\documentclass{aa}
%\documentclass[referee]{aa}
\usepackage{psfig}
\begin{document}
\thesaurus{(02.01.2; 02.13.1; 08.14.1; 08.13.1; 13.25.5)}
\title {ROSAT X-ray sources and 
exponential field decay in isolated neutron stars}

\author{S. B. Popov  \and
  M. E. Prokhorov
}
\offprints{S. B. Popov: polar@xray.sai.msu.su}

\institute{
Sternberg Astronomical Institute,
 Universitetskii pr.13, 119899, Moscow, Russia\\
e-mail: polar@xray.sai.msu.su; mike@sai.msu.su
}

\date{}

\authorrunning{S. Popov \& M. Prokhorov}
\titlerunning{Field decay in isolated neutron stars}

\maketitle

\begin{abstract}
The influence of 
exponential magnetic field decay on the spin evolution of 
isolated neutron stars is studied. 
%The ROSAT observations of several X-ray sources, which can be
%accreting old isolated neutron stars, are used to 
%constrain the exponential decay parameters.
ROSAT observations have revealed several 
X-ray sources which may be
accreting old 
isolated neutron stars. Assuming that this interpretation is
correct, the observations can be used to constrain parameters of the
exponential field decay.

We show that    
the range of minimum  value of magnetic moment, $\mu_b$,
and the characteristic  decay time, $t_d$, 
$\sim 10^{29.5}\ge \mu_b \ge 10^{28} \, {\rm G}\, {\rm cm}^3$ ,
$\sim 10^8\ge  t_d \ge 10^7\, {\rm yrs}$ are excluded  
assuming the standard 
initial magnetic momentum, $\mu_0=10^{30} \, {\rm G}\, {\rm cm}^3$. 
For these parameters,
neutron stars would never reach the stage of accretion from the interstellar
medium even for a low space velocity of the stars 
and a density of the ambient plasma. 
The range of excluded parameters increases for lower values of $\mu_0$.
\end{abstract}

\keywords{neutron stars -- magnetic fields -- stars: magnetic field --
X-rays: stars -- accretion}

\section{Introduction}

Many astrophysical manifestations of neutron stars (NSs) are determined by
their periods and magnetic fields.
Four main evolutionary stages of isolated NSs 
can be singled out (see e.g. Lipunov 1992 for
more detail): the {\it ejector}, when the star is observed as an active 
radio pulsar or a dead
pulsar, which  spins down according to magneto--dipole formula; 
the {\it propeller}, when
the gravitationally captured matter is stopped near the NS magnetosphere and
cannot get through the centrifugal
barrier; the {\it accretor}, when the matter can reach the surface 
and the NS appears as an
X-ray source; and the {\it georotator}, when the gravitational attraction 
becomes insignificant in comparison with
the magnetic pressure 
and the NS magnetosphere interaction with the interstellar medium (ISM) is
similar to the Earth magnetosphere interacting with the solar wind.

Magnetic field decay in NSs is a matter of controversy. Many models of the 
magnetric field decay have
been proposed starting from the first simple models (Gunn \& Ostriker 1970)
up to the recent calculations (Sang \& Chanmugam 1990, Urpin \& Muslimov
1992,
see also Ding et al., 1993; Jahan Miri  \& 
Bhattacharya, 1994 ). The strongest direct observational 
evidence seems to come from non-observing  decay effects in radio
pulsars (Lyne et al. 1998) 
for the exponential (or nearly exponential) decay
characteristic time scales $t_d$ shorter than  $\sim 10^7 \, {\rm yrs}$. 
Additional constraints for longer time
scales can be obtained from comparing the properties of 
NS calculated by the population synthesis method with those observed in  
radio pulsars (see e.g. Verbunt 1994,
Bhattacharya et al. 1992 and Hartman et al. 1996).

The magnetic field decay was used 
by Konenkov \& Popov (1997) and Wang (1997) to explain
properties of the source RX J0720-3125, which is considered to be a
candidate for old isolated accreting NSs. If this source 
really represents an old accreting NS and, assuming that 
it was born as  a normal radio pulsar
(with a small period of $\ll 1$ s  and a standard magnetic field of 
$\sim 10^{12}$ G), 
its properties can be explained only by field decay.

