Stellar-activity analysis of the nearby M dwarf GJ 526 - Multi-dimensional Gaussian-process modelling of RV, FWHM, and S-index

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Volume 703 (November 2025)

A&A, 703 (2025) A245

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Table B.2

Posterior comparison of our best model on all data (MEP; Table B.1) against planet-free models in our grid that used an ESP kernel with 2, 3, or 4 harmonics.

Parameter name	Symbol	Unit	Prior	Posterior

				MEP	ESP2	ESP3	ESP4
LTF parameters
Period	P_cyc	d	𝒰(1000, 2000)	$1680_{- 40}^{+ 50}$ $1680^{+50}_{-40}$	1680±50	1680±60	$1680_{- 50}^{+ 60}$ $1680^{+60}_{-50}$
RV phase	ϕ_{cyc, 0}	–	𝒰(0, 1)^(w)	${0.843}_{- 0.129}^{+ 0.118}$ $0.843^{+0.118}_{-0.129}$	${0.854}_{- 0.131}^{+ 0.123}$ $0.854^{+0.123}_{-0.131}$	${0.848}_{- 0.175}^{+ 0.156}$ $0.848^{+0.156}_{-0.175}$	${0.861}_{- 0.156}^{+ 0.140}$ $0.861^{+0.140}_{-0.156}$
RV semi-amplitude	k_{cyc, 0}	m s⁻¹	𝒰(0, 10)	${0.51}_{- 0.33}^{+ 0.35}$ $0.51^{+0.35}_{-0.33}$	${0.50}_{- 0.33}^{+ 0.36}$ $0.50^{+0.36}_{-0.33}$	${0.38}_{- 0.26}^{+ 0.34}$ $0.38^{+0.34}_{-0.26}$	${0.41}_{- 0.28}^{+ 0.36}$ $0.41^{+0.36}_{-0.28}$
FWHM phase	ϕ_{cyc, 1}	–	𝒰(0, 1)^(w)	${0.429}_{- 0.089}^{+ 0.085}$ $0.429^{+0.085}_{-0.089}$	${0.434}_{- 0.085}^{+ 0.087}$ $0.434^{+0.087}_{-0.085}$	0.425±0.101	${0.427}_{- 0.093}^{+ 0.100}$ $0.427^{+0.100}_{-0.093}$
FWHM semi-amplitude	k_{cyc, 1}	m s⁻¹	𝒰(0, 10)	${2.73}_{- 1.32}^{+ 1.51}$ $2.73^{+1.51}_{-1.32}$	${2.54}_{- 1.25}^{+ 1.27}$ $2.54^{+1.27}_{-1.25}$	${2.23}_{- 1.24}^{+ 1.25}$ $2.23^{+1.25}_{-1.24}$	${2.33}_{- 1.24}^{+ 1.26}$ $2.33^{+1.26}_{-1.24}$
S-index phase	ϕ_{cyc, 2}	–	𝒰(0, 1)^(w)	${0.548}_{- 0.248}^{+ 0.252}$ $0.548^{+0.252}_{-0.248}$	${0.577}_{- 0.280}^{+ 0.265}$ $0.577^{+0.265}_{-0.280}$	${0.664}_{- 0.287}^{+ 0.272}$ $0.664^{+0.272}_{-0.287}$	${0.681}_{- 0.255}^{+ 0.276}$ $0.681^{+0.276}_{-0.255}$
S-index semi-amplitude	k_{cyc, 2}	–	𝒰(0, 0.1)	${0.019}_{- 0.013}^{+ 0.019}$ $0.019^{+0.019}_{-0.013}$	${0.015}_{- 0.011}^{+ 0.016}$ $0.015^{+0.016}_{-0.011}$	${0.015}_{- 0.010}^{+ 0.016}$ $0.015^{+0.016}_{-0.010}$	${0.015}_{- 0.011}^{+ 0.016}$ $0.015^{+0.016}_{-0.011}$
RV zero-order correction	α₀	m s⁻¹	$N (μ_{lm}, 200 σ_{lm})$ $\mathcal{N}(\mu_\text{lm},200\sigma_\text{lm})$	${0.47}_{- 0.83}^{+ 0.85}$ $0.47^{+0.85}_{-0.83}$	${0.57}_{- 0.84}^{+ 0.86}$ $0.57^{+0.86}_{-0.84}$	0.45±0.82	0.43±0.82
