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\title{Old isolated neutron stars in the Galaxy: evolution and field decay}
\author{S. B. Popov$^1)$ and D. Yu. Konenkov$^2)$ \\
        1) Sternberg Astronomical Institute\\
         119899, Moscow, Russia \\
        {\it e-mail: polar@xray.sai.msu.su}\\
        2) A.F.Ioffe Institute of Physics and Technology \\
        194021 St.Petersburg, Russia, \\
        {\it e-mail: dyk@astro.ioffe.rssi.ru} \\
       }    
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\begin{document}
\maketitle

\begin{abstract} 

Old isolated neutron stars form a large population of the Galactic objects,
and at the present time most of them are unobserved. In this talk we
briefly describe their main properties and show some examples.

As an example we discuss 
possible evolution of the spin period and the magnetic
field of the X-ray source RX J0720.4-3125 assuming this source to be an 
isolated neutron star accreting interstellar medium. Magnetic field of the
source is estimated to be $10^6 - 10^9$ G, and it is difficult to explain
observed spin period  8.38 s without invoking hypothesis of the 
magnetic field decay. We used the model of ohmic decay of the crustal 
magnetic field. The estimates of accretion rate ($10^{-14} - 10^{-16} 
M_{\odot}/{\rm yr}$), velocity of the source relative to interstellar
medium ($10 - 50 $ km/s), neutron star age
($2\cdot 10^9 - 10^{10}$ yrs) are obtained.

Key words: netron stars, evolution, magnetic field decay

\end{abstract}

%\newpage

\section{Introduction} 

Among all astrophysical objects neutron stars (NS) (and black holes)
attract most attention of physicists. 
NS were initially predict by
Lev Landau immediatelly after neutron was discovered in 1932.
But as far as there were no ideas how to observe so small (about 10 km in
diameter) astronomical objects they were not a subject of intense search,
and were found as radiopulsars occasionally in 1967 in England.

Now we know more than 1000 NS as 
radiopulsars and more than 100 NS emmiting X-rays,
but the Galactic population of these objects is about $10^8$ -- $10^9$,
so only a tiny fraction of one of the most facinating astrophysical objects
is observed. Studying of the rest of population is crucially important
to obtain initial parameters of NS, and to understand their evolution
(for example, there are some argument for the proposition, that
not all NS pass through the stage of radiopulsar in their young years;
it can happen due to too long initial periods, too high (magnetars) or
too low magnetic fields).

Isolated neutron stars (INS) have received special attention in the last few 
years. The idea of observing such objects in the X-ray range has 
emerged in soon after discovery of the first X-ray sources. 
Later it was
supposed that INSs accreting matter of the interstellar medium (ISM) can be 
observed with the $ROSAT$ satellite in the UV and X-ray ranges. 
It was also proposed, that 
old INS can be observed  in 
supernovae remnants, for example in RCW103 (Popov, 1998), 
if they were components of binary systems.

The estimates of energy characteristics, spectra and possibilities of 
observing INSs were made in several works (see, for example, B\"ohringer 
et al., 1987; Treves and Colpi, 1991; Blaes and Madau, 1993). A large 
number of works is devoted to the study of the spatial distribution of INSs 
(see, for example, Blaes and Rajagopal, 1991). Spatial distribution
of accretion luminosity of INS (and black holes) was considered in
(Popov, Prokhorov, 1998).

$ROSAT$ observations showed, that only few INS are bright enough to appear
on the X-ray sky. It can be a result of high initial velocities of INS,
which they obtain during supernova explosion, as it appear in population
synthesis (Popov et al. 1999).

In 1996, Haberl et.al. reported the discovery of the pulsating 
source RX J0720.4--3125 with the $ROSAT$ satellite in the soft X-ray 
range. We shall use two observational characteristics of this source (Haberl 
et.al., 1997): the period $p = 8.38$ s and the blackbody temperature 
$T = (79 \pm 4)$ eV. 

Using the hypothesis that this source is an INS accreting matter from the 
ISM, we estimate the accretion rate and the magnetic field
strength of this source. 
We show that the INS could not increase its period to the observed value 
over the Universe lifetime if we assume that it was born with the present-day 
magnetic field strength. Finally we conclude that the 
magnetic field of this NS had to decay and calculate the
magnetorotational evolution of the NS.

