1 Introduction

After the discovery of accretion-powered X-ray binaries (Giacconi et al. 1971; Oda et al. 1971), the interaction between a magnetized star and a surrounding accretion disc has been extensively investigated by numerous researchers (Pringle & Rees 1972; Lamb et al. 1973; Ghosh et al. 1977; Ghosh & Lamb 1979b; Wang 1987, 1995; Li & Wang 1999; Erkut & Alpar 2004; Matt & Pudritz 2005; Dai & Li 2006; Zhang & Li 2010; Lai 2014; Bozzo et al. 2018). The basic picture of the interaction between the disc and the magnetic field of the star has been already known and can be explained in the following way. The accretion of matter onto the magnetic star can be divided into three parts: the magnetosphere of the star, the transition zone and the unperturbed or outer part of the disc. The transition zone is located where between the magnetosphere of the star and the unperturbed accretion disc. The inner radius of the transition zone is obtained from the balance of the magnetic and the plasma stresses, and its outer radius is located where the magnetic field of the star is screened to zero due to the currents induced by accreting material. In the transition zone, the magnetic field lines of the star thread the disc and are distorted due to the radial and azimuthal motions of the disc flow. The transition zone is also composed of two regions, the inner and the outer transition zones. The narrow inner transition zone is where the angular velocity of the disc due to the magnetic stress of the star decreases from the Keplerian value to corotational value, while the broad outer transition zone is where the magnetic field of the star has negligible influences on the plasma and so the disc rotates Keplerian in this region (Ghosh & Lamb 1979a,b).

The observations and the numerical simulations indicate that the mass outflows driven from accretion discs exist in many astrophysical systems, including active galactic nuclei (AGN), young stellar objects (YSOs) and many types of binary stars (see Proga 2003; Xie & Yuan 2008; Jiao & Wu 2011 and references therein). Three mechanisms can drive outflows in accretion discs. First, the magnetically driven outflows from discs can be seen in many astrophysical environments. The magnetically disc-outflows can occur when a sufficiently strong, large-scale, ordered magnetic field threading the disc. The ordered magnetic field can extract/channel mass, energy and angular momentum from the disc (Blandford & Payne 1982; Cannizzo & Pudritz 1988; Pelletier & Pudritz 1992; Romanova et al. 1997; Krasnopolsky et al. 1999; Kato et al. 2002; Lai 2003; Anderson et al. 2006; Vollmer et al. 2018). The radiation force and the thermal expansion are the other mechanisms can also produce the powerful outflows in accretion discs which are in systems such as X-ray binaries and AGNs (Begelman et al. 1983; Murray & Chiang 1996; Woods et al. 1996; Pereyra et al. 1997, 2000; Proga et al. 1998; Feldmeier et al. 1999; Proga 2003).

As mentioned above, the observational evidence shows that the mass outflow driven from disc exists in X-ray binaries and so reduces the accretion rate on the neutron star (Loeb et al. 2001). For example, Loeb et al. (2001) compared the luminosities of binary accreting neutron stars and white dwarfs under similar conditions. In spite of the gravitational potential difference between neutron stars and white dwarfs, Loeb et al. (2001) found that the accretion rate onto neutron stars were smaller by three orders of magnitude than that on the surface of white dwarfs. This property can be due to the outflows in accreting neutron stars. Moreover, in some cases of low-mass X-ray binaries, the outflow driven from the neutron star or accretion disc is important (Chakrabarty 1998; Díaz & Boirin 2016; Waters & Proga 2018; Tomaru et al. 2019). and can change the rotational velocities of the star and the disc (Zhang & Li 2010). The angular momentum of the mass outflow driven from the disc is taken away from or is deposited into the inflow so that we want to investigate the influences of disc-outflow on the rotational velocity of the accretion disc. Here, we apply a parametric model for disc-outflow. Although, a parametric model outflow provides a simple description for mass outflow driven from disc, it can be applied from many types of disc-outflow models, such as the radiation-driven disc outflows and centrifugally driven MHD disc winds (Knigge 1999).

The paper is organized as follows. In Section 2, the basic equations of a model for disc-outflow around neutron stars will be defined. In Section 3, the numerical method for solving equations, which govern the behavior of the accreting gas will be used, and the results of the present model are also brought in this section. We close with a summary and discussion of the results in Section 4.

