Long-term evolutions of the cyclotron line energies in Her X-1, Vela X-1 and Cen X-3 as observed with Swift/BAT

TL;DR

This study analyzes the long-term evolution of cyclotron line energies in three X-ray pulsars using Swift/BAT data, revealing significant decreases in some sources and stability in others, informing magnetic field models.

Contribution

It provides the first long-term observational analysis of CRSF energy evolution in Her X-1, Vela X-1, and Cen X-3 using Swift/BAT survey data.

Findings

Her X-1 shows a significant decrease in CRSF energy since 2004.

Vela X-1's first harmonic line energy decreased until 2012.

Cen X-3's CRSF energy remains stable over 14 years.

Abstract

We study the long-term evolution of the centroid energy of cyclotron lines - often referered to as Cyclotron Resonance Scattering Features (CRSF) - in Her X-1, Vela X-1 and Cen X-3, using survey observations of the Burst Alert Telescope (BAT) onboard Swift. We find a significant decrease of the fundamental CRSF energy in Her X-1 and the first harmonic line energy in Vela X-1, since the launch of Swift in 2004 and until 2010 and 2012, respectively.In both sources the decreases stopped at some time, with a quite stable centroid energy thereafter. Unlike in Her X-1 and Vela X-1, the CRSF energy in Cen X-3 does not show a long-term decrease. It is observed not to change for at least the past 14 years. The long-term variation of the line energy is a direct way to investigate the magnetic field structure in the polar regions of pulsars. Our results may stimulate the development of theoretical models, especially regarding to how the accreted mass accumulates in the accretion mound or how the magnetic field distorts around the polar cap.

Tables

Table 1. Table 1: Best-fitting parameters of Her X-1 observed with Swift /BAT. E cyc subscript 𝐸 cyc E_{\rm cyc} , σ 𝜎 \sigma and Depth are the energy, width and depth of the CRSF. The continuum is a powerlaw with a high energy cutoff ( highecut ), where the parameters are frozen at Γ Γ \Gamma =0.9, E cut subscript 𝐸 cut E_{\rm cut} =20.8 and E fold subscript 𝐸 fold E_{\rm fold} =10.2 (Fürst et al., 2013 ) . During the spectral analysis, we considered the influence of the variable flux on the CRSF (the right three columns, see text).

Time (MJD)	$E_{\mathrm{cyc}}$ (keV)	$\sigma$ (keV)	Depth	$E_{\mathrm{cyc}}$ (keV)	$\sigma$ (keV)	Depth
	No Flux Correction			Flux Correction
53440-53580	${39.38}_{-0.31}^{+0.33}$	${4.04}_{-0.24}^{+0.25}$	${6.44}_{-0.42}^{+0.44}$	${39.43}_{-0.33}^{+0.34}$	${4.08}_{-0.24}^{+0.26}$	${6.15}_{-0.42}^{+0.44}$
53610-53750	${38.86}_{-0.54}^{+0.55}$	${4.11}_{-0.38}^{+0.37}$	${4.74}_{-0.51}^{+0.52}$	${39.22}_{-0.54}^{+0.55}$	${4.11}_{-0.38}^{+0.37}$	${4.74}_{-0.51}^{+0.52}$
53790-54030	${38.39}_{-0.39}^{+0.41}$	${4.06}_{-0.25}^{+0.29}$	${5.71}_{-0.43}^{+0.47}$	${38.54}_{-0.39}^{+0.41}$	${4.08}_{-0.25}^{+0.29}$	${5.72}_{-0.43}^{+0.47}$
54070-54210	${37.94}_{-1.34}^{+1.69}$	${4.30}_{-0.97}^{+1.24}$	${3.57}_{-0.89}^{+1.14}$	${38.36}_{-1.35}^{+1.67}$	${4.29}_{-0.92}^{+1.21}$	${3.56}_{-0.87}^{+1.13}$
54240-54450	${38.04}_{-0.51}^{+0.54}$	${3.45}_{-0.38}^{+0.38}$	${4.34}_{-0.49}^{+0.52}$	${38.34}_{-0.51}^{+0.54}$	${3.42}_{-0.38}^{+0.39}$	${4.33}_{-0.49}^{+0.52}$
54480-54590	${38.19}_{-0.55}^{+0.66}$	${4.26}_{-0.30}^{+0.49}$	${5.05}_{-0.50}^{+0.65}$	${38.62}_{-0.56}^{+0.64}$	${4.30}_{-0.31}^{+0.47}$	${5.08}_{-0.51}^{+0.64}$
54620-54970	${38.27}_{-0.36}^{+0.36}$	${3.77}_{-0.25}^{+0.24}$	${5.03}_{-0.37}^{+0.38}$	${38.42}_{-0.36}^{+0.36}$	${3.76}_{-0.25}^{+0.24}$	${5.02}_{-0.37}^{+0.38}$
55010-55150	${38.45}_{-0.45}^{+0.46}$	${4.12}_{-0.32}^{+0.31}$	${4.95}_{-0.43}^{+0.44}$	${38.61}_{-0.45}^{+0.46}$	${4.12}_{-0.32}^{+0.31}$	${4.94}_{-0.43}^{+0.45}$
55180-55320	${38.40}_{-0.51}^{+0.48}$	${4.17}_{-0.37}^{+0.28}$	${4.72}_{-0.47}^{+0.44}$	${38.57}_{-0.55}^{+0.63}$	${4.07}_{-0.26}^{+0.41}$	${4.66}_{-0.40}^{+0.51}$
55360-55530	${38.08}_{-0.33}^{+0.34}$	${3.97}_{-0.23}^{+0.25}$	${5.44}_{-0.36}^{+0.38}$	${38.09}_{-0.33}^{+0.34}$	${3.99}_{-0.24}^{+0.23}$	${5.42}_{-0.36}^{+0.37}$
55570-55740	${37.43}_{-0.39}^{+0.42}$	${4.00}_{-0.25}^{+0.30}$	${4.49}_{-0.35}^{+0.38}$	${37.73}_{-0.40}^{+0.40}$	${4.02}_{-0.29}^{+0.27}$	${4.49}_{-0.36}^{+0.37}$
55780-55920	${37.98}_{-0.34}^{+0.34}$	${3.98}_{-0.24}^{+0.23}$	${5.79}_{-0.39}^{+0.40}$	${38.41}_{-0.34}^{+0.35}$	${3.98}_{-0.22}^{+0.25}$	${5.80}_{-0.38}^{+0.41}$
55950-56090	${37.50}_{-0.38}^{+0.39}$	${3.90}_{-0.28}^{+0.27}$	${5.15}_{-0.39}^{+0.41}$	${38.16}_{-0.38}^{+0.40}$	${3.92}_{-0.27}^{+0.29}$	${5.13}_{-0.39}^{+0.41}$
56120-56260	${37.49}_{-0.28}^{+0.28}$	${4.04}_{-0.20}^{+0.19}$	${5.49}_{-0.30}^{+0.30}$	${38.00}_{-0.28}^{+0.28}$	${4.03}_{-0.19}^{+0.21}$	${5.47}_{-0.30}^{+0.31}$
56300-56470	${38.60}_{-0.43}^{+0.45}$	${4.57}_{-0.28}^{+0.31}$	${6.48}_{-0.49}^{+0.53}$	${38.60}_{-0.43}^{+0.45}$	${4.58}_{-0.28}^{+0.31}$	${6.48}_{-0.49}^{+0.53}$
56510-56610	${37.78}_{-0.46}^{+0.47}$	${3.80}_{-0.34}^{+0.32}$	${4.93}_{-0.48}^{+0.49}$	${38.81}_{-0.46}^{+0.47}$	${3.79}_{-0.33}^{+0.34}$	${4.90}_{-0.47}^{+0.50}$
56650-56790	${38.26}_{-0.36}^{+0.38}$	${4.42}_{-0.23}^{+0.26}$	${6.02}_{-0.39}^{+0.42}$	${38.41}_{-0.37}^{+0.38}$	${4.45}_{-0.23}^{+0.26}$	${6.02}_{-0.39}^{+0.42}$
56820-57030	${37.09}_{-0.38}^{+0.40}$	${4.04}_{-0.28}^{+0.31}$	${4.19}_{-0.32}^{+0.34}$	${37.38}_{-0.37}^{+0.39}$	${4.04}_{-0.28}^{+0.30}$	${4.20}_{-0.32}^{+0.34}$
57060-57167	${37.97}_{-0.50}^{+0.54}$	${4.53}_{-0.32}^{+0.38}$	${6.38}_{-0.57}^{+0.65}$	${38.14}_{-0.51}^{+0.52}$	${4.55}_{-0.35}^{+0.35}$	${6.39}_{-0.60}^{+0.62}$
57200-57410	${36.99}_{-0.64}^{+0.70}$	${3.20}_{-0.49}^{+0.53}$	${6.42}_{-1.01}^{+1.13}$	${38.46}_{-0.64}^{+0.70}$	${3.19}_{-0.49}^{+0.53}$	${6.39}_{-1.00}^{+1.13}$
57440-57550	${37.39}_{-0.63}^{+0.72}$	${4.10}_{-0.49}^{+0.54}$	${5.05}_{-0.61}^{+0.68}$	${37.46}_{-0.63}^{+0.72}$	${4.10}_{-0.49}^{+0.55}$	${4.95}_{-0.59}^{+0.68}$
57585-57690	${36.62}_{-0.35}^{+0.36}$	${3.52}_{-0.28}^{+0.29}$	${4.66}_{-0.37}^{+0.39}$	${36.66}_{-0.35}^{+0.36}$	${3.50}_{-0.28}^{+0.29}$	${4.66}_{-0.37}^{+0.39}$
57725-57830	${37.36}_{-0.32}^{+0.34}$	${3.96}_{-0.24}^{+0.26}$	${5.57}_{-0.38}^{+0.40}$	${37.39}_{-0.32}^{+0.34}$	${3.96}_{-0.24}^{+0.26}$	${5.56}_{-0.38}^{+0.40}$
57865-58000	${37.49}_{-0.62}^{+0.66}$	${4.24}_{-0.39}^{+0.46}$	${5.45}_{-0.61}^{+0.69}$	${38.83}_{-0.63}^{+0.64}$	${4.40}_{-0.43}^{+0.41}$	${5.71}_{-0.65}^{+0.67}$
58030-58240	${36.69}_{-0.48}^{+0.51}$	${3.81}_{-0.33}^{+0.39}$	${4.84}_{-0.48}^{+0.52}$	${37.56}_{-0.48}^{+0.51}$	${3.82}_{-0.33}^{+0.39}$	${4.82}_{-0.47}^{+0.52}$

