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Restrictions on parameters of power­law magnetic field
decay for accreting isolated neutron stars
S.B. Popov, M.E. Prokhorov
Sternberg Astronomical Institute, Moscow, Russia
119899, Universitetskii pr. 13
polar@xray.sai.msu.su, mystery@sai.msu.su
Abstract
In this short note we discuss the influence of power­law magnetic field decay on the evolution of
old accreting isolated neutron stars. We show, that, contrary to exponential field decay (Popov &
Prokhorov 2000), no additional restrictions can be made for the parameters of power­law decay from
the statistics of isolated neutron star candidates in ROSAT observations.
We also briefly discuss the fate of old magnetars with and without field decay, and describe
parameters of old accreting magnetars.
Key words: neutron stars -- magnetic fields -- stars: magnetic field -- X­rays: stars -- accretion
1 Introduction
Isolated neutron stars (INSs), which don't show radio pulsar activity attract now much attention
of astrophysicists due to recent observations of several candidates with the ROSAT sattelite (see
Ne¨uhauser & Tr¨umper 1999 and a review in Treves et al. 2000). As we discussed in our previous
paper (Popov & Prokhorov 2000) INSs can be important for discussion of different models of magnetic
field decay (MFD) in NSs in general.
During its evolution an INS can pass through four phases: ``ejector'', ``propeller'', ``accretor'' and
``georotator''. At the first stage the INS is spinning down according to the magneto­dipole formula
till so­called ejector period is reached. At the second stage captured matter cannot penetrate down
to the surface of the INS, and the star continue to spin down faster than at the stage of ejection.
At last, so­called accretor period is reached, and matter can fall down: accretion starts. If the INS's
velocity (or magnetic field) is high enough, the star can appear as a georotator, where matter cannot
be captured, as far as the magnetosphere radius is large than the radius of gravitational capture.
Several models of MFD in NSs were suggested during the last 20­30 years (see for example a
recent brief review by Konar & Bhattacharya). Most of these models can be fitted by exponential or
power­law decay, or by their combination with some set of parameters. INSs can be an important
class of objects for verification of different theories of MFD, because in these sources accretion
rate is negligible, so it is not necessary to take into account the influence of accretion onto MFD
(Urpin et al. 1996). Spin­up/spin­down rates on the stage of accretion are also relatively low in
comparison with NSs in binary systems. It means, that in INSs MFD operates in the ``purest'' form
(Popov & Konenkov 1998). That's why these objects have a special importance, in our opinion, for
investigations of observational appearance of different effects of MFD.
Recently, Colpi et al. (2000) discussed power­law models of MFD in INSs and applied them to
highly magnetized NSs, ``magnetars''. Here we briefly discuss later stages of evolution of INSs with
the power­law MFD, and estimate if it is possible for them to reach the stage of accretion, and if
yes, what can be their properties at this stage.
Our analysis follows the papers Popov & Prokhorov (2000), and Colpi et al. (2000). So, we just
repeat calculations of Popov & Prokhorov (2000) but for the power­law decay, using some results of
Colpi et al. (2000). And we refer to these papers for all details of terminology, calculations etc.
1