Bhattacharya et al. (1992) performed 
population synthesis calculations to study the 
field decay in single radio pulsars and came to some important conclusions 
about NS properties on the time scales $\ge 100 $ Myr.
Recently the effect of the field decay in isolated NSs 
was studied by Colpi et al. (1998) and Livio et al. (1998).

Here we try to put some
limits on the parameters of the exponential field decay assuming that 
some accreting X-ray sources observed by ROSAT are indeed old isolated NSs  
(Walter, Wolk \& Ne\"uhauser 1996;
Haberl et al. 1996, 1998; Ne\"uhauser \& Tr\"umper 1999; Schwope et al.
1999). 
There are two main possibilities for the explanation of the nature of these
sources: accretion and cooling. 
We do not consider
the possibility that all of them are normal cooling NSs or
 highly magnetized NSs,
``magnetars'' (see Ne\"uhauser \& Tr\"umper (1999) and 
we gave a brief discussion on this subject in Popov et al. (2000)).
Cooling NSs have short lifetimes
($\sim 10^6$ yrs) in comparison with the age of a Galaxy ($\sim 10^{10}$ yrs)
and thus might be relatively rare objects.
Our calculations (Popov et al. 2000) show, that only for lifetimes of about
$10^7$ yrs, cooling NSs  and accreting
NSs can be observed in the solar vicinity with comparable probability.
Proper motion measurements are necessary to estimate spatial velocities
of these candidates. High velocities ($>40$-50 km/s) can exclude the
accretion interpretation. However  at the present time  the question of
interpretation is open, and one cannot exclude any variant,
but we discuss here only the  accretion interpretation.

On average, NSs should have high velocities due to an additional kick 
obtained during the supernova explosion (Lyne and Lorimer 1994, Lorimer et al.
1997). 
The ISM accretion rate for high velocity objects
should be rather low. However, recent population synthesis calculations
(Popov et al. 2000) indicate that several old accreting NSs can be observed in
the solar vicinity even for the 
space velocity distribution similar to one that has been derived from radio pulsar
observations. 
And since there is the theoretical possibility of accreting isolated NSs, 
we wish to discuss how they can be used (possibly in the future)
to put some limits on the models of field decay. 

\section{Calculations and results}

 The main idea 
is to calculate the ejector time, $t_E$, 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, $t_H$. 

The ejector time, $t_E$,  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:

\begin{equation}
t_E(\mu=const)\sim 10^9 \mu_{30}^{-1}n^{-1/2}\, v_{10} \,{\rm yrs}.
\end{equation}
Using a high mean
ISM density $n\sim 1 {\rm cm}^{-3}$ 
and a low space velocity of NSs (about the sound speed),
$v\sim 10 \, {\rm km s}^{-1}$, we arrive at the lower limit of $t_E$.
Any other value of density and velocity should increase $t_E$
(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 $t_P$ is poorly known 
(e.g. for a constant magnetic field $t_P$ is always less than $t_E$, 
see Lipunov and Popov 1995, 
for a decaying field these timescales can be comparable).
Therefore if for some parameters of an INS  
the lower limit of $t_E$ exceeds the Hubble time $t_H\simeq
10^{10}$ yrs, it can not come to the accretion stage and hence can not
underlie the ROSAT X-ray source.   

\begin{figure}
%\vbox{\psfig{figure=fig1.eps,width=9.0cm,angle=-90}}
\vbox{\psfig{figure=9016.f1,width=9.0cm,angle=-90}}
\caption[]{Tracks on P-B diagram. Tracks are plotted
for bottom polar magnetic field
$8\cdot 10^{10}\, {\rm G}$, initial polar field $2\cdot 10^{12}\, {\rm G}$,
NS velocity $10 \,
{\rm km s}^{-1}$, ISM density $1\, {\rm cm}^{-3}$ and different $t_d$.
The last point of tracks with different $t_d$ corresponds to 
the following NS ages: $10^{10}$ yrs for $t_d=10^7$ and $t_d=10^8$ yrs;
$1.5 \times 10^9$ yrs for $t_d=10^9$ yrs; 
$\sim 2 \cdot 10^9$ yrs for $t_d=10^{10}$ yrs.
The initial period is assumed to be $p_0=0.020$ s. The line with diamonds
shows the ejector period, $p_E$.}
\end{figure}