2 Basic equations

We consider a steady \((\partial /\partial t \equiv 0)\) and axisymmetric \((\partial /\partial \varphi \equiv 0)\) accretion disc with outflow in the potential of a neutron star of mass \(M_*\). We adopt a cylindrical coordinate system \((r, \varphi , z)\) centered on the neutron star and aligned with the star rotation axis. The magnetic field of the star has a dipolar configuration with the moment of \(\mu _*\). The magnetic lines of the star thread the disc plasma and are distorted by the radial and azimuthal motions of the accreting disc. Following Lovelace et al. (1994), the vertical component of magnetic field has an even dependence on z, i.e., \(B_r (r,z)=-B_r (r,-z)\), and the radial and azimuthal components of magnetic field are the odd functions of z, i.e., \(B_{r, \varphi } (r,z)=-B_{r, \varphi } (r,-z)\).

The induction equation of magnetic field is given by

$$\begin{aligned} \frac{\partial \mathbf{B}}{\partial t}=\nabla \times \left[ (\mathbf{v}\times \mathbf{B}) -\frac{4\pi \eta }{c}{} \mathbf{J}\right] , \end{aligned}$$
(1)

where \(\mathbf{B}\) is the magnetic field; \(\mathbf{v}\) is the velocity field; \(\mathbf{J} [=c(\nabla \times \mathbf{B} )/4\pi ]\) is the current density with being c as the speed of light; \(\eta\) is the resistivity diffusion. Under the above assumptions and the vertical hydrostatic equilibrium, i.e., \(v_z=0\), the radial and azimuthal components of the induction, the equation can be written as

$$\begin{aligned} r v_r B_z +\eta r \frac{\partial B_r}{\partial z}-\eta r \frac{\partial B_z}{\partial r}=0 \end{aligned}$$
(2)

and

$$\begin{aligned}&r \frac{\partial }{\partial r}\left[ v_r B_\varphi - r \Omega B_r -\frac{\eta }{r} \frac{\partial (r B_\varphi )}{\partial r} \right] \nonumber \\&\quad - \frac{\partial }{\partial z}\left[ r^2 \Omega B_z + \eta r \frac{\partial B_\varphi }{\partial z} \right] =0, \end{aligned}$$
(3)

where \(v_r\) is the radial velocity and \(\Omega\) is the angular velocity. Using the thin-disc limit and also assuming \(\vert B_r\vert \lesssim \vert B_z\vert\) and \(\vert B_\varphi \vert \lesssim \vert B_z\vert\) (Erkut & Alpar 2004), the vertical integration of the radial and azimuthal components of induction Equations (2) and (3) can be derived as

$$\begin{aligned} B_r^+= \beta _r B_z \end{aligned}$$
(4)

and

$$\begin{aligned} r B_z \frac{\partial \Omega }{\partial z}+\frac{\partial }{\partial z}\left( \eta \frac{\partial B_\varphi }{\partial z} \right) =0, \end{aligned}$$
(5)

where \(\beta _R [=-v_r H/\eta \le 1]\) is a constant of order unity, H is the half-thickness of the disc and \(B_r^+\) is the radial magnetic filed at the surface of the disc (for details see Lovelace et al. 1994, 1995). Assuming

$$\begin{aligned} \left| \frac{\partial \Omega }{\partial z}\right| = \epsilon | \Omega -\Omega _* | / H, \end{aligned}$$

where \(\epsilon <1\) is the shear reduction factor and \(\Omega _*\) is the rotational rate of neutron star (Wang 1987), the vertical integration of Equation (5) implies that

$$\begin{aligned} B_\varphi ^+ =\epsilon r(\Omega _* -\Omega )\frac{H}{\eta }B_z. \end{aligned}$$
(6)