2 Power­law decay
Power­law (as also exponential) MFD is a widely discussed variant of NSs' field evolution. Power­law
is a good fit for several different calculations of the field evolution (Goldreich & Reisenegger 1992,
Geppert et al. 2000). The power­law MFD can be described with the following simple formula (Colpi
et al. 2000):
dB
dt
= \GammaaB 1+ff : (1)
So, we have only two parameters of decay: a and ff. As far as this decay is relatively slow for the
most interesting values of ff & 1 (we use the same units as in Colpi et al. 2000), we don't specify
any bottom magnetic field, contrary to what we made for more rapid exponential decay (Popov &
Prokhorov 2000). Even for the Model C from Colpi et al. (2000) (see Table 1) with relatively fast
MFD the magnetic field can decrease only down to ¸ 10 8 G in 10 10 yrs (see Fig. 1). But for ff ! 1
the magnetic field can decay significantly during the Hubble time (we call here ``the Hubble time''
time interval 10 10 yrs, which is nearly equal to the age of our Galaxy) for any reasonable value of a.
And, probably, it is useful to introduce in the later case a bottom field.
At the stage of ejection an INS is spinning down according to the magneto­dipole formula:
P —
P ú bB 2 . Here (and everywhere below) b = 3, values of magnetic field, B, B1 and B 0 , are taken
in units 10 13 G and time, t, in units 10 6 yrs (as in Colpi et al. 2000).
In the table we show parameters of the Models A, B, C from Colpi et al. (2000). B1 is the
magnetic field calculated for t = t Hubble = 10 10 yrs and for the initial field B 0 = 10 12 G. Models A
and B correspond to ambipolar diffusion in the irrotational and the solenoidal modes respectively.
Model C describes MFD in the case of the Hall cascade.
Table 1: Models A,B,C from Colpi et al. (2000)
Model A B C
a 0.01 0.15 10
ff 5/4 5/4 1
B1 ú 1:9 \Delta 10 11 G ú 2:4 \Delta 10 10 G ú 10 8 G
In Fig. 2 we show dependence of the ejector period, p e , and the asymptotic period, p1 , on the
parameter a for ff = 1 for different values of the initial magnetic field, B 0 :
p e = 25:7 B 1=2
1 n \Gamma1=4 v 1=2
10 s; (2)
p 2
1 = 2
2 \Gamma ff
b
a
B 2\Gammaff
0 : (3)
Here v 10 is velocity (v 2
INS + v 2
s ) 1=2 in units 10 km/s; v INS is the spatial velocity of the INS and v s ­
sound velocity. n is the interstellar medium (ISM) number density. B 0 ­ initial magnetic field.
p e was calculated for t = t Hubble = 10 10 yrs, i.e. for the moment, when B = B1 .
It is clear from Fig. 2, that for the initial field & 10 11 G low velocity INSs are able to come to
the stage of accretion: for B 0 = 10 11 G lines for p1 and p e for the lowest possible velocity, 10 km/s,
coincides.
In Fig. 3 we show ``forbidden'' regions on the plane a--ff, where an INS for a given velocity for
sure cannot come to the stage of accretion in the Hubble time (compare with ``forbidden'' regions
2

10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
10 8
10 9
10 10
Time, yrs
10 8
10 9
10 10
10 11
10 12
Magnetic
field,
G
Model A
Model B
Model C
Figure 1: Power­law MFD. Model A: a = 0:01; ff = 1:25; solid line with circles. Model B: a =
0:15; ff = 1:25; dashed line with squares. Model C: a = 10; ff = 1; long­dashed line with diamonds.
Models were described in details in Colpi et al. (2000) (see also Table 1).
3

0.0 0.2 0.4 0.6 0.8
a
0.1
1
10
100
Period,
s
Log B = 11
p_e; v=10 km/s
p_infty; alpha=1
p_e; v = 40 km/s
0.0 0.2 0.4 0.6 0.8
0.1
1
10
100
Period,
s
Log B = 12
p_e; v=10 km/s
p_infty; alpha=1
p_e; v=40 km/s
0.0 0.2 0.4 0.6 0.8
a
0.1
1
10
100
Log B = 14
p_infty; alpha=1
p_e; v=40 km/s
0.0 0.2 0.4 0.6 0.8
0.1
1
10
100
Log B = 13
p_e; v=10 km/s
p_infty; alpha=1
p_e; v=40 km/s
Figure 2: Periods vs. parameter a for different values of the initial magnetic field:
10 11 ; 10 12 ; 10 13 ; 10 14 G.
4

