A&A 426, 267-277 (2004)
DOI: 10.1051/0004-6361:20040455
U. Geppert1 - M. Küker1 - D. Page2
1 - Astrophysikalisches Institut Potsdam
An der Sternwarte 16
14482 Potsdam, Germany
2 -
Instituto de Astronomía, UNAM,
04510 Mexico DF, Mexico
Received 16 March 2004 / Accepted 2 July 2004
Abstract
We investigate the influence of different magnetic field configurations on
the temperature distribution in neutron star crusts.
We consider axisymmetric dipolar fields which are either restricted to
the stellar crust, "crustal fields'', or allowed to penetrate the core,
"core fields''.
By integrating the two-dimensional heat transport equation in the crust,
taking into account the classical (Larmor) anisotropy of the heat conductivity,
we obtain the crustal temperature distribution, assuming an isothermal core.
Including classical and quantum magnetic field effects in the envelope as a
boundary condition, we deduce the corresponding surface temperature
distributions.
We find that core fields result in practically isothermal crusts unless the
surface field strength is well above 1015 G while for crustal fields
with surface strength above a few times 1012 G significant deviations
from crustal isothermality occur at core temperatures inferior or equal
to 108 K.
At the stellar surface, the cold equatorial region produced by the suppression
of heat transport perpendicular to the field by the Larmor rotation of the
electrons in the envelope, present for both core and crustal fields, is
significantly extended by that classical suppression at higher densities
in the case of crustal fields.
This can result, for crustal fields,
in two small warm polar regions which will have observational consequences:
the neutron star has a small effective thermally emitting area and the
X-ray pulse profiles are expected to have a distinctively different shape
compared to the case of a neutron star with a core field.
These features, when compared with X-ray data on thermal emission of young
cooling neutron stars, would provide a first step toward a new way of studying
the magnetic flux distribution within a neutron star.
Key words: stars: neutron - stars: magnetic fields - conduction - dense matter - X-rays: stars
The presence of strong magnetic fields in neutron stars is one of their distinctive characteristics. In typical neutron stars the observed and/or inferred surface fields are of the order of 1012...13 G; for magnetars they even reach 1014...15 G (Vasisht & Gotthelf 1997; Kouveliotou et al. 1998). The strength, structure and topology, and time evolution of the magnetic field is intimately related to its origin, which is still an open problem and about which several scenarios have been proposed. During the supernova core collapse which produces the neutron star, the magnetic field possibly present in the progenitor is amplified by flux conservation and may reach values compatible with the observed ones (Woltjer 1964). Moreover, during the convective phase of the proton-neutron star efficient dynamo processes are likely to take place (Thompson & Duncan 1993). In these two scenarios one can expect the magnetic field to have a very complicated topology, the currents supporting it certainly span the whole stellar interior, and significant toroidal components generated by differential rotation can be expected. An alternative scenario had been proposed by Blandford et al. (1983) in which a small seed field is amplified in the upper (liquid) layers of the star through thermomagnetic processes. The currents producing this field are then located entirely in the stellar crust. Notice that these three types of scenarios are not exclusive at all, may act almost simultaneously and/or in succession, and may produce magnetic fields with a double or even triple structure.
Beyond the existence of a strong dipolar component, very little is known about the exact structure of a neutron star magnetic field. Several lines of evidence, both observational and theoretical as described by Arons (1993), indicate that non-dipolar (quadrupolar, octopolar, ...) components in pulsars have to be smaller than the dipolar one. On the other hand, the four-peaked shape of the light curve of the giant flare of August 27, 1998 from SGR 1900+14 (Feroci et al. 2001) is probably strong evidence for the presence of a large quadrupolar component in this magnetar.
