Variations in energy of cyclotron lines with double structures formed in a line-forming region with bulk motion in accreting X-ray pulsars

Nishimura, Osamu

doi:10.1093/pasj/psac048

Abstract

We show that variations in the energy of a cyclotron resonant scattering feature with luminosity can be explained by considering a variation of the bulk velocity of infalling matter in the two-dimensional structure of an accretion column. Variations in the energy of a cyclotron line with luminosity are computed by taking into account the variation in gradient of the bulk velocity with luminosity in a line-forming region. We mainly discuss the positive correlation between the energy of the cyclotron line $E_{c y c}$ at the fundamental and the luminosity as observed in the spectra of GX 304−1, A0535+262, and Vela X-1, considering a change in gradient of the bulk velocity in the line-forming region with luminosity in addition to a change of altitude of the line-forming region with luminosity. Consequently, we demonstrate that the change in the observed line energy with luminosity can be successfully reproduced by a variation in bulk velocity due to radiation pressure as well as the altitude of an accretion mound.

Subject

stars: atmospheres stars: magnetic fields stars: neutron

Issue Section:

Paper

1 Introduction

Mihara, Makishima, and Nagase (1998) first reported that the observed energy of a cyclotron resonant scattering feature, E_cyc, changes with luminosity in the spectra of some accretion-powered X-ray pulsars. A negative correlation between the energy of the cyclotron line Ecyc at the fundamental and the luminosity (Negative, hereafter) has been observed in V0332+53 and X0115+63 (Mowlavi et al. 2006; Nakajima et al. 2006; Tsygankov et al. 2006). Negative is also observed in A0535+26 during the 2020 giant outburst (Kong et al. 2021). In contrast, a positive correlation between the energy of the cyclotron line E_cyc at the fundamental and the luminosity (Positive, hereafter) in GX 301−2 was reported in La Barbera et al. (2005). Furthermore, Staubert et al. (2007) also found that E_cyc in Her X-1 increases with increasing luminosity using only the data from Rossi X-ray Timing Explorer (RXTE). They showed the possibility that the change in E_cyc can be explained by a variation in the height of the emission region decelerated by Coulomb drag and collective plasma effects. In A0535+26, Positive was reported in a giant outburst in 2011 (Sartore et al. 2015) while Positive also may have appeared in a normal outburst in 2010 April (Müller et al. 2013). In GX 304−1, a clear Positive in which E_cyc increases from ∼50 keV to ∼60 keV with increasing luminosity is discovered with a significant varying pulse shape (Malacaria et al. 2015). In GRO J1008−57, E_cyc changes with luminosity during the Type II outburst in 2017 August observed by Insight-HXMT. The E_cyc was detected to be negatively correlated with the luminosity at 3.2 × 10³⁷ erg s⁻¹ < L < 4.2 × 10³⁷ erg s⁻¹ and positively correlated at L ≳ 5 × 10³⁷ erg s⁻¹ (Chen et al. 2021).

On the other hand, observed E_cyc tends not to change with luminosity (Mihara et al. 2007) as much as a variation of the shock height predicted by some theoretical models (cf. Basko & Sunyaev 1976; Langer & Rappaport 1982). Becker et al. (2012) considered Negative and Positive to be caused by the variation of the altitude of a radiation-dominated shock and the altitude at which Coulomb interactions begin to decelerate the plasma with luminosity, respectively. Nishimura (2014) also pointed out that the bulk motion of an infalling plasma can play a crucial role in changes of E_cyc with luminosity. However, a gradient in the bulk velocity as a function of altitude was ignored in his model. Mushtukov et al. (2015) took into account a variation of gradient in the bulk velocity to explain Positive of GX 304−1 in which the deceleration of the bulk velocity of an infalling plasma in the accretion column becomes stronger as luminosity increases.

In this work we consider the two-dimensional structure of an accretion column in which the behavior of the change in the bulk velocity with luminosity in the line-forming region can be considerably different from that in the continuum-forming region of the accretion mound, as Nishimura (2019) pointed out.

The paper is organized as follows. In section 2, we discuss the structures of the line-forming region employed in our calculations. Radiative transfer calculations are briefly explained in section 3. In section 4, we show the effect of the bulk motion of infalling plasma, gravitational light-bending, and beam pattern on cyclotron lines. We present the spectra calculated by our model for GX 304−1, A0535+262, and Vela X-1 in section 5. We compare our numerical results to the observations. In section 6, we summarize our results.

2 Line-forming region

An accretion mound, inside which is a sinking region, is expected to be prominent above L₃₇ ≡ L/(10³⁷ erg s⁻¹) ∼ 0.5, in which the line-forming region is expected to be located around the accretion mound. Thus, it is important to consider a two-dimensional structure of an accretion column (Lyubarskii & Syunyuaev 1988) for the line formation, although the infalling plasma is restricted to one-dimensional motion along the magnetic field (B field). Hence, in the present paper, we consider bulk velocity in a two-dimensional line-forming region, making use of the accretion column of a two-dimensional structure discussed by Arons (1992). In this case, the bulk motion of an infalling matter in the line-forming region can be significantly different from that in the continuum-forming region, unlike in one dimension. We will show that the line-forming region around a two-dimensional accretion mound is able to explain the variation of E_cyc with luminosity in the range of L₃₇ ∼ 0.2 to 1.0 in terms of variation in the bulk velocity as well as the altitude of an accretion mound.

The polar cap radius can be constrained by dynamical considerations related to the magnetically channeled accretion of the gas on to a neutron star (NS). By taking into account the effects of both the stellar rotation and the strong B field, Lamb, Pethick, and Pines (1973) find that the polar cap radius satisfies the constraint

\begin{array}{r} r_{c} ≲ R_{N S} {(\frac{R_{N S}}{r_{A}})}^{1 / 2}, \end{array}

(1)

where r_A is the Alfvèn radius. Thus, in this work, we assume the polar cap radius (Becker et al. 2012)

\begin{array}{rcl} r_{c} & = & 1.93 \times 10^{5} {(\frac{Λ}{0.1})}^{- 1 / 2} B_{12}^{- 2 / 7} {(\frac{R_{N S}}{10^{6} cm})}^{11 / 14} \\ \times {(\frac{M_{N S}}{1.4 M_{⊙}})}^{- 1 / 14} (L_{37})^{1 / 7} cm \end{array}

(2)

where the mass of NS is M_NS, M_⊙ is the solar mass, and the radius of NS is R_NS = 10⁶ cm. The constant Λ = 1 for the case of spherical accretion, and Λ < 1 for the case of disk accretion. In this work, we therefore assume Λ = 0.5 and M_NS = 1.4 M_⊙ for GX 304−1, A0535+262, and Λ = 1 and M_NS = 1.8 M_⊙ for Vela X-1 (Rawls et al. 2011).

