0.83}_{- 0.07}^{+ 0.16}$ $0.83_{ - 0.07}^{ + 0.16}$	-	${0.67}_{- 0.26}^{+ 0.29}$ $0.67_{ - 0.26}^{ + 0.29}$	-
ω₂ (rad)	𝒰 (0, 2π)	-	${3.29}_{- 0.32}^{+ 1.38}$ $3.29_{ - 0.32}^{ + 1.38}$	-	${3.74}_{- 0.97}^{+ 0.74}$ $3.74_{ - 0.97}^{ + 0.74}$	-	${2.52}_{- 0.81}^{+ 0.2}$ $2.52_{ - 0.81}^{ + 0.2}$	-
GProtρ (days)	𝒰 (1, 300)	-	-	-	-	-	-	${2.45}_{- 0.26}^{+ 0.04}$ $2.45_{ - 0.26}^{ + 0.04}$
GP_rotσ (m s^-1)	𝒰 (0, 7.1)	-	-	-	-	-	-	${1.16}_{- 0.25}^{+ 1.83}$ $1.16_{ - 0.25}^{ + 1.83}$
GProt f	𝒰 (0, 10)	-	-	-	-	-	-	${0.01}_{- 0.00}^{+ 0.09}$ $0.01_{ - 0.00}^{ + 0.09}$

Noise

MA ϕ	𝒰 (-1, 1)	-	-	${0.58}_{- 0.23}^{+ 0.16}$ $0.58_{ - 0.23}^{ + 0.16}$	${0.41}_{- 0.1}^{+ 0.05}$ $0.41_{ - 0.1}^{ + 0.05}$	-	${0.1}_{- 0.17}^{+ 0.2}$ $0.1_{ - 0.17}^{ + 0.2}$	-
MA τ (days)	𝒰 (1, 300)	-	-	${7.68}_{- 4.91}^{+ 16.18}$ $7.68_{ - 4.91}^{ + 16.18}$	${27.88}_{- 25.81}^{+ 7.65}$ $27.88_{ - 25.81}^{ + 7.65}$	-	${56.33}_{- 54.83}^{+ 2.06}$ $56.33_{ - 54.83}^{ + 2.06}$	-
A_T1	𝒰 (-1, 1)	-	-	-	-	${0.81}_{- 0.04}^{+ 0.07}$ $0.81_{ - 0.04}^{ + 0.07}$	${0.41}_{- 0.06}^{+ 0.05}$ $0.41_{ - 0.06}^{ + 0.05}$	-
A_PFS	𝒰 (-1, 1)	-	-	-	-	$- {0.09}_{- 0.06}^{+ 0.06}$ $- 0.09_{ - 0.06}^{ + 0.06}$	${0.11}_{- 0.18}^{+ 0.11}$ $0.11_{ - 0.18}^{ + 0.11}$	-
A_T2	𝒰 (-1, 1)	-	-	-	-	${0.58}_{- 0.07}^{+ 0.04}$ $0.58_{ - 0.07}^{ + 0.04}$	${0.48}_{- 0.15}^{+ 0.16}$ $0.48_{ - 0.15}^{ + 0.16}$	-

Notes. Models with K_n Keplerians in parenthesis, from left to right: White noise (WN), exponentially weighted moving average (MA), linearly correlated stellar-activity (SA), moving average with stellar-activity (SAMA), and GP with rotation kernel.

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