We expect the viscosity and the diffusivity are due to turbulent in the disc so that the diffusivity in analogy to \(\alpha\)-prescription of the standard model of the accretion disc (Shakura & Sunyaev 1973) can be obtained as

$$\begin{aligned} \nu = P_m \eta = \alpha \frac{c_s^2}{\Omega _K}, \end{aligned}$$
(7)

where \(\nu\) is the kinematic coefficient of viscosity, \(P_m\) is the Prandtl number assumed to be constant in the present model, \(\alpha\) is a constant less than unity (Shakura & Sunyaev 1973), \(c_s\) is the sound speed and \(\Omega _K[=\sqrt{G M_*/r^3}]\) is Keplerian angular velocity. Following Erkut & Alpar (2004), the sound speed can be estimated as \(c_s^2=\xi \Omega _K^2 H r\), so that the diffusivity can be written as

$$\begin{aligned} \eta = \eta _0 \Omega _K H r, \end{aligned}$$
(8)

where \(\eta _0 [{ =}\,\alpha \xi /P_m]\) is a constant. Using the Equations (6) and (8), the toroidal magnetic field can be driven as

$$\begin{aligned} B_\varphi ^+ =\gamma \left( \frac{\Omega _* -\Omega }{\Omega _K}\right) B_z, \end{aligned}$$
(9)

where \(\gamma [= \epsilon P_m /\alpha \xi ]\) is assumed to be a constant for simplicity (Livio & Pringle 1992). The Equations (4) and (9) indicate that the radial and the toroidal components of magnetic field depend on the vertical component of the magnetic field threading the disc plasma. The vertical magnetic field in the plane of the disc can be expressed by a dipole formula as

$$\begin{aligned} B_z = -s \frac{\mu _*}{r^3}, \end{aligned}$$
(10)

where s is the screening coefficient and can be assumed to be a constant less than unity (Ghosh & Lamb 1979a; Wang 1987). In the following, we will apply the obtained magnetic field (Equations 4, 9 and 10) in the momentum equation.

Under the assumptions, the mass continuity equation can be written as

$$\begin{aligned} \frac{1}{r}\frac{\partial }{\partial r}(r \rho v_r) + \frac{\partial }{\partial z}(\rho v_z)=0, \end{aligned}$$
(11)

where \(\rho\) is the mass density, \(v_r\) and \(v_z\) respectively, are the radial and vertical components of velocity. As mentioned before, the purpose of this paper is to investigate the influences of the outflow driven from disc on the rotation of the accretion disc and the neutron star. The mass outflow driven from the disc can transfer the mass and the angular momentum from the flow. Taking into account the effects of mass outflow driven from the disc, the vertical integration of the continuity equation from 0 to H becomes

$$\begin{aligned} \frac{d \dot{M}}{d r} = 4 \pi r \rho _w v_{z,w}, \end{aligned}$$
(12)

where H is the half-thickness of the disc, \(\dot{M}[=-2\pi r \Sigma v_r]\) is the mass accretion rate with being \(\Sigma [=2 \rho H]\) as the surface density of the disc, \(\rho _w\) and \(v_{z,w}\) respectively, are the mass density and the vertical velocity of the outflow when it is launched from the disc (Xie & Yuan 2008).

The angular momentum equation is

$$\begin{aligned}&\rho \left[ \frac{ v_r}{r} \frac{\partial }{\partial r} (r v_\phi ) + v_z \frac{\partial v_\phi }{\partial z} \right] = \frac{1}{r^2}\frac{\partial }{\partial r}\left( r^3\rho \nu \frac{\partial \Omega }{\partial r}\right) \nonumber \\&\quad +\frac{B_r}{4\pi r}\frac{\partial }{\partial r}(rB_\varphi ) +\frac{B_{z}}{4\pi }\frac{\partial B_\varphi }{\partial z}, \end{aligned}$$
(13)

where \(v_\varphi\) is the rotational velocity. Assuming \(\vert B_r\vert \lesssim \vert B_z\vert\) and \(\vert B_\varphi \vert \lesssim \vert B_z\vert\) and under thin-disc limit (Erkut & Alpar 2004), the second term in the right-hand side of Equation (13) is negligible in comparison to the last term,

$$\begin{aligned} \frac{\vert B_r \partial _r (r B_\varphi )\vert }{\vert r B_z \partial _z (B_\varphi ) \vert }\sim \left| \frac{B_r}{B_z}\right| \left( \frac{H}{r}\right) \ll 1. \end{aligned}$$
(14)