We adopt a model derived by Arons (1992) for an accretion mound shape as shown in figure 1. For 0 ≤ r_⊥ < r_c, the equality between the ram and radiation pressure yields the nominal location of the shock to be at the height

\begin{array}{r} h_{s} (r_{⊥}) = R_{N S} \frac{L_{a}}{L_{E D} H_{∥}} (1 - \frac{r_{⊥}^{2}}{r_{c}^{2}}) \equiv h_{t o p} (1 - \frac{r_{⊥}^{2}}{r_{c}^{2}}), \end{array}

(3)

where r_⊥ is the cylindrical radius with respect to the magnetic axis,

L_{a} = GM \dot{M} / R_{N S}

is the accretion luminosity of one pole (where G is the gravitational constant), and L_ED is the Eddington luminosity. H_∥ ∼ H_⊥ ∼ 1 is a good approximation over most of the mound’s volume (Arons 1992). In this work, we take H_∥ ∼ H_⊥ = 1.2 computed for 4U 0115−63 from the cyclotron energy E_cyc of the cyclotron absorption (CYAB) fit and the temperature kT of the negative and positive power laws with exponential cutoff (NPEX) continuum (Mihara et al. 2004).

Fig. 1.

Schematic of change of the line-forming region with luminosity in altitude and bulk motion. The region represented by oblique lines denotes the line-forming region while the shadowed region denotes the continuum-forming region. Solid arrows denote the velocity of bulk motion. Dashed arrows denote diffusive flux emerging from the continuum-forming region. (a) At L₃₇ ∼ 0.1, the bulk velocity is free-fall velocity in the line-forming region, while deceleration of bulk velocity due to radiation pressure in the accretion mound starts. (b) At L₃₇ ∼ 0.5, the bulk velocity is decelerated significantly in the continuum-forming region, while bulk velocity is nearly free-fall in the line-forming region. (c) At L₃₇ ∼ 1.0, the bulk velocity in the line-forming region is also decelerated significantly by radiation pressure due to resonant scattering.

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In this mound model, the central radiation pressure at the base in an accretion mound exceeds the confining magnetic pressure for

\begin{array}{rcl} (L / L_{1})^{2} > 16 + \ln [B_{12}^{2} {(\frac{M_{N S}}{M_{⊙}})}^{1 / 2} L_{37}^{- 1} {(\frac{R_{N S}}{10^{6} cm})}^{- 3 / 2} \\ \times {(\frac{r_{c}}{10^{5} cm})}^{2}] . \end{array}

(4)

Here,

L_{1} \equiv (θ_{c} / 4) L_{E D} \sqrt{H_{∥} H_{⊥}} \sim 3 \times 10^{36} (θ_{c} / 0.1) (M_{N S} / M_{⊙}) \sqrt{H_{∥} H_{⊥}}

erg s⁻¹, in which θ_c is the opening angle of the polar cap, and H_{∥, ⊥} = σ_T/σ_{R∥, ⊥} where σ_R∥ and σ_R⊥ are the Rosseland averaged cross-sections for radiation whose diffusive fluxes are parallel and normal to the field, respectively (Arons 1992). Therefore, magnetic confinement of the mound’s deep interior fails for L ≳ 10³⁷ erg s⁻¹. As a result, the model of the equilibrium structure of the accretion mound is not valid for L ≳ 10³⁷ erg s⁻¹. For such a high luminosity, a radiation-filled region of lowered density, which is a photon bubble, is likely to be formed. Thus, photon bubbles are expected to change the structure of an accretion mound for L > 10³⁷ erg s⁻¹. On the other hand, the Rosseland averaged cross-sections are applicable within an accretion mound for all computations, because the plasma is optically thick across and along the B field for

L ≫ L_{E D}^{(e f f)} \equiv [(π r_{c}^{2}) / (4 π R_{*}^{2})] H_{∥} L_{E D} \sim 4 - 6 \times 10^{34}

erg s⁻¹ for L₃₇ = 0.2 in this work. The Thomson optical depth parallel and normal to the field is given by

τ_{T ∥} = 2 (L / L_{E D}^{(e f f)}) [c / v_{f f} (0)] \exp [(L / L_{1})] \sim 100

and

τ_{T ⊥} = (8 / \sqrt{π}) τ_{T ∥} \sim 400

for L₃₇ = 0.2 (Arons 1992). Here,

v_{f f} (z) = - \sqrt{2 {GM}_{N S} / (R_{N S} + z)}

is free-fall velocity at a height z, and c is the speed of light. Thus, the model of the accretion mound can be applied to the range of the luminosity of 0.2 ≲ L₃₇ ≲ 1 we consider in the present paper.

In addition to continuum scattering, resonant scattering would contribute to the formation of a relatively high accretion mound even at a low luminosity L₃₇ ∼ 0.5, because cyclotron emission is expected to give a large contribution to the formation of the continuum spectrum (Arons et al. 1987; Farinelli et al. 2016). In fact, the value of cut-off energy E_cut increases with increasing E_cyc in observations (Makishima et al. 1999). As a result, a radiation pressure sufficient to form a relatively high accretion mound would be provided by resonant scattering as well as continuum. However, including resonant scattering for the formation of an accretion mound is beyond the scope of the paper. In this work, equation (3) is employed for an accretion mound.

The surface of an accretion mound is assumed to be an emission region of a continuum spectrum as in the work of Burnard et al. (1991) and the line-forming region is assumed to be the region outside an accretion mound in a cylindrical geometry as shown in figure 1. Therefore, the height of the line-forming region is assumed to be z_max = 1.1 × h_top, because we consider the line formation to occur in a region around the mound so that the height of the line-forming region is comparable to that of the mound in this work.

The bulk velocity is expected to be decelerated due to resonant scattering in the line-forming region as shown in figure 1c. We must solve radiation–hydrodynamic equations to determine the velocity profile. This however is beyond the scope of the present paper. Hence, we adopt the bulk velocity profile derived for subcritical luminosity (Mushtukov et al. 2015), in which infalling plasma is decelerated to some final velocity v₀ at the surface of NS, given by

\begin{array}{rcl} v (z) & = & - {v_{f f} (z_{m a x})^{2} - [v_{f f} (z_{m a x})^{2} - v_{0}^{2}] \\ \times [1 - \frac{2}{π} \tan^{- 1} (z / r_{c})]}^{1 / 2}, \end{array}

(5)

where z is the distance from the surface of an NS along the column axis; v(z) is decelerated from −v_ff(z_max) to −v₀ in the line-forming region.

The density profile, which is derived from the velocity profile via a continuity equation in a magnetically confined accretion column, at a height z in the line-forming region is given by

\begin{array}{r} ρ (z) \propto ρ_{0} / v (z), \end{array}

(6)

where ρ₀ is the density at the altitude of the bottom of the line-forming region.

