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Astron. Astrophys. 357, 164-168 (2000)

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2. Calculations and results

The main idea is to calculate the ejector time, [FORMULA], i.e. a time interval spent by an INS on the ejector stage, for different parameters of the field decay and using standard assumptions for the initial NS parameters, and to compare this time with the Hubble time, [FORMULA].

The ejector time, [FORMULA], monotonically increases with increasing velocity of NS v and density of the ISM n. For a constant NS magnetic field this relation takes the simple form:


Using a high mean ISM density [FORMULA] and a low space velocity of NSs (about the sound speed), [FORMULA], we arrive at the lower limit of [FORMULA]. Any other value of density and velocity should increase [FORMULA] (these quantities in fact are not independent since only low-velocity NSs remain for a long time inside the galactic disc where such high ISM density is observed). After the ejection stage has been over, the NS passes to the propeller stage and only after that can become an accreting X-ray source. The duration of the propeller stage [FORMULA] is poorly known (e.g. for a constant magnetic field [FORMULA] is always less than [FORMULA], see Lipunov & Popov (1995), for a decaying field these timescales can be comparable). Therefore if for some parameters of an INS the lower limit of [FORMULA] exceeds the Hubble time [FORMULA] yrs, it cannot come to the accretion stage and hence can not underlie the ROSAT X-ray source.

In addition, we assumed that NSs are born with sufficiently small rotational periods [FORMULA] and have the same parameters of the magnetic field decay. We shall consider different initial surface magnetic field values. The field decay is assumed to have an exponential shape:


where [FORMULA] is the initial magnetic moment ([FORMULA], here [FORMULA] is the polar magnetic field, [FORMULA] is the NS radius), [FORMULA] is the characteristic time scale of the decay, and [FORMULA] is the bottom value of the magnetic momentum which is reached at the time


and does not change after that.

In Fig. 1 we show as an illustration the evolutionary tracks of NSs on [FORMULA] diagram for [FORMULA] and [FORMULA]. Tracks start at [FORMULA] when [FORMULA] and [FORMULA] and end at [FORMULA] yrs (for [FORMULA], [FORMULA] yrs and for a constant magnetic field) or at the moment when [FORMULA] (for [FORMULA] yrs and [FORMULA] yrs). The line with diamonds shows [FORMULA].

[FIGURE] Fig. 1. Tracks on P-B diagram. Tracks are plotted for bottom polar magnetic field [FORMULA], initial polar field [FORMULA], NS velocity [FORMULA], ISM density [FORMULA] and different [FORMULA]. The last point of tracks with different [FORMULA] corresponds to the following NS ages: [FORMULA] yrs for [FORMULA] and [FORMULA] yrs; [FORMULA] yrs for [FORMULA] yrs; [FORMULA] yrs for [FORMULA] yrs. The initial period is assumed to be [FORMULA] s. The line with diamonds shows the ejector period, [FORMULA].

Since the accretion rate from the ISM is generally very small (even for our parameters), less than [FORMULA], no influence of the accretion on the field decay was taken into account (see Urpin et al. 1996).

The ejector stage ends when the critical ejector period [FORMULA] is reached:


where [FORMULA]. [FORMULA] is the NS space velocity, [FORMULA] and n are the sound velocity and density of the ISM, respectively. In the estimates below we shall assume [FORMULA] and [FORMULA].

The initial NS spin periods should be taken much smaller than [FORMULA]. To calculate the duration of the ejection stage here we assume [FORMULA] s. The actual value of [FORMULA], if much less than [FORMULA], has no effect on our results, i.e. [FORMULA] is determined only by [FORMULA] and the history of the field decay. We used the magnetodipole formula to compute this time, which in fact is appropriate for quite different specific ways of NS rotational energy loss (see Beskin et al. 1993 for a review):


where µ can be a function of time.

After a simple calculation we arrive at the following expression for [FORMULA]:


where the coefficient T (which would be simply [FORMULA] for [FORMULA]) is determined by the formula:


Here [FORMULA] can be formally determined according to the Bondi equation for the mass accretion rate even if the NS is not at the accretion stage:


The results of calculations of [FORMULA] for several values of [FORMULA] and [FORMULA] are shown in Fig. 2. The right end points of all curves are limited by the values [FORMULA]. These points correspond to the evolution of an INS with constant magnetic field (see Eq. (2)) and for them [FORMULA]. If [FORMULA] is small enough, the NS field has no time to reach the bottom value. In this case [FORMULA] is determined by the 1st branch by Eq. (6) and does not depend on [FORMULA]. In the Fig. 2 this situation corresponds to the left horizontal parts of the curves. At


the situation changes so that [FORMULA] starts depending on [FORMULA]. In this region two counter-acting factors operate. On the one hand, the NS braking becomes slower with decreasing µ (see Eq. (5)). On the other hand, the end period of the ejection [FORMULA] becomes shorter (4). Since [FORMULA] at the left hand side horizontal part and [FORMULA], the right hand side of the curve must have a maximum. The first factor plays the main role to the right of the maximum, the magnetic field there rapidly falls down to [FORMULA] at [FORMULA] and most often NS evolves with the minimum field [FORMULA] (this time increases with decreasing [FORMULA]). To the left of the maxium but before the horizontal part the NS magnetic field reaches [FORMULA] with the spin period close to [FORMULA] (the smaller [FORMULA], the closer) and soon after [FORMULA], the NS leaves the ejection stage.

[FIGURE] Fig. 2. Ejector time [FORMULA] (in billion years) vs. the bottom value of the magnetic momentum. The curves are shown for two values of the initial magnetic momentum: [FORMULA] (upper curves) and [FORMULA].

As seen from Fig. 2, for some combination of parameters, [FORMULA] is longer than the Hubble time. It means that such NSs never evolve further than the ejector stage.

We argue that if the soft ROSAT X-ray sources are accreting isolated neutron stars, the combinations of [FORMULA] and [FORMULA] for which no accreting isolated NS appear, can be excluded for their progenitors. The regions of excluded parameters are plotted in Figs. 3 and 4.

[FIGURE] Fig. 3. The characteristic time scale of the magnetic field decay, [FORMULA], vs. bottom magnetic moment, [FORMULA]. In the hatched region [FORMULA] is greater than [FORMULA]. The dashed line corresponds to [FORMULA], where [FORMULA] years. The solid line corresponds to [FORMULA], where [FORMULA]. Both the lines and hatched region are plotted for [FORMULA]. The dash-dotted line is the same as the dashed one, but for [FORMULA]. The dotted line shows the border of the "forbidden" region for [FORMULA].

[FIGURE] Fig. 4. The characteristic time scale of the magnetic field decay, [FORMULA], vs. bottom magnetic momentum, [FORMULA]. In the hatched region [FORMULA] is greater than [FORMULA]. The dashed line corresponds to [FORMULA], where [FORMULA] yrs. The solid line corresponds to [FORMULA], where [FORMULA]. Both lines and region are plotted for [FORMULA].

The hatched regions correspond to parameters for which [FORMULA] is longer than [FORMULA], so an INS with such parameters never reaches to the accretor stage and hence cannot appear as an accreting X-ray source. In view of observations of accreting old isolated NSs by ROSAT satellite, this region can be called "forbidden" for selected parameters of the exponential field decay with a given [FORMULA].

In the "forbidden" region in Fig. 3, which is plotted for [FORMULA], all NSs reach the bottom field in a Hubble time or faster, and the evolution at late stages proceeds with the minimal field. The left hand side of the forbidden region is determined approximately by the condition


A small difference between the line corresponding to the above condition and the left hand side of the "forbidden" region appears because a NS can slightly change its spin period even with the minimum magnetic moment [FORMULA]. However due to a small value of the field the angular momentum, losses are also small.

The right hand side of the region is roughly determined by the value of [FORMULA], with which an INS can reach the ejector stage for any [FORMULA], i.e. this [FORMULA] corresponds to the minimum value of [FORMULA] with which a NS reaches the ejector stage without field decay.

NSs to the right from the "forbidden" region leave the ejector stage, because the bottom magnetic momentum there is relatively high so that the spin-down is fast enough throughout the ejector stage. To the left of the "forbidden" region the situation is different. The NS spin-down is very small and they leave the ejector stage not because of the spin-down, but due to a decrease in [FORMULA], which depends upon the magnetic moment.

In Fig. 3 the "forbidden" region is also shown for [FORMULA] (dotted line). The dashed line in Fig. 3 shows that for all interesting parameters an INS with [FORMULA] reaches [FORMULA] in less than [FORMULA] yrs. The dash-dotted line shows the same for [FORMULA]. The solid line corresponds to [FORMULA], where [FORMULA]. The physical sense of this line can be described in the following way. This line divides two regions: in the upper left region [FORMULA] are relatively long and [FORMULA] relatively low, so the NS cannot reach the bottom field during the ejector stage; in the lower right region [FORMULA] are short and [FORMULA] relatively high, so the NS reaches [FORMULA] at the stage of ejection.

Fig. 4 is plotted for [FORMULA]. For long [FORMULA] ([FORMULA] yrs) the NS cannot leave the ejector stage for any [FORMULA]. This is the reason why in the upper part of the figure a horizontal "forbidden" region appears.

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000