FWHM zero-order correction	α₁	m s⁻¹	$N (μ_{lm}, 200 σ_{lm})$ $\mathcal{N}(\mu_\text{lm},200\sigma_\text{lm})$	${3.89}_{- 3.23}^{+ 3.28}$ $3.89^{+3.28}_{-3.23}$	${4.28}_{- 3.12}^{+ 3.19}$ $4.28^{+3.19}_{-3.12}$	${3.95}_{- 3.00}^{+ 2.98}$ $3.95^{+2.98}_{-3.00}$	${3.96}_{- 2.86}^{+ 2.89}$ $3.96^{+2.89}_{-2.86}$
S-index zero-order correction	α₂	–	$N (μ_{lm}, 200 σ_{lm})$ $\mathcal{N}(\mu_\text{lm},200\sigma_\text{lm})$	0.050±0.048	0.056±0.048	0.053±0.044	${0.052}_{- 0.042}^{+ 0.043}$ $0.052^{+0.043}_{-0.042}$
Dataset parameters
ESPRESSO19 RV offset	O_1,0	m s⁻¹	$N (0, 5 σ_{0})$ $\mathcal{N}(0,5\sigma_0)$	-1.06±0.74	$- {1.17}_{- 0.86}^{+ 0.87}$ $-1.17^{+0.87}_{-0.86}$	$- {1.02}_{- 0.83}^{+ 0.82}$ $-1.02^{+0.82}_{-0.83}$	$- {0.95}_{- 0.84}^{+ 0.82}$ $-0.95^{+0.82}_{-0.84}$
HIRES RV offset	O_2,0	m s⁻¹	$N (0, 5 σ_{0})$ $\mathcal{N}(0,5\sigma_0)$	${0.00}_{- 1.01}^{+ 1.00}$ $0.00^{+1.00}_{-1.01}$	$- {0.30}_{- 1.02}^{+ 1.01}$ $-0.30^{+1.01}_{-1.02}$	$- {0.23}_{- 0.97}^{+ 0.96}$ $-0.23^{+0.96}_{-0.97}$	-0.19±0.96
HARPS RV offset	O_3,0	m s⁻¹	$N (0, 5 σ_{0})$ $\mathcal{N}(0,5\sigma_0)$	$- {0.05}_{- 1.12}^{+ 1.13}$ $-0.05^{+1.13}_{-1.12}$	$- {0.10}_{- 1.09}^{+ 1.13}$ $-0.10^{+1.13}_{-1.09}$	${0.07}_{- 1.04}^{+ 1.07}$ $0.07^{+1.07}_{-1.04}$	${0.09}_{- 1.05}^{+ 1.06}$ $0.09^{+1.06}_{-1.05}$
ESPRESSO19 FWHM offset	O_1,1	m s⁻¹	$N (0, 5 σ_{1})$ $\mathcal{N}(0,5\sigma_1)$	$- {4.63}_{- 2.21}^{+ 2.30}$ $-4.63^{+2.30}_{-2.21}$	$- {4.97}_{- 2.95}^{+ 3.14}$ $-4.97^{+3.14}_{-2.95}$	$- {4.57}_{- 2.63}^{+ 2.72}$ $-4.57^{+2.72}_{-2.63}$	-4.50±2.57
HARPS FWHM offset	O_3,1	m s⁻¹	$N (0, 5 σ_{1})$ $\mathcal{N}(0,5\sigma_1)$	$- {0.19}_{- 4.10}^{+ 4.30}$ $-0.19^{+4.30}_{-4.10}$	$- {1.13}_{- 3.79}^{+ 3.87}$ $-1.13^{+3.87}_{-3.79}$	$- {0.64}_{- 3.77}^{+ 3.78}$ $-0.64^{+3.78}_{-3.77}$	$- {0.63}_{- 3.65}^{+ 3.76}$ $-0.63^{+3.76}_{-3.65}$
ESPRESSO19 S-index offset	O_1,2	–	$N (0, 5 σ_{2})$ $\mathcal{N}(0,5\sigma_2)$	$(- {6.22}_{- 3.38}^{+ 3.44}) \times 10^{- 2}$ $\left(-6.22^{+3.44}_{-3.38}\right)\times 10^{-2}$	$(- {6.69}_{- 4.49}^{+ 4.77}) \times 10^{- 2}$ $\left(-6.69^{+4.77}_{-4.49}\right)\times 10^{-2}$	$(- {6.26}_{- 3.88}^{+ 4.05}) \times 10^{- 2}$ $\left(-6.26^{+4.05}_{-3.88}\right)\times 10^{-2}$	$(- {6.07}_{- 3.90}^{+ 3.85}) \times 10^{- 2}$ $\left(-6.07^{+3.85}_{-3.90}\right)\times 10^{-2}$