3 Radiative transfer

We use a Monte Carlo radiative transfer code developed by Nishimura (2019). In the present work, we include variation of density and bulk velocity with altitude. The basic radiation transfer method is the same as was described in Nishimura (2019). We also include gravitational light-bending as Nishimura (2019) in which two symmetrically opposite polar caps are assumed, as shown later in figure 4.

3.1 Continuum spectra

The observed continuum spectra of accretion powered X-ray pulsars have been traditionally modeled by some functions as follows. In our computations, we take a standard power-law/cut-off model for the injected continuum spectra of GX 304−1 and A0535+262

\begin{array}{r} N (E) = E^{- Γ} \exp (- \frac{E}{E_{f o l d}}) \end{array}

(7)

where Γ is the photon index and E_fold is the e-folding energy of exponential roll-off. We also use a power-law continuum with a Fermi–Dirac cut-off for the injected continuum spectra of Vela X-1

\begin{array}{r} N (E) = E^{- Γ} / [1 + \exp (\frac{E - E_{c u t}}{E_{f o l d}})] \end{array}

(8)

where E_cut is the cut-off energy (Tanaka 1986). We employ the observed values of Malacaria et al. (2015), Fürst et al. (2014), Müller et al. (2013), and Sartore, Jourdain, and Rocques (2015) for these parameters of GX 304−1, Vela X-1, and a normal and a giant outburst in A0535+262, respectively.

3.2 Beam patterns

In the subcritical regime (L < L_crit), a pencil-like beam or both a pencil-like and a fan beam are expected to appear. Here, L_crit = 1.49 × 10³⁷ erg s $^{- 1} (Λ / 0.1)^{- 7 / 5} (M_{N S} / 1.4 M_{⊙})^{29 / 30} B_{12}^{16 / 15}$ for R_NS = 10⁶ cm and w = 1, where the mean photon energy $\bar{E}$ is given by $\bar{E} = w k T_{e f f}$ ⁠; T_eff is the effective temperature of the radiation in the post-shock region and k is Boltzmann’s constant (Becker et al. 2012). We therefore consider a beam pattern that has a beam with a peak intensity of θ_p1 = 45° to be a pencil-like beam, and a beam pattern composed of two beam patterns that have one beam with a peak intensity of θ_p1 = 45° and one with a peak intensity of θ_p2 = 90° to be a fan beam. Here we consider the angle-dependence of the intensity of the continuum spectra, i.e., beam patterns, as follows (Kraus 2001):

\begin{array}{rcl} I (θ) & = & N (E) (B_{1} \exp [- \frac{(θ - θ_{p 1})^{2}}{d^{2}}] \\ + B_{2} \exp [- \frac{(θ - θ_{p 2})^{2}}{d^{2}}]) . \end{array}

(9)

Here, θ is the angle between the direction of photon’s propagation and the B field, and B₁ and B₂ are coefficients that represent the intensity of two beam patterns. We use 45° for parameter θ_p1 and 90° for θ_p2, respectively, and fix 5° for parameter d, since a pencil-like plus fan beam is considered in this work.

3.3 Cyclotron resonant scattering

When B ∼ B_c = 4.414 × 10¹³ G, a relativistic quantum treatment is necessary for cyclotron resonant scattering regardless of the electron temperature, because the electron’s momentum eigenvalue perpendicular to the B field is given by p_⊥/m_ec = n(B/B_c) ≡ nb_c, where n represents the quantum number in the Landau levels. We use the cross-section for relativistic electrons obtained by summing over the final spin state and averaging over the photon polarization (Harding & Daugherty 1991);

\begin{array}{r} σ_{a b s}^{n} (θ) = σ_{c y c}^{n} L_{n} \frac{e^{- Z} Z^{n - 1}}{(n - 1)!} [(1 + μ^{r 2}) + \frac{Z}{n} (1 - μ^{r 2})], \end{array}

(10)

where

\begin{array}{r} σ_{c y c}^{n} \equiv \frac{α π^{2} ℏ^{2} c^{2}}{E_{n}^{r}}, \end{array}

(11)

\begin{array}{r} L_{n} \equiv \frac{Γ_{n} / 2 π}{(ℏ ω^{r} - ℏ ω_{n}^{r})^{2} + Γ_{n}^{2} / 4}, \end{array}

(12)

\begin{array}{r} Z \equiv \frac{1}{2} {(\frac{ℏ ω^{r}}{m_{e} c^{2}})}^{2} \frac{(1 - μ^{r 2})}{b_{c}}, \end{array}

(13)

\begin{array}{r} μ \equiv \cos θ, \end{array}

(14)

and the quantized electron energy

\begin{array}{r} E_{n}^{r} = m_{e} c^{2} {[1 + (ℏ ω^{r} μ^{r} / m_{e} c^{2})^{2} + 2 n b_{c}]}^{1 / 2} . \end{array}

(15)

Here

Γ_{n} = (4 n α E_{B}^{2}) / (3 m_{e} c^{2})

is the radiative width for the nth harmonic, where E_B = 11.6 keV B₁₂ is the cyclotron energy and α is the fine structure constant. The superscript r denotes quantities measured in the frame of reference in which the electron is at rest before scattering.

The exact relativistic resonant energy for the nth harmonic is given by

\begin{array}{r} ℏ ω_{n}^{r} = \frac{2 n b_{c} m_{e} c^{2}}{1 + \sqrt{1 + 2 n b_{c} (1 - μ^{r 2})}} . \end{array}

(16)

We use the approximate form as the cross-section for cyclotron resonant scattering,

\begin{array}{r} σ_{s c}^{n} ≃ \frac{σ_{a b s}^{n}}{L_{n}} \frac{E_{n}^{2} Γ_{n} / 2 π m_{e}^{2} c^{4}}{(ℏ ω^{r} - ℏ ω_{n}^{r})^{2} + (E_{n} Γ_{n} / 2 m_{e} c^{2})^{2}} . \end{array}

(17)

This expression is a much more accurate approximation to resonant scattering derived by replacing the Lorentz profiles in the absorption cross-section (Harding & Daugherty 1991). The resonant energy for the nth harmonic in the lab frame is given by

\begin{array}{r} ℏ ω_{n} = ℏ ω_{n}^{r} γ (1 + β μ^{r}), \end{array}

(18)

where γ = (1 − β²)^−1/2 is the Lorentz factor and β is the bulk flow velocity in units of c.

We assume that the initial and final electron state is Landau ground state (n = 0) in each resonant scattering like Nishimura (2019). We include excitations to the n = 1, 2, and 3 Landau levels and photon spawning followed by decay of these levels in cyclotron resonant scattering. The differential transition rate for cyclotron emission from an excited Landau level n to the ground state, averaged over initial and summed over final spin states, is given by

\begin{array}{rcl} \frac{d Γ_{n \to 0}}{d μ} (μ) & = & \frac{α}{(n - 1)!} \frac{1}{(E - p_{z} μ)^{2}} \frac{2 n b_{c}}{E} e^{- Z} Z^{n - 1} \frac{1}{(1 + ξ)^{2}} \\ \times (1 + 2 n b_{c} - \frac{1}{ξ}), \end{array}

(19)

where

\begin{array}{r} ξ \equiv \frac{1 - 2 n b_{c} (1 - μ^{2})}{(E - p_{z} μ)^{2}}, \end{array}

(20)

Here, E is the electron’s energy and p_z is the electron’s momentum in the field direction (Araya & Harding 1999).