1 2 3 4 5 6 7 8 9
a
0.0
0.5
1.0
1.5
alpha
200 km/s
1 2 3 4 5 6 7 8 9
0.0
0.5
1.0
1.5
alpha
40 km/s
1 2 3 4 5 6 7 8 9
a
0.0
0.5
1.0
1.5
400 km/s
1 2 3 4 5 6 7 8 9
0.00
0.50
1.00
1.50
100 km/s
Figure 3: ``Forbidden'' regions for the initial field 10 13 G and different INS's spatial velocities: 40
km/s, 100 km/s, 200 km/s and 400 km/s. In the filled regions NSs never leave the ejector stage.
5

in Popov & Prokhorov 2000). In a forbidden region an INS for specified parameters cannot leave
the stage of ejector even after 10 10 years of evolution. If one also takes into account the stage of
propeller (between ejector and accretor stages) it becomes clear, that ``forbidden''' regions for an INS
which cannot reach the stage of accretion is even larger. We note, that the propeller stage can be
shorter (probably much shorter, especially for constant field) than the stage of ejection (see Lipunov
& Popov 1995 for detailed arguments), so the ``forbidden'' regions on Fig. 3 cannot become much
larger if one also takes into account the stage of propeller. It is also important, that we take very low
INS's velocity and high ISM density. For most part of INSs all plotted ``forbidden'' regions should
be larger.
One can see, that for the most interesting cases (Models A, B, C from Colpi et al. 2000) and
v ! 200 km/s INSs can reach the stage of accretion. It is an important point, that fraction of low
velocity NSs is very small (Popov et al. 2000) and most of them have velocities about 200 km/s.
3 Evolved magnetars
In the last several years a new class of objects ­ highly magnetized NSs, ``magnetars'' (Duncan &
Thompson 1992) -- became very popular in connection with soft fl­repeaters (SGR) and anomalous
X­ray pulsars (AXP) (see Mereghetti & Stella 1995, Kouveliotou et al. 1999, Mereghetti 1999 and
recent theoretical works Alpar 1999, Marsden et al. 2000, Perna et al. 2000).
Magnetars come to the propeller stage with periods ¸ 10 -- 100 s in the Models A, B, C (see
Fig. 2 in Colpi et al. 2000). Then their periods quickly increase, and NSs come to the stage of
accretion with significantly longer periods, and at that stage they evolve to a so­called equilibrium
period (Lipunov & Popov 1995, Konenkov & Popov 1997) due to accretion of the turbulent ISM:
p eq ¸ 2800B 2=3 I 1=3
45 n \Gamma2=3 v 13=3
10 v \Gamma2=3
t 10
M \Gamma8=3
1:4 s (4)
Here v t is a characteristic turbulent velocity, I -- moment of inertia, M -- INS's mass.
Isolated accretor can be observed both with positive and negative sign of —
p (Lipunov & Popov
1995). Spin periods of INSs can differ significantly from p eq contrary to NSs in disc­fed binaries,
and similar to NSs in wide binaries, where accreted matter is captured from giant's stellar wind. It
happens because spin­up/spin­down moments are relatively small.
As the field is decaying the equilibrium period is decreasing, coming to 28 sec when the field is
equal to 10 10 G (we note here recently discovered objects RX J0420.0­5022 (Haberl et al. 2000) with
spin period ¸ 22:7 s).
It is important to discuss the possibility, that evolved magnetar can appear as georotator (see
Lipunov 1992 for detailed description or Popov et al. 2000 for short description of different INSs'
stages). It happens if:
v & 300B \Gamma1=5 n 1=10 km=s: (5)
For all values of a and ff that we used (see Fig. 3) NSs, at the end of their evolution (t = 10 10 yrs),
have magnetic fields . 10 12 G for wide range of initial fields, so they never appear as georotators
if v ! 580 km/s for n = 1cm \Gamma3 . But without MFD magnetars with B & 10 15 G and velocities
v & 100 km/s can appear as georotators.
In Popov et al. (2000) it was shown, that georotator is a rare stage for INSs, because an INS can
come to the georotator stage only from the propeller or accretor stage, but all these phases require
relatively low velocities, and high velocity INSs spend most of their lives as ejectors. This situation
6

