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Magnetic field decay and period evolution of the
source RX J0720.4-3125
by S.B. Popov and D.Yu. Konenkov.
Izvestiya VUZov ''Radiofizika'' V.41 P.28-35 (1998)
(English translation: Radiophysics and quantum electronics)
Presented at the summer school Volga-97 (1-11 June 1997)
Abstract
We studied possible evolution of rotational period and magnetic field
of the X-ray source RX J0720.4-3125 assuming this source to be an iso-
lated neutron star accreting from interstellar medium. Magnetic field of
the source is estimated to be 10 6 \Gamma 10 9 Gs (most probably Я 2 \Delta 10 8 Gs),
and it is difficult to explain the observable rotational period 8.38 s with-
out invoking hypothesis of the magnetic field decay. We used the model
of ohmic decay of crustal magnetic field. The estimates of the accretion
rate (10 \Gamma14 \Gamma 10 \Gamma16 M fi =yr), velocity of the source relative to the interstellar
medium (10 \Gamma 50 km/s), the neutron star age (2 \Delta 10 9 \Gamma 10 10 yrs) are obtained.
(remark added in proof: As we found out in July 1997 when this paper was
already submitted, common results were recieved independently by John
C.L. Wang // Astrophys. J., 1997, V.486. P.L119)
1

1 Introduction
Recently isolated neutron stars (INSs), which are not observed as radiopul-
sars, became of great interest both for observers and theorists. The idea
of observations of such sources was proposed more than 25 years ago [6],
and in 1991 Treves and Colpi [8] suggested, that INSs, accreting from inter-
stellar medium (ISM), can be observed in great number in UV and X-rays
by ROSAT . Here we present the results of the work on the source RX
J0720.4-3125 observed by Haberl et al. [12]. In [12] it was proposed, that
RX J0720.4-3125 is an accreting INS with the rotational period 8.38 s.
There are four possible stages of an INS in low density plasma: Ejector
(E), Propeller (P), Accretor (A) and Georotator (G). The stage is deter-
mined by relations between specific radii: R l -- light cylinder radius, R st --
stopping radius, RG = ( 2GM
v 2
1
)-- the radius of the gravitational capture and
R co = ( GM
! 2
) 1=3 -- the corotation radius.
As a result we have two critical periods: PE and PA , separating different
stages. If p ! PE , we have an Ejector (i.e. a pulsar), if PE ! p ! PA ,
the NS is on the Propeller stage, and if p ? PA and R st ! RG -- we have
an accreting NS. It is possible, that p ? PA , but R st ? RG . In this case a
geo-like magnitosphere is formed, and we call it Georotator.
For describtion of period evolution it is useful to use a gravimagnetic
parameter, y, (it is described in [4]).
On the figure 1 three examples of evolutionary tracks of an INS on p-y-
diagram are presented. All of them are ended at the Accretor stage.
I. E \Gamma! P \Gamma! A-- evolution of an INS in the ISM with the constant density
without magnetic field decay.
II. E \Gamma! P \Gamma! A \Gamma! P \Gamma! A-- evolution of an INS, passing through a
giant molecular cloud without magnetic field decay.
III. Evolution with magnetic field decay.
We are especially interested in the third case.
2 Analitical estimates of the X-ray pulsar's pa-
rameters
In this part we show mainly estimates which were not included into the
article [3].
Lets consider the situation, when the Alfven radius, RA , and the coro-
tation radius, R co , are equal. This equation gives us the accretion period,
PA :
2