A way to observe the structure of the star's surface magnetic field is provided with the observation of surface thermal emission in the soft X-ray band. The structure of the field in the envelope, a shallow layer below the surface, has a direct effect on the surface temperature and leads to a non-uniform distribution (Greenstein & Hartke 1983) with observational consequences (Page 1995) if its strength is above 1010 G. Modeling of the thermal X-ray pulse profile of the Geminga pulsar and PSR 1055-52 (Page & Sarmiento 1996) seemed to require the presence of a quadrupolar component with a strength compatible with the constraint that Arons (1993) had obtained for millisecond pulsars. (See, however, Harding & Muslimov 1998 for a different analysis with a purely dipolar field.) Moreover, Page & Sarmiento (1996) found that the superposition of an arbitrary quadrupolar component with a strength comparable to the dipole generally reduced the observable modulation of the soft thermal X-ray emission, much below observed values, simply due to the fact that it introduced more (up to four) magnetic poles and hence more warm regions at the surface. Together with the considerations of the previous paragraph, these results show that the surface and external field of radio pulsars could be, as a first approximation, reasonably well approximated by a dipolar field. In contrast, almost nothing in observationally known about the structure of the magnetic field inside the neutron star.
Considering physical processes in the interior of a neutron star,
transport processes are the most affected by a magnetic field.
The effects of the magnetic field onto the transport processes can be
roughly divided into classical and quantum ones (see, e.g.,
Yakovlev & Kaminker 1994 for a review).
The classical effects are due to the Larmor rotation of the electrons,
the main carriers of charge and heat, and are determined by the
magnetization parameter
where
is the gyrofrequency
of the electrons,
being their relaxation time and
their
effective mass.
Quantized motion of the electrons transverse to the magnetic field
causes the quantum effects which are of importance only if few Landau
levels are occupied.
Quantum effects play an important role for strong magnetic fields in
the outermost layers of the neutron star crust - the thin low density
shell of the envelope - but are negligible in the deeper layers.
The deeper layers of the crust, as well as the envelope, can, however,
be affected by classical effects in case of strong fields, i.e., large
,
and/or low temperature, i.e., high
.
In such a case the usual assumption of an isothermal crust could be
questionable and this is the issue we want to address in this paper.
As a first step in this study, in the present paper we will compare two extreme field configurations referring to the old dichotomy of "core'' and "crustal'' fields. In the first case we have a field penetrating the whole star while the latter is characterized by having the field and its supporting currents restricted to the stellar crust (see, e.g., Chanmugam 1992). Moreover, for simplification, we also restrict ourselves to axisymmetric poloidal dipolar fields. For identical external field structure and strength, in the case of a crustal magnetic field its strength in the crust unavoidably exceeds its surface value by one to two orders of magnitude (see, e.g., Page et al. 2000) and its effects on heat transport can naturally be expected to be much stronger than in the case of a field permeating the whole star whose internal strength can perfectly be everywhere inferior to its external strength. To date, there is still no compelling observational evidence in favor of or against either of these two hypotheses of core or crustal field, but recently Link (2003) argued that long period pulsar precession, as observed in PSR B1828-11 (Stairs et al. 2000), may be impossible if the magnetic field penetrates regions of the core where neutrons are superfluid and proton superconducting (see, however, Jones 2004b for a different point of view).
Our results will show that the structure of the magnetic field deep in the crust can potentially control the distribution of temperature at the stellar surface. This may open a way to study the structure of the magnetic field in the crust and provide observational features which may allow us to discriminate between the above mentioned two types of hypothesized field structure, crustal vs. core. More complex field structure, as, e.g., superposition of core and crustal poloidal dipolar fields, toroidal components, quadrupolar poloidal components, will considered in future papers.
The paper is organized as follows: in the next Sect. 2 the basic equations are introduced which describe the magnetic field and the heat transport influenced by it. The components of the heat conductivity tensor are given and the outer boundary condition is discussed. The physical input as well as the numerical method are shortly described. In Sect. 3 the results of the numerical calculations are presented. For different core temperatures, magnetic field strengths and geometries the crustal temperature profiles, the surface temperature distributions and the corresponding luminosities are calculated. Section 4 is devoted to the discussion of the consequences of the magnetic effects on the crustal and surface temperature distributions.
The thermal evolution of the crust is determined by
the energy balance equation
In this paper we intend to consider only the effect of the crustal
magnetic field onto the stationary temperature distribution in the
crust, which, in a first approximation, is assumed to be free of
heat sources and sinks.
The cooling process itself as well as the back reaction of the now
non-spherically symmetric temperature distribution upon the magnetic
field decay are beyond the scope of this work.