\begin{figure}
%\vbox{\psfig{figure=fig2.eps,width=9.0cm,angle=-90}}
\vbox{\psfig{figure=9016.f2,width=9.0cm,angle=-90}}
\caption{ Ejector time $t_E$ (in billion years) vs. the bottom value of the 
magnetic momentum.
The curves are shown for two values of the initial magnetic momentum:
$10^{30} {\rm G} \, {\rm cm}^3$ (upper curves)
and $10^{31} {\rm G} \, {\rm cm}^3$. }
\end{figure} 

In addition, we assumed that NSs are born with sufficiently small
rotational periods $p_0$ 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:

\begin{equation}
\mu=\mu_0\cdot e^{-t/t_d}, \, for \, \mu > \mu_b
\label{eq:mu(t)}
\end{equation} 
where $\mu_0$ is the initial magnetic moment 
($\mu=\frac12 B_p R_{NS}^3$, here $B_p$ is the polar magnetic field,
$R_{NS}$ is the NS radius), $t_d$ is the characteristic time
scale of the decay, and $\mu_b$ is the bottom value of the 
magnetic momentum which is reached at the time 

\begin{equation}
t_{cr}=t_d\cdot \ln\left( \frac{\mu_0}{\mu_b} \right)
\end{equation}
and does not change after that. 

In Fig. 1 we show as an illustration the evolutionary tracks of NSs on
$P-B$ diagram for $v=10 \, {\rm km} \, {\rm s}^{-1}$ and
$n=1\, {\rm cm}^{-3}$. 
Tracks start at $t=0$ when $p=20\, {\rm ms}$ and $\mu=10^{30} {\rm
G} \, {\rm cm}^3$ and end at 
$t=t_H=10^{10}$
yrs (for $t_d=10^{9} {\rm yrs}$, $t_d=10^{10}$ yrs and for a constant magnetic 
field) 
or at the moment when $p=p_E$ (for $t_d=10^7$ yrs and $t_d=10^8$ yrs). 
The line with diamonds shows $p=p_E (B)$.

Since the accretion rate from the ISM is generally very 
small (even for our parameters),
less than $\sim 10^{12} \, {\rm g} \, {\rm s}^{-1}$, 
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 $p_E$ is reached:

\begin{equation}
p_E=11.5 \, \mu_{30}^{1/2} n^{-1/4} v_{10}^{1/2} \, {\rm s},
\label{eq:p_E}
\end{equation}
where  $v_{10}=\sqrt{v_p^2+v_s^2}/10 \, {\rm km} \, {\rm s}^{-1}$.
$v_p$ is the NS space velocity, $v_s$ and $n$ are the sound velocity and
density of the ISM, respectively.
In the  estimates  below we shall assume $v=10~{\rm km\,s}^{-1}$ and
$n=1~{\rm cm}^{-3}$.

The initial NS spin periods should be taken much smaller than $p_E$. To
calculate the duration of the ejection stage here we assume
$p_0=0$ s. The actual value of $p_0$, if much less than $p_E$, has no effect 
on our results, i.e. 
$t_E$ is determined only by $p_E$ 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):

\begin{equation}
\frac{dp}{dt}=\frac{2}{3}\frac{4\pi ^2 \mu^2}{pIc^3},
\label{eq:dp/dt}
\end{equation}
where $\mu$ can be a function of time.