\section{Evolution of the spin period and the magnetic field of the 
neutron star} 

\subsection{Spin period}

In the low-density plasma an INS may have four possible evolutionary states 
(Lipunov, 1987):  ejector (young quickly spinning NS, spinning down due to 
losses of the kinetic energy onto emission of the magnitodipolar waves), 
propeller (NS ejecting plasma from alfven radius), accretor (NS spinning
slow enough to allow plasma from ISM to accrete onto its surface), and 
georotator (usially fastly moving NS, 
which is not important for our purposes). 
Two critical periods $P_E$ 
and $P_A$ separate different stages of the NS evolution. These 
periods as well as period derivatives on ejector and propeller stages
can be estimated using the formulae from (Lipunov, 1987). 
If $p < P_E$, then NS is at the ejector stage; if $P_E < p <
P_A$, we have the NS at the propeller stage; and if $p > P_A$
and, then NS is an accretor.

The change in the period of an accreting INS is due to its
interaction with the turbulized ISM. This introduces specific
features into the problem of the period evolution. If we adopt
the hypothesis of spin acceleration of the NS in the turbulized
ISM (Lipunov and Popov, 1995), the new characteristic period
emerges,

\begin{equation}
P_{eq}= 3450 k_t^{1/3} \mu_{30}^{2/3} I_{45}^{1/3} \dot M_{-15}^{-2/3}
v_{\infty_6}^{7/3} v_{t_6}^{-2/3} M_{1.4}^{-4/3}\,\, {\rm sec}, 
\end{equation}
where $\mu_{30}$ is the magnetic dipole moment in units of $10^{30}$ Gs
cm$^2$, $I_{45}$ is the moment of inertia in units of $10^{45}$ g cm$^2$, 
 $\rho_{-24}$ is the ISM density in units of $10^{-24}$ g cm$^{-3}$, 
$v_{\infty_6}$ is the 
NS velocity relative to the ISM in units $10^6$ cm/s, and $v_{t_6}$ 
is the turbulent velocity in units of
$10^6$ cm/s. The period $P_{eq}$ corresponds to the NS rms
rotation rate obtained from the solution of the corresponding
Fokker--Planck equation.  Here
take into account the three-dimensional character of turbulence,
i.e. the fact that the vortex can be oriented not only in the
equatorial plane but also at any angle to this plane.

\subsection{Ohmic dissipation of the crustal magnetic field
of the neutron star}

The magnetic field decay in the neutron star crust has been investigated in 
many papers (see, Urpin, Muslimov, 1992; Urpin, Konenkov, 1997) at which we
address the reader. The model contains two parameters, which determine the
field evolution: the impurity parameter $Q$, and the density at the bottom
of the layer initially occupied by the field, $\rho_0$.

Figure 1 shows the evolution of the NS surface magnetic field
for various parameters $\rho_0$ and $Q$. We perfomed
our calculations for the model of the NS based on the
Friedman--Pandharipande (1981)  equation of state
in the star core, with the mass of the neutron star $M = 1.4M_{\odot}$, 
the radius $R = 10.6\,$km, and the crust thickness $\Delta R =
940\,$m (Van Riper, 1988).  It is apparent that the decay at the
initial stage ($t < 10^6\,$yr) is determined by the initial depth
of the current layer and that the subsequent decay rate depends
on the impurity concentration, characterized by $Q$.

Accretion may affect the field evolution. First, it heats the NS
crust  and, therefore, decreases the
conductivity.  Second, the flux of matter toward the star center
arises; it tends to bring the field to deeper layers.
However, calculations show (Urpin et al., 1996) that accretion with the
rate $\dot M < 10^{-14}M_{\odot}\,$yr$^{-1}$ speeds up
insignificantly the field decay.


\section{ Estimates of parameters of RX J0720.4--3125}


Using the hypothesis that we observe an accreting isolated  NS and taking the
observed period $p = 8.38$ s and the temperature $T = (79 \pm
4)$ eV (Haberl et. al., 1997), we can obtain the constraints
on the magnetic field, accretion rate, and luminosity. 

