Thermal radiation from magnetic neutron star
surfaces
J.F. P ÉREZ –A ZOR ÍN , J.A. M IRALLES AND J.A. P ONS
Departament de Fı́sica Aplicada, Universitat d’Alacant, 03690 Alacant,Spain
3 Solid surface model
, -/.130 2452*6578:9 -/.130 2452*657 -. ;7
9 - . <0 2=52*657>@? - . A0 2=526@7
9 -CB
-CB
J
H
I
- KMLN . DEGF HO;7P?Q
EGF
Suppose the solid surface emits according to Kirchoff’s law
[2]:
where
emissivity;
5 ION’S EFFECT
4.1 Emissivity
The effect of the ions is visible in the next figure. We compare the observed flux from two different NS models: a single temperature BB and the solid surface model described in
4.3 (with and without ions). In both cases, we take into account interstellar medium absorption (
hydrogen column density).
Normalized emissivity integrated in all possible incident
angles as a function of energy for different orientations of
the magnetic field (B
G and T
K):
reflectivity
1
0.7
0.6
They seem to have high magnetic fields (B
G).
The optical counterparts have been detected in four
cases, showing a systematic excess flux of about a factor 5 10.
Light element atmosphere models are ruled out by the
multiwavelenght observations.
Heavy element atmosphere models don’t explain the
absence of spectral features.
Single component models cannot account for the optical excess observed in 4 sources.
Two component models (i.e. two BB) can reconcile the
optical and X-ray observations.
To explain the lack of spectral features and the 2–component
spectrum we consider the EMISSION FROM A SOLID
SURFACE with NON UNIFORM TEMPERATURE, but
HOW CAN A NS BE LEFT WITHOUT AN ATMOSPHERE?
keV
keV
e cyclotron frequency
keV
ion cyclotron frequency
ion plasma frequency
– Critical temperature (Fe):
"!$#%'&'()
* ,
,
W ] g ˆ » 5? [ ] g ] † ¥M¦ ž 7 9 =**¿¾
. ] † W ] g ½¼
D ¹º
9‰—sÀ ]‡ÁŠÂQ] † ] ( ] g
9‰Và À ]‡Á…ÂÄ] † ] ( :
9-+ ]½ÅÆ] g
– POLE (
):
For
):
).
. 24Ç7 †É È . W ’8¥M¦ ž ÊÇs7kË
V
*
j
j
j
j
,
"
Í
Í
Ç
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Ç
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Í
Ç
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=
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Ÿ
ž
=
ž
¤
Ÿ
[ \1Ì •ÏÎsÐ • Î •ÑÎsÐÉÒ • Î Ò Ò Ò
Ç 2Ð B
Ò Ò
4.2 Spectral energy distribution
(
)
K)
– S T —O S O T ?ŠO U S T W ] I R S T
à
à
0.1
à
z¬ 0 0
˜0 0
˜ Œ h

Œ
¯
®
%
®
U
7
®
%
®
U
˜C­ Œ KLN . Y  ˜ Œ W  KMLN . Y  7 ˜ 

Œ
¯
®
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U
7
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U

