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 D.4

GJ 699 FWHM ESPRESSO parameter comparison.

Parameter	Prior	WN(K₃)	𝒢𝒫 _grot(K₀)	𝒢𝒫 _rot(K₀)	𝒢𝒫 _rot(K₁)	𝒢𝒫 _rot+Mag(a)	𝒢𝒫 _rot+Mag(b)


Offsets and Jitter
γ_E18	~ 𝒰 (-10.5,10..5)	${5.25}_{- 1.24}^{+ 0.96}$ ${5.25_{ - 1.24}^{ + 0.96}}$	${1.17}_{- 1.29}^{+ 0.46}$ ${1.17_{ - 1.29}^{ + 0.46}}$	${2.33}_{- 1.59}^{+ 0.34}$ ${2.33_{ - 1.59}^{ + 0.34}}$	${2.75}_{- 1.32}^{+ 0.72}$ ${2.75_{ - 1.32}^{ + 0.72}}$	${0.14}_{- 0.36}^{+ 1.74}$ ${0.14_{ - 0.36}^{ + 1.74}}$	${0.62}_{- 2.33}^{+ 2.39}$ ${0.62_{ - 2.33}^{ + 2.39}}$
γ_E19	~ 𝒰 (-10.5,10.5)	${0.15}_{- 0.13}^{+ 0.26}$ ${0.15_{ - 0.13}^{ + 0.26}}$	${0.91}_{- 0.25}^{+ 0.31}$ ${0.91_{ - 0.25}^{ + 0.31}}$	${0.89}_{- 0.17}^{+ 0.38}$ ${0.89_{ - 0.17}^{ + 0.38}}$	${0.94}_{- 0.13}^{+ 0.27}$ ${0.94_{ - 0.13}^{ + 0.27}}$	${1.19}_{- 0.19}^{+ 0.17}$ ${1.19_{ - 0.19}^{ + 0.17}}$	${0.17}_{- 1.72}^{+ 1.41}$ ${0.17_{ - 1.72}^{ + 1.41}}$
σ_E18	~ 𝒩 (0.5,1)	${0.98}_{- 0.37}^{+ 0.34}$ ${0.98_{ - 0.37}^{ + 0.34}}$	${0.54}_{- 0.07}^{+ 0.15}$ ${0.54_{ - 0.07}^{ + 0.15}}$	${0.53}_{- 0.08}^{+ 0.13}$ ${0.53_{ - 0.08}^{ + 0.13}}$	${0.50}_{- 0.05}^{+ 0.18}$ ${0.50_{ - 0.05}^{ + 0.18}}$	${0.57}_{- 0.08}^{+ 0.16}$ ${0.57_{ - 0.08}^{ + 0.16}}$	${0.55}_{- 0.08}^{+ 0.14}$ ${0.55_{ - 0.08}^{ + 0.14}}$
σ_E19	~ 𝒩 (0.5,1)	${1.43}_{- 0.09}^{+ 0.10}$ ${1.43_{ - 0.09}^{ + 0.10}}$	${0.32}_{- 0.03}^{+ 0.02}$ ${0.32_{ - 0.03}^{ + 0.02}}$	${0.34}_{- 0.03}^{+ 0.01}$ ${0.34_{ - 0.03}^{ + 0.01}}$	${0.36}_{- 0.03}^{+ 0.02}$ ${0.36_{ - 0.03}^{ + 0.02}}$	${0.38}_{- 0.04}^{+ 0.01}$ ${0.38_{ - 0.04}^{ + 0.01}}$	${0.37}_{- 0.06}^{+ 0.04}$ ${0.37_{ - 0.06}^{ + 0.04}}$

