Radiophysics and Quantum Electronics, Vol. 41, No. 1, 1998
RX J0720.4-3125 AS A POSSIBLE IN NEUTRON
EXAMPLE
OF MAGNETIC
FIELD
DECAY
STARS"
D. Yu. Konenkov
a n d S. B. P o p o v
UDC 524.354.6
We consider the possible evolution of the rotation period and magnetic field of the X-ray source R X J0720.~-3125, assuming that this source is an isolated neutron star aecretin 9 from the interstellar medium. The magnetic field of the source is estimated to be 10e - 1.00 G (the most probable value is about 2.10 s G}, and it is di.~cult to ezplain the observed rotational period 8.38 s without invoking the hypothesis of magnetic field decay. For calculations we used the model of ohmic dissipation of the field in the core of the neutron star. Estimates for the accretion rate (10 -14 - lO-lSM| velocity of the source through the interstellar medium (10 - 50 km/s}, and neutron star age (2.100 - 101~ yrs) are obtained.
1. INTI~ODUCTION Recently particular attention has been drawn to isolated neutron stars (INS) which are not observed as radio pulsars and are virtually unaccessible for observations in other ranges of the electromagnetic spectrum because of their small luminosity. The idea itself of observation of such objects was born long ago [1], and in 1991 Treves and Colpi [2] assumed that the INS accreting from the interstellar m e d i u m (ISM) can be observed in UV and X-ray ranges from the KOSAT satellite. Here we present the results of the work devoted to the object ILK J0720.4-3125 discovered by Haberl and Pietsch [3]. Presumably, this object is an accreting INS with period 8.38 s. There are four possible states of a neutron star (NS) in a low-density plasma: ejector (E), propeller (P), accretor (A), and georotator (G). The stage is determined by the relationships between four characteristic 2GM radii: Rh the radius of the light cylinder, Rst, the stopping radius, RG - - - , the radius of gravitational
vL
trapping, and Rr -- {G M ~ l / 3 the radius of corotation. \w2
J
,
As a result, we have two critical periods, P~ and PA, which separate different stages of the NS. If P < PE, then the NS is in the stage of an ejector (i. e., a pulsar radiating rotational energy). If PE < P < PA, then the NS is in the propeller stage and ifp > PA and Rst < RG, then we have the stage of an accretor. A situation is also conceivable in which p > PA but Rst > RG. Accretion is not possible in this case since a magnetosphere similar to terrestrial is formed under such conditions. To describe the evolution of the period it is convenient to introduce the so-called gravimagnetic parameter, which represents a combination of quantities often met in the evolution equations. The gravimagnetic parameter y was described in [4]. Figure 1 shows three examples of evolution tracks for the NS ending its evolution as an accretor. I. E --* P --~ A is the evolution of the NS in an ISM of constant density in the absence of magnetic field dissipation. * At the end of July 1977, after this paper was completed and submitted for publication, similar results were obtained independently by John C. L. Wang, Astrophys. J., 486, Ll19 (1997).
Lomonosov State University, Moscow, Russia; Ioffe Physical and Engineering Institute, St. Petersburg, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 1, pp. 28-35, January, 1998. Original article submitted June 10, 1997'. 16
0033-8443/98/4101-0016520.00 ~1998 Plenum Publishing Corporation
Fig. 1.
P
- y-diagram.
H. E ~ P --* A ~ P ~ A is the evolution of the NS passing through a giant molecular cloud (GMC) in the absence of magnetic field dissipation. III. Evolution with magnetic field dissipation. The third case is of particular interest for us. 2. ANALYTICAL ESTIMATES OF PULSAR PARAMETERS In this section we give the estimates of the source parameters which were not included in [5]. Consider the condition of equality of the Alfv~n radius RA and the corotation radius Rco. From this condition we find the accretion period
PA ~ 6. ln~,,e/L'-s/~'-,g/7 (' ~ '~-111~ -.. ,..~o ,..-, ooo, ~--oo J
[s],
(~)
where ~30 is the magnetic dipole moment in units of 1030 G-cm z, P-24 is the density of the ISM in units of 10= 24 g / c m 3, and voo6 is the velocity of the NS relative to the ISM in units of 10 e cm/s. For accretion p > IDA. If these periods are equal, then ~.~/7
8.38
3/~' -o17 ( M '~ II17
3o - C~.p-24'.'oo. ....