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Figure 1:
Thermal conductivity ![]() ![]() ![]() ![]() ![]() ![]() |
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In the relaxation time approximation the components of
parallel and perpendicular to the magnetic field,
and
respectively, as well as the Hall component,
,
are related to the scalar heat conductivity
and to the magnetization parameter
by (Yakovlev & Kaminker 1994)
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(5) |
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Figure 2:
Magnetization parameter
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Given the conductivities described above,
the magnetization parameter
varies
strongly throughout the crust by many orders of magnitude.
In
the values of both B and
span
a large range of
values throughout the crust and in
there is also a strong dependence
on the temperature T, chemical composition and the thermodynamic
phase of the matter.
For illustration we show, in Fig. 2,
in the crust at various uniform temperatures and a
uniform field strength:
however, neither T nor B will be uniform in our realistic
calculation presented below.
Note that values of
at T= 105 and
106 K
are very close to each other because at such low temperatures,
is
dominated by
which is temperature independent while
when going to increasingly higher temperatures
contributes more and more and hence
decreases.
Finally, Fig. 3 shows the resulting
:
notice its very different T-
behavior compared to
due to the simple fact that at high
we obtain
while
.
Note, that the contribution of electron-impurity collisions to the
collisional frequency is far from being understood.
Moreover, Jones (2001, 2004a) argues that the crust
is a glass rather than a bcc lattice, resulting in
.
However, calculation of the effect of electron-impurity collisions
(Flowers & Itoh 1976) assumes that impurity collisions are
uncorrelated, which requires that impurities are randomly located.
This raises the issue of how the impurities are arranged, besides their
concentration.
While well ordered impurities do not disturb significantly the electron
motion, highly disordered impurities will
(Yakovlev 2004, private communication).
In the former case the conductivity would be close to the perfect
lattice one while in the latter case it would approach the
conductivity of a liquid, i.e. be dominated by electron-ion
collisions.
These qualitative different behaviors are not reflected by the
"impurity parameter'' Q.
In this investigation, as described in the previous paragraph, we will
consider the crust to be a Coulomb crystal which is so far the only case
studied in detail.
Conductive processes which are not affected by magnetic fields can increase
the heat flux perpendicular to .
Recently, Yakovlev (2004, private communication) pointed out that there are
at least three processes which have to be considered carefully:
the convective counterflow of superfluid neutrons in the inner crust,
the diffusive transport by non-superfluid neutrons which may be present in
certain regions of the crust and the phonon transport.
These transport processes have not yet been considered in the context of
cooling calculations of neutron stars up to now.
They are beyond the scope of the present manuscript but should be
seriously investigated.
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Figure 3:
Thermal conductivity, in cgs units, perpendicular to the magnetic field,
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A dipolar poloidal magnetic field can be conveniently described in
terms of the (possibly time dependent) Stokes stream function
S = S(r,t).
The vector potential
is written as
,
where r,
and
are spherical coordinates.
The field components are then expressed as
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(8) |
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(9) |
For the present investigation we select for the crustal field
configuration a snapshot of the evolution of S(x,t).
The Stokes function S(x,t) was calculated by solving the induction
equation, applying the above boundary
conditions and an electric conductivity which reflects the same
microphysics as the heat conductivity does for the model under consideration
(see, e.g., Geppert & Urpin 1994 or Page et al. 2000).
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Figure 4:
Magnetic field lines of the core field
( left panel) and of the crustal field as applied in our
calculations. Both field configurations match for
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The initial value S(x,t=0) is a priori unknown and depends
on the field generation process.
For a crustal magnetic field, the function S(x,t=0) initially
vanishes in the core and, due to proton superconductivity, the
Meissner-Ochsenfeld effect prevents the field from penetrating into
the core (see, e.g. Page et al. 2000).
A typical crustal field structure is shown in the right panel of
Fig. 4.
In case the magnetic field is maintained by electric currents
circulating in the core the field penetrates the crust too but has a
qualitatively different structure.
Let us assume that for the core field there are no currents in the crust
and the field is maintained in the core by axisymmetric currents
circulating around the center of the star.