Table 2. Table 2: Best-fitting parameters of Vela X-1 observed with Swift /BAT. E cyc ​ _ ​ H subscript 𝐸 cyc _ H E_{\rm cyc\_H}  (keV), σ H ​ ( k ​ e ​ V ) subscript 𝜎 H 𝑘 𝑒 𝑉 \sigma_{\rm H}\,(keV) and Depth H are the energy, width and depth of the harmonic CRSF, respectively. k ​ T Comp 𝑘 subscript 𝑇 Comp kT_{\rm Comp} and τ Comp subscript 𝜏 Comp \tau_{\rm Comp} are the temperature and the optical depth of the Comptonized continuum spectra, respectively.

Time (MJD)	$E_{\mathrm{cyc}\_H}$ (keV)	${\sigma }_H\left(keV\right)$	DepthH	$k{T}_{\mathrm{Comp}}$ (keV)	${\tau }_{\mathrm{Comp}}$
53351-53375	${55.83}_{-0.56}^{+0.93}$	${7.70}_{-0.48}^{+0.53}$	${17.61}_{-1.75}^{+2.86}$	${6.41}_{-0.10}^{+0.10}$	${16.78}_{-1.04}^{+1.14}$
53375-53403	${58.25}_{-1.20}^{+0.82}$	${8.50}_{-0.63}^{+0.44}$	${22.05}_{-3.01}^{+1.94}$	${7.68}_{-0.19}^{+0.18}$	${10.64}_{-0.53}^{+0.56}$
53403-53454	${59.43}_{-0.59}^{+0.64}$	${8.31}_{-0.38}^{+0.41}$	${18.67}_{-1.42}^{+1.60}$	${6.78}_{-0.04}^{+0.04}$	${15.57}_{-0.17}^{+0.16}$
53454-53588	${61.55}_{-1.47}^{+1.46}$	${8.37}_{-0.98}^{+1.00}$	${20.43}_{-4.42}^{+2.64}$	${7.53}_{-0.15}^{+0.16}$	${10.94}_{-0.61}^{+0.48}$
53648-53696	${58.83}_{-0.74}^{+0.87}$	${7.70}_{-0.67}^{+0.59}$	${16.27}_{-1.97}^{+2.08}$	${7.26}_{-0.12}^{+0.15}$	${11.65}_{-0.61}^{+0.55}$
53696-53855	${59.44}_{-0.76}^{+1.01}$	${8.00}_{-0.54}^{+0.81}$	${18.08}_{-0.75}^{+2.34}$	${7.17}_{-0.06}^{+0.15}$	${12.43}_{-0.75}^{+0.29}$
53855-53886	${57.91}_{-1.43}^{+0.49}$	${7.75}_{-0.72}^{+0.33}$	${15.98}_{-2.87}^{+1.63}$	${6.85}_{-0.09}^{+0.06}$	${14.52}_{-0.45}^{+0.66}$
53886-53945	${58.33}_{-1.19}^{+1.15}$	${8.82}_{-0.85}^{+0.74}$	${22.68}_{-4.51}^{+4.37}$	${7.80}_{-0.28}^{+0.20}$	${10.17}_{-0.45}^{+0.62}$
53945-53990	${61.41}_{-1.07}^{+1.22}$	${9.17}_{-0.62}^{+0.70}$	${24.97}_{-3.00}^{+3.70}$	${7.09}_{-0.06}^{+0.07}$	${12.83}_{-0.20}^{+0.18}$
54028-54051	${56.30}_{-0.52}^{+0.73}$	${7.66}_{-0.38}^{+0.68}$	${16.09}_{-1.81}^{+2.49}$	${6.94}_{-0.11}^{+0.14}$	${13.70}_{-0.82}^{+0.63}$
54051-54064	${57.20}_{-0.84}^{+1.41}$	${8.88}_{-0.83}^{+0.93}$	${19.99}_{-3.94}^{+3.82}$	${7.33}_{-0.21}^{+0.17}$	${12.51}_{-0.60}^{+0.88}$
54064-54079	${57.93}_{-0.90}^{+0.49}$	${7.48}_{-0.59}^{+0.36}$	${19.10}_{-2.51}^{+1.93}$	${6.93}_{-0.11}^{+0.13}$	${13.32}_{-0.77}^{+0.61}$
54079-54116	${57.04}_{-0.71}^{+0.85}$	${9.08}_{-0.61}^{+0.77}$	${21.58}_{-2.67}^{+3.87}$	${7.49}_{-0.13}^{+0.19}$	${11.63}_{-0.39}^{+0.29}$
54116-54215	${57.89}_{-1.09}^{+0.93}$	${9.40}_{-0.58}^{+0.89}$	${22.63}_{-3.53}^{+4.05}$	${7.24}_{-0.19}^{+0.16}$	${12.85}_{-0.53}^{+0.81}$
54215-54338	${57.42}_{-0.84}^{+1.24}$	${8.69}_{-0.58}^{+0.87}$	${21.03}_{-2.57}^{+3.60}$	${6.78}_{-0.10}^{+0.13}$	${15.17}_{-0.79}^{+1.15}$
54349-54366	${56.15}_{-0.94}^{+0.58}$	${7.01}_{-1.09}^{+0.36}$	${14.47}_{-2.85}^{+1.13}$	${6.94}_{-0.14}^{+0.10}$	${13.24}_{-0.68}^{+0.73}$
54366-54420	${57.69}_{-1.71}^{+1.14}$	${8.94}_{-1.01}^{+0.74}$	${21.54}_{-5.73}^{+2.93}$	${7.23}_{-0.21}^{+0.22}$	${12.41}_{-0.69}^{+0.78}$
54420-54451	${55.06}_{-0.62}^{+0.50}$	${6.81}_{-0.45}^{+0.74}$	${15.02}_{-1.81}^{+1.85}$	${6.91}_{-0.10}^{+0.13}$	${13.75}_{-0.94}^{+0.76}$
54451-54493	${55.90}_{-1.12}^{+1.14}$	${8.02}_{-0.74}^{+0.46}$	${15.83}_{-2.89}^{+2.36}$	${6.59}_{-0.13}^{+0.10}$	${15.90}_{-1.03}^{+1.13}$
54494-54519	${57.40}_{-0.64}^{+0.71}$	${8.25}_{-0.46}^{+0.51}$	${18.19}_{-1.64}^{+1.92}$	${6.78}_{-0.05}^{+0.06}$	${14.80}_{-0.22}^{+0.20}$
54519-54637	${56.00}_{-1.33}^{+0.82}$	${8.09}_{-0.92}^{+0.63}$	${17.75}_{-2.95}^{+3.08}$	${6.93}_{-0.14}^{+0.15}$	${13.94}_{-0.91}^{+0.85}$
54637-54691	${55.77}_{-0.83}^{+1.07}$	${8.36}_{-0.66}^{+0.75}$	${17.67}_{-2.62}^{+2.90}$	${6.91}_{-0.12}^{+0.13}$	${14.07}_{-0.94}^{+0.65}$
54691-54769	${56.71}_{-1.21}^{+0.43}$	${10.30}_{-1.06}^{+0.69}$	${30.25}_{-6.79}^{+0.86}$	${8.05}_{-0.40}^{+0.32}$	${10.27}_{-0.48}^{+0.67}$
54769-54809	${58.31}_{-0.81}^{+0.86}$	${9.76}_{-0.74}^{+0.68}$	${25.70}_{-4.49}^{+4.20}$	${7.29}_{-0.25}^{+0.25}$	${12.19}_{-0.88}^{+0.93}$
54809-54858	${57.20}_{-0.33}^{+1.35}$	${9.27}_{-0.29}^{+0.74}$	${19.55}_{-1.60}^{+3.39}$	${6.64}_{-0.10}^{+0.13}$	${16.15}_{-0.90}^{+0.99}$
54858-54871	${56.97}_{-0.82}^{+1.09}$	${8.76}_{-0.48}^{+0.89}$	${19.64}_{-4.34}^{+3.34}$	${6.67}_{-0.12}^{+0.11}$	${16.83}_{-1.14}^{+1.30}$
54871-54932	${56.91}_{-0.92}^{+1.07}$	${8.35}_{-0.54}^{+0.69}$	${19.34}_{-2.02}^{+3.68}$	${6.89}_{-0.11}^{+0.15}$	${14.16}_{-0.94}^{+0.66}$
54932-54986	${59.56}_{-1.25}^{+0.46}$	${9.32}_{-0.79}^{+0.54}$	${21.93}_{-3.10}^{+2.89}$	${6.69}_{-0.09}^{+0.10}$	${15.23}_{-0.88}^{+0.58}$
54986-55068	${57.71}_{-0.99}^{+0.75}$	${9.99}_{-0.77}^{+1.31}$	${26.63}_{-4.05}^{+2.71}$	${7.40}_{-0.16}^{+0.38}$	${11.84}_{-0.58}^{+0.37}$
55068-55145	${57.43}_{-0.65}^{+0.58}$	${10.00}_{-0.48}^{+0.88}$	${25.29}_{-2.42}^{+1.58}$	${7.01}_{-0.08}^{+0.17}$	${13.92}_{-0.34}^{+0.25}$
55145-55202	${55.64}_{-0.81}^{+0.92}$	${8.81}_{-0.64}^{+0.75}$	${17.58}_{-2.31}^{+3.06}$	${7.01}_{-0.09}^{+0.12}$	${13.22}_{-0.33}^{+0.28}$
55240-55311	${56.35}_{-0.72}^{+0.82}$	${9.14}_{-0.71}^{+0.69}$	${22.45}_{-3.56}^{+2.96}$	${7.11}_{-0.20}^{+0.16}$	${12.94}_{-0.82}^{+0.79}$
55312-55388	${55.85}_{-0.45}^{+0.62}$	${9.46}_{-0.51}^{+0.35}$	${25.22}_{-2.93}^{+2.30}$	${7.09}_{-0.19}^{+0.09}$	${13.81}_{-0.60}^{+1.20}$
55388-55446	${57.39}_{-0.97}^{+1.23}$	${8.95}_{-0.51}^{+0.91}$	${22.29}_{-2.80}^{+5.00}$	${7.28}_{-0.15}^{+0.24}$	${12.28}_{-0.84}^{+0.59}$
55447-55522	${55.45}_{-0.72}^{+0.81}$	${8.18}_{-0.54}^{+0.61}$	${18.28}_{-2.02}^{+2.46}$	${6.89}_{-0.07}^{+0.09}$	${13.95}_{-0.28}^{+0.25}$