is opposite to binary systems, where a lot of georotators are expected for fast stellar winds (wind
velocity can be much faster than INS's velocity relative to ISM).
Without MFD magnetars also can appear as accreting sources. In that case they can have very
long periods and very narrow accretion columns (that means high temperature). Such sources are
not observed now. Absence of some specific sources associated with evolved magnetars (binary or
isolated) can put some limits on their number and properties (dr. V. Gvaramadze drew our attention
to this point).
At the accretion part of INSs' evolution periods stay relatively close to p eq (but can fluctuate
around this value), and INSs' magnetic fields decay down to ¸ 10 10 \Gamma 10 11 G in several billion years
for the Models A and B. It corresponds to the polar cap radius about 0.15 km and temperature about
250 -- 260 eV, higher than for the observed INS candidates with temperature about 50 -- 80 eV. We
calculate the polar cap radius, R cap = R
p
(R=RA ), with the following formula:
R cap = 6 \Delta 10 3 B \Gamma2=7 n 1=7 v \Gamma3=7
10 cm: (6)
Here RA ' 1:8 \Delta 10 10 n \Gamma2=7 v 6=7
10 B 4=7 cm is the Alfven radius. The temperature can be even larger, than
it follows from the formula above as far as for very high field matter can be channeled in a narrow
ring, so the area of the emitting region will be just a fraction of the total polar cap area.
As the field is decreasing the radius of the polar cap is increasing, and the temperature is falling.
Sources with such properties (temperature about 250­260 eV) are not observed yet (Schwope et
al. 1999). But if the number of magnetars is significant (about 10% of all NSs) accreting evolved
magnetars can be found in the near future, as far as now we know about 5 accreting INS candidates
(Treves et al. 2000, Ne¨uhauser & Trumper 1999), and their number can be increased in future. —
p
measurements are necessary to understand the nature of such sources, if they are observed.
Recently discovered object RX J0420.0­5022 (Haberl et al. 2000) with the spin period ¸ 22:7 s,
can be an example of an INS with decayed magnetic field accreting from the ISM, as previously
RX J0720.4­3125. Due to relatively low temperature, 57 eV, its progenitor cannot be a magnetar
for power­law MFD (Models A,B,C) or similar sets of parameters, because a very large polar cap is
needed, which is difficult to obtain in these models. Of course RX J0420.0­5022 can be explained
also as a cooling NS. The question ''are the observed candidates cooling or accreting objects?'' is still
open (see Treves et al. 2000). If one finds an object with p & 100 s and temperature about 50 --
70 eV can be a strong argument for its accretion nature, as far as such long periods for magnetars can
be reached only for very high initial magnetic fields (see Fig. 2 in Colpi et al. 2000) for reasonable
models of MFD and other parameters.
4 Conclusions
Our main result means, that for power­law MFD (contrary to exponential decay) we cannot put
serious limits on the parameters of decay with the ROSAT observations of INS candidates as far as
for all plausible models of power­law MFD INSs from low velocity tail are able to become accretors.
For more detailed conclusions a NS census for power­law MFD is necessary, similar to non­decaying
and exponential cases (Popov et al. 2000).
An interesting possibility of observing evolved accreting magnetars appear both for the case of
MFD and for constant field evolution. These sources should be different from typical present day INS
candidates observed by ROSAT. Existence or absence of old accreting magnetars is very important
for the whole NS astrophysics.
7

Acknowledgments
We thank drs. Monica Colpi, Vasilii Gvaramadze and Roberto Turolla for discussions. S.P. thanks
University of Milan, University of Padova and Astronomical Observatory of Brera for their hospitality.
This work was supported by the RFRB, INTAS and NTP ``Astronomy'' grants.
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