1. If we really observe the accretor, then $p > P_A(B, \dot M)$, where 
$P_A$ is defined by (2). This yields the first constraint on $B$
and $\dot M$: 

\begin{equation}
B < 4.6 \cdot 10^9 M_{1.4}^{5/6} R_6^{-3} \dot M_{-15}^{1/2}\,
                  {\rm G}, 
\end{equation}		  

2. The pulsation of X-ray emission means that the accreting matter is 
funneled by the magnetic field onto the polar caps. Using the known relation 
$R_{\rm cap}= \sqrt{(R/R_A)} \cdot R,$ connecting the polar cap
radius $R_{\rm cap}$, the Alfven radius $R_A$, and the NS radius, we
can estimate $R_{\rm cap}$ and obtain another constraint from the condition 
$R_{cap} < R $: 

\begin{equation}
B>3.4
\cdot 10^4 M_{1.4}^{1/4}R_6^{-5/4} \dot M_{-15}^{1/2}\,{\rm G}.
\end{equation}

3. And finally, we know the polar cap temperature. Assuming that all heat
released in the polar caps is radiated away we find the additional relation 
between $B$ and $\dot M$: 

\begin{equation}
B=2.5\cdot 10^7 \left(T/79 \,\, {\rm eV}\right)^7 R_6^4 M_{1.4}^{-3/2} 
                             \dot M_{-15}^{-5/4}\,{\rm G}.
\end{equation}
Combining inequalities (2) and (3) and condition (4) yields the allowed 
range of the $B$ and $\dot M$ values: 
\begin{eqnarray}
& 2\cdot 10^5< B[{\rm G}] < 10^9,\nonumber \\
& 6\cdot 10^{-17}< \dot M [M_\odot /{\rm yr}] < 4 \cdot 10^{-14}.
\end{eqnarray}


Now it is easy to show that spin-down time to $p=8.38$ s greatly exceeds
the Hubble time if one assumes that magnetic field of RXJ 0720.4-3125
has the same strength in the past. This means that in the past the given 
INS had a larger magnetic field that has decayed over the time of evolution. 

Since the NS presently accretes the matter from the ISM, we can estimate
its period derivative:
$$
\dot p \approx p/t_{su} = p\; \frac{\dot M v_t R_G}{I\omega}<10^{-12} {\rm s/s}.
$$

\section{The evolutionary tracks of neutron star on the 
$B-P$ diagram }

The evolution of the spin period was calculated
taking into account the decay of the
magnetic field.  Since the
treatment of the spin-up rate in the turbulized ISM at the
accretor stage is not quite a simple matter, we applied to the
description of the period evolution the following simplifying
model. When the NS gets to the accretor stage, the spin-down
momentum substantially exceeds the spin-up momentum because the
acceleration occurs in the turbulized ISM and there is no
constant spin-up momentum, unlike in a binary system. However, we
can obtain an analog of the equilibrium period
which corresponds to the stationary solution of the
Fokker--Planck equation (Lipunov, 1987). 
Therefore the evolution of the spin period
at this stage was treated as a stationary spindown from
$P_A$ to $P_{eq}$ after which the period was set to be $P_{eq}$,
which, in its turn, changed due to the magnetic field decay.

We calculated the magnetic and spin evolution of the NS
with mass $M = 1.4 M_{\odot}$ for the accretion rates $10^{-15}
M_{\odot}$yr$^{-1}$ and $10^{-16} M_{\odot}$yr$^{-1}$,
moving through the ISM with density $\rho = 10^{-24}$ g/cm$^3$.
Such accretion rates at the given ISM density correspond to the
velocities of motion $19$ and $41$ km/s. We assumed the NS to be
born as an ordinary pulsar with a short period ($p_0 = 0.01$ s)
and a "standard" initial magnetic field $B_0 = 10^{13}$ G. The
model of the ohmic decay of magnetic field allows one to obtain
both the high and low dissipation rate. However, in this case we
should fit the parameters of the model to obtain the
accreting INS with a period of 8.38 s and a field of $2 \cdot
10^7$ G for the accretion rate $\dot M = 10^{-15} M_{\odot}$
yr$^{-1}$ or with a field of $4 \cdot 10^8$ G for the accretion
rate $10^{-16} M_{\odot}$ yr$^{-1}$ (4). Therefore, the field
should decay with a moderate rate to give the NS enough time to
spin down to $p = 8.38$ s. Moreover, it is desirable at the
initial stage of evolution to provide the agreement with
magnetic fields and periods of the observed radio pulsars. Basing
on these assumptions we can obtain the estimates of the
parameters $\rho_0$ and $Q$, as well as the lower limit on the
INS age.