55522-55596	${55.66}_{-0.73}^{+0.26}$	${10.00}_{-0.49}^{+0.28}$	${27.39}_{-3.10}^{+1.93}$	${7.58}_{-0.24}^{+0.13}$	${11.77}_{-0.34}^{+0.73}$
55597-55649	${53.95}_{-0.47}^{+0.69}$	${8.56}_{-0.68}^{+0.56}$	${18.96}_{-3.67}^{+2.64}$	${7.06}_{-0.21}^{+0.11}$	${12.65}_{-0.44}^{+0.74}$
55650-55739	${56.00}_{-0.82}^{+0.28}$	${9.99}_{-0.67}^{+1.02}$	${27.89}_{-3.91}^{+0.96}$	${7.31}_{-0.17}^{+0.34}$	${12.17}_{-0.46}^{+0.39}$
55740-55809	${55.13}_{-0.70}^{+0.82}$	${9.02}_{-0.60}^{+0.72}$	${21.51}_{-2.70}^{+3.71}$	${7.15}_{-0.12}^{+0.17}$	${12.86}_{-0.41}^{+0.33}$
55809-55874	${54.16}_{-0.91}^{+1.08}$	${8.78}_{-0.81}^{+0.94}$	${18.41}_{-3.53}^{+4.87}$	${6.98}_{-0.21}^{+0.27}$	${13.49}_{-1.03}^{+1.18}$
55874-55933	${55.14}_{-0.70}^{+1.08}$	${9.80}_{-0.62}^{+0.84}$	${24.14}_{-2.78}^{+5.04}$	${7.49}_{-0.14}^{+0.30}$	${11.61}_{-0.72}^{+0.36}$
55933-55976	${53.84}_{-0.87}^{+1.02}$	${8.84}_{-0.74}^{+0.87}$	${19.40}_{-3.56}^{+4.84}$	${6.71}_{-0.19}^{+0.24}$	${15.46}_{-1.38}^{+1.68}$
55976-56044	${53.68}_{-0.58}^{+0.75}$	${8.05}_{-0.39}^{+0.70}$	${15.59}_{-2.20}^{+2.27}$	${6.65}_{-0.13}^{+0.13}$	${15.91}_{-0.79}^{+1.30}$
56044-56126	${55.83}_{-0.71}^{+0.47}$	${9.96}_{-0.61}^{+0.77}$	${26.86}_{-3.41}^{+1.41}$	${7.10}_{-0.14}^{+0.19}$	${13.60}_{-0.35}^{+0.39}$
56126-56203	${55.68}_{-0.88}^{+1.04}$	${9.50}_{-0.59}^{+0.79}$	${23.05}_{-3.75}^{+4.31}$	${6.89}_{-0.15}^{+0.19}$	${14.06}_{-0.98}^{+0.81}$
56204-56283	${55.87}_{-0.88}^{+0.42}$	${8.18}_{-0.71}^{+0.53}$	${16.98}_{-2.25}^{+1.76}$	${6.44}_{-0.14}^{+0.08}$	${17.21}_{-0.94}^{+1.46}$
56283-56375	${53.12}_{-1.07}^{+0.54}$	${8.12}_{-0.76}^{+0.46}$	${16.30}_{-3.50}^{+2.76}$	${6.79}_{-0.21}^{+0.15}$	${14.78}_{-0.91}^{+1.54}$
56375-56466	${54.55}_{-0.89}^{+1.04}$	${8.86}_{-0.76}^{+0.86}$	${19.67}_{-3.55}^{+4.72}$	${7.39}_{-0.23}^{+0.28}$	${11.77}_{-0.73}^{+0.81}$
56466-56525	${55.19}_{-0.93}^{+0.53}$	${10.01}_{-0.76}^{+0.94}$	${25.07}_{-4.44}^{+1.68}$	${6.93}_{-0.21}^{+0.04}$	${14.44}_{-0.66}^{+1.27}$
56525-56591	${53.49}_{-0.92}^{+1.07}$	${8.64}_{-0.80}^{+0.90}$	${17.82}_{-3.41}^{+4.58}$	${6.95}_{-0.20}^{+0.25}$	${13.57}_{-0.99}^{+1.15}$
56592-56647	${54.75}_{-1.11}^{+0.86}$	${9.60}_{-0.90}^{+1.07}$	${22.80}_{-4.79}^{+3.87}$	${7.27}_{-0.26}^{+0.20}$	${12.31}_{-0.73}^{+0.99}$
56647-56742	${53.45}_{-0.58}^{+0.99}$	${8.52}_{-0.44}^{+0.89}$	${16.98}_{-2.87}^{+2.38}$	${6.72}_{-0.17}^{+0.16}$	${15.06}_{-1.06}^{+1.11}$
56742-56839	${55.88}_{-1.28}^{+0.99}$	${10.00}_{-0.74}^{+1.03}$	${23.96}_{-5.29}^{+4.93}$	${6.96}_{-0.23}^{+0.22}$	${14.18}_{-0.82}^{+1.12}$
56839-56879	${53.35}_{-0.50}^{+0.56}$	${8.23}_{-0.43}^{+0.48}$	${18.22}_{-1.69}^{+2.05}$	${6.81}_{-0.08}^{+0.09}$	${14.23}_{-0.29}^{+0.27}$
56880-56951	${55.98}_{-0.87}^{+0.71}$	${9.99}_{-0.71}^{+1.63}$	${23.34}_{-3.34}^{+1.03}$	${7.06}_{-0.13}^{+0.35}$	${13.32}_{-0.46}^{+0.36}$
56951-57003	${55.95}_{-1.15}^{+0.43}$	${10.00}_{-0.74}^{+0.45}$	${21.34}_{-3.52}^{+3.24}$	${6.78}_{-0.15}^{+0.19}$	${15.74}_{-1.15}^{+1.27}$
57003-57040	${54.09}_{-0.68}^{+0.92}$	${9.54}_{-0.67}^{+0.77}$	${20.51}_{-3.68}^{+2.72}$	${7.27}_{-0.21}^{+0.17}$	${12.31}_{-0.65}^{+0.80}$
57040-57085	${54.51}_{-0.98}^{+0.85}$	${9.67}_{-0.80}^{+0.91}$	${26.24}_{-4.91}^{+2.62}$	${7.24}_{-0.26}^{+0.14}$	${12.51}_{-0.75}^{+1.01}$
57085-57197	${55.66}_{-1.23}^{+0.44}$	${10.76}_{-1.07}^{+0.56}$	${31.56}_{-7.02}^{+1.02}$	${7.59}_{-0.32}^{+0.21}$	${12.09}_{-0.46}^{+0.70}$
57197-57235	${55.36}_{-1.18}^{+0.59}$	${8.14}_{-0.73}^{+0.46}$	${19.24}_{-3.39}^{+2.83}$	${7.30}_{-0.20}^{+0.23}$	${12.29}_{-0.89}^{+0.70}$
57235-57298	${55.51}_{-1.22}^{+1.10}$	${8.47}_{-0.90}^{+0.89}$	${19.48}_{-3.44}^{+3.84}$	${7.14}_{-0.21}^{+0.19}$	${13.11}_{-0.80}^{+1.08}$
57298-57386	${55.79}_{-0.57}^{+0.78}$	${8.64}_{-0.28}^{+0.88}$	${23.03}_{-2.30}^{+4.83}$	${7.52}_{-0.10}^{+0.20}$	${11.50}_{-0.65}^{+0.46}$
57386-57458	${55.79}_{-0.68}^{+0.83}$	${9.68}_{-0.81}^{+0.67}$	${24.54}_{-3.42}^{+3.68}$	${7.18}_{-0.25}^{+0.14}$	${13.30}_{-0.62}^{+1.07}$
57459-57506	${54.31}_{-0.67}^{+0.74}$	${7.35}_{-0.47}^{+0.52}$	${16.14}_{-1.64}^{+1.90}$	${6.61}_{-0.05}^{+0.06}$	${16.63}_{-0.28}^{+0.27}$
57506-57560	${55.12}_{-0.56}^{+1.23}$	${8.55}_{-0.44}^{+1.07}$	${23.06}_{-2.45}^{+4.65}$	${7.05}_{-0.20}^{+0.17}$	${13.59}_{-0.83}^{+0.85}$
57560-57610	${55.88}_{-0.47}^{+1.40}$	${9.02}_{-0.49}^{+0.97}$	${21.97}_{-3.05}^{+5.02}$	${7.11}_{-0.22}^{+0.21}$	${12.59}_{-0.78}^{+0.92}$
57610-57687	${53.56}_{-0.48}^{+0.91}$	${7.60}_{-0.44}^{+0.76}$	${15.55}_{-2.33}^{+3.25}$	${6.72}_{-0.14}^{+0.17}$	${15.43}_{-1.20}^{+1.12}$
57687-57765	${54.94}_{-0.73}^{+1.10}$	${8.50}_{-0.43}^{+0.76}$	${19.22}_{-3.69}^{+4.44}$	${6.97}_{-0.20}^{+0.18}$	${13.76}_{-0.86}^{+0.94}$
57765-57800	${53.77}_{-0.46}^{+0.78}$	${6.99}_{-0.31}^{+0.64}$	${14.30}_{-1.36}^{+2.14}$	${6.74}_{-0.07}^{+0.15}$	${14.85}_{-1.11}^{+0.61}$
57821-57898	${56.19}_{-0.83}^{+0.50}$	${9.00}_{-0.56}^{+0.45}$	${22.75}_{-4.43}^{+2.85}$	${7.09}_{-0.19}^{+0.19}$	${13.42}_{-0.82}^{+1.03}$
57898-57969	${57.38}_{-1.02}^{+1.01}$	${9.20}_{-0.84}^{+0.58}$	${23.64}_{-4.43}^{+5.23}$	${6.97}_{-0.16}^{+0.20}$	${13.79}_{-0.96}^{+0.84}$
57969-58017	${56.84}_{-1.39}^{+1.26}$	${8.47}_{-1.08}^{+0.89}$	${17.44}_{-3.47}^{+2.77}$	${6.45}_{-0.11}^{+0.09}$	${19.17}_{-1.32}^{+1.90}$
58018-58108	${55.46}_{-0.84}^{+1.39}$	${7.80}_{-0.47}^{+1.12}$	${17.33}_{-2.48}^{+3.87}$	${6.75}_{-0.12}^{+0.17}$	${14.74}_{-1.13}^{+0.96}$
58108-58181	${54.52}_{-0.96}^{+0.59}$	${7.33}_{-1.03}^{+0.54}$	${16.23}_{-3.31}^{+2.59}$	${6.84}_{-0.14}^{+0.12}$	${14.25}_{-0.89}^{+0.73}$
58181-58260	${55.05}_{-1.33}^{+0.69}$	${8.36}_{-1.11}^{+0.82}$	${17.86}_{-4.21}^{+3.24}$	${6.83}_{-0.18}^{+0.17}$	${14.35}_{-1.01}^{+0.93}$