Then a dipolar field will penetrate the crust with the components
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(10) |
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(11) |
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(12) |
Denoting the components of the temperature gradient
parallel and perpendicular to the unit vector of the magnetic field,
as well as the Hall component by
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(13) | |
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= | ![]() |
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(14) |
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(15) |
Axial symmetry implies that, along the polar axis,
and
vanish while x2 Fx is
constant.
Mirror symmetry at the magnetic equator implies that
vanishes (but
can be very large).
Our aim in this paper will be to find stationary solutions for the
temperature distribution in the neutron star crust, i.e. to solve the
equation
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(19) |
In the lowest-density layers, close to the surface, matter is no longer degenerate and the magnetic field affects the equation of state. Appropriate treatment of this layer requires solving for hydrostatic equilibrium simultaneously with heat transport. Moreover, quantum effects of the magnetic field become important. To avoid these problems, we separate this layer, called envelope, from the crust and incorporate it in the outer boundary condition.
Quantum effects become significant (see, e.g., Yakovlev & Kaminker
1994, for a review) when electrons occupy only few
Landau levels, requiring thus densities
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The thinness of this envelope justifies a study of heat transport in a
plane parallel, one dimensional approximation and many such
calculations have been performed (see, e.g., for the most recent one,
Potekhin & Yakovlev 2001, hereafter PY01, and references therein).
In such 1D envelope approximation, a surface temperature ,
and hence
an out-coming flux
,
is chosen and
hydrostatic equilibrium and heat transport are solved toward
increasing densities up to
giving the temperature
at that density.
Varying
gives a "
relationship''.
Two-dimensional calculations of heat transport with magnetic
field have been presented by Schaaf (1990a,b) who
however restricted himself to the thin envelope
and an uniform magnetic field.
Tsuruta (1998) has presented results of two-dimensional cooling
calculations of neutron stars which included the quantizing effect of
a dipolar magnetic field in the envelope.
These 2D calculations showed that the 1D approximation is indeed very
good when the field affects heat transport only in the thin envelope.
We will use the 1D results of PY01.
We stop the interior integration at an outer boundary density
,
at radius
,
and, for each latitude
,
obtain a radial flux
and a temperature
which we match to the envelope.
At each point
we apply an envelope with a magnetic field
equal to the field we have at that point
(hence the "
relationship'' should be called a
"
relationship'').
The matching of the envelope with our interior calculation is simply
obtained by imposing
and
which is our outer boundary condition.
YP01 place the bottom of the envelope at the neutron drip
density, i.e.,
g cm-3, while we
intend to apply lower values down to
g cm-3
in order to extend the 2-dimensional transport calculation as far as
possible while still being safely in the non-quantizing regime.
Since the temperature profile in the 1D calculations of YP01 is quite flat
between
g cm-3 and 1010 g cm-3 the same
relationship can be applied for the lower
which we
consider a good approximation.
Explicitly, we apply their Eqs. (26) to (30) but replace their
,
which they assume to be spherically symmetric, by our
calculated angle-dependent
.
To illustrate the main features of this boundary condition, a good
approximation (Greenstein & Hartke 1983; YP01) is to write it as
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Figure 5:
The crustal temperature T (normalized on
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We will consider a
neutron star whose structure
is obtained by integrating the Tolman-Oppenheimer-Volkoff equation
of hydrostatic equilibrium.
The core matter is described by the equation of state (EOS)
calculated by Wiringa et al. (1988).
We separate the crust from the core at the density
g cm-3(Lorenz et al. 1993)
and use the EOS of Negele & Vautherin (1973) for the inner crust,
at
g cm-3,
and Haensel et al. (1989) for the outer crust.
We assume that the chemical composition is that of cold catalyzed matter,
as is the case for these two crustal EOSs.
The star so produced has a radius
km and the
crust-core boundary is at radius
km.
We solve the heat transport equation in its time-dependent form,
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(21) |
For the discretization of the spatial derivatives a staggered mesh
method as described in Stone & Norman (1992) is used in
spherical polar coordinates (r and ).
For the integration in time the fully explicit method turned out to
be prohibitively slow because the diffusion coefficients vary
strongly in space in the case of strong magnetic fields.
