How young the accretion-powered pulsars could be?

A question about the age of accretion-powered X-ray pulsars has recently been reopened by a discovery of the X-ray pulsar SXP 1062 in the SMC. This High Mass X-ray Binary (HMXB) contains a neutron star rotating with the period of 1062 s and is associated with a supernova remnant of the age ∼ 104 yr. An attempt to explain the origin of this young long-period X-ray pulsar within the traditional scenario of three basic states (ejector, propeller and accretor) encounters difficulties. Even if this pulsar were born as a magnetar the spin-down time during the propeller stage would exceed 104 yr. Here we explore a more circuitous way of the pulsar spin evolution in HMXBs, in which the propeller stage in the evolutionary track is avoided. We find this way to be possible if the stellar wind of the massive companion to the neutron star is magnetized. The geometry of plasma flow captured by the neutron star in this case differs from spherically symmetrical and the magnetospheric radius of the neutron star is smaller than that evaluated in the convention accretion scenarios. We show that the age of an accretion-powered pulsar in this case can be as small as ∼ 104 years without the need of invoking initial magnetic field in excess of 1013 G.


Introduction
The age of neutron stars (NSs) identified with accretion-powered pulsars in High Mass X-ray Binaries (HMXBs) is expected to exceed 10 5 yr [13]. A newly formed NS needs this time to spin-down from its initially short period up to a critical period, P cr , at which the accretion of material onto its surface starts. Observations, however, suggest that the accretion-powered NS in HMXBs can be significantly younger. In particular, the 1062 s X-ray pulsar SXP 1062 has recently been identified with a Supernova Remnant of the age ∼ 10 4 yr [2,3,12]. This discrepancy cannot be resolved within the framework of traditional scenario without invoking additional assumptions about properties of the NS itself [1,11], or its environment [7]. In this paper, we show that under certain conditions evolution of a NS can follow a simplified scenario, skipping the propeller stage. The time required for a NS to come into accretor stage this way is significantly smaller than that expected in traditional scenarios.

Spin-down phase
A newly formed NS is presumed to rotate rapidly and be strongly magnetized. Initially it appears to be a powerful ejector of relativistic wind which prevents the surrounding material from penetrating to within the Bondi radius of the NS and, correspondingly, from reaching the stellar surface. The NS, therefore, starts its evolution with a non-accreting spin-down phase at which its spin period monotonically increases. As the spin period of the NS reaches a critical value, P cr , the accretion of material onto its surface starts and the star switches on as an accretion-powered pulsar. The critical period, P cr , is usually defined by equating the magnetospheric radius, r m , with the corotation radius, r cor = (G M NS /ω 2 s ) 1/3 , where M NS is the mass and ω s is the angular velocity of the NS. The initial spin-down phase in the general case can be divided into ejector and propeller states (see e.g. [6] and references therein).

Ejector state
The rotational rate of NS in the ejector state decreases by the conventional spin-powered pulsar energy-loss mechanism expressed by equation for the magneto-dipole radiation. The spindown power in this case is converted predominantly into the energy of relativistic wind ejected from the magnetosphere of the NS. The star remains in the ejector state as long as the pressure of the relativistic wind at its the Bondi radius exceeds the ram pressure of the surrounding matter. The spin-down time of NS in a course of the ejector state can be evaluated as (see e.g. [7] and references therein) where f m is a dimensionless parameter of order unity which is according for plasma contribution into the processes of energy release in the magnetosphere of NS, I 45 is the moment of inertia of the NS in units 10 45 g cm 2 ,Ṁ 16 is the mass transfer rate between the system components in units 10 16 g s −1 , v 7 is the velocity of NS in the frame of surrounding material in units 10 7 cm s −1 and μ 30 is the dipole magnetic moment of NS in units 10 30 G cm 3 . This indicates that the spindown time of a strongly magnetized (μ 10 30 G cm 3 ) NS, which moves through a relatively fast stellar wind of its massive companion can be significantly shorter than a million years.

