X-ray and optical observations of the millisecond pulsar binary PSR J1431–4715

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Table 2.

Posterior parameters obtained from the fits of our multi-colour light curves and radial velocity curve from Strader et al. (2019).

Model	DH	C	D+C	DH+Spot	DH	C	D+C	DH+Spot
β	0.25	0.25	0.25	0.25	0.08	0.08	0.08	0.08

Fitted parameters
E(B − V)	$0 . 45_{- 0.03}^{+ 0.03}$ $0.45_{-0.03}^{+0.03}$	$0 . 45_{- 0.03}^{+ 0.03}$ $0.45_{-0.03}^{+0.03}$	$0 . 45_{- 0.03}^{+ 0.03}$ $0.45_{-0.03}^{+0.03}$	$0 . 45_{- 0.03}^{+ 0.02}$ $0.45_{-0.03}^{+0.02}$	$0 . 18_{- 0.03}^{+ 0.03}$ $0.18_{-0.03}^{+0.03}$	$0 . 19_{- 0.03}^{+ 0.03}$ $0.19_{-0.03}^{+0.03}$	$0 . 19_{- 0.03}^{+ 0.03}$ $0.19_{-0.03}^{+0.03}$	$0 . 17_{- 0.03}^{+ 0.03}$ $0.17_{-0.03}^{+0.03}$
K_c (km s⁻¹)	$278_{- 3}^{+ 2}$ $278_{-3}^{+2}$	$279_{- 3}^{+ 2}$ $279_{-3}^{+2}$	$279_{- 3}^{+ 2}$ $279_{-3}^{+2}$	$280_{- 3}^{+ 1}$ $280_{-3}^{+1}$	$278_{- 2}^{+ 2}$ $278_{-2}^{+2}$	$279_{- 2}^{+ 1}$ $279_{-2}^{+1}$	$279_{- 2}^{+ 1}$ $279_{-2}^{+1}$	$278_{- 2}^{+ 2}$ $278_{-2}^{+2}$
d (kpc)	$3 . 9_{- 0.2}^{+ 0.2}$ $3.9_{-0.2}^{+0.2}$	$3 . 9_{- 0.2}^{+ 0.2}$ $3.9_{-0.2}^{+0.2}$	$3 . 9_{- 0.2}^{+ 0.2}$ $3.9_{-0.2}^{+0.2}$	$4 . 1_{- 0.2}^{+ 0.2}$ $4.1_{-0.2}^{+0.2}$	$2 . 9_{- 0.3}^{+ 0.2}$ $2.9_{-0.3}^{+0.2}$	$3 . 3_{- 0.2}^{+ 0.2}$ $3.3_{-0.2}^{+0.2}$	$3 . 2_{- 0.2}^{+ 0.2}$ $3.2_{-0.2}^{+0.2}$	$3 . 0_{- 0.2}^{+ 0.2}$ $3.0_{-0.2}^{+0.2}$
i (°)	$48_{- 1}^{+ 1}$ $48_{-1}^{+1}$	$48_{- 1}^{+ 1}$ $48_{-1}^{+1}$	$48_{- 1}^{+ 1}$ $48_{-1}^{+1}$	$45 . 5_{- 0.9}^{+ 0.9}$ $45.5_{-0.9}^{+0.9}$	$62_{- 5}^{+ 7}$ $62_{-5}^{+7}$	$55_{- 3}^{+ 3}$ $55_{-3}^{+3}$	$56_{- 3}^{+ 3}$ $56_{-3}^{+3}$	$60_{- 3}^{+ 5}$ $60_{-3}^{+5}$
T_base (K)	$10200_{- 70}^{+ 70}$ $10200_{-70}^{+70}$	$10210_{- 60}^{+ 70}$ $10210_{-60}^{+70}$	$10210_{- 70}^{+ 60}$ $10210_{-70}^{+60}$	$10200_{- 70}^{+ 70}$ $10200_{-70}^{+70}$	$7540_{- 70}^{+ 60}$ $7540_{-70}^{+60}$	$7600_{- 100}^{+ 100}$ $7600_{-100}^{+100}$	$7670_{- 100}^{+ 90}$ $7670_{-100}^{+90}$	$7510_{- 90}^{+ 60}$ $7510_{-90}^{+60}$
T_irr (K)	$6700_{- 100}^{+ 100}$ $6700_{-100}^{+100}$	$6700_{- 100}^{+ 100}$ $6700_{-100}^{+100}$	$6700_{- 100}^{+ 100}$ $6700_{-100}^{+100}$	$6700_{- 100}^{+ 100}$ $6700_{-100}^{+100}$	$3860_{- 70}^{+ 70}$ $3860_{-70}^{+70}$	$4000_{- 70}^{+ 70}$ $4000_{-70}^{+70}$	$4040_{- 70}^{+ 60}$ $4040_{-70}^{+60}$	$3880_{- 60}^{+ 60}$ $3880_{-60}^{+60}$