We therefore use the scheme in an operator-split implementation that
treats the radial diffusion terms implicitly while the horizontal
diffusion and the
term in
remain explicit.
This keeps the computational effort per time step small as the
equations to be solved are tridiagonal, but allows for a sufficiently
large time step to reach a stationary state within several hours of
computing time.
For the evaluation of the transport coefficients and the outer
boundary condition the temperature resulting from the last time step
is always used, i.e., only the radial derivatives are treated
implicitly.
The importance of the magnetic field-induced non-isothermality of the
whole crust is illustrated in Figs. 5-8.
The temperature profiles of Fig. 5,
when compared with Fig. 6,
show clearly the difference between a crustal and a core field,
the latter inducing temperature variations in the crust of much less
than 1% at
G while the former can result
in variations of a factor two for the same dipolar external field
strength and
K, and even much larger at lower
.
For strong fields, when
,
one has
and hence heat flows
essentially along the field lines and, given the large values of
,
no large temperature gradient can build up along them,
as illustrated in Fig. 7.
Only extremely strong core fields
may cause significant deviations from the isothermality of the crust;
for
B0 = 1016 G we could observe a difference of only 10% for
between pole and equator.
The qualitative difference between core and crustal magnetic fields is then
easily understood by observing that field lines are essentially
radial in most of the crust in the case of a core field while they are
predominantly meridional for a crustal field inhibiting radial
heat flow in a large part of the crust.
We could find significant differences to the isothermal crust model only if the
polar surface field strength exceeds 1012 G.
Additionally, the core temperature should be smaller than 108 K.
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Figure 6:
Same as Fig. 5 but for
a dipolar core field with
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In the case of a star-centered core field, Fig. 6 and the right panel of Fig. 7 show that the stellar equator is very slightly warmer than the pole. This is a direct consequence of the outer boundary condition, i.e., the envelope: magnetic-field-induced anisotropies are weak within the crust for such fields but are large in the envelope which is more insulating around the equator than around the poles.
A 3D representation of the crust temperature is shown in
Fig. 8:
it may be surprising in the sense that heat flows into the crust from the
core, at a fixed
,
and out of the crust at the surface
and most of the heat comes out in the polar regions which are, first,
warmer than the equator and where, second, the envelope is less insulating.
So heat must be flowing from the equatorial regions toward the polar regions,
i.e., apparently from cold regions toward warmer ones!
That this situation cannot violate the second law of thermodynamics
is built into the heat conductivity tensor
which is
positive definite and guarantees that
always.
Figure 9 illustrates this situation: at a point in the northern
hemisphere
is almost perpendicular to
and
is clearly pointing toward the equator.
However, since
,
,
and
we have
and the resulting
is negative, i.e., pointing toward the pole:
heat is flowing from the equator toward the poles but does it along
the magnetic field lines from warmer regions toward colder ones.
Our numerical results show that, e.g., in the northern hemisphere,
is negative in large regions and is positive elsewhere
depending on the orientation of
.
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Figure 7:
Representation of both field lines and temperature
distribution in the crust whose radial scale
(
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Figure 8:
3-D presentation of the temperature distribution in the
crust for
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Figure 9: Illustration of the magnetic-field-induced anisotropy (see text for details). |
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The results shown in Fig. 5 for the crustal field
configuration with the typical strength
G confirms the
statement that with increasing magnetization parameter the anisotropy of the
temperature distribution within the crust increases too. The magnetization
parameter increases with increasing magnetic field strength and with
decreasing crustal temperature, i.e. with decreasing
,
because the relaxation time of electron-phonon collisions grows
strongly in the course of cooling. Therefore, while the temperature profile
along the poles shows practically no gradient with decreasing
the ratio
decreases
from 0.95 to 0.5 and 0.2 when
decreases from 108 K to
107 and 106 K, respectively.
Also, an increase of the magnetic field strength amplifies that difference.
Applying the same field structure but
B0=1013 G
the temperature ratio becomes smaller than 0.1 for
K.
Note also that the highly unknown parameter of the impurity concentration (Q=0.1 throughout this paper) affects the relaxation time: the more impurities the shorter the relaxation time of electron-impurity collisions. Therefore, while in a highly impure crust the magnetic field effects on the crustal temperature distribution are reduced, in a very pure neutron star crust these effects will be even more pronounced.