Table 3. Table 3: Best-fitting parameters of Cen X-3 observed with Swift /BAT. E cyc subscript 𝐸 cyc E_{\rm cyc} , σ 𝜎 \sigma and Depth are the energy, width and depth of the CRSF. The continuum model is a powerlaw modified by a Fermi-Dirac cutoff, where Γ Γ \Gamma and E fold subscript 𝐸 fold E_{\rm fold} are the powerlaw index and the e-folding parameter.

Time (MJD)	$E_{\mathrm{cyc}}$ (keV)	$\sigma \left(keV\right)$	Depth	$\Gamma$	$E_{\mathrm{fold}}$
53422-53595	${32.18}_{-0.46}^{+0.30}$	${7.17}_{-0.84}^{+0.70}$	${12.44}_{-2.17}^{+2.37}$	${1.35}_{-0.04}^{+0.07}$	${8.55}_{-0.26}^{+0.39}$
53596-53707	${31.70}_{-0.94}^{+2.07}$	${7.73}_{-1.39}^{+2.42}$	${11.88}_{-4.95}^{+9.29}$	${1.00}_{-0.13}^{+0.19}$	${6.85}_{-0.33}^{+0.62}$
53717-53889	${31.61}_{-1.41}^{+1.05}$	${5.35}_{-1.17}^{+1.17}$	${5.87}_{-2.11}^{+2.06}$	${1.84}_{-0.13}^{+0.16}$	${8.72}_{-0.13}^{+0.53}$
53897-53955	${31.67}_{-0.61}^{+2.01}$	${4.36}_{-0.63}^{+1.68}$	${4.80}_{-0.95}^{+3.44}$	${1.76}_{-0.12}^{+0.15}$	${8.10}_{-0.31}^{+0.78}$
53956-54015	${32.66}_{-1.73}^{+1.61}$	${8.69}_{-1.52}^{+0.97}$	${15.46}_{-8.04}^{+3.64}$	${1.11}_{-0.14}^{+0.18}$	${7.22}_{-0.23}^{+0.84}$
54022-54071	${30.24}_{-0.88}^{+0.55}$	${2.51}_{-1.19}^{+0.85}$	${1.85}_{-0.84}^{+0.40}$	${2.60}_{-0.02}^{+0.05}$	${12.96}_{-0.60}^{+1.50}$
54076-54135	${30.69}_{-1.70}^{+0.87}$	${9.96}_{-0.93}^{+1.98}$	${18.50}_{-4.72}^{+5.13}$	${0.95}_{-0.11}^{+0.16}$	${7.14}_{-0.35}^{+0.56}$
54136-54195	${30.26}_{-0.38}^{+1.09}$	${6.32}_{-0.62}^{+0.65}$	${8.19}_{-1.56}^{+1.23}$	${1.49}_{-0.09}^{+0.11}$	${8.25}_{-0.17}^{+0.33}$
54230-54366	${31.94}_{-1.44}^{+0.60}$	${8.04}_{-1.54}^{+3.77}$	${14.14}_{-7.06}^{+9.64}$	${0.93}_{-0.03}^{+0.12}$	${7.36}_{-0.17}^{+0.61}$
54403-54495	${30.18}_{-1.05}^{+2.35}$	${6.22}_{-0.12}^{+2.25}$	${9.02}_{-0.71}^{+5.67}$	${0.98}_{-0.04}^{+0.05}$	${7.54}_{-0.79}^{+1.01}$
54496-54545	${34.69}_{-1.39}^{+1.64}$	${9.99}_{-1.31}^{+1.49}$	${26.18}_{-11.52}^{+7.61}$	${0.90}_{-0.10}^{+0.17}$	${7.81}_{-0.23}^{+0.93}$
54558-54675	${32.13}_{-0.76}^{+0.45}$	${6.10}_{-0.73}^{+0.24}$	${8.08}_{-2.25}^{+0.62}$	${1.72}_{-0.01}^{+0.03}$	${8.79}_{-0.30}^{+0.60}$
54679-54777	${33.40}_{-1.22}^{+3.27}$	${9.98}_{-1.83}^{+4.13}$	${20.20}_{-10.62}^{+17.24}$	${0.86}_{-0.08}^{+0.16}$	${7.17}_{-0.20}^{+0.92}$
54800-54854	${30.55}_{-2.50}^{+2.99}$	${9.95}_{-0.37}^{+0.60}$	${15.90}_{-7.90}^{+5.83}$	${0.92}_{-0.17}^{+0.24}$	${6.94}_{-0.35}^{+0.87}$
54087-54975	${31.84}_{-0.66}^{+0.90}$	${5.41}_{-1.02}^{+0.82}$	${5.66}_{-1.35}^{+0.88}$	${1.86}_{-0.13}^{+0.15}$	${9.52}_{-0.30}^{+0.49}$
54986-55151	${30.41}_{-0.82}^{+2.36}$	${9.98}_{-3.01}^{+5.32}$	${17.45}_{-11.09}^{+22.57}$	${0.93}_{-0.13}^{+0.23}$	${7.34}_{-0.37}^{+0.54}$
55161-55215	${29.53}_{-0.40}^{+0.71}$	${3.37}_{-0.53}^{+0.33}$	${3.02}_{-0.45}^{+0.55}$	${2.22}_{-0.11}^{+0.13}$	${11.00}_{-0.40}^{+0.63}$
55216-55272	${32.45}_{-0.81}^{+1.69}$	${9.62}_{-1.84}^{+3.17}$	${18.29}_{-8.05}^{+12.77}$	${0.81}_{-0.03}^{+0.12}$	${6.98}_{-0.32}^{+0.52}$
55278-55452	${31.37}_{-0.72}^{+0.42}$	${6.19}_{-0.96}^{+0.67}$	${8.10}_{-2.17}^{+1.14}$	${1.57}_{-0.04}^{+0.08}$	${8.55}_{-0.41}^{+0.53}$
55465-55515	${32.85}_{-0.59}^{+0.96}$	${7.69}_{-1.19}^{+1.48}$	${15.18}_{-3.71}^{+3.66}$	${1.28}_{-0.03}^{+0.08}$	${8.20}_{-0.30}^{+0.40}$
55517-55631	${34.02}_{-1.14}^{+1.95}$	${10.02}_{-1.81}^{+1.57}$	${23.03}_{-10.27}^{+9.37}$	${0.94}_{-0.10}^{+0.19}$	${8.04}_{-0.29}^{+1.03}$