HIRES S-index offset	O_2,2	–	$N (0, 5 σ_{2})$ $\mathcal{N}(0,5\sigma_2)$	$(- {4.63}_{- 5.44}^{+ 5.39}) \times 10^{- 2}$ $\left(-4.63^{+5.39}_{-5.44}\right)\times 10^{-2}$	$(- {5.14}_{- 5.16}^{+ 5.15}) \times 10^{- 2}$ $\left(-5.14^{+5.15}_{-5.16}\right)\times 10^{-2}$	$(- {4.74}_{- 4.84}^{+ 4.76}) \times 10^{- 2}$ $\left(-4.74^{+4.76}_{-4.84}\right)\times 10^{-2}$	$(- {4.74}_{- 4.68}^{+ 4.67}) \times 10^{- 2}$ $\left(-4.74^{+4.67}_{-4.68}\right)\times 10^{-2}$
HARPS S-index offset	O_3,2	–	$N (0, 5 σ_{2})$ $\mathcal{N}(0,5\sigma_2)$	$({0.07}_{- 5.77}^{+ 6.09}) \times 10^{- 2}$ $\left(0.07^{+6.09}_{-5.77}\right)\times 10^{-2}$	$(- {1.63}_{- 5.55}^{+ 5.54}) \times 10^{- 2}$ $\left(-1.63^{+5.54}_{-5.55}\right)\times 10^{-2}$	$(- {1.09}_{- 5.35}^{+ 5.31}) \times 10^{- 2}$ $\left(-1.09^{+5.31}_{-5.35}\right)\times 10^{-2}$	$(- {1.03}_{- 5.21}^{+ 5.14}) \times 10^{- 2}$ $\left(-1.03^{+5.14}_{-5.21}\right)\times 10^{-2}$
ESPRESSO18 RV jitter	J_0,0	m s⁻¹	$\exp [N (\ln σ_{0}, \ln σ_{0})]$ $\exp[\mathcal{N}(\ln\sigma_0,\ln\sigma_0)]$	${1.82}_{- 0.35}^{+ 0.47}$ $1.82^{+0.47}_{-0.35}$	${1.82}_{- 0.37}^{+ 0.48}$ $1.82^{+0.48}_{-0.37}$	${1.90}_{- 0.35}^{+ 0.47}$ $1.90^{+0.47}_{-0.35}$	${1.94}_{- 0.36}^{+ 0.48}$ $1.94^{+0.48}_{-0.36}$
ESPRESSO19 RV jitter	J_1,0	m s⁻¹	$\exp [N (\ln σ_{0}, \ln σ_{0})]$ $\exp[\mathcal{N}(\ln\sigma_0,\ln\sigma_0)]$	${0.85}_{- 0.22}^{+ 0.21}$ $0.85^{+0.21}_{-0.22}$	0.92±0.20	${0.85}_{- 0.22}^{+ 0.20}$ $0.85^{+0.20}_{-0.22}$	0.71±0.23
HIRES RV jitter	J_2,0	m s⁻¹	$\exp [N (\ln σ_{0}, \ln σ_{0})]$ $\exp[\mathcal{N}(\ln\sigma_0,\ln\sigma_0)]$	${1.20}_{- 0.65}^{+ 0.71}$ $1.20^{+0.71}_{-0.65}$	${1.48}_{- 0.84}^{+ 0.80}$ $1.48^{+0.80}_{-0.84}$	${1.63}_{- 0.83}^{+ 0.65}$ $1.63^{+0.65}_{-0.83}$	${1.58}_{- 0.78}^{+ 0.64}$ $1.58^{+0.64}_{-0.78}$
HARPS RV jitter	J_3,0	m s⁻¹	$\exp [N (\ln σ_{0}, \ln σ_{0})]$ $\exp[\mathcal{N}(\ln\sigma_0,\ln\sigma_0)]$	${2.21}_{- 0.36}^{+ 0.43}$ $2.21^{+0.43}_{-0.36}$	${2.32}_{- 0.40}^{+ 0.45}$ $2.32^{+0.45}_{-0.40}$	${2.24}_{- 0.42}^{+ 0.47}$ $2.24^{+0.47}_{-0.42}$	${2.19}_{- 0.44}^{+ 0.48}$ $2.19^{+0.48}_{-0.44}$
ESPRESSO18 FWHM jitter	J_0,1	m s⁻¹	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	${1.02}_{- 0.27}^{+ 0.34}$ $1.02^{+0.34}_{-0.27}$	${1.05}_{- 0.25}^{+ 0.34}$ $1.05^{+0.34}_{-0.25}$	${1.09}_{- 0.28}^{+ 0.35}$ $1.09^{+0.35}_{-0.28}$	${1.09}_{- 0.29}^{+ 0.35}$ $1.09^{+0.35}_{-0.29}$