PA Я 6 \Delta 10 2 ? 6=7
30
ae \Gamma3=7
\Gamma24 v 9=7
1 6
`
M
M fi
' \Gamma11=7
s (1)
here ? 30 -- magnetic momentum in units 10 30 Gs \Delta cm 2 , ae \Gamma24 -- density of the
ISM in units 10 \Gamma24 g=cm 3 , v 1 6
-- velocity of the INS relative to the ISM in
units 10 6 cm=s.
During accretion p ? PA . If these two periods are equal:
? 6=7
30
= 8:38
6 \Delta 10 2
ae 3=7
\Gamma24 v \Gamma9=7
1 6
`
M
M fi
' 11=7
So, ? 30 = 0:007ae 1=2
\Gamma24 v \Gamma3=2
1 6
i
M
M fi
j 11=6
for PA = p = 8:38 s (the period of
RX J0720.4-3125). For the NS's radius R = 10 km we have B ! 7 \Delta 10 9 Gs.
If we use the hypothesis of the acceleration of an INS from the turbu-
lizated ISM, we have an equation, which gives an estimate of the magnetic
field of the INS based on the equilibrium period, P eq , [3, 5]:
P eq = 2355k 1=3
t ? 2=3
30
I 1=3
45
ae \Gamma2=3
\Gamma24 v 13=3
1 6
v \Gamma2=3
t 6
`
M
M fi
' \Gamma8=3
s (2)
Here I 45
-- the momentum of inertia in units 10 45 g \Delta cm 2 , v t 6
-- the tur-
bulent velocity in units 10 6 cm=s:
So we have:
? 30 =
` 8:38
2355
' 3=2
k \Gamma1=2
t I \Gamma1=2
45
ae \Gamma24 v \Gamma13=2
1 6
v t 6
`
M
M fi
' 4
And finaly B Я 2:1 \Delta 10 8 Gs.
In [3] we showed, that the magnetic field can't be less than 10 5 \Gamma 10 6 Gs,
because in the opposite case we can't observe pulsations of the X-ray flux.
Lets suppose, that the INS was born with a period about 0.01 -- 0.02 s
(the exact value is not very important, because the only necessary condition
is P initial !! 8.38 s, and initially the INS was on the Ejector stage). So, the
star had to decelerate till 8.38 s. Lets estimate the time of this deceleration.
The time of deceleration is determined by the final, not by the initial
period!
PE Я 10(k t ) 1=4 ? 1=2
30
ae \Gamma1=4
\Gamma24 v \Gamma1=2
1 6
s
dI!
dt
= \Gammak t
? 2
R 3
l
; R l = c=!; ! = 2Я=p
3

Figure 1: P-y diagram
4

2 4 6 8 10
0,0001
0,001
0,01
0,1
1
B/B
0
lg t, годы
1
2
3
Figure 2: Changes of the surface magnetic field of an isolated neutron star
with time for the model of standard cooling. Curves 1, 2, and 3 correspond
to the initial depths of the current layer, 10 11 , 10 12 , and 10 13 g cm \Gamma3 ,
respectively. The solid curves correspond to Q = 0.001; the dashed curves,
to Q = 0.01; the dot-dashed curves, to Q = 0.1.
5

0,01 0,1 1 10 100
7
8
9
10
11
12
13
9.5
9.4
9.3
9 u
u
u
u
u
u
u
u
u
u
9.7
9.5
9.4
9.3
9
8
7
6
5
4
3
u
u
u
u
u
u
u u
u
u
u
u
u
u а)
2
1
P, c
0,01 0,1 1 10 100
7
8
9
10
11
12
13
9.7
9.5
9.3
9
8
7
6
5
4
9.7
u
u
u
u u u
u
u
u
u
9.5
9.3
9
8
7
6
5
4
3
P, c
б)
2
1
Figure 3: The evolutionary tracks of the neutron star for the accretion
rates -
M = 10 \Gamma15 M fi yr \Gamma1 (a) and -
M = 10 \Gamma16 M fi yr \Gamma1 (b). The model
parameters are described in the text. The dashed lines correspond to p =
PE ; the dot-dashed lines, to p = PA . 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.
6