After a simple calculation we arrive at the following expression for $t_E$:
\begin{equation}
t_E=\left\{
 \begin{array}{cl}
 \ & \displaystyle -t_d\cdot \ln \left[ \displaystyle\frac{T}{t_d}\left(\sqrt{
\displaystyle 1+\frac{t_d^2}{T^2}}-1
\right) \right], \, t_E< t_{cr}\\
 \ & \displaystyle t_{cr}+\displaystyle T\frac{\mu_0}{\mu_b}-
\displaystyle t_d\frac{1}{2}\left(\displaystyle \frac{\mu_0}{\mu_b}\right)^2
\left(\displaystyle 1-e^{-2t_{cr}/t_d}\right),\\
\multicolumn{2}{r}{t_E> t_{cr}} \\
 \end{array}
 \right.
\label{eq:t_E}
\end{equation}
where the coefficient $T$ (which would be simply $t_E$ for
$\mu=\mu_0=const$)  
is determined by the formula:

\begin{equation}
T=\frac{3Ic}{2\mu_0\sqrt{2v\dot M}}\simeq 10^{17} I_{45} \mu_{0_{30}}^{-1}
v_{10}^{-1/2}\dot M_{11} ^{-1/2} \, {\rm s}.
\end{equation}
Here $\dot M$ can be formally determined according to the Bondi equation 
for the mass accretion rate 
even if the NS is not at the accretion stage:
\begin{equation}
\dot M\simeq 10^{11}\,  n v_{10}^{-3} \, {\rm g}\, {\rm s}^{-1}.
\end{equation}

The results of calculations of $t_E$
for several values of $\mu_0$ and $t_d$ are shown in Fig.~2. 
The right end points of all curves are limited by the values 
$\mu_b=\mu_0$. These points correspond to the evolution of an 
INS with constant magnetic field (see Eq. (\ref{eq:mu(t)}))
and for them $t_E=T$. If 
$\mu_b$ is small enough, the NS field has no time to reach
the bottom value. In this case $t_E$ is determined by the 1st 
branch by Eq. (\ref{eq:t_E}) and does not depend on $\mu_b$. 
In the Fig. 2 this situation corresponds to the left horizontal parts
of the curves. 
At 
$$
\mu_b > \mu_0 \left[\frac{T}{t_d}\left(\sqrt{1-\frac{t_d^2}{T^2}}-1\right)\right]
$$
the situation changes so that 
$t_E$ starts depending on $\mu_b$. In this region two counter-acting
factors operate. On the one hand, the NS braking becomes 
slower with decreasing $\mu$ (see Eq. (\ref{eq:dp/dt})). On the other
hand, the end period of the ejection $p_E$ becomes shorter (\ref{eq:p_E}).
Since $t_E<T$ at the left hand side horizontal part and 
$\left.(dT_E/d\mu_b)\right|_{\mu_0}<0$, 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 $\mu_b$ at $p\ll P_E$ and most often 
NS evolves with the minimum field $\mu=\mu_b$ (this time increases 
with decreasing $\mu_b$). To the left of the maxium but before 
the horizontal part the NS magnetic field reaches 
$\mu=\mu_b$ with the spin period close to 
$p_E$ (the smaller $\mu_b$, the closer) and soon after 
$t=t_{cr}$, the NS leaves the ejection stage.

\begin{figure} 
%\vbox{\psfig{figure=fig3.eps,width=9.0cm,angle=-90}}
\vbox{\psfig{figure=9016.f3,width=9.0cm,angle=-90}}
\caption{ The characteristic time scale of the magnetic field decay, $t_d$, vs.
bottom magnetic moment, $\mu_b$.
In the hatched region $t_E$ is greater than $10^{10} {\rm yrs}$.
The dashed line corresponds to $t_H=t_d\cdot \ln \left( \mu_0/\mu_b
\right)$, where $t_H=10^{10}$ years. The solid line corresponds to
$p_E(\mu_b)=p(t=t_{cr})$, where $t_{cr}=t_d\cdot \ln \left(
\mu_0/\mu_b \right)$. Both the lines and hatched region
are plotted for $\mu_0=10^{30} {\rm G} \, {\rm cm}^{-3}$. 
The dash-dotted line is the same as the dashed one, 
but for $\mu_0=5\cdot 10^{29} 
\, {\rm G} \, {\rm cm}^3$.
The dotted line shows the border of the ``forbidden'' region for $\mu_0=5\cdot
10^{29}  \, {\rm G} \, {\rm cm}^3$.}
\end{figure} 