55638-55804	${31.68}_{-0.74}^{+1.71}$	${9.52}_{-1.92}^{+3.10}$	${16.99}_{-8.80}^{+10.86}$	${1.20}_{-0.09}^{+0.15}$	${7.92}_{-0.30}^{+0.69}$
55833-55875	${31.75}_{-0.65}^{+0.75}$	${6.33}_{-1.52}^{+2.91}$	${7.99}_{-3.36}^{+12.19}$	${1.30}_{-0.10}^{+0.12}$	${7.74}_{-0.13}^{+0.38}$
55876-55935	${32.68}_{-0.51}^{+0.56}$	${9.99}_{-1.15}^{+0.85}$	${20.19}_{-6.31}^{+2.80}$	${1.16}_{-0.06}^{+0.12}$	${7.67}_{-0.36}^{+0.44}$
55996-56055	${30.92}_{-0.69}^{+0.34}$	${5.48}_{-1.17}^{+0.51}$	${6.01}_{-2.12}^{+0.80}$	${1.44}_{-0.13}^{+0.16}$	${7.77}_{-0.16}^{+0.48}$
56056-56163	${31.71}_{-1.23}^{+0.36}$	${6.13}_{-1.09}^{+0.32}$	${7.40}_{-2.85}^{+0.44}$	${1.58}_{-0.12}^{+0.14}$	${8.57}_{-0.25}^{+0.67}$
56207-56278	${31.49}_{-1.03}^{+2.20}$	${10.00}_{-1.39}^{+3.98}$	${17.24}_{-4.24}^{+14.38}$	${0.83}_{-0.02}^{+0.09}$	${6.96}_{-0.20}^{+0.46}$
56307-56355	${31.84}_{-0.49}^{+0.75}$	${7.30}_{-0.60}^{+0.37}$	${10.87}_{-2.05}^{+1.60}$	${1.35}_{-0.02}^{+0.02}$	${7.96}_{-0.22}^{+0.38}$
56356-56413	${32.07}_{-0.54}^{+0.49}$	${6.62}_{-0.81}^{+0.42}$	${8.33}_{-1.40}^{+0.46}$	${1.66}_{-0.13}^{+0.14}$	${8.49}_{-0.24}^{+0.38}$
56416-56595	${31.85}_{-1.17}^{+0.52}$	${7.19}_{-1.04}^{+0.45}$	${11.47}_{-3.15}^{+0.65}$	${0.69}_{-0.09}^{+0.12}$	${6.83}_{-0.16}^{+0.37}$
56596-56631	${32.50}_{-0.83}^{+2.41}$	${9.37}_{-1.65}^{+1.95}$	${17.95}_{-8.29}^{+10.68}$	${1.22}_{-0.13}^{+0.17}$	${7.65}_{-0.28}^{+0.84}$
56702-56835	${31.78}_{-1.02}^{+0.82}$	${9.58}_{-1.71}^{+1.48}$	${16.70}_{-6.88}^{+5.51}$	${0.98}_{-0.07}^{+0.08}$	${7.03}_{-0.15}^{+0.29}$
56838-57008	${32.42}_{-1.17}^{+2.02}$	${10.00}_{-2.07}^{+3.05}$	${18.32}_{-9.06}^{+11.84}$	${0.74}_{-0.07}^{+0.12}$	${6.94}_{-0.28}^{+0.60}$
57020-57067	${31.62}_{-0.45}^{+1.02}$	${8.78}_{-1.26}^{+0.32}$	${14.55}_{-3.79}^{+1.24}$	${0.97}_{-0.09}^{+0.11}$	${7.08}_{-0.18}^{+0.32}$
57102-57246	${31.37}_{-0.81}^{+0.60}$	${6.74}_{-1.47}^{+0.94}$	${9.04}_{-3.56}^{+2.39}$	${1.50}_{-0.11}^{+0.14}$	${8.73}_{-0.24}^{+0.38}$
57276-57359	${32.65}_{-0.67}^{+0.87}$	${9.14}_{-1.47}^{+1.62}$	${17.65}_{-7.07}^{+5.56}$	${1.34}_{-0.15}^{+0.18}$	${8.51}_{-0.22}^{+0.59}$
57379-57405	${32.81}_{-0.90}^{+1.47}$	${10.00}_{-1.20}^{+0.23}$	${19.82}_{-3.95}^{+1.78}$	${1.16}_{-0.04}^{+0.08}$	${8.14}_{-0.41}^{+0.60}$
57436-57615	${29.75}_{-0.25}^{+0.53}$	${6.35}_{-0.60}^{+1.07}$	${6.86}_{-0.99}^{+1.98}$	${1.22}_{-0.14}^{+0.16}$	${7.38}_{-0.25}^{+0.41}$
57616-57735	${30.79}_{-0.61}^{+1.31}$	${9.32}_{-1.67}^{+2.76}$	${15.12}_{-4.98}^{+8.55}$	${1.31}_{-0.13}^{+0.17}$	${8.09}_{-0.31}^{+0.65}$
57736-57795	${28.89}_{-1.45}^{+1.42}$	${10.00}_{-1.00}^{+1.37}$	${14.38}_{-4.25}^{+4.97}$	${0.71}_{-0.18}^{+0.27}$	${6.25}_{-0.44}^{+0.70}$
57614-57851	${31.65}_{-1.05}^{+0.79}$	${10.02}_{-2.82}^{+3.67}$	${14.44}_{-7.35}^{+10.37}$	${0.69}_{-0.11}^{+0.20}$	${6.48}_{-0.25}^{+0.60}$
57856-57915	${31.44}_{-0.33}^{+0.27}$	${7.09}_{-1.04}^{+0.75}$	${11.03}_{-3.07}^{+1.87}$	${1.34}_{-0.04}^{+0.09}$	${8.27}_{-0.30}^{+0.39}$
57916-57956	${33.31}_{-0.94}^{+0.96}$	${9.26}_{-1.09}^{+1.78}$	${16.41}_{-5.46}^{+6.83}$	${1.10}_{-0.10}^{+0.15}$	${7.54}_{-0.23}^{+0.41}$
57984-58009	${32.44}_{-1.33}^{+1.39}$	${8.51}_{-0.77}^{+0.61}$	${14.50}_{-4.60}^{+4.45}$	${1.03}_{-0.10}^{+0.13}$	${7.39}_{-0.25}^{+0.63}$
58037-58155	${33.12}_{-1.61}^{+1.88}$	${9.96}_{-1.41}^{+2.33}$	${19.24}_{-7.89}^{+8.02}$	${1.12}_{-0.03}^{+0.07}$	${7.91}_{-0.44}^{+0.85}$
58159-58267	${34.31}_{-0.58}^{+0.90}$	${8.28}_{-1.37}^{+0.50}$	${16.14}_{-4.72}^{+1.78}$	${1.27}_{-0.13}^{+0.16}$	${8.24}_{-0.36}^{+0.67}$