_P1	~ 𝒰 (50,300)	${165.38}_{- 0.11}^{+ 0.55}$ $165.38_{ - 0.11}^{ + 0.55}$	-	-	${249.73}_{- 3.36}^{+ 4.40}$ $249.73_{ - 3.36}^{ + 4.40}$	-	-
K₁	~ 𝒰 (0,6.60)	${2.75}_{- 0.30}^{+ 0.22}$ $2.75_{ - 0.30}^{ + 0.22}$	-	-	${3.57}_{- 0.32}^{+ 0.52}$ $3.57_{ - 0.32}^{ + 0.52}$	-	-
ϕ₁	~ 𝒰 (0,2π)	${4.41}_{- 0.26}^{+ 0.15}$ $4.41_{ - 0.26}^{ + 0.15}$	-	-	${2.54}_{- 1.2}^{+ 0.63}$ $2.54_{ - 1.2}^{ + 0.63}$	-	-
e₁	~ 𝒩 (0,0.1)	${0.23}_{- 0.09}^{+ 0.07}$ $0.23_{ - 0.09}^{ + 0.07}$	-	-	${0.02}_{- 0.02}^{+ 0.03}$ $0.02_{ - 0.02}^{ + 0.03}$	-	-
ω₁	~ 𝒰 (0,2π)	${5.24}_{- 0.06}^{+ 0.39}$ $5.24_{ - 0.06}^{ + 0.39}$	-	-	${1.28}_{- 0.42}^{+ 1.29}$ $1.28_{ - 0.42}^{ + 1.29}$	-	-
_P2	~ 𝒰 (50, 300)	${244.41}_{- 1.31}^{+ 1.50}$ $244.41_{ - 1.31}^{ + 1.50}$	-	-	-	-	-
K₂	~ 𝒰 (0, 6.60)	${4.39}_{- 0.01}^{+ 0.19}$ $4.39_{ - 0.01}^{ + 0.19}$	-	-	-	-	-
ϕ₂	~ 𝒰 (0, 2π)	${1.02}_{- 0.33}^{+ 0.40}$ $1.02_{ - 0.33}^{ + 0.40}$	-	-	-	-	-
e₂	~ 𝒩 (0, 0.1)	${0.15}_{- 0.04}^{+ 0.06}$ $0.15_{ - 0.04}^{ + 0.06}$	-	-	-	-	-
ω₂	~ 𝒰 (0, 2π)	${2.47}_{- 0.29}^{+ 0.23}$ $2.47_{ - 0.29}^{ + 0.23}$	-	-	-	-	-
P₃	~ 𝒰 (1,5000)	$2384_{- 888}^{+ 2616}$ $2384_{ - 888}^{ + 2616}$	-	-	-	-	-
_K3	~ 𝒰 (0, 6.60)	${1.58}_{- 0.15}^{+ 0.21}$ $1.58_{ - 0.15}^{ + 0.21}$	-	-	-	-	-
ϕ₃	~ 𝒰 (0, 2π)	${5.13}_{- 0.15}^{+ 1.15}$ $5.13_{ - 0.15}^{ + 1.15}$	-	-	-	-	-
e₃	~ 𝒩 (0, 0.1)	${0.05}_{- 0.05}^{+ 0.06}$ $0.05_{ - 0.05}^{ + 0.06}$	-	-	-	-	-
ω₃	~ 𝒰 (0, 2π)	${4.01}_{- 1.00}^{+ 0.64}$ $4.01_{ - 1.00}^{ + 0.64}$	-	-	-	-	-

𝒢𝒫 _grotρ	~ 𝒰 (50, 300)	-	${221.98}_{- 10.41}^{+ 8.55}$ $221.98_{ - 10.41}^{ + 8.55}$	-	-	-	-
𝒢𝒫 _grotτ	~ 𝒰 (50, 600)	-	${28.65}_{- 3.11}^{+ 1.28}$ $28.65_{ - 3.11}^{ + 1.28}$	-	-	-	-
𝒢𝒫 _grotA₁	~ 𝒩 (3.3, 3.3)	-	${2.60}_{- 0.61}^{+ 0.22}$ $2.60_{ - 0.61}^{ + 0.22}$	-	-	-	-
𝒢𝒫 _grotA₂	~ 𝒩 (3.3, 6.6)	-	${3.70}_{- 0.05}^{+ 0.21}$ $3.70_{ - 0.05}^{ + 0.21}$	-	-	-	-