\begin{figure} 
%\vbox{\psfig{figure=fig4.eps,width=9.0cm,angle=-90}}
\vbox{\psfig{figure=9016.f4,width=9.0cm,angle=-90}}
\caption{The characteristic time scale of the magnetic field decay, $t_d$, vs. bottom
magnetic momentum, $\mu_b$.
In the hatched region $t_E$ is greater than $10^{10}\,  {\rm yrs}$.  
The dashed line corresponds to $t_H=t_d\cdot \ln \left( \mu_0/\mu_b   
\right)$, where $t_H=10^{10}$ yrs. The solid line corresponds to  
$p_E(\mu_b)=p(t=t_{cr})$, where $t_{cr}=t_d\cdot \ln \left(    
\mu_0/\mu_b \right)$. Both lines and region are plotted for 
$\mu_0=10^{29}\, {\rm G} \, {\rm cm}^{-3}$.}
\end{figure} 

As seen from  Fig. 2 , for some combination of
parameters, $t_E$ 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,  
% since accreting isolated NSs are really observed, 
the combinations
of $t_d$ and $\mu_b$ for which no accreting isolated NS appear, can be
excluded for their progenitors.
% progenitors ROSAT X--ray sources.
The regions of excluded parameters are plotted in Figs. 3 and 4.
 
The hatched regions correspond to parameters for which
$t_E$ is longer than $10^{10}\, {\rm yrs}$, 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 $\mu_0$. 

In the ``forbidden'' region in Fig. 3, which is plotted for
$\mu_0=10^{30}\, {\rm G} \, {\rm cm}^3$, 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
\begin{equation}
 p_E(\mu_b)=p(t=t_{cr}).
\end{equation}
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 $\mu=\mu_b$. 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 $\mu_b$, with which
an INS can reach the ejector stage for any $t_d$, i.e. 
this $\mu_b$ corresponds to the minimum value of $\mu_0$ 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 $p_E$, which depends upon the magnetic moment. 

In Fig. 3   the ``forbidden'' region  is also shown 
for $\mu_0=0.5\cdot 10^{30}\, {\rm G} \, {\rm cm}^3$ (dotted line).
The dashed line in Fig. 3 shows that for all interesting parameters an  INS
with $\mu_0=10^{30}\, {\rm G}\,{\rm cm}^3$
reaches $\mu_b$ in less than $10^{10}$ yrs.  The dash-dotted 
line shows the same
for $\mu_0=0.5\cdot 10^{30} \, {\rm G}\,{\rm cm}^3$.
The solid line corresponds to
$p_E(\mu_b)=p(t=t_{cr})$, where $t_{cr}=t_d\cdot \ln \left(
\mu_0/\mu_b \right)$. The physical sense of this line can be described in
the following way. This line divides two regions: in the upper left region
$t_d$ are relatively long and $\mu_b$ relatively low, so the NS cannot reach 
the bottom field during the ejector stage; in the lower right region $t_d$ are short
and $\mu_b$ relatively high, so the NS reaches $\mu_b$ at the stage of ejection.

Fig. 4 is plotted for $\mu_0=10^{29} \, {\rm G} \, {\rm cm}^3$.
For long $t_d$ ($>4\cdot 10^9$ yrs) the NS cannot leave the
ejector stage for any $\mu_b \le \mu_0$. This is the reason 
why in the upper part of the
figure a horizontal ``forbidden'' region appears.
 

\section{Discussion and conclusions}

 We tried to evaluate the region of parameters which are
excluded for models of the exponential
magnetic field decay in NSs using 
the possibility that several ROSAT 
soft X-ray sources are indeed old accreting isolated
NSs in X-rays.  
Of course we are not absolutely sure that all of the observed candidates
are accreting NSs. Some of them can be cooling NSs or magnetars
(this appears plausible, for example, for the source RX J0720-3125 as
 it has a spin period similar to soft gamma-ray repeaters).
Future observations (especially the proper motion and $\dot p$
measurements) are required.