Equations

$$\displaystyle{\tilde{E}_{\rm cyc}}(t)=\left\{\begin{array}[]{l}{E_{\rm 0}}+b\times(t-{t_{\rm 0}}),t\leq{t_{\rm crit}}\\ const={E_{\rm 0}}+a\times({t_{\rm crit}}-{t_{\rm 0}}),t>{t_{\rm crit}}\end{array}\right.$$

Full text

††pubyear: 2019††pagerange: Long-term evolutions of the cyclotron line energies in Her X-1, Vela X-1 and Cen X-3 as observed with Swift/BAT–A

1 Introduction

Accretion powered X-ray pulsars are some of the brightest sources in our Galaxy. Their magnetic fields are believed to be of the order of $\sim$ $10^{11}$-$10^{13}$ G. The accreting matter is funnelled from the magnetospheric radius to a small region on the surface of the neutron star (the polar cap, see, e.g., Basko & Sunyaev, 1976). In the presence of a strong magnetic field, the energies of the electrons with respect to their movement perpendicular to the magnetic field are quantized into discrete Landau levels. Resonant scattering of photons on such electrons in the line-forming region, results in cyclotron resonance scattering features (CRSFs) simply referred to as cyclotron lines. The centroid cyclotron line energy is $E_{\rm cyc}$ = $11.6nB_{12}(1+z)^{-1}$, where the $B_{12}$ is the magnetic field strength in units of $10^{12}$ Gauss and $z$ is the gravitational redshift in the line-forming region. n=1 and n=2,3,4… correspond to the fundamental and harmonic cyclotron lines, respectively 111In this paper, we quote the fundamental and the first harmonic as $E_{\rm cyc}$ and $E_{\rm cyc\_H}$, respectively.. Cyclotron lines provide a direct measure of the magnetic field strength in the line-forming region, and its variability reflects the changes of the accretion geometry and/or the re-arrangement of the magnetic field configuration (see, e.g., Becker et al., 2012; Mushtukov et al., 2015).

It has been found that the CRSF energy generally depends on pulse phase, often on luminosity (see, e.g., Staubert et al., 2007; Klochkov et al., 2011; Vasco et al., 2013; Fürst et al., 2014; Vybornov et al., 2017). In addition, the long-term time dependence was discovered by Staubert et al. (2014); Staubert et al. (2016) in Her X-1, in which the cyclotron line energy decreases by $\sim$ 5 keV over 20 years. They used the data obtained with several X-ray observatories (RXTE, Beppo-SAX, INTEGRAL, Suzaku and NuSTAR), in the time period from 1996 to 2015. Subsequently, Klochkov et al. (2015) independently confirmed this result by using monitoring Swift/BAT observations. What is more interesting is that recently Staubert et al. (2017) proposed that the 20-year $E_{\rm cyc}$ decrease has ended and an inverse trend could start soon. In this paper, we have started a detailed re-analysis of Swift/BAT data extending to the most recent observations, in order to follow the evolution of the cyclotron line energies over time. In addition to Her X-1, a second source was found - Vela X-1 - showing a similar long-term decrease of its first harmonic cyclotron line energy (La Parola et al., 2016). Swift/BAT data of Vela X-1 were analysed with the software bat_imager (Segreto et al., 2010), and it was found that the first harmonic cyclotron line decreased by $\sim$ 0.72 ${\rm keV\,yr^{-1}}$ between December 2004 and June 2010, and then remained constant. Additionally, there are two other candidates showing CRSF variations with time: V0332+53 and 4U 1538-22. The former shows, in its 2015 outburst, a systematically lower cyclotron line energy in the declining phase of the outburst compared to the rising phase for equal levels of luminosity (Cusumano et al., 2016; Doroshenko et al., 2017; Vybornov et al., 2017). For 4U 1538-22 a possible increase by $\sim$1.5 keV may have happened between the RXTE and Suzaku observations, which are about $\sim$ 8.5 years apart (Hemphill et al., 2016).