Thus, the angular momentum equation using the above estimation can be written as

$$\begin{aligned}&\rho \left[ \frac{ v_r}{r} \frac{\partial }{\partial r} (r v_\phi ) + v_z \frac{\partial v_\phi }{\partial z} \right] = \frac{1}{r^2}\frac{\partial }{\partial r}\left[ r^3\rho \nu \frac{\partial \Omega }{\partial r}\right] +\frac{ B_{z}}{4\pi }\frac{\partial B_\varphi }{\partial z}. \end{aligned}$$
(15)

Taking into account the influences of mass outflow driven from the disc, the vertical integration of Equation (15) from the equatorial to the above surface of the disc can be written as

$$\begin{aligned}&\frac{\rho v_r}{r} \frac{d}{dr}(r v_\varphi )+ \rho _w v_{z,w} \left[ \frac{v_{\varphi ,w}-v_\varphi }{H}\right] \nonumber \\&\quad =\frac{1}{r^2}\frac{d}{d r}\left[ r^3\rho \nu \frac{d \Omega }{d r}\right] +\frac{1}{4\pi }\frac{B_z B_\varphi ^+}{H}, \end{aligned}$$
(16)

where \(v_{\varphi ,w}\) is the azimuthal velocity of the outflow when it is launched from the surface of the disc (Xie & Yuan 2008). Using the Equation (12) and some simplifications, the the angular momentum Equation (16) becomes

$$\begin{aligned} \frac{d}{d r} \left[ \dot{M} r^2 \Omega +2\pi r^3\Sigma \nu \frac{d \Omega }{d r} \right] = r v_{\varphi ,w} \frac{d\dot{M}}{d r} -r^2 B_z B_\varphi ^+. \end{aligned}$$
(17)

Because of the outflow emanating from the disc surface, the mass accretion \(\dot{M}\) is not constant and varies by radius. In the case of without outflow, the solutions of Equation (17) reduce to the solutions of Erkut & Alpar (2004).

Here, we apply a power-law dependence for mass accretion rate (Blandford & Begelman 1999; Knigge 1999),

$$\begin{aligned} \dot{M}(r)=\dot{M}_0 \left( \frac{r}{r_\mathrm{in}}\right) ^\lambda , \end{aligned}$$
(18)

where \(r_\mathrm{in}\) is the inner radius of the disc and \(\dot{M}_0\) is the mass accretion rate at \(r_\mathrm{in}\). In the absence of outflow, \(\lambda =0\), and when the mass outflow is launched from the surface of the disc \(\lambda >0\). Here, we also introduce a parameter to evaluate the azimuthal velocity of the outflow in terms of the inflow,

$$\begin{aligned} v_{\varphi ,w} = \chi v_\varphi , \end{aligned}$$
(19)

where \(\chi\) is a constant of the order of unity (Xie & Yuan 2008).

Using Equations (4), (9), (10), (18) and (19), and the nondimensional transformations for r, \(\Omega\) and \(\nu \Sigma\), i.e., \(x=r/r_\mathrm{in}\), \(\omega (x)=\Omega /\Omega _K(r_\mathrm{in})\) and \(\Gamma (x)=3\pi \nu \Sigma /\dot{M}_0\) as a dimensionless function for the dynamical viscosity, Equation (17) can be written as

$$\begin{aligned}&\frac{d}{dx} \left[ x^{\lambda +2} \omega (x)+\frac{2}{3}\Gamma (x)x^{3+\lambda }\frac{d \omega (x)}{dx} \right] \nonumber \\&\quad = \lambda \chi x^{\lambda +1} \omega (x)-\beta \frac{\omega _*-\omega (x)}{x^{5/2}}, \end{aligned}$$
(20)

where \(\omega _*=\Omega _*/\Omega _K(r_\mathrm{in})=(r_\mathrm{co}/r_\mathrm{in})^{-3/2}\) with \(r_\mathrm{co}[=(G M_*/\Omega _*^2)^{1/3}]\) as the corotation radius and \(\beta = \gamma s^2 \mu _*^2 /\sqrt{r_\mathrm{in}^7 \dot{M}_0^2 G M_*}\).