,/~-3/,
"
~Moo/
(Y_~) ~/6
We have/~30 = u.uulp_24voo~
for p = PA = 8.38 s (the p e r i o d of ~
J0720.4-3125). For
the neutron star radius R = 10 b-~ we have B < 7 9109 G.
If we adopt the hypothesis of neutron star acceleration from a turbulent interstellar medium, then we arrive at the expression imposing a limit on the magnetic field [5, 6]: .....
1/3 2 / 3 . 1 / 3 - 2 / 3
13/3-2/3 /~______~)-8/3 b].
P~q = ~,~oo,~, ~3o ~,~ p-2.~ %0, ~,,~
~
V
(2)
~
Here, I4s is the moment of inertia in units of 104s g-cm 2 and re6 is the turbulent velocity in units of 10 6 cm/s. Hence, #30 = \ 2 3 5 5 ) "
"t
"45
/~-24"oo,
t,
MOO 17
T h u s , B .~, 2.1 9 10 s G. In [5] it was shown t h a t this field cannot be smaller t h a n 106 G since o t h e r w i s e the p u l s a t i o n phen o m e n o n will not occur. Assume t h a t initially the n e u t r o n star had a period of the order of 0.01 s ( t h e exact value o f this initial period is not o f interest to us; it is only i m p o r t a n t t h a t this period be m u c h smaller t h a n 8 s a n d t h a t initially the star be in the ejector stage). This means t h a t a spinning down to 8.38 s was necessary. Let us estimate the time required for such a deceleration. T h e deceleration t i m e is determined by the final r a t h e r t h a n initial period! "ill/#.. ~t/4. 1/2 - 1 / 4 -1/2 RE = ~vt~t] /'L30P-24 Voo,
dIw _ dt
#2 R,
_sAP
p2
=
=
cl<,,,
-
[S], 2
lp,
k<#2(2") 2
At
c3p a
Thus, for A p = p fP 2c3
7 2 --1 --2
At _ kd~2(27r)2 _ 3.10 P~k t #ao 14s [yrs]= lCL9L--1/2 .--1 - - 1 / 2
= 3. ~u ,~
--lr
~so P-24 %o~14s [yrs].
For #so < 0.01 At > 3 9 i0 tt yrs > > tHubble. This means that initiallythe star could not have a small field. Since the neutron star is in the accretion stage, its period must be much greater than the ejection period: p > PPropeller > PEW e have PE ~. 10 s for standard parameters; thus, the fieldmust be much smaller than its standard value (B < < 1012 G) to ensure that the star withp = 8.38 s be in the accretion stage. Let us try to estimate the characteristictime of spinning up and spinning d o w n of such a neutron star: _ _
tsu : tsa =
l~/lvtRo
=
/,
vs
-I
iv
-t
vt
~-i
w h e r e tsu and tsd are the characteristic times of period variation. For p = 10 s the time tsu = 2 9 10 s years. T h e decrease of vt f u r t h e r increases this time.
Thus,
8.38 ~ p/tsu<~. 2- 10 s 9 3. I07 ~ lO-t2 s/s. 3. C A L C U L A T I O N S OF T H E M A G N E T I C FIELD DECAY
Tl~us, it can be said with confidence that it was exactly the field dissipation that took place and the characteristictime of dissipation, td, was fairlysmall: td < tE (tE is the lifetimeof the ejector). This is the only reason why we observe a fast-rotatingneutron star in the accretion stage. W e calculated the magneto-rotating evolution of a N S of mass M -- 1.4M| for the accretion rates 10 -Is Me/yr and 10 -16 M e / y r when the star moved through an I S M of density p = 10 -24 g/cm s. Such accretion rates correspond to the velocities~ 20 and ~ 40 km/s. The decay of the magnetic fieldin the crust of a neutron star was studied by m a n y authors (see [7l). The analytical expression for the impurity conductance was obtained by Yakovlev and Urpin [8] and the phonon conductance was taken from [9]. The initialfield is assumed to be localized in a surface layer of certain thickness. The density of the matter P0 at the inner boundary of this layer is the parameter of the problem. The value of the impurity parameter Q is assumed to be independent of the depth.