1
p 2
I \Deltap
\Deltat = k t
? 2 (2Я) 2
c 3 p 3
So for \Deltap = p:
\Deltat = Ip 2 c 3
k t ? 2 (2Я) 2
=
= 3 \Delta 10 7 P 2
E k \Gamma1
t ? \Gamma2
30
I 45 yrs =
= 3 \Delta 10 9 k \Gamma1=2
t ? \Gamma1
30
ae \Gamma1=2
\Gamma24 v \Gamma1
1 6
I 45 yrs
For ? 30
! 0:01 \Deltat ? 3 \Delta 10 11 yrs ?? t Hubble . I.e. the star couldn't
initially have low magnetic field.
As far as the NS is on the Accretor stage, its period should be much
greater than the ejection period, PE :
p ? P P ropeller ? PE
PE Я 10s for standart parameters. It means, that if the star is on the
accretor stage with p=8.38 s, the field is much less than the standart value:
B !! 10 12 Gs.
Lets try to estimate the characteristic time of acceleration and deceler-
ation of such a NS:
t su = t sd = I!
-
Mv t RG
=
= 20yrs \Delta I 45
v 5
1 6
ae \Gamma1
\Gamma24
`
p
10 5 s
' \Gamma1 `
v t
10 6 cm=s
' \Gamma1
where t su ; t sd - characteristic times of the period changes.
For p=10 s t su = 2 \Delta 10 5 yrs. If v t is less than 10 6 cm=s, than t su is greater.
So, -
p Я p=t su !Я 8:38
2\Delta10 5 \Delta3\Delta10 7
Я 10 \Gamma12 s=s
3 Calculations of the magnetic field decay
So, we can say, that the magnetic field of the source dissipated, and the
characteristic time of the dissipation , t d , was short enough: t d ! t E (t E --
Ejector's time). Because of that we have rapidly rotating INS on the Accre-
tor stage.
The detailes of calculations one can find in [3].
7

We calculated the magneto-rotational evolution of the star with the
mass M = 1:4M fi for accretion rates 10 \Gamma15 M fi =yr and 10 \Gamma16 M fi =yr passing
through the ISM with the density ae = 10 \Gamma24 g=cm 3 . These rates correspond
to velocities Я 20 and Я 40 km/s.
The magnetic field decay was studied by different authors (see [10]).For
conductivity we used formulae from [13] and [2]. Initially the magnetic field
was localized in the surface layer. Low accretion rates don't have much
influence on the magnetic field decay [1, 9].
On the figure 2 we show the decrease of the surface magnetic field for
different parameters. We used the model of the NS structure from [11].
On figure 3 the evolutionary tracks for the accretion rates 10 \Gamma15 M fi =yr
(fig. 3a) and 10 \Gamma16 M fi =yr (fig. 3b) are shown.
4 Conclusions
Observations of the accreting INSs can be an important test for theories of
the magnetic field decay. In that case we have a very ''pure'' example of the
decay, because a lot of effects of the interaction of the magnetic field, the
surface of the NS and surrounding plasma are not important.
Observations of the source RX J0720.4-3125 [12] showed the existence
of the INS with the rotation period which can be easily explained with the
assumption of the magnetic field decay. It can be important for estimates
of the total number of observable INS and for their appearence as periodic
X-ray sources [7].
We want to thank dr.F. Haberl for the information about the source,
drs. M.E. Prokhorov, V.A. Urpin, D.G. Yakovlev and V.M. Lipunov for
helpful discussion, and participantes of the summer school ''The many faces
of neutron stars'', where the idea of this article was born.
The work of D.K. was supported by RFFI 96-02-16905a, the work of S.P.
-- by RFFI 95-02-06053, INTAS 93-3364, ISSEP a96-1896.
5 References
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J. 1992. V.384. P.129.
2. Itoh N., Hayashi H., Kohyama Y. // Astrophys. J. 1993. V.418. P.405.
3. Konenkov D.Yu, Popov S.B.// Astronomy Letters. 1997. V.23, P. 200
(astro-ph/9707318)
4. Lipunov V.M.// Astrophysics of neutron stars, Springer, 1992.
8

5. Lipunov V.M., Popov S.B.// Astron. Zhur.. 1995. V.72, P. 711. (see
also astro-ph/9609185)
6. Ostriker J.P., Rees M. J., Silk J. // Astrophys. J. Letters 1970. V.6,
P.179
7. Popov S.B.//Astron. Circ 1994. No.1556. P.1.
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10. Urpin V.A., Muslimov A.G.// Astron. Zhur. 1992. V.69. P.1028.
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13. Yakovlev D.G., Urpin V.A.// Astron. Zhur. 1980. V.24. P.303.
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