3 Numerical solutions

The rotational velocity of accretion disc with mass outflow surrounding a magnetic star can be described through the solutions of Equation (20). In the absence of outflow which \(\lambda = 0\), the Equation (20) reduces to the angular momentum equation in the paper of Erkut & Alpar (2004), which denotes the discs without outflow surrounding the magnetic stars. In the other case of no outflow and weakly magnetized system, i.e., \(\lambda =0\) and \(\beta \simeq 0\), the solution of Equation (20) reduces to the standard model of accretion discs (Shakura & Sunyaev 1973), which the disc rotates Keplerian in all radii except in a narrow boundary layer located at the inner edge of the disc, where the angular velocity of accretion disc reduces from \(\Omega _K(R_*)\) to \(\Omega _*\). In the following, we solve Equation (20), taking into account both the effects of mass outflow driven from the disc and the magnetic field of the star, i.e., \(\lambda \ne 0\) and \(\beta \ne 0\).

From Equation (20), the angular velocity of the disc strongly depends on the dimensionless dynamical viscosity \(\Gamma (x)\) and its first derivative \(\Gamma '(x)\). The integration of Equation (20) for Keplerian angular velocity results

$$\begin{aligned} \Gamma (x)=&\,\frac{1-2(\chi -1)\lambda }{2\lambda +1} -\frac{2}{3}\beta \omega _* x^{-(\lambda +2)}\nonumber \\&+\frac{\beta }{3}x^{-(\lambda +7/2)} -j x^{-(\lambda +1/2)}, \end{aligned}$$
(21)

where \(j\equiv \dot{J}/\dot{M}_0 r_{\mathrm{in}}^2\Omega _K({r_{\mathrm{in}}})\) is the net angular momentum flux into the neutron star, with being \(\dot{J}\) as the inflow rate of angular momentum onto the neutron star through the inner edge of the disc. Thus, here, the parameter j is the torque applied by the disc on the neutron star, i.e., the star spins up when \(j > 0\), while it spins down for \(j < 0\).

Following Erkut & Alpar (2004), here to avoid the uncertainties which there are in the calculations of the torque in the inner disc, we calculate the net torque applied by the disc on the star using a closed surface through the corotation radius (Rappaport et al. 2004; Bhattacharyya & Chakrabarty 2017)

$$\begin{aligned} N_* = N_\mathrm{mat} + N_\mathrm{vis} + N_\mathrm{mag} \end{aligned}$$
(22)

where \(N_\mathrm{mat}\) is the flux of the angular momentum carried by the material stress,

$$\begin{aligned} N_\mathrm{mat} = \dot{M} r_\mathrm{co}^2\Omega _K(r_\mathrm{co}). \end{aligned}$$
(23)

The angular momentum flux due to the viscous stress is given by

$$\begin{aligned} N_\mathrm{vis}&= \frac{\dot{M}}{3\pi } \Gamma (x_\mathrm{co}) r_\mathrm{co}^2 \left( \frac{d \Omega _K}{d r}\right) _\mathrm{co} 2 \pi r_\mathrm{co} \nonumber \\&= -\Gamma (x_\mathrm{co}) \dot{M} r_\mathrm{co}^2\Omega _K(r_\mathrm{co}). \end{aligned}$$
(24)

The interaction of the stellar magnetic field with the disc plasma also yields a torque as

$$\begin{aligned} N_\mathrm{mag}&= -\!\int _{r_\mathrm{co}}^{\infty }\! r^2 B_z B_\varphi ^+ dr\nonumber \\&= -\frac{1}{3} \beta x_\mathrm{co}^{-7/2} \dot{M} r_\mathrm{co}^2 \Omega _K(r_\mathrm{co}). \end{aligned}$$
(25)

The contribution of the torques (Equations 23–25) in Equation (22) gives \(N_* =j \dot{M} r_\mathrm{co}^2\Omega _K(r_\mathrm{co})\). This explains the interpretation of j as the dimensionless torque on the star in the unit of the angular momentum flux carried by matter through the inner edge of the disc (Erkut & Alpar 2004).

