Rotational modulation and long-term evolution of the small-scale magnetic fields of M dwarfs observed with SPIRou

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Volume 702 (October 2025)

A&A, 702 (2025) A111

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Table A.1

GP hyper-parameters for DS Leo (Gl 410).

DS Leo	Value	Prior
GP fit on 〈B〉
μ (kG)	${0.90}_{- 0.02}^{+ 0.02}$ $0.90^{+0.02}_{-0.02}$	𝒰(−10, 10)
σ (kG)	${0.01}_{- 0.01}^{+ 0.01}$ $0.01^{+0.01}_{-0.01}$	𝒰(0, 1000)
α (kG)	${0.08}_{- 0.01}^{+ 0.01}$ $0.08^{+0.01}_{-0.01}$	𝒰(0, 10)
l (d)	$54_{- 7}^{+ 8}$ $54^{+8}_{-7}$	𝒰(10, 500)
β	${0.57}_{- 0.08}^{+ 0.10}$ $0.57^{+0.10}_{-0.08}$	𝒰(0, 20)
P_rot (d)	${13.953}_{- 0.086}^{+ 0.088}$ $13.953^{+0.088}_{-0.086}$	𝒰(5, 20)
χ²_r	0.72
RMS	0.02 (kG)
GP fit on B_ℓ
μ_ℓ (G)	$- {0.60}_{- 3.28}^{+ 3.24}$ $-0.60^{+3.24}_{-3.28}$	𝒰(−500, 500)
σ_ℓ (G)	${1.94}_{- 1.16}^{+ 0.94}$ $1.94^{+0.94}_{-1.16}$	𝒰(0, 1000)
α_ℓ (G)	${15.19}_{- 1.60}^{+ 1.92}$ $15.19^{+1.92}_{-1.60}$	𝒰(0, 1000)
l_ℓ (d)	$57_{- 7}^{+ 9}$ $57^{+9}_{-7}$	𝒰(10, 500)
β_ℓ	${0.45}_{- 0.05}^{+ 0.06}$ $0.45^{+0.06}_{-0.05}$	𝒰(0, 5)
P_rot,ℓ (d)	${14.001}_{- 0.075}^{+ 0.077}$ $14.001^{+0.077}_{-0.075}$	𝒰(5, 20)
χ²_{r, ℓ}	0.66
RMS_ℓ	3.93 (G)
GP fit on 〈B〉 and B_ℓ
μ (kG)	${0.90}_{- 0.02}^{+ 0.02}$ $0.90^{+0.02}_{-0.02}$	𝒰(−10, 10)
σ (kG)	${0.01}_{- 0.01}^{+ 0.01}$ $0.01^{+0.01}_{-0.01}$	𝒰(0, 1000)
α (kG)	${0.08}_{- 0.01}^{+ 0.01}$ $0.08^{+0.01}_{-0.01}$	𝒰(0, 10)
l (d)	$54_{- 7}^{+ 8}$ $54^{+8}_{-7}$	𝒰(10, 500)
β	${0.57}_{- 0.08}^{+ 0.10}$ $0.57^{+0.10}_{-0.08}$	𝒰(0, 20)
P_rot (d)	${13.980}_{- 0.058}^{+ 0.059}$ $13.980^{+0.059}_{-0.058}$	𝒰(5, 20)
μ_ℓ (G)	$- {0.76}_{- 3.34}^{+ 3.31}$ $-0.76^{+3.31}_{-3.34}$	𝒰(−500, 500)
σ_ℓ (G)	${1.90}_{- 1.14}^{+ 0.97}$ $1.90^{+0.97}_{-1.14}$	𝒰(0, 1000)
α_ℓ (G)	${15.48}_{- 1.61}^{+ 1.98}$ $15.48^{+1.98}_{-1.61}$	𝒰(0, 1000)
l_ℓ (d)	$58_{- 7}^{+ 9}$ $58^{+9}_{-7}$	𝒰(10, 500)
β_ℓ	${0.45}_{- 0.05}^{+ 0.06}$ $0.45^{+0.06}_{-0.05}$	𝒰(0, 20)
χ²_r	0.72
χ²_{r, ℓ}	0.66
RMS	0.02 (kG)
RMS_ℓ	3.94 (G)
GP fit on dTemp
μ_dTemp (K)	$- {1.18}_{- 1.61}^{+ 1.57}$ $-1.18^{+1.57}_{-1.61}$	𝒰(−50, 50)
σ_dTemp (K)	${0.71}_{- 0.12}^{+ 0.13}$ $0.71^{+0.13}_{-0.12}$	𝒰(0, 1000)
α_dTemp (K)	${7.46}_{- 0.81}^{+ 0.93}$ $7.46^{+0.93}_{-0.81}$	𝒰(0, 100)$
l_dTemp (d)	$38_{- 3}^{+ 3}$ $38^{+3}_{-3}$	𝒰(10, 500)
β_dTemp	${0.63}_{- 0.06}^{+ 0.07}$ $0.63^{+0.07}_{-0.06}$	𝒰(0, 20)
P_{rot, dTemp} (d)	${14.214}_{- 0.102}^{+ 0.106}$ $14.214^{+0.106}_{-0.102}$	𝒰(3, 200)
χ²_{r, dTemp}	0.98
RMS_dTemp	0.58 (K)

Notes. Hyper-parameters obtained when applying our process to the small-scale magnetic field measurements (〈B〉), the longitudinal magnetic field (B_t), or 〈B〉 and B_t simultaneously. Error bars on 〈B〉 and dTemp measurements were adjusted to ensure a minimum reduced χ² close to one for the best fit.

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