Since the scattering cross-section is computed in the electron rest frame, the photon frequency ω and angle μ must be transformed to the rest frame by

\begin{array}{r} ω^{r} = γ (1 - β μ) ω, \end{array}

(21)

\begin{array}{r} μ^{r} = \frac{μ - β}{(1 - β μ)}, \end{array}

(22)

where

γ = 1 / \sqrt{1 - β^{2}}

and β = v/c is the velocity in units of c. The resonant energy for the nth harmonic in the lab frame is given by

\begin{array}{r} E_{n} = \frac{E_{n}^{r}}{γ (1 - β μ)} . \end{array}

(23)

Our main goal in this paper is to examine changes of E_cyc, i.e., the peak energy of an absorption feature, with luminosity. We therefore use a non-relativistic scattering profile (Wasserman & Salpeter 1980) to reduce CPU time, since the use of a non-relativistic scattering profile would not affect the present main conclusions considerably. In the line core, the scattering profile ϕ_n(x_n, μ) for photons is given by

\begin{array}{r} ϕ_{n} (x_{n}, μ) \approx ϕ_{n}^{c} (x_{n}, μ) \propto \exp [- {(\frac{x_{n}}{μ})}^{2}], \end{array}

(24)

in which the thermal electron distribution dominates the profile, while in the line wings

\begin{array}{r} ϕ_{n} (x_{n}, μ) \approx ϕ_{n}^{w} (x_{n}, μ) \propto \frac{a_{n}}{x_{n}^{2}}, \end{array}

(25)

in which the tail of the Lorentzian distribution dominates. Here

a_{n} = Γ_{n} / 2 E_{d}^{n}

is the natural width of the cyclotron line divided by its Doppler width

E_{d}^{n} = n E_{B} \sqrt{2 k T_{e} / m_{e} c^{2}}

⁠, and

x_{n} \equiv (E - E_{n}) / E_{d}^{n}

is the dimensionless frequency shift in units of

E_{d}^{n}

⁠. The core–wing division is defined as the point where ϕ_c(x, μ) = ϕ_w(x, μ), i.e., where

\begin{array}{r} {[\frac{x_{1}}{μ}]}_{C} \equiv x_{C} \approx 2.67 - 0.19 \ln (\frac{10^{2} a_{1}}{| μ |}) . \end{array}

(26)

3.4 Polarization

Wang, Wasserman, and Salpeter (1988) argue that the polarization-averaged cross-sections are suitable for first-harmonic scattering in optically thick media when the vacuum contribution to the dielectric tensor dominates the plasma contribution, i.e.,

\begin{array}{r} \frac{w}{δ} = 3 \times 10^{- 6} (\frac{n_{e}}{10^{23} {cm}^{- 3}}) b_{c}^{- 4} ≪ 1, \end{array}

(27)

where w ≡ (ω_p/ω_cyc)² is the plasma frequency parameter, ω_p is the plasma frequency, n_e is the electron number density, and δ is the magnetic vacuum polarization parameter (Adler 1971).

In this work, the optical depth is assumed to be $τ = ρ r_{c} σ_{c y c}^{1} \sim 1000.0$ in the line-forming region that is optically thick to line photons but optically thin to continuum photons. The electron density is also adopted from ρ ∼ 10⁻⁵ to ∼10⁻⁴ g cm⁻³ in the line-forming region. Thus, n_e ∼ 10¹⁸–10¹⁹ cm⁻³ and $b_{c}^{- 4} ≲ 2.9 \times 10^{5}$ ⁠. The condition of equation (27) therefore generally holds in this work such that a vacuum polarization cross-section is employed.

3.5 Magnetic field and temperature distribution

We assume a dipole B field given by B(z) = B_s[R_NS/(R_NS + z)]³, where B_s the B-field strength at the surface of an NS. The equilibrium distribution of the electrons is Maxellian in the frame of reference moving with the accretion flow (cf. Isenberg et al. 1998). A resonant scattering is computed in the electron rest-frame such that we can apply the Compton equilibrium temperature (i.e., the electron temperature at equilibrium) approximated by kT_e(z) ∼ ℏω_cyc(z)/4, which is derived by Lamb, Wang, and Wasserman (1990) from a balance of the cooling and heating that arise from cyclotron resonant and non-resonant scattering, to an electron temperature. The temperature varies with height via a variation of the B-field strength.

3.6 Monte Carlo method

We basically employ the Monte Carlo method described by Araya and Harding (1999). However, in this work, magnetic field strength, bulk velocity, plasma density, and temperature change with height, and these variations are incorporated into the Monte Carlo method as follows. A photon with an initial energy ω and angle μ propagates from height z₀ to z through the plasma in the line-forming region over a mean free path which is given by

\begin{array}{r} λ (ω, μ, z) = \frac{1}{⟨ n_{e} (z) σ (ω, μ, z) ⟩}, \end{array}

(28)

where

⟨ n_{e} (z) σ (ω, μ, z) ⟩ = \int_{z_{0}}^{z} \int_{- \infty}^{+ \infty} f (p^{'}) γ (z^{'}) [1 - β (z^{'}) μ] σ_{n}^{r} (ω^{r}, μ^{r}, z^{'}) n_{e} (z^{'}) d p^{'} d z^{'}

⁠. Here z–z₀ represents the z-component of the distance that a photon travels before being scattered, and the cross-section

σ_{n} (ω, μ, z) = \int d Ω_{s}^{r} (d σ_{n}^{r} / d Ω_{s}^{r})

for a photon with a frequency of ω and an angle of μ at height z. We take the distribution of the electron momenta along the field, f(p), to be a one-dimensional non-relativistic Maxwellian f(p)dp = Nexp ( − u²)du, where u = p/(2m_ekT_e)^1/2 and

N = \sqrt{m_{e} / 2 π k T_{e}}

⁠. Thus, the optical depth τ that a photon travels before scattering is sampled by exp ( − τ/λ) and then scattered with an electron.

The z-component of the target electron’s momentum, p_z, scattered by a photon at height z is sampled by

\begin{array}{r} ζ = \frac{\int_{- \infty}^{p_{z}} F (p^{'}, ω, μ, z) d p^{'}}{\int_{- \infty}^{+ \infty} F (p^{'}, ω, μ, z) d p^{'}}, \end{array}

(29)

where ζ is a random number sampled uniformly from the range 0 to 1 and

F (p^{'}, ω, μ, z) = f (p^{'}) γ (z) [1 - β (z) μ] σ_{n}^{r} (ω^{r}, μ^{r}, z)

⁠. Here the lower and higher limit of the electron momentum are set to be ±413 keV, which correspond to ±0.9 c of electron velocity.