The magnetic field permeating the envelope induces a non-uniform
surface temperature distribution, mostly due to quantizing effects
of the field at low densities, even in the case of a uniform
crustal temperature (Schaaf 1990a,b; Page 1995).
The non-isothermality of the crust produced by a crustal magnetic
field will result in an even more pronounced non-uniformity of the
surface temperature.
These effects are shown quantitatively in Fig. 10
where the two cases of core and crustal fields are compared.
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Figure 10:
The surface temperature ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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The different field structures do not only affect the relation between
polar and equatorial surface temperature but also the setup and the
extension of the warm polar regions.
In Fig. 10 it is seen that the crustal field can
cause a much smaller warm polar region than a star-centered core field of the
same polar surface strength would do.
While for the core field geometry with
B0=1013 G and
K the surface temperature at a polar angle of
is reduced only to 0.93 of its polar value, for the crustal field
configuration the corresponding value is 0.82.
With decreasing
this difference becomes larger:
thus for
K the corresponding values for the core and
the crustal field are 0.93 and 0.77, respectively.
The setup of a clearly distinct warm polar region becomes more and
more pronounced with an increasing magnetization parameter (i.e. increasing
field strength and/or decreasing crustal temperature and/or decreasing impurity
concentration) for a crustal field configuration, while the shape of the surface
temperature distribution is almost unaffected by the magnetization parameter in
case of a core magnetic field.
Having a non-uniform surface temperature distribution,
,
the effective temperature
can be calculated, from its
definition, as
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Table 1:
Photon luminosities for a neutron star with
K. The
differences between isothermal crust and star-centered field are
negligible but significant between them and crustal fields larger
than 1013 G.
Table 2:
Photon luminosity for a neutron star with
K.
The almost identical luminosities obtained for the isothermal crust
and a crust penetrated by a star-centered core field (Table 1)
reflect the little effect even strong fields of that structure have
onto the crustal temperature distribution.
While for a star-centered magnetic field (both with isothermal and
non-isothermal crust) the photon luminosity increases with increasing
field strength, in neutron stars possessing a crustal magnetic
field above a certain strength (1012 G) the luminosity is
reduced since then over the major part of the surface the heat
insulating effect of such a field configuration dominates;
its
-component causes strong meridional heat fluxes toward the
polar region whose area, however, becomes smaller with increasing field
strength.
This effect impedes the radial heat transport strongly, finally less
heat can be irradiated away from the surface and the photon cooling process
will be decelerated significantly in comparison to a non-magnetized
neutron star (here
G) or even to a strongly
magnetized neutron star with a star-centered core field.
Table 3:
Photon luminosity for a neutron star with
K.
A sufficiently strong magnetic field modifies the thermal insulation of the
envelope by increasing the longitudinal thermal conductivity due to the
Landau quantization of the electron motion and increases the insulation
perpendicular to
because of the classical electron Larmor rotation
which may diminish the transversal thermal conductivity considerably.
Here, importance of the classical (Larmor) effect for the heat transport through the whole crust has been demonstrated in the particular case of a specific magnetic field configuration, exclusively maintained by electric currents circulating in the crust, which does not penetrate the core of the star and, consequently, has a large tangential component in most of the crust. Besides its insulating effect, the tangential crustal magnetic field creates a meridional heat flux from the equatorial regions towards the high latitude ones. It transports the heat, which is dammed in the equatorial region, towards the poles where it can be much more easily irradiated away. Eventually this leads to an extended equatorial belt which is much cooler than the poles.
In case the magnetic field is allowed to penetrate the core of the star, and assuming a star-centered dipolar geometry in the crust, we have shown that the stationary thermal state of the crust is very close to isothermality. We are thus confirming, for this field geometry, the assumptions of the models of surface temperature distribution of magnetized neutron stars (e.g., Page 1995; Page & Sarmiento 1996; Shibanov & Yakovlev 1996) which considered only the effects of different magnetic field structures and strengths in the envelope and assumed the rest of the crust to be isothermal.