f_RL	$0 . 77_{- 0.01}^{+ 0.02}$ $0.77_{-0.01}^{+0.02}$	$0 . 77_{- 0.01}^{+ 0.02}$ $0.77_{-0.01}^{+0.02}$	$0 . 77_{- 0.01}^{+ 0.02}$ $0.77_{-0.01}^{+0.02}$	$0 . 79_{- 0.01}^{+ 0.01}$ $0.79_{-0.01}^{+0.01}$	$0 . 70_{- 0.03}^{+ 0.03}$ $0.70_{-0.03}^{+0.03}$	$0 . 74_{- 0.02}^{+ 0.03}$ $0.74_{-0.02}^{+0.03}$	$0 . 73_{- 0.02}^{+ 0.02}$ $0.73_{-0.02}^{+0.02}$	$0 . 71_{- 0.02}^{+ 0.02}$ $0.71_{-0.02}^{+0.02}$
C_ampl (W K⁻¹ m⁻²)	–	$- 35000_{- 2000}^{+ 1000}$ $-35000_{-2000}^{+1000}$	$- 35000_{- 2000}^{+ 1000}$ $-35000_{-2000}^{+1000}$	–	–	$- 15000_{- 800}^{+ 800}$ $-15000_{-800}^{+800}$	$- 15000_{- 1000}^{+ 1000}$ $-15000_{-1000}^{+1000}$	–
κ_diff (W K⁻¹ m⁻²)	–	–	$300_{- 300}^{+ 800}$ $300_{-300}^{+800}$	–	–	–	$2000_{- 1000}^{+ 2000}$ $2000_{-1000}^{+2000}$	–
ϕ_spot (°)	–	–	–	$67_{- 3}^{+ 2}$ $67_{-3}^{+2}$	–	–	–	$- 71_{- 2}^{+ 2}$ $-71_{-2}^{+2}$
θ_spot (°)	–	–	–	$132_{- 9}^{+ 10}$ $132_{-9}^{+10}$	–	–	–	$111_{- 6}^{+ 9}$ $111_{-6}^{+9}$
R_spot (°)	–	–	–	$16_{- 2}^{+ 2}$ $16_{-2}^{+2}$	–	–	–	$9_{- 2}^{+ 3}$ $9_{-2}^{+3}$
T_spot (K)	–	–	–	$2800_{- 700}^{+ 2000}$ $2800_{-700}^{+2000}$	–	–	–	$- 500_{- 400}^{+ 200}$ $-500_{-400}^{+200}$
Derived parameters
q	$10 . 44_{- 0.1}^{+ 0.08}$ $10.44_{-0.1}^{+0.08}$	$10 . 47_{- 0.1}^{+ 0.06}$ $10.47_{-0.1}^{+0.06}$	$10 . 47_{- 0.1}^{+ 0.06}$ $10.47_{-0.1}^{+0.06}$	$10 . 50_{- 0.1}^{+ 0.05}$ $10.50_{-0.1}^{+0.05}$	$10 . 43_{- 0.08}^{+ 0.07}$ $10.43_{-0.08}^{+0.07}$	$10 . 48_{- 0.08}^{+ 0.05}$ $10.48_{-0.08}^{+0.05}$	$10 . 48_{- 0.08}^{+ 0.05}$ $10.48_{-0.08}^{+0.05}$	$10 . 44_{- 0.08}^{+ 0.07}$ $10.44_{-0.08}^{+0.07}$
M_p (M_⊙)	$2 . 9_{- 0.2}^{+ 0.2}$ $2.9_{-0.2}^{+0.2}$	$3 . 0_{- 0.2}^{+ 0.2}$ $3.0_{-0.2}^{+0.2}$	$3 . 0_{- 0.2}^{+ 0.2}$ $3.0_{-0.2}^{+0.2}$	$3 . 4_{- 0.2}^{+ 0.2}$ $3.4_{-0.2}^{+0.2}$	$1 . 8_{- 0.3}^{+ 0.3}$ $1.8_{-0.3}^{+0.3}$	$2 . 2_{- 0.2}^{+ 0.3}$ $2.2_{-0.2}^{+0.3}$	$2 . 2_{- 0.2}^{+ 0.3}$ $2.2_{-0.2}^{+0.3}$	$1 . 8_{- 0.2}^{+ 0.2}$ $1.8_{-0.2}^{+0.2}$
M_c (M_⊙)	$0 . 28_{- 0.02}^{+ 0.02}$ $0.28_{-0.02}^{+0.02}$	$0 . 29_{- 0.02}^{+ 0.02}$ $0.29_{-0.02}^{+0.02}$	$0 . 28_{- 0.01}^{+ 0.02}$ $0.28_{-0.01}^{+0.02}$	$0 . 32_{- 0.02}^{+ 0.02}$ $0.32_{-0.02}^{+0.02}$	$0 . 17_{- 0.03}^{+ 0.03}$ $0.17_{-0.03}^{+0.03}$	$0 . 21_{- 0.02}^{+ 0.03}$ $0.21_{-0.02}^{+0.03}$	$0 . 21_{- 0.02}^{+ 0.02}$ $0.21_{-0.02}^{+0.02}$	$0 . 18_{- 0.02}^{+ 0.02}$ $0.18_{-0.02}^{+0.02}$