3.7 Gravitational light-bending

We can obtain the observed azimuthal angle θ of a photon with azimuthal angle θ₀ emitting from the wall of an accretion column at a height r₀ − R_NS using K(r₁, r₀, μ₀) by r₁ → ∞ as a result of gravitational light-bending, since the observer is at infinity (Riffert & Mészáros 1988; Poutanen 2020). Here, K(r₁, r₀, μ₀) is the angle between the radius vector pointing toward the emission point Q(r₀, μ₀) and the radius vector pointing to the observation point P(r₁) for a photon emitted with μ₀, as shown in figure 2. The quantities K(r₁, r₀, μ₀) and Q(r₀, μ₀) are given by

Schematic of the angle K(r1, r0, μ0) between the emission point Q(r0, μ0) and the observation point P(r1) for a photon emitted with an angle μ0. When θ0 ≥ π − θmax, the photon hits the surface of the NS.

Fig. 2.

Schematic of the angle K(r₁, r₀, μ₀) between the emission point Q(r₀, μ₀) and the observation point P(r₁) for a photon emitted with an angle μ₀. When θ₀ ≥ π − θ_max, the photon hits the surface of the NS.

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\begin{array}{rcl} K (r_{1}, r_{0}, μ_{0}) & = & b \int_{r_{0}}^{r_{1}} {[x^{2} - b^{2} (1 - \frac{1}{x})]}^{- 1 / 2} \frac{d x}{x} \\ + Q (r_{0}, μ_{0}), \end{array}

(30)

\begin{array}{r} Q (r_{0}, μ_{0}) = {\begin{cases} 0, & μ_{0} \geq 0, \\ 2 b \int_{r_{m}}^{r_{0}} {[x^{2} - b^{2} (1 - \frac{1}{x})]}^{- 1 / 2} \frac{d x}{x}, & μ_{0} < 0. \end{cases} \end{array}

(31)

Here b is the relativistic impact parameter given by

\begin{array}{r} b = \frac{r_{0}}{(1 - 1 / r_{0})^{1 / 2}} (1 - μ_{0}^{2})^{1 / 2}, \end{array}

(32)

and r_m is a solution of the cubic equation

\begin{array}{r} r_{m}^{3} - b^{2} (r_{m} - 1) = 0. \end{array}

(33)

Here, r_m is the distance of the minimum approach of the photon to the star. We obtain a positive solution for r_m, in which the photon can travel without hitting the surface of an NS, when

\begin{array}{r} μ_{0} > - {[1 - \frac{27}{4 r_{0}^{2}} (1 - \frac{1}{r_{0}})]}^{1 / 2} . \end{array}

(34)

A photon also hits the surface of a star when θ₀ ≥ π − θ_max, in which θ_max is given by

\begin{array}{r} \sin θ_{m a x} = \frac{s (R_{N S})}{s (r)} \sqrt{\frac{1 - 1 / s (r)}{1 - 1 / s (R_{N S})}} . \end{array}

(35)

Here s(r) = r/R_s, in which R_s is the Schwarzschild radius

2 GM / c^{2}

(Poutanen et al. 2013). Therefore, considering case of r = R_NS, a photon with 0 < θ₀ < π/2 is observed without hitting the surface of an NS, because θ_max = π/2. Thus, mixing of photons from two viewing angles will occur even when luminosity is low, L₃₇ ∼ 0.1, since the location of the emission region is higher than r = R_NS for all computations.

Here, the photon of a hard component which hits the surface of an NS is assumed to turn to a soft component via Comptonization in the atmosphere of the NS surface (White et al. 1988). For simplicity, we take two symmetrically located polar caps and we assume that all photons except those hitting the surface of an NS are observed.

4 Results

Figure 3 shows cyclotron lines computed for cases of bulk velocity β = 0 and 0.64 in a cylindrical geometry with a uniform magnetic field B₁₂ = 1.7 and a uniform column density N_e,21 ≡ N_e/(10²¹ electron cm⁻²) = 1.2 between the photon source and the surface of a cylinder that Isenberg, Lamb, and Wang (1998) adopted. These spectra are in agreement with Isenberg, Lamb, and Wang (1998)’s results for β = 0 in which strong emission wings tend to be formed at small μ. For β = 0.64, a strong angle-dependence of E_cyc is generated and an absorption feature is formed for almost all viewing angles, in which the lines become deeper and E_cyc becomes lower as the viewing angle μ becomes smaller, because photons scattered by electrons with high velocity tend to be scattered downward, as discussed in Schwarm et al. (2014).

Fig. 3.

Angle-dependent fluxes of photons for 0 < μ < 0.125, 0.375 < μ < 0.5, 0.875 < μ < 1 for cases of bulk velocity β = 0 and 0.64 in a cylindrical geometry.

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In a line-forming region with high bulk velocity, gravitational light-bending tends to produce the structure of double fundamental lines via a mixing of two lines from two angles with respect to the B field (Nishimura 2019). This is because the strong angle-dependence of E_cyc is generated by high bulk velocity via equation (23). One fundamental line at a lower energy (1st fundamental line, hereafter) comes from an angle parallel to the B field, while the other fundamental line at a higher energy (2nd fundamental line, hereafter) comes from an angle normal to the B field and parallel to the B field, as shown in figure 4. Thus, in most phase-resolved spectra, double fundamental lines are formed by gravitational light-bending in the line-forming region with high bulk velocity, as shown in figure 5.

$Schematic of the effect of gravitational light-bending for the direction of photon’s propagation. Photons emitted at angles θ1 and $\theta _1^{\prime }$ with respect to the B field from one pole propagate in the same direction as photons emitted at angles θ2 and $\theta _2^{\prime }$ from another pole as a result of gravitational light-bending, respectively.$

Fig. 4.

Schematic of the effect of gravitational light-bending for the direction of photon’s propagation. Photons emitted at angles θ₁ and $θ_{1}^{'}$ with respect to the B field from one pole propagate in the same direction as photons emitted at angles θ₂ and $θ_{2}^{'}$ from another pole as a result of gravitational light-bending, respectively.

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Fig. 5.

Angle-dependent fluxes of photons for two angles 0 < μ < 0.125 and 0.625 < μ < 0.75 without gravitational light-bending for the case of only a pencil-like beam (θ_p1 = 45°) in the line-forming region with high constant bulk velocity β = 0.73. These two spectra are mixed as a result of gravitational light-bending such that a spectrum around angle 0.375 < μ′ < 0.5 is formed. Assuming that the surface magnetic field B₀ = 7.3 × 10¹² G, using continuum (Obs.I) observed in Vela X-1 Fürst et al. (2014).