Propeller state
The ejector spin-down ceases as the pressure of relativistic wind at the Bondi radius decreases up to the ram pressure of surrounding material. The surrounding gas under these conditions penetrates to within the Bondi radius and moves towards the NS forming the accretion flow. The interaction between the flow and the magnetic field of the NS leads to formation of the magnetosphere. The radius of the magnetosphere, r m , depends on the parameters of the NS, the mass accretion rate and on the geometry and parameters of the accretion flow itself. If r m < r cor the star switches on as an accretion-powered pulsar. Otherwise, it remains to spin-down in the propeller state up to a moment when the corotation radius reaches the radius of the magnetosphere.
The NS in the propeller state is spinning-down due to the interaction between its magnetosphere and surrounding matter. The mechanism of this interaction is rather complicated. However, the spin-down power of NS in this state can be limited to L ns ≤ L max pr , where ω k (r m ) = GM ns /r 3 m 1/2 is the Keplerian angular velocity at the magnetospheric radius of NS and K sd = Iω s is the spin-down torque exerted to NS from matter surrounding its magnetospheric boundary (note, that the spin-down power of a star with the moment of inertia I which rotates with the angular velocity ω s and brakes at the rateω s is L sd = Iω sωs ).
The spin-down timescale of the NS in the propeller state can be evaluated as τ pr ≥ E rot (ω ej )/L max pr , where E rot = I ω 2 ej /2 is the rotational energy of the NS and ω ej is the angular velocity of the NS at the moment when it switches into the propeller state. Hence, τ pr ≥ 10 6 yr × I 45Ṁ

The direct ejector to accretor state transition
For a NS to switch its state directly from ejector to accretor the spin period at which the matter starts to penetrate to within the Bondi radius should be equal the critical period P cr . In the conventional quasi-spherical or Keplerian disk accretion scenarios in which the magnetospheric radius of the NS is close to the Alfvén radius r A = μ 4 /2GM nsṀ 2 1/7 , this situation can unlikely be realized (see fig. 1). It could occur only if the mass accretion rate onto the NS were in excess of 10 24 g s −1 (see eq. (10) in [6]), which significantly exceeds the typical mass-loss rate of O/B-type stars.
The direct transition of a NS from the ejector to accretor state could be also expected if accretion onto a NS were realized according to a scenario in which the magnetospheric radius of the NS is smaller than the Alfvén radius. Such a scenario has recently been proposed for wind-fed HMXBs in which the stellar wind of massive component is magnetized.
As recently shown in [9], the structure of the accretion flow onto a NS would significantly deviate from the traditional quasi-spherical and Keplerian disk if the magnetic field in the stellar wind of optical companion at the Bondi radius is B ≥ B min , where Here ξ 0.2 = ξ/0.2 is a dimensionless parameter accounting for density and velocity gradient in the material which the NS captures at its Bondi radius and P 100 is the orbital period of the binary system in units 100 days. If this condition is satisfied the NS would accrete matter from a non-keplerian magnetic disk which is referred to as Magnetic self-Levitating Disk (see e.g. [7,8,9] and references therein). Numerical study of such accretion flow has been made in [5] (see also references therein). The magnetospheric radius of the NS in this case is [9] r ma = c m For α B ∼ 0.01 (which is a typical case of the Earth magnetosphere, see [8] and references therein) the mass accretion rate at which the direct ejector to accretor state transition is expected lies in the rangeṀ ∼ 10 15 ÷ 10 16 g s −1 , which is typical for HMXBs.

Conclusions
Our basic conclusion is that the direct transition of a NS from the ejector to accretor state can be realized if the NS accretes matter from the ML-disk. In this case the spin-down time of the NS to a moment when its switches as an accretion-powered pulsar is comparable to the spin-down time in the ejector state (for a discussion see also [7,8,9]). It, therefore, appears that the age of accretion-powered pulsars in HMXBs can be as young as τ ej provided the magnetic field in their plasma environment exceeds B min expressed by Eq. (4).