In this work, we re-analyze Swift/BAT monitoring observations of Her X-1 and Vela X-1 by using a procedure and software developed by Klochkov et al. (2015), which had led to the confirmation of the long-term decrease of the cyclotron line energy in Her X-1. We improve previous studies in four aspects:

1. Data after 2015, especially in Her X-1 when the decrease trend ended, are included.

2. The flux-correction (see below) is taken into account in Her X-1.

3. Updated calibration files are employed, which improves the results for all sources.

4. In addition to Her X-1 and Vela X-1, we searched for a long-term variation of cyclotron line energies in other sources by using archived Swift/BAT data. We considered all the source in Table 1 published by Maitra (2017), however, only in Cen X-3 was it possible to detect the cyclotron line (see, e.g., Santangelo et al., 1998) with sufficient significance.

Therefore, in this paper we present the evolution of the cyclotron line energies in Her X-1, Vela X-1 and Cen X-3. The paper is organized as follows: the detailed data reduction and the corresponding results are shown in Section 2 and 3, respectively. We discuss the implication of our results in Section 4.

2 OBSERVATIONS AND DATA ANALYSIS

Swift/BAT is a coded aperture telescope operating in the 15–150 keV range (Barthelmy et al., 2005). The data we selected have been taken in the survey mode, for which events were collected in the detector plane histograms (DPHs), typically with a five-minute exposure time. All data available since the launch of the mission in 2004 have been used. In this paper, we generally followed the data reduction of Klochkov et al. (2015). Here we briefly summarize the procedures. We reconstructed the sky map in each observation with the tools "batbinevt" and "batfftimage" from the heasoft ver. 6.21 package 222https://heasarc.gsfc.nasa.gov/docs/software/lheasoft/. We extracted the spectra only if the source could be identified in the sky map. We used the "beterebin" tool to correct the gain/offset of detectors with the latest CALDB that was released in October 2017. As suggested by the BAT team, we added the energy-dependent systematic errors by using the "batphasyserr" tool. In order to model the cyclotron line, we used a Gaussian absorption line ("gabs" in xspec, i.e., $exp\left\{-\frac{\rm Depth}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\frac{(E-E_{\rm cyc})^{2}}{{\sigma}^{2}}}\right\}$). This model has been widely used to describe cyclotron lines (e.g., Staubert et al., 2007; Fürst et al., 2013). For the continuum, different functions were used for different sources (see below). The energy band used for the fits is 15-70 keV. To enhance the statistics in the spectral analysis, we jointly fitted tens to hundreds of spectra in a given time interval, for which the spectral shapes were assumed to be the same while normalizations were variable. The validity of this method has been verified by Klochkov et al. (2015). In this paper, all uncertainties quoted correspond to a 68% confidence level.

3 RESULTS

3.1 Her X-1

We summed the BAT spectra into time intervals based on Table 1 in Klochkov et al. (2015), and extended to new observations. We fitted the spectra with a highecut 333https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node238.html model, i.e., a power-law continuum with an exponential cut-off, modified by the CRSF component (gabs). This continuum model has been widely used for the long-term CRSF evolution (see, e.g., Staubert et al., 2014). Thanks to the stable continuum shape of Her X-1, we froze the parameters at the e-folding energy $E_{\rm fold}$ =10 keV, the cutoff energy $E_{\rm cut}$=21 keV, and the powerlaw index $\Gamma$ = 0.9 (Fürst et al., 2013). We show the best-fitting CRSF energy values in Figure 1, together with previous results of Klochkov et al. (2015) and Staubert et al. (2014); Staubert et al. (2016, 2017). We found that the line energy in this paper was systematically different from that reported by Klochkov et al. (2015), although the same data and fitting method were used. We note that there are two main reasons:

1. the continuum models have a slight influence on the line detection. We fitted the spectra with two continuum models, i.e., the highecut and the cutoffpl (a power law with high energy exponential rolloff) used by Klochkov et al. (2015), and found that on average the latter resulted in a systematically higher $E_{\rm cyc}$ by $\delta E_{sys}$ = 1.04 $\pm$ 0.10 keV. We show the comparison in Figure 2.
2. we used the updated detector gain calibration of Swift/BAT in this paper, which had $\sim$ 4% gain shift during 2004-2011. Therefore, the resulting shift of the cyclotron line energy is $\delta{\rm gain}\times E_{\rm cyc}$. We compared the expected shift with observations by using a $\chi^{2}$-test, which leads to a reduced-$\chi^{2}$ of 1.2 (18 dof) with a p-value of 0.25. This suggests that the CRSF energy reported in this paper is well in agreement with that detected by Klochkov et al. (2015) after considering the above discussed two effects.

In addition, it is worth noting that the CRSF energy starts to significantly deviate from the downwards trend since $\sim$ MJD 56500, which is consistent with NuSTAR observations, i.e., the last two green points in Figure 1.

It is well-known that in Her X-1 the CRSF energy is related to the luminosity ($L_{\rm x}$) (Staubert et al., 2007; Staubert et al., 2016). Therefore, in the following spectral analysis we took the $E_{\rm cyc}/L{\rm x}$ correlation into account, by using the maximum flux of the individual 35d Main-On cycle from which the data were taken as a reference. Historically, the maximum Main-On flux, as measured by RXTE/ASM in the energy range of 2–10 keV, was taken as a measure of $L_{\rm x}$ of this particular 35d cycle (see, e.g., Staubert et al., 2007). In this paper we follow this approach, using flux measurements by Swift/BAT, which are converted into units of ASM-cts/s according to the following formula: (2-10 keV ASM-cts/s) = 93 $\times$ (15-50 keV BAT-cts/s $\rm{cm}^{-2}\,s^{-1}$) (for details see Appendix A.2. in Staubert et al., 2016). The $E_{\rm cyc}$/luminosity correlation is $\delta E_{\rm cyc}$ $\propto$ $f\times\delta$ Flux, where $f$ is a scaling factor allowing to normalize the measured cyclotron line energy to a reference flux. Following Staubert et al. (2014), the reference flux is $\rm Flux_{ref}$ = 6.8 ASM-cts/s, where the CRSF energy is assumed to be $E_{\rm ref}$. For data until 2012 (i.e., MJD 55927) we used the scaling factor 0.44 keV/ASM-cts/s as stated by Staubert et al. (2016, 2017). For data after 2012 we used a scaling factor of 0.70 keV/ASM-cts/s, which was found to describe the data from 2012 to February 2018 with high precision (private communication with Staubert).

Generally, one point shown in Figure 1 comprises tens to hundreds of spectra from several 35d cycles. Since different 35d cycles have different maximum Main-On fluxes, the combined fitting of those spectra was done by applying appropriate scaling factors to spectra from different 35d cycles (which can easily be done within the XSPEC fitting software). So, for a spectrum ($S_{\rm i}$), its CRSF energy could be written as $E_{\rm cyc\_i}$ = $E_{\rm ref}$ + ($\rm Flux_{\rm i}-Flux_{ref}$)$\times f$, where the $\rm Flux_{\rm i}$ is the maximum flux of the corresponding 35d Main-On. The $E_{\rm ref}$ can be regarded as a fitting parameter in the spectral analysis. In this way we have obtained the long-term evolution of the cyclotron line energy for the reference flux ($\rm Flux_{ref}$ = 6.8 ASM-cts/s), i.e., the flux-corrected $\tilde{E}_{\rm cyc}$. We show the result in Fig. 3. For comparison, we also include the line energies measured by other satellites as reported by Staubert et al. (2014); Staubert et al. (2016, 2017). It is evident that the line energy decreases until a certain time $t_{\rm crit}$, which is in good agreement with previous reports (the blue dashed line). After that, the line energy significantly deviates from the linear decreasing trend, and in general remains unchanged. We note that an alternative continuum model cutoffpl only leads to a systematically higher line energy by 1.08 $\pm$ 0.10 (Figure 2), and does not have an influence on the trend. We used a break line to fit the flux-corrected $\tilde{E}_{\rm cyc}$ evolution of Swift observations as

[TABLE]

where $t_{\rm 0}$=MJD 53500, $E_{\rm 0}$=39.88 keV and b=$\rm-7.22\times{10}^{-4}\,keV/d$ by following Staubert et al. (2014). The resulting $t_{\rm crit}$ is MJD 55400$\pm$ 200, and the corresponding $\tilde{E}_{\rm cyc}$ after $t_{\rm crit}$ is at $37.94\pm 0.12$ keV, showing as the horizontal blue line in Figure 3. We note, that this value is systematically higher (by 0.32 keV) than the mean of NuSTAR and Suzaku observations (i.e., the green points) after $t_{\rm crit}$ 444The comparison of the blue and the green points in Fig. 3 later that MJD 55400 may point to a calibration issue between NuSTAR/Suzaku and Swift/BAT.