The evolutionary tracks of the NS for the accretion rates 
$\dot M = 10^{-15} M_{\odot}$ yr$^{-1}$ and 
$\dot M = 10^{-16} M_{\odot}$ yr$^{-1}$ are shown
in panels a) and b) of the Fig.2. 

Tracks 1 in both panels of Fig. 2 illustrate the evolution with
the maximum possible rate of magnetic field dissipation with
time. The model parameters for the accretion rate $\dot M =
10^{-15} M_{\odot}$ yr$^{-1}$ are: $\rho_0 = 3 \cdot 10^{13}$
g/cm$^3$, $Q = 0.02$, and $v = 10^6$ cm/s. In the first $10_6$ yrs, 
the field decays by about 20
times, while the NS slows down to $p \approx 0.3$ s. At that
time the NS represents a typical radio pulsar.  Subsequently,
due to the cooling, the field decay slows down. In the next
$8\cdot 10^7$ yrs the field decays to $4.5 \cdot 10^{11}$ G, the
period increases, and the radiopulsar dies.  However, the
ejector stage continues for $1.2 \cdot 10^9$ years.  During this
stage, the field decays to $2.5 \cdot 10^{10}$ G, the period increases to
$2.1$ s, and the NS goes over to the propeller stage (1) which
lasts $\sim 10^9$ yrs. At this stage, further deceleration
of the NS occurs, according to (4), to $p = 5.7$ s, and the field
decays to $3 \cdot 10^9$ G. The star goes over to the accretor stage
and becomes the source of periodic X-ray radiation. The core
heating speeds up insignificantly the field decay (Urpin et.al.,
1996). The field decays to $2 \cdot 10^7$ G over $4 \cdot 10^9$ yrs.
The period does not increase because the field is weak enough.
As soon as $P_{eq}$ becomes equal to the current period, the
spin-up of the NS due to the interaction with the ISM may
occur. The period fluctuates around $P_{eq}$, but we do not
estimate here the amplitude of these fluctuations.

Tracks 2 illustrate the evolution with slower field decay. Such
a decay can be obtained, for example, for the crust with lower
impurity content. For $\dot M = 10^{-15} M_{\odot}$ yr$^{-1}$,
we set $Q=0.01$, while the other parameters remained the same as
for track 1 in Fig. 2a. Tracks 1 and 2 coincide at the initial
stage of evolution, when the dissipation rate does not depend on
the impurity concentration. However, in $10^7$ yrs the tracks
diverge. As a result, the star resides at the ejector stage $2
\cdot 10^9$ yrs going over to the propeller stage with $p = 3$
s and $B =4 \cdot 10^{10}$ G. The spin-down rate at the
propeller stage is much higher than in the first case for three
reasons: due to the longer spin period, the stronger
magnetic field during the NS transition from the ejector to the
propeller stage, and to the lower rate of the field decay caused by the
lower value of the $Q$ parameter. At the accretor stage, the NS
spin period decreases to 190 s over $2 \cdot 10^9$ yrs, and when
the field decays to $4 \cdot 10^8$ G, the effect of acceleration in
the turbulized ISM may begin to act. The track in the absence of
such acceleration ($v_t = 0$) is shown by the dashed line. In this
case the final period turns out to be about 200 s. The turbulent
acceleration may decrease this period. However, the
self-consistent calculation of the evolution of the NS period at
the accretor stage with allowance for the magnetic field decay
would require the solution of the Fokker--Planck equation for
the distribution function in the space of angular velocities,
which is beyond the scope of this work.