Using the values of a typical X-ray binary pulsar and torque expression as \(N_*=I \dot{\Omega }_*=j\dot{M}_0\sqrt{G M_* r_\mathrm{in}}\) with being \(I=0.4 M_* R_*^2\) as the moment inertia, the parameters \(\beta\) and j can be written as

$$\begin{aligned} \beta \simeq 12.04 \gamma \dot{M}_{17}^{-1} \mu _{*30}^2 \omega _s^{-7/3} P_*^{-7/3} \left( \frac{M_*}{1.4 M_\odot }\right) ^{-5/3} \end{aligned}$$
(26)

and

$$\begin{aligned} j\simeq -0.4 \omega _*^{-1/3} R_{*6}^2 P_*^{-7/3} \dot{P}_{*12} \dot{M}_{17}^{-1} \left( \frac{M_*}{1.4 M_\odot }\right) ^{1/3}, \end{aligned}$$
(27)

where \(\mu _{*30}\) is the magnetic dipole moment in unit of \(10^{30}\) G cm\(^3\), \(\dot{M}_{17}\) is the inner mass accretion rate in unit of \(10^{17} \text {g s}^{-1}\), \(R_{*6}\) is the neutron star radius in unit of \(10^6\) cm and \(\dot{P}_{*12}\) is the time rate of change of the pulsation period of star in unit of \(10^{-12}\) s s\(^{-1}\).

To solve the second-order differential Equation (20) using Equation (21), we require appropriate boundary conditions. As mentioned before, the disc rotates Keplerian in the outer boundary, i.e., \(\omega (x_\mathrm{out})=\omega _K(x_\mathrm{out})\) with \(x_\mathrm{out}\) as the outer radius of the disc. For \(x<x_\mathrm{out}\), the angular velocity of the disc departs significantly from the Keplerian due to the magnetic torque, as in the inner edge of disc, \(x_\mathrm{in}[=1]\), it rotates with \(\omega _*[=\omega (x_\mathrm{in})]\). Using these boundary conditions and the typical values of X-ray binary pulsar, the example solutions of Equations (20) and (21) for the angular velocity \(\omega (x)\) and the dynamical viscosity \(\Gamma (x)\) can be seen in Figures 1 and 2.

Figure 1
figure 1

The angular velocity \(\omega (x)\) and the dynamical viscosity \(\Gamma (x)\) as the functions of radius. In each panel the solid, dashed and dotted curves correspond to \(\omega (x)\), \(\Gamma (x)\) and the Keplerian angular velocity \(\omega (x)=x^{-3/2}\), respectively. The left panel corresponds to a disc without outflow, while the right panel corresponds to a disc with outflow that the parameters are set to \(\lambda =0.5\) and \(\chi =2.0\). The period of the neutron star is \(P_*=1s\) and the input parameters are set as \(s=1.0\), \(\beta =3.0\) and \(\omega _*=0.4\).

Full size image

In Figure 1, we consider the angular velocity and the dynamical viscosity of an X-ray binary pulsar in two cases of with and without outflow driven from the disc. The system parameters are set to \(P_*=1\text {s}\), \(\beta = 3.0\) and \(\omega _*=0.4\). In both cases, the angular velocity of the disc is Keplerian for the regions far from the star. While in the regions close to the star due to the domination of the magnetic stress on the shear stress, the angular velocity of the disc reduces from Keplerian to the angular velocity of the star. From the left and the right panels of Figure 1 which respectively denote to the disc with and without outflow, the inner transition zone changes and becomes larger in the presence of the outflow with the parameters of \(\lambda =0.5\) and \(\chi =2.0\). This property can be explained in the following way. The values of \(\lambda =0.5\) and \(\chi =2.0\) denote to a strong outflow which takes away the angular momentum from the inflow. In such situations, the angular velocity of the disc decreases due to disc-outflow so that the discrepancy between the angular velocities of the star and the disc increases, and so the width of the inner transition zone becomes larger. We have also calculated the dimensionless torque applied by the disc on the star (j) and the time rate of change of the pulsation period of the star (\(\dot{P}_*\)) for both cases of with and without outflows. In the case of without outflow, the left-hand panel of Figure 1, we obtained \(j=1.04\), which corresponds to \(\dot{P}_*=-1.92\times 10^{-12}\,\text {s}\), and implies that the star spins up. However, in the presence of outflow, the right-hand panel of Figure 1, the solutions show that \(j=-0.18\) and so \(\dot{P}_*=0.33\times 10^{-12}\,\text {s}\), which imply the star spins down. This property represents that the spin rate of the star strongly depends on the outflow which drives from its surrounding disc.