3.2 Vela X-1

Following La Parola et al. (2016), we employed a Comptonization model (compTT in xspec) to describe the continuum of Vela X-1. We note that the fundamental line around 28 keV cannot be detected by BAT in Vela X-1, and in this paper we only concentrate on the first harmonic $E_{\rm cyc\_H}$. Following the above procedures, we extracted the spectra of Vela X-1, and did the spectral analysis. Although observed by La Parola et al. (2016) and Fürst et al. (2013) that the $E_{\rm cyc\_H}$ is related to the luminosity in Vela X-1, their relation has not been well constrained yet (other than in Her X-1). Therefore, in the following analysis, we did not consider the flux-correction. Actually, as shown above, the flux-correction has little influence on the trend of the cyclotron line evolution detected by BAT because the stochastic variability of the luminosity is expected to be mitigated. During the spectral analysis, we fixed the temperature of seed photons at $kT_{\rm seed}$ = 1 keV, because of no energy coverage below 15 keV for BAT. We show the results in Table LABEL:tab_vela and Figure 4. For the sake of comparison, we divided the observations into five epochs, the first four of which were already defined in La Parola et al. (2016). The line energy decreased significantly in the first two epochs, remaining almost unchanged thereafter. Therefore, we tried to fit the line evolution with a piecewise function as mentioned above. The critical point is around MJD 55980 (February 2012). Before MJD 55980, the decrease rate of $E_{\rm cyc\_H}$ is -0.51 $\pm$ 0.09 keV per year. The $E_{\rm cyc\_H}$ after MJD 55980 is 54.96 $\pm$ 0.19 keV. In addition, it seems that there is a hump around MJD $\sim$ 55000. We fitted the $E_{\rm cyc\_H}$ variation with a multi-segment function (the black dashed line in Figure 4), however, the confidence level of the presence of a hump is only at 1.9 $\sigma$, estimated by an F-test. Apart from the $E_{\rm cyc\_H}$, we also found a hint that the width $\sigma_{\rm H}$ might be variable. We fitted the $\sigma_{\rm H}$ evolution with a constant and a quadratic function, respectively. An F-test shows that the latter is better at a 3.5$\sigma$ confidence level. We confirmed that the source of the variability of $\sigma_{\rm H}$ is not instrumental because such a trend did not appear in other sources.

3.3 Cen X-3 and other sources

We tried to apply the above method in more sources. We considered the source list in Table 1 in Maitra (2017) as a reference. We only considered persistent sources that could show the smooth long-term evolution of CRSFs. However, we found that most of sources (except for Cen X-3 and GX 301-2) could not be identified from the mosaic which was a prerequisite for extracting spectra, mainly because of the sensitivity of BAT. We found that the minimum flux to be detected in one DPH in the survey mode is approximately 0.02 $\rm cts\ s^{-1}\ {cm}^{-2}$, i.e., 90 mCrab. In GX 301-2, we could not constrain the cyclotron line, even if different continuum models were tested. This might be due to the dramatic changes of the CRSF in GX 301-2 with different orbital phases and pulse phases (see, e.g., Kreykenbohm et al., 2004; La Barbera et al., 2005; Suchy et al., 2012), which might wipe out the absorption if stacking hundreds of spectra during fits.

In Cen X-3, the cyclotron line can be constrained well. Here we used a Fermi-Dirac ("fdcut") function to describe the continuum following Suchy et al. (2008), where $fdcut(E)=AE^{-\Gamma}\frac{1}{1+e^{(E-E_{\rm cut})/E_{\rm fold}}}$ . We show the best-fitting results in Table LABEL:tab_cen and Figure 5. Unlike the variability in Her X-1 and Vela X-1, we found that the centroid line energy in Cen X-3 was very stable over the past 14 years, at approximately 31.6$\pm$0.2 keV.

4 summary and discussion

We searched for the long-term evolution of cyclotron line parameters in persistent sources by using archived Swift/BAT data. Because of the regular visiting and the large field-of-view, BAT provides nearly homogeneously spaced observations without long time gaps, which is a proper instrument for monitoring CRSFs in bright sources. In Her X-1, Vela X-1 and Cen X-3, we detected cyclotron lines well, and found significant line decreases in the first two sources.

For Her X-1, we confirmed the results of Klochkov et al. (2015), while including two improvements: 1) We considered the flux-correction in the spectral analysis, which turned out to slightly influence the $E_{\rm cyc}$ values. 2) We analysed the data in the recent 4 years, which implied that the decrease of the CRSF centroid energy had ended around 2010 and remained constant around 37.94 $\pm$0.12 keV afterwards. Staubert et al. (2016, 2017) reported the continued decrease of the line energy until August 2015, when a particularly low flux was measured. With the above mentioned change of the flux correction factor (to 0.7 instead of 0.44 keV/ASM-cts/s) the flux corrected $\tilde{E}_{\rm cyc}$ for August 2015 is shifted upwards, leading to an estimate for the end of the decrease around 2012 to 2013 (Staubert et al., in preparation), which is consistent with the result derived here. This, in turn, indicates that the long-term $\tilde{E}_{\rm cyc}$ decrease observed by BAT is robust because it is not so sensitive to the flux-correction factor by averaging many spectra in different fluxes. In Vela X-1, we independently confirmed the decrease of the energy of the first cyclotron harmonic reported by La Parola et al. (2016) by using different methods. The decrease rate is $0.51\pm 0.09$ $\rm keV\,yr^{-1}$, which is similar to the decrease rate of $\sim$ 0.72 $\rm keV\,yr^{-1}$ stated by La Parola et al. (2016). In addition to the centroid energy, we also found a long-term variation of the width of the cyclotron line, although the validity should be further confirmed by other observatories. We also searched for the cyclotron lines in other sources, but only in Cen X-3 it is well detected. The line energy in Cen X-3 is very stable at least since 2004.

The variation of the cyclotron line energy (and width) observed in Her X-1 and Vela X-1 is believed to be caused by a local effect around the magnetic polar cap. As summarized by Staubert et al. (2014), the variation might be due to a geometric displacement of the line-forming region, or a change of the local magnetic configuration. For example, the accreting matter accumulated in the accretion mound, gradually resulting in a more extended line-forming region that corresponds to a lower magnetic field. On the other hand, the magnetic field might be changed because of the drag by the accretion material (Cheng & Zhang, 1998; Zhang & Kojima, 2006), and the Hall drift and the Ohmic dissipation (Goldreich & Reisenegger, 1992). However, as far as we know, no model gives a conclusive explanation of the decrease of the cyclotron line energy. More observations and theoretical works should be accumulated to investigate the complex magnetic field around the polar cap.

The time scale of the decrease of the line energy ($-E_{\rm cyc}$/$\dot{E_{\rm cyc}}$) in Her X-1/Vela X-1 is $\sim$ 100 years which is significantly shorter than the characteristic time scale of the magnetic filed evolution in pulsars (Bhattacharya et al., 1992). Therefore, as suggested by Staubert et al. (2014), the centroid energy of $E_{\rm cyc}$ may be cyclic on time scales of a few tens to hundreds of years, which may comprise of declining, stable and rising phases. The results presented in this work show that BAT is a proper instrument to observe CRSFs in relatively bright sources such as Her X-1, Vela X-1, and Cen X-3, and supports the need for more sensitive and regular long-term monitoring of CRSFs in other accreting pulsars with observatories such as NuSTAR, INTEGRAL, HXMT and Astrosat.

Acknowledgements

JL thanks the support from the Chinese NSFC 11733009. ZS thanks the support from XTP project XDA 04060604, the Strategic Priority Research Programme ’The Emergence of Cosmological Structures’ of the Chinese Academy of Sciences, Grant No.XDB09000000,the National Key Research and Development Program of China (2016YFA0400800) and the Chinese NSFC 11473027 and 11733009. We acknowledge discussions with Dr. Klochkov Dmitry, Dr. Markwardt Craig and Dr. Amy Lien, and help from the Swift Helpdesk.

Appendix A Best-fitting spectral parameters in Her X-1, Vela X-1 and Cen X-3