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On the other hand, in an angle-averaged spectrum, gravitational light-bending does not produce double fundamental lines when a continuum emission is isotropic (Nishimura 2019). This is because gravitational light-bending changes only the propagation direction of a photon: averaging angle-dependent spectra over an angle eliminates the effect of gravitational light-bending.

However, when a continuum emission is anisotropic, such as a pencil-like beam, double fundamental lines can be formed even in an angle-averaged spectrum via the superposition of the lines with distinct energies generated by high bulk velocity, as shown in figure 6, regardless of gravitational light-bending. This is because the 1st fundamental line originates in a pencil beam component such that the relatively deep line is formed when a pencil beam dominates over a fan beam, like in figure 5. On the other hand, when a fan beam dominates, the 1st fundamental line will disappear, as shown in figure 6. When a pencil beam is comparable to a fan beam, the shallow 1st fundamental line at a low energy, ∼25 keV, can exist. In this case, the 1st fundamental line at a low energy tends to be shallow and broad as a result of the superposition of a broad feature from an angle parallel to the B field and no feature from an angle normal to the B field. On the other hand, the 2nd fundamental line at a high energy, ∼55 keV, is constantly formed for any beam pattern, because it is formed as a result of the superposition of a broad feature from an angle parallel to the B field and a narrow feature from an angle normal to the B field. As a result, the 2nd fundamental line at a high energy, ∼55 keV, tends to appear readily.

Fig. 6.

Angle-integrated fluxes of photons for four beam patterns, in which only a pencil-like beam (θ_p1 = 45°), beam patterns composed of a pencil-like beam (θ_p1 = 45°) and a fan beam (θ_p2 = 90°) for B₁/B₂ = 2.0 and 1.0, and only a fan beam (θ_p2 = 90°) are computed under the same conditions as figure 5 but fixing z_max = r_c = 10⁵ cm.

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5 Discussions

5.1 GX 304−1

The larger the velocity of bulk motion of an infalling matter, the lower the peak energy of the fundamental cyclotron line (= first harmonic line). This is because the resonant energy in laboratory frame is given by equation (23) (Mushtukov et al. 2015). We demonstrate that the observed change in E_cyc with luminosity can be explained by considering a change of bulk velocity in the line-forming region as well as a change in geometry of the line-forming region.

In this work, only a pencil-like beam with θ_p1 = 45° is assumed to be formed at a low luminosity of L₃₇ = 0.2 in which the height of the accretion mound is h_top ∼ 110 m for polar cap diameter 2r_c = 770 m. On the other hand, the height of the accretion mound is a relatively high h_top ∼ 220 m for 2r_c = 856 m at L₃₇ ∼ 0.42. In addition, the observed pulse profile significantly changes. We therefore consider that the beam pattern is composed of a pencil-like beam (θ_p1 = 45°) in the upper part of the accretion mound (z > h_top z_fan where z_fan = 0.25) and a fan beam (θ_p2 = 90°) in the lower part (z < h_top z_fan) for L₃₇ ≥ 0.4, which is an emission pattern of the continuum from an accretion mound as in Burnard, Arons, and Klein (1991). Figure 7 shows computed spectra for L₃₇ = 0.2, 0.4, 0.8, and 1.0 in which bulk velocity β in the line-forming region is assumed to be constant, v_ff/c = 0.63, decreasing with decreasing altitude from 0.63 to 0.55, from 0.62 to 0.4, and from 0.62 to 0.1, respectively. The strength of the B field on the surface of an NS is assumed to be B₀ = 6.6 × 10¹² G, which corresponds to E_cyc = 77 keV.

Angle-integrated flux of photons at L37 = 0.2, 0.4, 0.8, and 1.0, using continua observed around each luminosity in GX 304−1.

Fig. 7.

Angle-integrated flux of photons at L₃₇ = 0.2, 0.4, 0.8, and 1.0, using continua observed around each luminosity in GX 304−1.

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In observations, the observed E_cyc increases from ∼50.6 keV to ∼54.4 keV with luminosity increasing from L₃₇ ∼ 0.19 to 0.42. This change of E_cyc can be explained by the idea that radiation pressure due to resonant scattering in the line-forming region decelerates infalling matter with increasing luminosity.

Secondly, the observed E_cyc decreases from ∼54.4 keV to ∼54.2 keV, or remains unchanged, although luminosity increases from L₃₇ ∼ 0.42 to 0.85. At L₃₇ ∼ 0.42, the height of the accretion mound h_top ∼ 220 m, while at L₃₇ ∼ 0.85 the height of an accretion mound h_top ∼ 445 m. The mean strength of the B field in the line-forming region therefore decreases as luminosity increases from L₃₇ ∼ 0.42 to 0.85 while the bulk velocity will decrease. As a result, the value of E_cyc in the angle-averaged spectrum at L₃₇ ∼ 0.85 could be somewhat lower than that at L₃₇ ∼ 0.42, although the luminosity is high. Consequently, in this regime, E_cyc can decrease or unchange with increasing luminosity.

Finally, the observed E_cyc starts increasing again from ∼54.2 keV to ∼59.3 keV as luminosity increases from L₃₇ ∼ 0.85 to 1.66 via significant deceleration of the bulk velocity from β = 0.62 to 0.1. Consequently, E_cyc mostly increases due to the decrease in the mean velocity of the bulk motion. These characteristics are consistent with the observations (Malacaria et al. 2015).

5.2 Vela X-1

In observation with NuSTAR (Fürst et al. 2014), E_cyc seems to decrease once and then increase with increasing luminosity in Vela X-1. On the other hand, the cyclotron line energy at the second harmonic (E_{cyc, 2nd}) in Vela X-1 mostly continues to increase with increasing luminosity like GX 304−1. In addition, the observed fundamental line is much shallower than that in GX 304−1.

In this work, at L₃₇ = 0.2, only a pencil-like beam with θ_p1 = 45° is assumed to be formed. On the other hand, for high luminosity L₃₇ ≥ 0.4 we assume that the beam pattern is composed of a pencil-like beam (θ_p1 = 45°) in the upper part of the accretion mound (z > h_top z_fan, where z_fan = 0.25) and a fan beam (θ_p2 = 90°) in the lower part (z < h_top z_fan), as in our simulations for GX 304−1. The strength of the B field on the surface of an NS is assumed to be B₀ = 7.3 × 10¹² G, which corresponds to E_cyc = 85 keV. We demonstrate that high bulk velocity in the line-forming region in two dimensions is able to produce cyclotron line features around E_cyc = 25 keV. Figure 8 shows computed spectra for L₃₇ = 0.2, 0.4, 0.8, and 1.0 in which bulk velocity β in the line-forming region is assumed to be constant free-fall velocity v_ff/c = 0.73, decreasing with decreasing altitude from 0.72 to 0.67, from 0.72 to 0.4, and from 0.72 to 0.1, respectively. The 1st fundamental line is somewhat deep at L₃₇ = 0.2, because the beam pattern is composed of only a pencil-like beam.