ESPRESSO19 FWHM jitter	J_1,1	m s⁻¹	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	${1.82}_{- 0.29}^{+ 0.34}$ $1.82^{+0.34}_{-0.29}$	${1.95}_{- 0.31}^{+ 0.38}$ $1.95^{+0.38}_{-0.31}$	${1.75}_{- 0.33}^{+ 0.40}$ $1.75^{+0.40}_{-0.33}$	${1.78}_{- 0.33}^{+ 0.39}$ $1.78^{+0.39}_{-0.33}$
HARPS FWHM jitter	J_3,1	m s⁻¹	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	${4.25}_{- 0.62}^{+ 0.77}$ $4.25^{+0.77}_{-0.62}$	${4.24}_{- 0.61}^{+ 0.76}$ $4.24^{+0.76}_{-0.61}$	${4.29}_{- 0.62}^{+ 0.78}$ $4.29^{+0.78}_{-0.62}$	${4.29}_{- 0.62}^{+ 0.79}$ $4.29^{+0.79}_{-0.62}$
ESPRESSO18 S-index jitter	J_0,2	–	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	$({1.27}_{- 0.45}^{+ 0.55}) \times 10^{- 2}$ $\left(1.27^{+0.55}_{-0.45}\right)\times 10^{-2}$	$({1.54}_{- 0.47}^{+ 0.58}) \times 10^{- 2}$ $\left(1.54^{+0.58}_{-0.47}\right)\times 10^{-2}$	$({0.87}_{- 0.32}^{+ 0.55}) \times 10^{- 2}$ $\left(0.87^{+0.55}_{-0.32}\right)\times 10^{-2}$	$({0.84}_{- 0.31}^{+ 0.53}) \times 10^{- 2}$ $\left(0.84^{+0.53}_{-0.31}\right)\times 10^{-2}$
ESPRESSO19 S-index jitter	J_1,2	–	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	$({4.77}_{- 0.54}^{+ 0.62}) \times 10^{- 2}$ $\left(4.77^{+0.62}_{-0.54}\right)\times 10^{-2}$	$({4.62}_{- 0.55}^{+ 0.64}) \times 10^{- 2}$ $\left(4.62^{+0.64}_{-0.55}\right)\times 10^{-2}$	$({4.70}_{- 0.55}^{+ 0.63}) \times 10^{- 2}$ $\left(4.70^{+0.63}_{-0.55}\right)\times 10^{-2}$	$({4.76}_{- 0.56}^{+ 0.64}) \times 10^{- 2}$ $\left(4.76^{+0.64}_{-0.56}\right)\times 10^{-2}$
HIRES S-index jitter	J_2,2	–	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	$({3.14}_{- 0.59}^{+ 0.75}) \times 10^{- 2}$ $\left(3.14^{+0.75}_{-0.59}\right)\times 10^{-2}$	$({3.20}_{- 0.62}^{+ 0.83}) \times 10^{- 2}$ $\left(3.20^{+0.83}_{-0.62}\right)\times 10^{-2}$	$({3.24}_{- 0.62}^{+ 0.85}) \times 10^{- 2}$ $\left(3.24^{+0.85}_{-0.62}\right)\times 10^{-2}$	$({3.26}_{- 0.63}^{+ 0.84}) \times 10^{- 2}$ $\left(3.26^{+0.84}_{-0.63}\right)\times 10^{-2}$
HARPS S-index jitter	J_3,2	–	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	$({0.43}_{- 0.27}^{+ 0.64}) \times 10^{- 2}$ $\left(0.43^{+0.64}_{-0.27}\right)\times 10^{-2}$	$({0.43}_{- 0.28}^{+ 0.64}) \times 10^{- 2}$ $\left(0.43^{+0.64}_{-0.28}\right)\times 10^{-2}$	$({0.40}_{- 0.25}^{+ 0.65}) \times 10^{- 2}$ $\left(0.40^{+0.65}_{-0.25}\right)\times 10^{-2}$	$({0.42}_{- 0.26}^{+ 0.69}) \times 10^{- 2}$ $\left(0.42^{+0.69}_{-0.26}\right)\times 10^{-2}$