18
BIBo '
.
0.1
.
.
.
.
logB ....... 4:. '-:'. ! " 13 ...~-~4...., ":~'.~ .... ~ 12 ~ ~
.
\ ~ ~ 2 3 ~ ,
~
"-~..'~',
11
9.4'
,,o
0.001 0.0001
...... 2
. . . . . . 4
6
8
8 7
./ ""
!"
13 ~4. "-'i, 12 !11
:".
io
",
t
9.3
".3 9$
1
10 100
'"
9
/' 9.s
7
:: % log
0.01 0.1
1
10 100 0.01 0.1
P, s
Fig. 2. Change of the surface magnetic field of an isolated neutron star with time for standard cooling of the star. Curves 1, 2, and 3 correspond to the initial depths of occurrence 1011, 1012, and 1013 g/cm s. The solid curves correspond to Q = 0.001, the dashed curves correspond to Q = 0.01, and the dash-dot curves correspond to Q=0.1.
P, s
Fig. 3. Evolution tracks of the NS for 3~ = 10-1SMo/Yr (a) and .~r = 10-1SM| (b). The model parameters for each track are described in the text. The dotted lines correspond to p = PE and the dash-dot lines correspond to p = P.4- The dashed line in Fig. 3a shows the evolution of the NS for the second track without allowance for the acceleration in a turbulent ISM. The numbers at the track marks denote the logarithm of the NS age expressed in years. The observed radio pulsars are shown by dots.
Figure 2 shows the decrease in the surface magnetic field of a neutron star with time for different parameters p0 and Q. For the calculations we used a neutron star model with a Friedman-Pandharipande moderately hard equation of state [10] in the core of a star of mass M = 1.4 M o , radius R = 10.6 kin, and crust depth A R = 940 m. Accretion influences the evolution of the field. Firstly, it heats the crust of the neutron star [11], thereby decreasing the conductance. Secondly, this produces a flow of matter which is directed to the star center and tends to transfer the field to the deeper layers. Calculations show [12] that accretion with rate /~ < 10 -14 M| accelerates only slightly the decay of the field. Since the description of the acceleration process in a turbulent ISM is difficult, we used a simplified model to examine the evolution of the period in the accretor stage. When the NS enters the accretor stage the decelerating torque exceeds considerably the accelerating torque since the acceleration occurs in a turbulent ISM where, unlike the binary system, the constant accelerating torque is absent. However, we can obtain an analogue of the equilibrium period Peq which corresponds to the stationary solution of the Fokker-Planck equation (see [4, 6]). Hence, the evolution of the period in this stage was considered a stationary deceleration from PA to Peq and then the period was assumed to be equal to Peq, which, in turn, was changed owing to the decay of the magnetic field. In Fig.3 we show the evolution tracks of the NS for the accretion rates 10 - i s M o / y r (Fig.3a) and 10 -16 M| (Fig. 3b). The tracks marked by 1 in both parts of Fig. 3 illustrate the period evolution with the maximum possible rate of magnetic field dissipation. For the accretion rate 10 -15 M| the model parameters were as follows: Po = 3-1013 g / c m 3, Q = 0.02, and vt = l0 s cm/s. Once Pea becomes equal to the current period, the NS begins to be accelerated by the turbulized ISM. The period fluctuates about Peq. The fluctuation amplitude was not estimated. The tracks marked by 2 illustrate the evolution with a slower decay of the field. For ,~r = 10 -15 M| we put Q = 0.01 and the values of the remaining parameters were the same as for the first track in Fig. 3a. Tracks 1 and 2 coincide in the initial stage of the evolution when the dissipation rate does not depend on 19