Recently, Svidzinsky (2003) argued that the accumulation of magnetic field lines along the proton superconductor at the crust-core boundary, due to the Meissner-Ochsenfeld effect, produces an insulating barrier preventing heat to flow between the crust and the core. Our results are in the same line of thought but they do not confirm his claims of insulation of the crust from the core. The field geometries, i.e., the Stokes stream functions (Eqs. (6), (7)), we used in our calculations come from field evolution models (Page et al. 2000) of crustal fields in which the migration, and accumulation, of the currents and the field lines toward the crust-core superconducting boundary was modeled in detail and, as illustrated in Fig. 7, they allow heat diffusion through the crust-core boundary. Nevertheless, other field evolution scenarios, e.g. the possible expulsion of the magnetic flux from the core by a proton type I superconductor (Link 2003; Buckley et al. 2004) may produce a much stronger piling up of field lines tangentially to the crust-core boundary and result in more efficient thermal insulation.
Our results have several observational consequences which will be explored in details in forthcoming papers. For both magnetic field structures considered here, the thermal emission of cooling neutron stars will show low amplitude pulsations due to the non-uniform surface temperature exhibited in Fig. 10. Even if the crust remains (almost) isothermal the effect of the magnetic field in the envelope still creates a significant meridional temperature gradient. However, in the presence of a purely crustal field, that temperature gradient will be considerably steeper than for a core field and pulse profiles of different shape and amplitude can be expected. Generally, the steeper meridional temperature gradient will enhance the pulsed fraction of the light curve. However, since it is strongly affected both by the inclination of the line of sight and by the gravitational light bending (see e.g. Page 1995; Heyl & Hernquist 2002), these influences on the light curve have to be taken into account. Clearly, the effect of a crustal magnetic field on the surface temperature distribution will counteract the reduction of the pulsed fraction by gravitational light bending.
A second observational consequence comes from the warm polar region
which in case of a strong crustal field is much
smaller than for star-centered and/or weak magnetic fields
(see Fig. 10).
This may open a new way to distinguish between crustal and core
magnetic fields:
a strong crustal magnetic field implies a smaller effective
area for thermally emitting cooling neutron stars.
For example, the "Three Musqueteers''
(PSR 0656+14, PSR 1055-52 and Geminga: Becker & Trümper1997)
as well as RX J185635-3754 (Pons et al. 2002) all
have
km when their thermal spectra are fitted
with blackbody spectra.
If these radii would coincide with the radius of the neutron star,
the equation of state describing the state of the core matter would
have to be extremely soft.
A relatively small warm polar region, created by a strong
crustal field and emitting almost all the thermal radiation would be
a reasonable explanation for such small
.
The differences in the photon luminosities for a star-centered or a crustal field will also affect the long term cooling of neutron stars. A neutron star having a magnetic field confined to its crust will stay warmer for a longer time, due to its lower photon luminosity, than a neutron star with a field penetrating its core. The strength of this effect has to be explored by 2D cooling calculations.
We also mention the consequences the non-isothermality of the crust may have for the crustal field itself. Since the electric conductivity in the hot polar region is much smaller than in the equatorial layer, the field decay will be affected too and may cause differences in the field structure. This "back reaction'' of the field onto its own decay via a field driven non-spherical symmetric crustal temperature distribution will be subject of future investigations.
As discussed in the introduction, the dichotomy "core'' versus "crustal'' magnetic field is probably an over simplification. More realistic cases, comprising a superposition of a core and a crustal field as well as including toroidal components will be considered in future work. The inclusion of higher order multipolar components will also be needed. Nevertheless, one can expect that any field configuration which presents strong meridional components in the crust will affect the surface temperature.
Acknowledgements
The authors thank the (anonymous) referee as well as D. G. Yakovlev whose comments helped to improve the first version of this manuscript. U.G. is grateful to M. Rheinhardt for discussions and to G. Ruediger, whose engagement enabled the realization of the DFG-project "The interaction of thermal and magnetic effects in neutron stars'' (RU 488/18-1). Part of this work is supported by a binational grant from DGF-Conacyt #444MEX113/4/0-2. D.P.'s work is partially supported by grants from UNAM-DGAPA (#IN112502) and Conacyt (#36632-E).