Stellar-activity hyperparameters
Timescale	τ	d	$U_{\log} (20, 10^{4})$ $$\mathcal{U}_{\log}\left(20, 10^{4})$	$277_{- 82}^{+ 109}$ $277^{+109}_{-82}$	$68_{- 19}^{+ 22}$ $68^{+22}_{-19}$	$79_{- 20}^{+ 23}$ $79^{+23}_{-20}$	$81_{- 18}^{+ 22}$ $81^{+22}_{-18}$
Period	P_rot	d	$U (40.1, 64.1)$ $\mathcal{U}(40.1,64.1)$	48.7±0.3	${49.2}_{- 1.5}^{+ 1.7}$ $49.2^{+1.7}_{-1.5}$	${49.2}_{- 0.9}^{+ 1.2}$ $49.2^{+1.2}_{-0.9}$	${49.3}_{- 0.9}^{+ 1.1}$ $49.3^{+1.1}_{-0.9}$
Sinescale (harmonic complexity)	η	–	$U_{\log} (10^{- 2}, 10^{2})$ $$\mathcal{U}_{\log}\left(10^{-2}, 10^{2})$	${0.53}_{- 0.10}^{+ 0.13}$ $0.53^{+0.13}_{-0.10}$	${0.60}_{- 0.14}^{+ 0.16}$ $0.60^{+0.16}_{-0.14}$	0.58±0.08	${0.55}_{- 0.07}^{+ 0.08}$ $0.55^{+0.08}_{-0.07}$
RV amplitude	A₀	m s⁻¹	$U (- 10^{3}, 10^{3})$ $\mathcal{U}(-10^3,10^3)$	${1.45}_{- 0.26}^{+ 0.30}$ $1.45^{+0.30}_{-0.26}$	${1.42}_{- 0.25}^{+ 0.28}$ $1.42^{+0.28}_{-0.25}$	${1.33}_{- 0.23}^{+ 0.26}$ $1.33^{+0.26}_{-0.23}$	${1.33}_{- 0.22}^{+ 0.25}$ $1.33^{+0.25}_{-0.22}$
RV gradient amplitude	B₀	m s^-1 d^-1	$U (- 10^{3}, 10^{3})$ $\mathcal{U}(-10^3,10^3)$	${19.9}_{- 3.2}^{+ 4.0}$ $19.9^{+4.0}_{-3.2}$	${19.6}_{- 3.3}^{+ 3.9}$ $19.6^{+3.9}_{-3.3}$	${17.8}_{- 3.0}^{+ 3.6}$ $17.8^{+3.6}_{-3.0}$	${17.3}_{- 2.8}^{+ 3.4}$ $17.3^{+3.4}_{-2.8}$
FWHM amplitude	A₁	m s⁻¹	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	${7.26}_{- 0.86}^{+ 1.13}$ $7.26^{+1.13}_{-0.86}$	${6.72}_{- 0.73}^{+ 0.87}$ $6.72^{+0.87}_{-0.73}$	${6.78}_{- 0.71}^{+ 0.83}$ $6.78^{+0.83}_{-0.71}$	${6.73}_{- 0.71}^{+ 0.82}$ $6.73^{+0.82}_{-0.71}$
S-index amplitude	A₂	m s⁻¹	$U_{\log} (10^{- 3}, 10^{3})$ $$\mathcal{U}_{\log}\left(10^{-3}, 10^{3})$	${0.106}_{- 0.012}^{+ 0.015}$ $0.106^{+0.015}_{-0.012}$	${0.101}_{- 0.010}^{+ 0.012}$ $0.101^{+0.012}_{-0.010}$	${0.100}_{- 0.010}^{+ 0.011}$ $0.100^{+0.011}_{-0.010}$	${0.099}_{- 0.010}^{+ 0.011}$ $0.099^{+0.011}_{-0.010}$

Notes. ^(w)Wrapped parameter. Reported uncertainties reflect the 16^th and the 84^th percentiles. Offsets are defined relative to ESPRESSO18. The standard deviation of measurements in physical quantity j is denoted σ_j.

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