T_day (K)	$9620_{- 70}^{+ 70}$ $9620_{-70}^{+70}$	$9620_{- 60}^{+ 60}$ $9620_{-60}^{+60}$	$9620_{- 60}^{+ 60}$ $9620_{-60}^{+60}$	$9730_{- 90}^{+ 100}$ $9730_{-90}^{+100}$	$7450_{- 70}^{+ 60}$ $7450_{-70}^{+60}$	$7500_{- 100}^{+ 100}$ $7500_{-100}^{+100}$	$7570_{- 100}^{+ 90}$ $7570_{-100}^{+90}$	$7420_{- 90}^{+ 60}$ $7420_{-90}^{+60}$
T_night (K)	$9410_{- 70}^{+ 60}$ $9410_{-70}^{+60}$	$9420_{- 60}^{+ 60}$ $9420_{-60}^{+60}$	$9420_{- 60}^{+ 60}$ $9420_{-60}^{+60}$	$9400_{- 70}^{+ 70}$ $9400_{-70}^{+70}$	$7370_{- 70}^{+ 60}$ $7370_{-70}^{+60}$	$7400_{- 100}^{+ 100}$ $7400_{-100}^{+100}$	$7490_{- 100}^{+ 90}$ $7490_{-100}^{+90}$	$7350_{- 90}^{+ 60}$ $7350_{-90}^{+60}$
R_c (R_⊙)	$0 . 68_{- 0.02}^{+ 0.02}$ $0.68_{-0.02}^{+0.02}$	$0 . 69_{- 0.02}^{+ 0.02}$ $0.69_{-0.02}^{+0.02}$	$0 . 68_{- 0.02}^{+ 0.02}$ $0.68_{-0.02}^{+0.02}$	$0 . 72_{- 0.02}^{+ 0.02}$ $0.72_{-0.02}^{+0.02}$	$0 . 54_{- 0.05}^{+ 0.04}$ $0.54_{-0.05}^{+0.04}$	$0 . 61_{- 0.03}^{+ 0.04}$ $0.61_{-0.03}^{+0.04}$	$0 . 59_{- 0.03}^{+ 0.03}$ $0.59_{-0.03}^{+0.03}$	$0 . 56_{- 0.04}^{+ 0.03}$ $0.56_{-0.04}^{+0.03}$
Goodness of fit
log Z	−2097.5	−1606.0	−1611.6	−1568.2	−1891.3	−1451.5	−1453.7	−1419.0
N_d.o.f.	8703	8702	8701	8699	8703	8702	8701	8699
$χ_{ν}^{2}$ $\chi^{2}_{\nu}$	0.591	0.529	0.528	0.524	0.566	0.511	0.511	0.507

Notes. Different models with β = 0.25 and β = 0.08 grouped from left to right: direct heating (DH), convection (C), diffusion+convection (D+C) and direct heating with a spot (Spot), applying gravity darkening after irradiation (post-IGD). The derived parameters are listed below the fit parameters. These are the mass ratio q, the pulsar M_psr and companion M_c mass, the day T_day and night T_night temperatures, which are obtained averaging the temperature over the surface elements of the day and night side hemispheres (i.e. in the plane of the orbit) according to T⁴ = ∑_i(T_i⁴ s_i, eff)/∑_i s_i, eff, with T_i the temperature and s_i, eff the projected area of a surface element. log Z is the Bayesian evidence of the fit with more positive values indicating better fits. The uncertainties are the 1σ confidence intervals.

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