To study the influences of the parameter \(\beta\) on the angular velocity of the discs with outflow, we apply two cases of \(\beta =4.0\) (left-hand panel) and \(\beta =10.0\) (right-hand panel) in Figure 2. Here, the increase of the parameter \(\beta\) corresponds to the larger magnetic dipole moment of the star so that the magnetic field of the star can dominate on the accreting plasma for the larger radii. Thus, the inner transition zone increases by adding the parameter \(\beta\) which the angular velocity profiles in the left-hand and the right-hand panels of Figure 2 confirm it. In Figure 2, we also consider the effects of the outflow parameters \(\lambda\) and \(\chi\) on the angular velocity of the disc. We apply \(\lambda =0\), 0.3 and 0.5 which respectively correspond to no outflow, the moderate and strong outflows. From Equations (16) and (19), the outflow term in the angular momentum equation is proportional to \((\chi -1)\) which implies that for \(\chi < 1\) (or \(\chi > 1\)) the angular momentum is deposited into (or taken away from) the inflow by the outflow. Thus, in the cases with outflow, we apply the values of 0.5 and 1.5 for \(\chi\), which respectively denote the angular momentum of the outflow is deposited into and is taken away from the inflow. In Figure 2, the Keplerian angular velocity profile has been shown by the dotted line and the angular velocity of the disc without outflow by the solid line. Although the inner transition zone exists in both cases of with or without outflow but the width of the inner transition zone strongly depends on the values of the parameters \(\lambda\) and \(\chi\). The angular velocity profiles imply that the \(\lambda\) parameter, which denotes to stronger outflow, increases the influences of the parameter \(\chi\). As seen in Figure 2, the angular velocity increases for \(\chi = 0.5\) due to the deposit of the angular momentum of the outflow into the inflow, while it decreases for \(\chi = 1.5\) because the angular momentum is taken away from the inflow by the outflow.

Figure 2
figure 2

The angular velocity \(\omega (x)\) as a function of radius for the several values of the parameters \(\lambda\) and \(\chi\). The period of the neutron star is \(P_*=1\text{s}\) and the input parameters are set as \(s=1.0\), \(\beta =5.0\) and \(\omega _*=0.5\).

Full size image
Figure 3
figure 3

The inner transition zone (\(\delta r/r_0\)) and the spin rate (\(\dot{P}_*\)) as the functions of the parameter \(\beta\) for the several values of outflow parameters, i.e., \((\lambda , \chi )=(0.0,0.0)\), (0.2, 0.5), (0.2, 1.5), (0.3, 0.5) and (0.3, 1.5) which correspond to solid, dot-dashed, long-dashed, and short-dashed lines.

Full size image
Figure 4
figure 4

The inner transition zone (\(\delta r/r_0\)) and the spin rate (\(\dot{P}_*\)) as the functions of the parameter \(\omega _*\) for the several values of outflow parameters, i.e., \((\lambda , \chi )=(0.0,0.0)\), (0.2, 0.5), (0.2, 1.5), (0.3, 0.5) and (0.3, 1.5) which correspond to solid, dot-dashed, long-dashed, and short-dashed lines.

Full size image

In the left-hand panel of Figure 3, we have shown the spin rate of the star (\(\dot{P}_*\)) as a function of the parameter \(\beta\) for the several values of the outflow parameters \(\lambda\) and \(\chi\). The \(\dot{P}_*\) profiles imply that the spin rate of the star decreases by adding the parameter \(\beta\). This property can be explained in the following way. The stellar magnetic field becomes stronger for the larger values of \(\beta\), so that the stronger coupling occurs between the stellar magnetic field lines and the rotating plasma. Consequently, the larger torque is exerted on the star by the disc. In the left-hand panel of Figure 3, we have also considered the influences of the outflow parameters on the spin rate of the star. The solutions show that the spin rate of the star decreases in the presence of the mass outflow from the disc and for all values of \(\chi\) parameter. But, the spin rate of the star decreases more in the case of \(\chi > 1\), i.e., when the angular momentum of outflow is taken away from the inflow. As discussed before, the width of the transition zone increases by adding the parameter \(\beta\). Because a larger value of the parameter \(\beta\) denotes to stronger magnetic field of the star so that the magnetic stress dominates to shear stress in the large radii in contrary to the lower values of the parameter \(\beta\). This property can be seen in the transition zone profiles in the right-hand panel of Figure 3. In the cases in which the outflow launches from the disc, the width of the transition zone strongly depends on the amount of the angular momentum of the outflow which is deposited into or is taken away from the inflow. The solutions show that the width of the transition zone increases for the cases in which the angular momentum of outflow is deposited into the inflow, (i.e., \(\chi >1\)), while it decreases when the angular momentum of the outflow is taken away from the inflow (i.e., \(\chi <1\)). Moreover, the influences of the parameter \(\chi\) increase for the stronger outflows, i.e., for the larger values of the parameter \(\lambda\).