Angle-integrated flux of photons for luminosity L37 = 0.2, 0.4, 0.8, and 1.0. For luminosity L37 = 0.2 and 0.4, continuum (Obs.I) observed in Vela X-1 (Fürst et al. 2014) is used, while for luminosity L37 = 0.8 and 1.0 continuum (Obs.II) is used.

Fig. 8.

Angle-integrated flux of photons for luminosity L₃₇ = 0.2, 0.4, 0.8, and 1.0. For luminosity L₃₇ = 0.2 and 0.4, continuum (Obs.I) observed in Vela X-1 (Fürst et al. 2014) is used, while for luminosity L₃₇ = 0.8 and 1.0 continuum (Obs.II) is used.

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Our simulations show that E_cyc of the 2nd fundamental line clearly increases with increasing luminosity like GX 304−1 as shown in figure 8. On the other hand, E_cyc of the 1st fundamental line tends to decrease once from ∼23 keV to ∼20 keV and then increase to ∼25 keV with increasing luminosity. The trend of this variation in E_cyc with luminosity is consistent with the observations (Fürst et al. 2014). The observed fundamental line in Vela X-1 is therefore likely to be one of double fundamental lines. Thus, the second harmonic line observed in Vela X-1 would correspond to the fundamental line observed in GX 304−1.

The 1st fundamental line at a low energy ∼25 keV tends to be shallow and broad as a result of the superposition of a broad feature from an angle parallel to the B field and no feature from an angle normal to the B field. On the other hand, the 2nd fundamental line around a high energy ∼50 keV tends to be deep and narrow as a result of the superposition of a broad feature from an angle parallel to the B field and a narrow feature from an angle normal to the B field, as shown in figure 5.

The 1st fundamental line is primarily formed around the upper region of an accretion mound because it comes from a pencil-like beam component. The line-forming region for the 1st fundamental line is therefore near free-fall bulk velocity. The decrease in the energy of the 1st fundamental line around L₃₇ ∼ 0.4 can be caused by an ascent accretion mound. This is because a pencil-like beam is formed primarily in the upper region of an accretion mound, as shown in figure 1, such that E_cyc of the 1st fundamental line decreases as an accretion mound rises. On the other hand, the 2nd fundamental line, which comes primarily from a fan beam, is primarily formed around the lower region with a fan beam component. The bulk velocity in the line-forming region tends to change significantly via the deceleration due to radiation pressure. Thus, E_cyc of the 2nd fundamental line can clearly increase with decreasing bulk velocity. Consequently, in observations, E_cyc at the second harmonic line in Vela X-1 shows variation similar to that at the fundamental line in GX 304−1.

Next, let us consider angle-dependent spectra. Vela X-1 is estimated to have a higher mass of M = 1.8 M_⊙, which generates a large free-fall velocity v_ff ∼ 0.72c at a height of several hundred meters around the top of an accretion mound. Therefore, a larger variation in the ratio of the second harmonic line to the fundamental in cyclotron line energy, E_{cyc, 2nd}/E_cyc, in phase-resolved spectra can be generated, because the E_cyc can be influenced considerably by bulk velocity, i.e., v_ff (Nishimura 2019). The observed ratio E_{cyc, 2nd}/E_cyc can be larger than 2 at some pulse phases while it can be smaller than 2 at other pulse phases. The large ratio E_{cyc, 2nd}/E_cyc ∼ 2.3 can result from β ∼ 0.72 at viewing angle 0.625 < μ < 0.75, as shown in figure 9, because the angle-dependence of resonant energy becomes larger via equation (23) as β becomes larger. In addition, as E_{cyc, 2nd} becomes higher with decreasing μ, E_cyc tends to become lower. This characteristic is consistent with observations (Maitra & Paul 2013). Thus, double fundamental lines via gravitational light-bending are able to reproduce successfully the observed properties of two lines in phase-resolved spectra of Vela X-1.

Angle-dependent fluxes of photons for 0 < μ < 0.125, 0.125 < μ < 0.25, 0.375 < μ < 0.5, 0.5 < μ < 0.625, 0.625 < μ < 0.75, and 0.875 < μ < 1, at L37 = 0.8 in Vela X-1. The values of Ecyc, 2nd/Ecyc, 1st change significantly from ∼1.7 to ∼2.3 with viewing angle. The 1st fundamental line can be too shallow to be seen at some viewing angles. Each spectrum is offset with respect to 0 < μ < 0.125 for clarity.

Fig. 9.

Angle-dependent fluxes of photons for 0 < μ < 0.125, 0.125 < μ < 0.25, 0.375 < μ < 0.5, 0.5 < μ < 0.625, 0.625 < μ < 0.75, and 0.875 < μ < 1, at L₃₇ = 0.8 in Vela X-1. The values of E_{cyc, 2nd}/E_{cyc, 1st} change significantly from ∼1.7 to ∼2.3 with viewing angle. The 1st fundamental line can be too shallow to be seen at some viewing angles. Each spectrum is offset with respect to 0 < μ < 0.125 for clarity.

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Consequently, the second harmonic line observed in Vela X-1, which is expected to be one of double fundamental lines, is likely to correspond to the fundamental line observed in GX 304−1. As a fan beam component becomes strong, the shallow fundamental line at a low energy ∼25 keV becomes shallower. This is because the shallow fundamental line comes from a pencil beam component, i.e., a large μ. This is consistent with observations (Fürst et al. 2014; Wang 2014) in which the fundamental line becomes shallower as luminosity becomes higher. Moreover, the surface magnetic field is ∼1.07 × 10¹³ G, on taking into account the gravitational redshift z_g = 0.47 for M = 1.8 M_⊙ and R_NS = 10⁶ cm.

5.3 A0535+262

In a normal outburst in 2010 April (Müller et al. 2013), E_cyc at ∼45 keV did not change considerably with luminosity. Furthermore, E_cyc seems to decrease somewhat with increasing luminosity in contrast with GX 304−1 and then return to ∼45 keV. The observed E_cyc thus does not change like GX 304−1, although the observed E_cyc ∼ 45 keV, i.e., the strength of the magnetic field, is close to E_cyc ∼ 50 keV at L₃₇ ∼ 0.1 in GX 304−1 such that the same behavior is expected. If the observed second harmonic line is one of the fundamental lines like Vela X-1, L_crit is larger than GX 304−1. In fact, in a giant outburst of A0535+262 in 2011 (Sartore et al. 2015), E_{cyc, 2nd} increases with increasing luminosity above L₃₇ ∼ 0.8. Furthermore, the depth of the fundamental line is nearly constant during the normal outburst, while it significantly decreases around L₃₇ ≳ 3 during the giant outburst. This is probably because the spectrum formed by a fan beam has no feature at E_cyc ∼ 45 keV, such that a feature at E_cyc ∼ 45 keV tends to become shallow as a fan beam dominates over a pencil-like beam with increasing luminosity. Thus, a pencil-like beam is expected to dominate until a higher luminosity L₃₇ ∼ 0.8 because of a larger L_crit, such that the 1st fundamental line in A0535+262 remains deep compared to Vela X-1. Consequently, the observed second harmonic line is likely to be one of the double fundamental lines. In addition, E_cyc would decrease slightly around L₃₇ ∼ 0.4 due to an ascending accretion mound. This small change is also seen in observations.