At the accretion rate $\dot M = 10^{-16} M_{\odot}$ yr$^{-1}$ we
used for track 1 the following parameters: $\rho_0 = 3 \cdot
10^{13}$ g/cm$^3$, $Q = 0.01$, and $v_t = 10^6$ cm/s. The NS
comes to the propeller stage in $2.7
\cdot 10^9$ yrs having a period of 3.1 s and a magnetic field of 
$2 \cdot 10^{10}$ G.  The transition to the accretor stage
occurs when the NS has a period of 4.9 s and a magnetic field of
$7 \cdot 10^8$ G. Deceleration at this stage does not occur because
the magnetic field is low. The full time taken for the field decay to
$4 \cdot 10^8$ G is $6 \cdot 10^9$ yrs. In this case $p = 5$ s. The second
track in Fig. 2b was calculated for the greater initial depth of
the current layer corresponding to $\rho_0 = 6 \cdot 10^{13}$
g/cm$^3$. As a consequence, at the initial stage of evolution the
magnetic field decays with a slower rate, the ejector stage
lasts $2.1 \cdot 10^9$ yrs, and the propeller stage lasts $1.9 \cdot
10^9$ yrs. For the field $4 \cdot 10^8$ G, the period is 63 s.

\section{Conclusion} 

The observed period and the temperature of the X-ray source RX
J0720.4-- 3125 can be accounted for by using the hypothesis for
the ISM accretion onto an old INS. We showed that the NS
magnetic field is low in this case ($B < 10^9 $ G). The time
taken for deceleration to $p = 8.38$ s with such a magnetic field
exceeds the age of the Universe. We supposed that the NS was
born with higher strenght of magnetic field and that the field
strength has substantially decreased during the evolution. Using
the model of ohmic dissipation of the magnetic field in the NS
crust, we calculated the possible evolution of the NS on the
$B-P$ diagram. The observed rotational period can be obtained at $Q=
0.01-0.05$. However, the evolution of the period depends on the
field dissipation rate and, therefore, on the parameters of the
decay model. Thus, the twofold change in the impurity parameter
$Q$ caused the spin period to change more than by the order of
magnitude at the accretor stage (Fig. 2a, tracks 1 and 2). For
this reason, observations of periods of old INSs may
become an important test for the validity of the model of
evolution of neutron stars.

Decay of the magnetic field may affect the estimate of the
total number of the observed accreting INSs. In particular,
because the formation of a periodic X-ray source at the
propeller stage calls for rather strong magnetic field, 
its decay may reduce the number of sources of this kind.
At the same time, the absence of pulsations for other INS
candidates may suggest that their field has already decayed to
such extent that it cannot funnel the plasma toward the INS
polar caps.

Due to decay, depending upon its timescale
and minimum (bottom) magnetic field, relative
numbers of different stages of INS  
(Ejectors, Propeller or Accretors) can  be increased or decreased
(Popov et al., 1999). Here we didn't addressed the problem of the whole
population's properties, but constructed the evolutionary track of the
single object. 


\section*{Acknowledgements}

The authors express their gratitude 
to M.E. Prokhorov, V.A.Urpin, V.M. Lipunov, M. Colpi, R. Turolla, A. Treves 
and D.G. Yakovlev.
The work of D. Konenkov was supported 
by the Russian Foundation for Basic Research (grant 97-02-18096a), and by
INTAS (grant 96-0154),
the work of S. Popov -- by the Russian Foundation for Basic Research (grant 
98-02-16801), and by INTAS (grant 96-0315). 

%\newpage

\section{References}

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onto neutron stars'' 
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\noindent
B\"ohringer, H., Morfill, G.E. and Zimmermann, H.U., ``Characteristics of
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\noindent
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\noindent
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\noindent
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\noindent
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%\newpage

Figure captions.

Figure 1.

The evolution of the surface magnetic field of the isolated neutron star 
for the standard cooling scenario. Curves 1, 2, and 3 correspond 
to the initial depths of the current layer, $10^{11}$, $10^{12}$, and
$10^{13}\,$ g cm$^{-3}$, respectively. The solid curves correspond to 
Q = 0.001; the dashed curves, to Q = 0.01; the dot-dashed curves, to Q = 0.1. 


Figure 2.

 The evolutionary tracks of the neutron star for the accretion rates
$\dot M = 
10^{-15} M_{\odot}\,$yr$^{-1}$ (a) and $\dot M = 10^{-16} M_{\odot}\,$
yr$^{-1}$ (b). The model 
parameters are described in the text. The dashed lines correspond to $p = 
P_E$; the dot-dashed lines, to $p = P_A$. The dashed line in Fig. 2a 
shows for the second track the neutron star evolution with no acceleration in 
the turbulized intestellar medium. The numbers near the marks in tracks 
denote the logarithm of the neutron star age in years. The observed radio 
pulsars are indicated by dots (Taylor et al., 1993). 



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