𝒢𝒫 _rotρ	~ 𝒰 (50, 300)	-	-	${170.95}_{- 1.98}^{+ 22.01}$ ${170.95_{ - 1.98}^{ + 22.01}}$	${151.46}_{- 0.81}^{+ 10.27}$ ${151.46_{ - 0.81}^{ + 10.27}}$	${155.3}_{- 6.04}^{+ 2.44}$ ${155.3_{ - 6.04}^{ + 2.44}}$	${156.3}_{- 34.51}^{+ 100.99}$ ${156.3_{ - 34.51}^{ + 100.99}}$
𝒢𝒫 _rotσ	~ 𝒩 (3.3, 3.3)	-	-	${3.66}_{- 0.14}^{+ 0.28}$ ${3.66_{ - 0.14}^{ + 0.28}}$	${2.80}_{- 0.52}^{+ 0.63}$ ${2.80_{ - 0.52}^{ + 0.63}}$	${2.47}_{- 0.51}^{+ 0.63}$ ${2.47_{ - 0.51}^{ + 0.63}}$	${3.44}_{- 0.59}^{+ 0.99}$ ${3.44_{ - 0.59}^{ + 0.99}}$
𝒢𝒫 _rotQ0	~𝒰 (0, 100)	-	-	${1.37}_{- 0.97}^{+ 0.62}$ ${1.37_{ - 0.97}^{ + 0.62}}$	${4.13}_{- 3.60}^{+ 2.01}$ ${4.13_{ - 3.60}^{ + 2.01}}$	${2.81}_{- 1.72}^{+ 2.74}$ ${2.81_{ - 1.72}^{ + 2.74}}$	${1.93}_{- 1.56}^{+ 0.56}$ ${1.93_{ - 1.56}^{ + 0.56}}$
𝒢𝒫 _rotδQ	~ 𝒰 (0, 100)	-	-	${0.20}_{- 0.20}^{+ 0.13}$ ${0.20_{ - 0.20}^{ + 0.13}}$	${0.74}_{- 0.74}^{+ 0.59}$ ${0.74_{ - 0.74}^{ + 0.59}}$	${3.90}_{- 0.63}^{+ 1.06}$ ${3.90_{ - 0.63}^{ + 1.06}}$	${0.62}_{- 0.04}^{+ 1.34}$ ${0.62_{ - 0.04}^{ + 1.34}}$
𝒢𝒫 _{rot f}	~ 𝒰 (0, 1)	-	-	${0.23}_{- 0.08}^{+ 0.18}$ ${0.23_{ - 0.08}^{ + 0.18}}$	${0.51}_{- 0.28}^{+ 0.24}$ ${0.51_{ - 0.28}^{ + 0.24}}$	${0.43}_{- 0.18}^{+ 0.67}$ ${0.43_{ - 0.18}^{ + 0.67}}$	${0.23}_{- 0.23}^{+ 0.77}$ ${0.23_{ - 0.23}^{ + 0.77}}$

P_mag	~ 𝒰 (1,5000)	-	-	-	-	${506.21}_{- 8.03}^{+ 4.71}$ ${506.21_{ - 8.03}^{ + 4.71}}$	${809.26}_{- 9.26}^{+ 4190.46}$ ${809.26_{ - 9.26}^{ + 4190.46}}$
A1_mag	~ 𝒰 (0, 6.6)	-	-	-	-	${1.07}_{- 0.29}^{+ 0.53}$ ${1.07_{ - 0.29}^{ + 0.53}}$	${1.47}_{- 1.47}^{+ 1.36}$ ${1.47_{ - 1.47}^{ + 1.36}}$
A2_mag	~ 𝒰 (0, 6.6)	-	-	-	-	${0.10}_{- 0.10}^{+ 0.54}$ ${0.10_{ - 0.10}^{ + 0.54}}$	${1.64}_{- 0.57}^{+ 1.03}$ ${1.64_{ - 0.57}^{ + 1.03}}$
ϕ1_mag	~ 𝒰 (0, 2π)	-	-	-	-	${3.75}_{- 0.75}^{+ 1.04}$ ${3.75_{ - 0.75}^{ + 1.04}}$	${2.80}_{- 2.80}^{+ 1.27}$ ${2.80_{ - 2.80}^{ + 1.27}}$
ϕ2_mag	~ 𝒰 (0, 2π)	-	-	-	-	${4.13}_{- 0.27}^{+ 0.28}$ ${4.13_{ - 0.27}^{ + 0.28}}$	${5.30}_{- 1.50}^{+ 0.49}$ ${5.30_{ - 1.50}^{ + 0.49}}$

Notes. Runs from different models: (K_n) denotes the number of Keplerian signals. In descending order: white noise only (WN), Gaussian process with a rotation kernel (𝒢𝒫 _rot), and GP with a rotation kernel and a magnetic cycle (𝒢𝒫_rot+Mag). This last model was tested with two different priors for the magnetic cycle: (a) 𝒰 (0.5, 5000), and (b) 𝒰 (800, 5000).

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