The intermediate values of $t_d$ ($\sim 10^7-10^8 \, {\rm yrs}$)
in combination with the intermediate values of
$\mu_b$ ($\sim 10^{28}-10^{29.5} \, {\rm G} \, {\rm cm}^3$) 
for $\mu_0=10^{30} \, {\rm G}\, {\rm cm}^3$
can be excluded for progenitors of isolated accreting NSs
because NSs with such parameters would always remain
on the ejector stage and never pass to the accretion stage.
Even if all modern candidates are not accreting objects, 
the possibility of limitations of magnetic field
decay models based on future observations of isolated accreting NSs
should be addressed. 

As seen in Fig. 2, for higher $\mu_0$ NSs should reach $t_E$ even
for $t_d < 10^8 $ yrs, for weaker fields 
the ``forbidden'' region becomes
wider. The results are dependent on the initial magnetic field $\mu_0$, 
the ISM density $n$, and NS velocity $v$. So here different ideas can be
investigated. For example, 
this implies that the observed accreting isolated NSs can come from
objects with a high initial magnetic field, and the others never appear
as accreting objects
because their parameters lie in the forbidden region. 
To explore this idea in details, the
population synthesis of NSs for realistic distributions of $v$, $\mu_0$ and
$n$ is needed. It is clear, however,  that 
accreting old isolated NSs can hardly be formed from 
the initially high-field objects
because the fraction of the high-field NSs cannot be large (as follows
from radio pulsars observations). Since 
the fraction of the low velocity NSs
is not more than several percent (Popov et al. 2000) and the volume
fraction filled with relatively high density ISM is also small, accreting
old isolated NSs should come from the ``typical'' population, i.e. from NSs
with $\mu_0 $ about $ 10^{30} \, {\rm G} \, {\rm cm}^3$.

In fact the limits obtained are even stronger than 
they could be in nature, because we did not take into account
that NSs can spend 
some significant time (in the case with field decay)
at the propeller stage (the spin-down rate at this stage is very
uncertain, see the list of formulae, for example, in Lipunov \& Popov 1995
or Lipunov 1992). 
The calculations of this effect for
different models of non-exponential field decay will be studied separately.

 We cannot derive parameters of the field decay in accreting NSs
in close binaries, because there the situation is completely
different due to  high accretion rates 
that can significantly affect 
the process of magnetic field decay. So 
our results cannot be applied to millisecond radio pulsars
or other objects which were formed in close binary systems.

Note that there is another reason for which
a very fast decay down to small values of $\mu_b$ can also be
excluded, because this would lead to a huge amount of accreting isolated
NSs in drastic contrast with observations. This situation
is similar to the ``turn-off'' of the magnetic field of an INS
(i.e., quenching any magnetospheric effect on the accreting matter). So
for any velocity and density distributions we should expect 
significantly more accreting isolated NSs than we know from ROSAT observations
(of course, for high velocities X-ray sources will be very dim, but close
NSs can be observed even for velocities $\sim 100$ km s$^{-1}$).

We conclude that 
the existence of several old isolated accreting NSs observed by ROSAT
(if it is the correct interpretation of observations),
can put important bounds on the models of the magnetic field decay for
isolated NSs (without influence of accretion, which can stimulate field
decay).
These models should 
explain observations of $\sim 10$ accreting isolated NSs in the
solar vicinity. Here we can not fully discuss the relations between
decay parameters and X-ray observations of isolated NSs without detailed
calculations. What we showed is that this connection should be taken
into account and made some illustrations of it, 
and future investigations in that field RE desirable.

\begin{acknowledgements}
 We thank Monica Colpi, Denis Konenkov, Konstantin Postnov
and Roberto Turolla 
for numerous comments on the text, suggestions and discussions. 
We  want to thank Vladimir Lipunov and Aldo Treves for advice and
attention to our work and the unknown referee for numerous useful 
comments.

This work was supported by the RFBR (98-02-16801) and
the INTAS (96-0315) grants.
\end{acknowledgements}

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