The dependence of the spin period rate of the star (\(\dot{P}_*\)) and the transition zone of the disc (\(\delta r/r_0\)) on the magnetosphere velocity (\(\omega _*\)) can be seen in Figure 4. From the left-hand panel of Figure 4, we find that the spin period rate of the star (\(\dot{P}_*\)) increases by adding the parameter \(\omega _*\). In other words, the spin period rate of the star increases for the faster rotating star. Moreover, the influences of the outflow parameter imply that the spin period rate of the star increases in the presence of the disc-outflow. The right-panel profiles of Figure 4 represent that the transition zone decreases by adding the parameter \(\omega _*\). Because the corotation radius decreases by adding the parameter \(\omega _*\) and so the inner transition zone becomes narrow. We have also investigated the influences of the outflow parameters \(\lambda\) and \(\chi\) on the width of the transition zone. The \(\delta r/r_0\) profiles in Figure 4 represent that the inner transition zone becomes narrow (or wide) for \(\chi <1\) (or \(\chi > 1\)).

4 Discussion and summary

The existence of the mass outflow driven from the disc in accreting neutron stars have been confirmed by the observational evidence (e.g., see Díaz & Boirin 2016 and references therein). Thus, in this paper, we especially focused on investigating the influences of disc-outflow in such systems. We adopted the solutions presented by Erkut & Alpar (2004), Xie & Yuan (2008). We assumed a parametric model for disc-outflow which can be applied for many types of disc-outflow models, such as the radiation-driven disc outflows and centrifugally driven MHD disc winds (Knigge 1999; Xie & Yuan 2008). In this paper, the magnetic field of neutron star has a symmetry axis aligned with the rotation axis of accreting matter. The magnetic field lines of the star threads through the accreting plasma. For the regions close to the star, the magnetic field of the neutron star dominates the flow and enforces it to corotate with the star. For the regions far from the star, the magnetic field of the star has little effect on the accreting matter.

In the presence of disc-outflow, the solutions showed that the disc-outflow plays an important role in accreting neutron stars systems, especially on the change of angular velocities of the neutron star and the accretion disc. Here, we applied two cases for the angular momentum of matter emanated from the inflow. The angular momentum of disc-outflow can be taken away from or can be deposited into the inflow. The solution represents that the angular velocity of the inner region of the disc changes in the presence of the outflows. If the angular momentum of the outflow is deposited into (or is taken away from) the inflow, the angular velocity the disc increases (or decreases) in contrary to the discs without outflows. The solutions imply that the spin period rate of star increases in the presence of the mass outflow is driven from the disc, and also increases for the larger values of the magnetosphere velocity and the magnetic field of the neutron star. The width of the inner transition zone decreases (or increases) in contrary to the discs without outflows, if the angular momentum of the outflow is deposited into (or is taken away from) the inflow. The width of the inner transition zone decreases for the stars with faster magnetosphere velocity, while it increases for the stars with the larger magnetic field.

In the present model, we used a one-and-a-half-dimensional description of the accretion flow and applied a simplified parametric model for mass-loss rate. However, a two and three-dimensional approach can be useful to know under which conditions the outflow occurs in an accreting neutron star (Jiao & Wu 2011; Jiao et al. 2015; Waters & Proga 2018; Gholipour 2019; Tomaru et al. 2019 and references therein). Moreover, in this paper, we supposed that the inner radius of the disc lies within the corotation radius, but there are situations that the inner edge penetrates to the corotation radius (Rappaport et al. 2004; Bhattacharyya & Chakrabarty 2017 and references therein). Thus, such studies for the present model can be interesting subjects for future research.