Thus, in our computations, the strength of the magnetic field of A0535+262 is assumed to be B₀ = 1.3 × 10¹³ G, which corresponds to E_cyc = 155 keV. Figure 10 shows computed spectra for L₃₇ = 0.2, 0.4, 0.8, and 1.0. The bulk velocity β in the line-forming region are assumed to be constant free-fall velocity v_ff/c = 0.64 and 0.63 for L₃₇ = 0.2 and 0.4, respectively, and decreasing with decreasing altitude from 0.63 to 0.5 and from 0.63 to 0.3 for L₃₇ = 0.8 and 1.0, respectively.

Angle-integrated flux of photons for luminosities L37 = 0.2, 0.4, 0.8 and 1.0, using continua observed around each luminosity in A0535+262. For comparison, each spectrum is offset with respect to L37 ∼ 0.2.

Fig. 10.

Angle-integrated flux of photons for luminosities L₃₇ = 0.2, 0.4, 0.8 and 1.0, using continua observed around each luminosity in A0535+262. For comparison, each spectrum is offset with respect to L₃₇ ∼ 0.2.

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In these simulations for A0535+262, at L₃₇ = 0.2 and 0.4, only a pencil-like beam at θ_p1 = 45° is assumed to be formed, such that the 1st fundamental line remains somewhat deep. In L₃₇ ≳ 0.8, a beam pattern is composed of a pencil-like beam at θ_p1 = 45° and a fan beam at θ_p2 = 90°. The beam pattern is composed of a pencil-like beam (θ_p1 = 45°) in the upper part of the accretion mound (z > h_top z_fan where z_fan = 0.25) and a fan beam (θ_p2 = 90°) in the lower part (z < h_top z_fan).

Moreover, in observations, the width of the fundamental line can be more than two times that of the second harmonic line. This is contrast with the theoretical prediction from the Doppler width of the cyclotron line $E_{d}^{n} = n E_{c} \sqrt{2 k T_{e} / m_{e} c^{2}}$ ⁠, which suggests that the width of the fundamental line is half that of the second harmonic line. This observation therefore favors our present model, because the 1st fundamental line tends to become broader than the 2nd fundamental line in double fundamental lines.

In addition, our present model, in which E_cyc can change primarily via a change in the bulk velocity, suggests that the rate of change of E_cyc with luminosity is dependent of viewing angle, i.e., pulse phase, because it changes with β through equation (23); the effect of bulk motion becomes smaller as the viewing angle moves from μ = 1 to μ = 0. In fact, in phase-resolved spectra, the rate of change of E_cyc with luminosity was observed to vary with pulse phase, i.e., angle (Müller et al. 2013).

6 Summary

For L ≪ L_crit in which only a pencil-like beam is expected to be formed, in an angle-averaged spectrum, clear double fundamental lines tend to be formed. When a pencil beam predominates over a fan beam for L < L_crit, a shallow 1st and a deep 2nd fundamental line can be formed. As a fan beam becomes strong compared to a pencil beam, the 1st fundamental line would therefore become weaker and weaker. This is because the shallow 1st fundamental line comes from a pencil beam component. In some phase-resolved spectra, however, the 1st fundamental line can still appear.

In GX 304−1, the observed fundamental line would correspond to the 2nd fundamental line, such that E_cyc clearly increases with increasing luminosity as the bulk velocity decreases due to increase in radiation pressure. In Vela X-1, on the other hand, the observed fundamental line would correspond to the 1st fundamental line such that E_cyc would decrease once and then increase with increasing luminosity as an accretion mound rises. Thus, E_cyc at the observed second harmonic line increases with increasing luminosity like GX 304−1, as shown in figure 11. In A0535+262, E_cyc of the fundamental line does not clearly increase with increasing luminosity unlike that of the fundamental line in GX 304−1. Thus, the fundamental line observed in A0535+262 is also likely to correspond to the 1st fundamental lines like Vela X-1, such that E_{cyc, 2nd} is expected to increase with increasing luminosity below a critical luminosity higher than that in GX 304−1 and Vela X-1 because of a higher magnetic field strength. This idea of double fundamental lines could be applied to 4U 1538−522 and 4U 1907+09, which possess a relatively shallow fundamental line and do not indicate clearly Positive.

Fig. 11.

Trends of variation of observed E_cyc as a function of luminosity for GX 304−1, A0535+262 and Vela X-1. Error bars are ignored. The solid line denotes E_cyc of the fundamental line in GX 304−1 (Malacaria et al. 2015). Dashed and double-dot–dashed lines denote E_cyc of the second harmonic and the fundamental line in Vela X-1 (Fürst et al. 2014), respectively. Dotted and dot–dashed lines denote E_cyc of the fundamental line for normal outbursts (Ballhausen et al. 2017) and a giant outburst Sartore et al. (2015) in A0535+262, respectively.

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Acknowledgements

I thank the anonymous referee for their constructive comments. This work was supported by JSPS KAKENHI Grant Number JP19K03934.

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Article Contents

Variations in energy of cyclotron lines with double structures formed in a line-forming region with bulk motion in accreting X-ray pulsars

Abstract

1 Introduction

2 Line-forming region

3 Radiative transfer

3.1 Continuum spectra

3.2 Beam patterns

3.3 Cyclotron resonant scattering

3.4 Polarization

3.5 Magnetic field and temperature distribution

3.6 Monte Carlo method

3.7 Gravitational light-bending

4 Results

5 Discussions

5.1 GX 304−1

5.2 Vela X-1

5.3 A0535+262

6 Summary

Acknowledgements

References

Citations

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Article Contents

Variations in energy of cyclotron lines with double structures formed in a line-forming region with bulk motion in accreting X-ray pulsars

Abstract

1 Introduction

2 Line-forming region

3 Radiative transfer

3.1 Continuum spectra

3.2 Beam patterns

3.3 Cyclotron resonant scattering

3.4 Polarization

3.5 Magnetic field and temperature distribution

3.6 Monte Carlo method

3.7 Gravitational light-bending

4 Results

5 Discussions

5.1 GX 304−1

5.2 Vela X-1

5.3 A0535+262

6 Summary

Acknowledgements

References

Citations

Views

Altmetric

Email alerts

Astrophysics Data System

Citing articles via

Latest

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Most Cited

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