˜CŒ­ Œ KLN . Y Œ ˜ Œ W KMLN . Y Œ 7 ˜ 
Imposing boundary conditions at the surface (conand , and
tinuity of normal components of
tangential components of
and ), we obtain the
components of the reflected wave in terms of the
):
components of the incident wave (
Parallel:
à
.  2=Çs78 [ g . W ’8¥¦ ž Çs7
g —!šÓ‰ g ,
Í
Ê
Ô
Õ
§
-Ì
- ¥¦ ž Ç Ò ž=Ÿ¤ Ç Î Ç ÎÐ
k
§B
ÖØ‹× k
with
G and
FINALLY THE EMISSIVITY
9‰@? ±
K.
The size of the emitter can be understimated
by a large factor if a simplified BB model is
used. No need to appeal to strange stars to
explain the apparent small radius of some
NS [3, 9].
Depending on the magnetic field’s strength
and including the effects of the ions, the optical flux is about 5 times larger than the corresponding BB extrapolation of the best Xray fit. This is similar to the observed spectra
from INS.
scale factor
1000
100
BB o
θo=0
θo=45o
θo=90o
References
10
[1] Burwitz, V. et al., 2001, A&A 379, L35
1
[2] Brinkmann, W. 1980, A&A 82, 352352
0.1
[3] Drake, J.J., et al., 2002, ApJ, 572, 996
0.01
[4] Geppert, U., et al., A&A, submitted, astro-ph/0403441.
[5] Lai, D. 2001, Rev. of Mod. Phys. 73, 629
0.001
[6] Pavlov, G.G., et al., 1996, ApJ 472, L33
1e-04
0.001
0.01
– Transverse:
Total reflectivity
à
Assuming the following anisotropic temperature distribution:
For each incident polarization we obtain the reflectivity:
 °® %  ® W ®  ® Œ ¯® % Œ ® W ® Œ ® ± .  W Œ 7*H D
1
4.3 Observed flux
Transverse:
– Parallel:
0.1
At low energies, the behavior is similar for different
magnetic field strenghts
Fν (arbitrary units)
3.3 Boundary conditions and emissivity
0.01
Spectrum is essentially featureless
) is
The presence of the magnetic field produces
a large anisotropy in the surface temperature
distribution.
E (keV)
– S T˜™T :š1› z `œ . – S T 78V
The apparent estimated value of the radius (
3.4 times smaller for a single BB.
µß H z
Spectrum is almost featureless. Minor differences only at energies where the interstellar
medium absorption makes difficult to distinguish between models.
1
0.001
considering that
Optical flux including ion effects (green) is 4.5 times
larger, similary to what has been observed in INS’s.
Solid surface models show a significantly depressed spectrum (up to factor 10) compared
to a single temperature BB with the same effective temperature.
BB
Bp=1012G
Bp=5 1012G
Bp=1013G
Bp=5 1013G
1e-04
0.001
1
6 CONCLUSIONS
1000
10
0.1
E(keV)
and con-
define the plane of incidence
100
0.01
X-ray spectrum is practically indistinguishable (notice
that
= 11.5).
and
the emissivity approaches to BB (
Dipolar magnetic field:
stant temperature (
0.1
§
and
– EQUATOR (
BB fit
θo=90oo
θo=90 w/ ions
1
0.001
0.001
] ¸ ] †
Introducing the Maxwell tensor,
where
+
K,
0.01
0.01
€
b i . { WŠ™ž4Ÿ 9‡7 W b .¢¡  ? D { x WŠ£;ž4Ÿ 9‡7 W {  W ¡ £ ž=Ÿ¤ @Yž=Ÿ¤ . D 9‡7¥¦ ž/§ . b ? ž=Ÿ¤ Y 7 =4 . £¨W b  7
€
 x ?…{©Ž £ V{ x ? 
¡ ž4Ÿ¤ Yª 5? ž=Ÿ¤ 9 . W ¥¦ ž § 7l«
Thermal conductivity is different in the directions
parallel and perpendicular to B temperature varies
over the surface. Need to solve the 2D diffusion equation in the condensed envelope [8, 4].
10
The plasma resonance depends on the orientation of
the magnetic field (neglecting damping):
and applying the complex Snell law , the dispersion relation
leads to a quartic equation
1
]Q² ‡] D´† ³ µ? VW ¶ W [ . ] g H ]‡† 7 ·
damping frequencies
eV
– Zero-pressure density:
gr/cm
0.1
Emissivity is strongly reduced for low energies because one mode developes a large imaginary part for
e plasma frequency
effective relaxation times
0.01
E (KeV)
3.2 Dispersion relation
Highly magnetized NS can undergo a phase transition that turns the gaseous atmosphere into a solid
[5].
G,
10
0
0.001
Fν (arbitrary units)
Model:
.
0.1
a www.ioffe.rssi.ru/astro/conduct/condmag.html
2 Effects of the magnetic field
g
K,
0.2
M
b
c
I
c
]
J
g
h
S^ T'_a` R*dfe S T e S h T
[
\
] gJh c a i*j o lc k4n mn p/q rs&'= i B (
e S hT B
vwyx ? x Ylz |}
Rt$u Yz { ~€‚ 5?
]cM7 gJ h c W ] g$ h S 7
. ]„ƒ…]‡† h . ]‰ˆ…]‡† h S WŠY ]Œ‹ ]
c
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g
h
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g
{@? ] WŠY ]>] ‹h S
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x .  W 7Ž z .  ? 7
D
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] † h o † n !r5 i B (
c

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] † h S oX‘ † Vq #’! i B
] gJ h S Vq D !%“(1=4&5”l• *i B
] ‹ $H e  Ž ] Œ‹ $H e Œ B
keV
WHAT ROLE DOES THE MAGNETIC FIELD PLAY
IN ALL OF THIS?
Blackbody:
0.3
where the conductivity tensor(a ) is
Most of them show a featureless thermal spectrum
well fit by a blackbody (BB), with low temperatures
eV).
(T
0.5
bÛ B
—Vr©Ó‰ b1Û"pq D Ӆ=•™IM܄()
=
•
)
(
Ù
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1
b
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q
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g
V!©Ó‰ § q ! [
0.4
RS T VU S TXWZY[]:\ ^ S T
In the last few years, the thermal emission of about 20 isolated neutron stars (NS) has been detected in the X-ray
band. The observations can be summarized as follows:
o
0.8
3.1 Dielectric tensor
1 Introduction
Ú
α=4.5
α=18oo
α=36o
α=45o
α=54o
α=63o
α=72 o
α=85.5
0.9
αν
We investigate the thermal emission from magnetic neutron
star surfaces in which the cohesive effects of the magnetic
field have produced the condensation of the atmosphere
and the external layers [7, 5, 11]. This may happen for sufficiently cool (
) atmospheres with moderately intense magnetic fields (about
G ). The thermal emission
from an isothermal bare surface of a neutron star shows no
remarkable spectral features, in agreement with recent observations. However, the presence of the magnetic field is
expected to produce a highly anisotropic temperature distribution, resulting in an observed flux very similar to a BB
spectrum, but depressed in a nearly constant factor at all
energies. This may lead to a systematic underestimation of
the area and size of the emitter by a factor 5-10.
4 Emission properties(without ions)
Fν (arbitrary units)
ABSTRACT
0.1
E(keV)
Red line is the observed flux for a single BB (
[7] Pérez–Azorı́n, J.F., Miralles J.A., & Pons J.A. 2004, submitted
A&A.
1
Vs
K).
ÇÙ
Optical band is not very much altered for different observation angles ( ), but high energy tail is significantly depressed.
Broadband spectrum mimics the BB, but with an overall reduced flux.
[8] Pérez–Azorı́n, J.F., Miralles J.A., & Pons J.A. 2004, in preparation
[9] Pons, J.A, et al., 2002, ApJ, 564, 981
[10] Potekhin, A.Y., 1999, A&A 351, 787
[11] Turolla, R., Zane, S. & Drake, J.J. 2004, ApJ, in press
[12] Walter, F.M. & Lattimer, J.M., 2002, ApJL 575, L145
[13] Zavlin, V.E., et al., 1995, A&A, 297, 441