the impurity concentration. However, the tracks diverge over I0 v yrs. As a result, the star is in the ejector stage for 2- 109 years and then passes to the propeller stage for p "- 3 s and B -- 4.101~ G. The deceleration in the propeller stage is faster than in the first case, and there are three reasons for that: the longer the period, the greater the value of the magnetic field in the transition from the ejector stage to the propeller stage, and the slower the decay of the field because of the low value of the parameter Q. In the accretor stage the N S spins down to 190 s over 2 9 109 yrs, and the acceleration effect in a turbulized I S M can be significant when the field decays clown to 4. I0 s G. In the absence of such acceleration (vt = 0) the track is shown by a dashed line. The final period is of the order of 200 s. Turbulent acceleration can decrease this period. For M = 10 -I~ M | and for the firsttrack we chose the following parameters: P0 = 3- 1013 g / c m a, Q = 0.01, and vt = 106 cm/s. The transition to the propeller stage is over 2.7 9 109 yrs for the period 3.1 s and magnetic field 2 9 10 l~ G. The NS changes to the accretor stage with period 4.9 s and magnetic field 7 9 10 s G. The spinning down is absent in the accretor stage since the magnetic field is small. The time of field decay to 4 9 106 G amounts to 6 - 109 yrs. In this case, p -- 5 s. The second track in Fig. 3b was calculated for the greater initialdepth of occurrence of currents corresponding to po ---6 9 1013 g / c m 3. Hence, in the initialstage the decay of the magnetic field is slower and the ejector stage lasts 2.1 9 109 yrs and the propeller stage lasts 1.9.109 yrs. For the field 4- 10 s G the period is 63 s.
4. CONCLUSION The observation of INS periods can become an important test for NS magnetic field decay theories. The absence of external effects (for example, the absence of a companion star in the binary system) allows the decay to be considered in "pure" form. The observations of R X J0720.4-3125 [3] show the presence of a period which cannot be explained without invoking the hypothesis of magnetic field decay. Such a decay can be critical for the estimation of the total number of observed accreting INS. The observed period and temperature of the X-ray source R X J0720.4-3125 can be explained within the framework of the hypothesis of I S M accretion to an old INS. W e showed that the magnetic field of the N S is small in this case (B < 109 G). For such a magnetic field the time of deceleration to p -- 8.38 s exceeds the age of the Universe. W e assumed that the N S was born with a higher value of the magnetic field and that the field decreased strongly during the evolution. Using the model of ohmic dissipation of the magnetic field in the crust of a NS, we calculated the possible evolution of the N S in the B - P diagram. The observed value of the rotation period can be obtained for Q ~ 0.01 - 0.05. However~ the evolution of the period depends on the rate of field dissipation and, therefore, on the parameters of the field decay model. For example, the change in the impurity parameter Q by two times led to a change in the period in the accretor stage by more than two orders of magnitude (tracks 1 and 2 in Fig. 3a). That is w h y observations of the rotation periods of old isolated N S can be an important test for N S magnetic field evolution models. The decay of the magnetic field can affect the estimate of the total number of observed accreting NS. In particular, since the formation of a periodic X-ray source in the propeller stage requires a strong magnetic field [13], the decay of this field can decrease the number of such sources. The absence of pulsations in other candidate INS can be an indication of decay of the field to such an extent that it carmot guide the plasma moving toward the INS poles.
We express gratitude to F. Haberl for information on the source and to M. E. Prokhorov, D. G. Yakovlev, V. A. Urpin, and V. M. Lipunov for discussions as well as to the participants of the conference "The Many Faces of Neutron Stars" which provide the impetus to write this paper. The work of D.Yu. Konenkov was supported by Russian Foundation for Fundamental Research Grant No. 96-02-16905a and the work of S. B. Popov was supported by Russian Foundation for Fundamental Research Grant No. 95-02-06053a, INTAS Grant No. 93-3364, and ISSEP Grant No. a96-1896.
20
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