Thermal radiation from magnetic neutron starsurfaces

J.F. PEREZ–AZORIN, J.A. MIRALLES AND J.A. PONSDepartament de Fısica Aplicada, Universitat d’Alacant, 03690 Alacant,Spain

ABSTRACT

We investigate the thermal emission from magnetic neutronstar surfaces in which the cohesive effects of the magneticfield have produced the condensation of the atmosphereand the external layers [7, 5, 11]. This may happen for suf-ficiently cool (

���������) atmospheres with moderately in-

tense magnetic fields (about�����

G ). The thermal emissionfrom an isothermal bare surface of a neutron star shows noremarkable spectral features, in agreement with recent ob-servations. However, the presence of the magnetic field isexpected to produce a highly anisotropic temperature dis-tribution, resulting in an observed flux very similar to a BBspectrum, but depressed in a nearly constant factor at allenergies. This may lead to a systematic underestimation ofthe area and size of the emitter by a factor 5-10.

1 Introduction

In the last few years, the thermal emission of about 20 iso-lated neutron stars (NS) has been detected in the X-rayband. The observations can be summarized as follows:� Most of them show a featureless thermal spectrum

well fit by a blackbody (BB), with low temperatures(T� �����

eV).� They seem to have high magnetic fields (B � ������� G).� The optical counterparts have been detected in fourcases, showing a systematic excess flux of about a fac-tor 5 � 10.� Light element atmosphere models are ruled out by themultiwavelenght observations.� Heavy element atmosphere models don’t explain theabsence of spectral features.� Single component models cannot account for the op-tical excess observed in 4 sources.� Two component models (i.e. two BB) can reconcile theoptical and X-ray observations.

To explain the lack of spectral features and the 2–componentspectrum we consider the EMISSION FROM A SOLIDSURFACE with NON UNIFORM TEMPERATURE, but� HOW CAN A NS BE LEFT WITHOUT AN ATMO-

SPHERE?� WHAT ROLE DOES THE MAGNETIC FIELD PLAYIN ALL OF THIS?

2 Effects of the magnetic field� Highly magnetized NS can undergo a phase transi-tion that turns the gaseous atmosphere into a solid[5].

– Critical temperature (Fe):������������� ��� ��� ���� eV

– Zero-pressure density:�� �"!$#���%'&'()�� � � �*� ���� gr/cm

� Thermal conductivity is different in the directionsparallel and perpendicular to B + temperature variesover the surface. Need to solve the 2D diffusion equa-tion in the condensed envelope [8, 4].

3 Solid surface model

Suppose the solid surface emits according to Kirchoff’s law[2]: , -/.10�324�52*6578�:9 -/.10�324�52*657�� -. �;7where 9 - . 0�<2=�52*657>���@? � - . 0�A2=�52�6@79 -CB emissivity; � -CB reflectivity� - � D�EGF �HJI��KMLN . EGF H�O��;7P?Q�3.1 Dielectric tensorR�S T �VU S TXWZY�[�\]:^ S Twhere the conductivity tensor(a) is^ S T'_a` �Mb�c�I��R*dfe S T � ] �gJh c[�\ e S h T] �gJh c �ai*j clk4m�no n �p���/q rs&'�=� � � �� �� i keV B e

(plasma frequencye S h T B effective relaxation times

R�t$u � vwyx ? Ylz �Y�z x �� � { |}~ ���� ���5? ] �gJh c W ] �g$h S. ]���]�� h cM7 . ]���]�� h S 7 W�Y ]��� ]

{�� �@? ] �g$h c W ] �gJh S] � W�Y ]>] � �x � �D . � W � 7�� z � �D . � ? � 7

] � h c � c �o n � ������!�r5� � i keV B e(

cyclotron frequency] � h S ��� c �oX� � �V�q #���!��� � i keV B ion cyclotron frequency] �gJh S �V�q D !�%�(1�=�4��&5�����l��� *� �� i keV B ion plasma frequency] � � � �$H e � � ] �� � �$H e � B damping frequencies

awww.ioffe.rssi.ru/astro/conduct/condmag.html

3.2 Dispersion relation

Introducing the Maxwell tensor,� S T ��O S O T ?�O � U S T W ] �I � R S Tconsidering that� S T���T �:���1� z `�� . � S T 78�V�and applying the complex Snell law , the dispersion relationleads to a quartic equationb i . { W����4�   � 9�7 W b � .¢¡ � ? D { x W�£;�4�   � 9�7 W { � � W ¡ £ ��=�¤ @Y��=�¤  . D 9�7�¥�¦ �/§ . b � ? �=�¤  � Y 7 �=�4� . £¨W b � � 7where � � x ?�{©� £ �V{ x ? � �¡ � �4�¤  � Y�ª �5? �=�¤  � 9 . � W ¥�¦ � � § 7l«3.3 Boundary conditions and emissivity� Imposing boundary conditions at the surface (con-

tinuity of normal components of 0z and 0� , andtangential components of 0� and 0¬ ), we obtain thecomponents of the reflected wave in terms of thecomponents of the incident wave ( � � h � ):Parallel: �C­� �¯® % � ® K�LN . Y U � � 7 � � W ® % � ® KMLN . Y U �� 7 � �Transverse: �C­� �¯® � � ® K�LN . Y U �� 7 � � W ® � � ® KMLN . Y U �� 7 � �� For each incident polarization we obtain the reflectiv-ity:

– Parallel: � � �°® % � ® � W ® � � ® �– Transverse: � � �¯® % � ® � W ® � � ® �

Total reflectivity ��± � . � � W � � 7*H DFINALLY THE EMISSIVITY9����@? � ±

4 Emission properties(without ions)

4.1 Emissivity

Normalized emissivity integrated in all possible incidentangles as a function of energy for different orientations ofthe magnetic field (B

��������G and T

�������K):

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.01 0.1 1 10

α ν

E (KeV)

α=4.5o

α=18o

α=36o

α=45o

α=54o

α=63o

α=72o

α=85.5o

� Emissivity is strongly reduced for low energies be-cause one mode developes a large imaginary part for]Q² ]��D´³ ?µ� WV¶ � W [ . ] g H ]�� 7 ��·� The plasma resonance depends on the orientation ofthe magnetic field (neglecting damping):] �¸ � ] �� W ] �gD ¹º � ��» �5? [ ] �g ] �� ¥M¦ � � 9. ] �� W ] �g 7 �½¼ �=�*�*¾¿

– POLE (9����sÀ

): ]�Á�ÂQ] � and ] ( � ] g– EQUATOR (

9��VÃ�� À): ]�Á�ÂÄ] � and ] ( �:�� For ]½ÅÆ] g the emissivity approaches to BB (

9 - + �).

4.2 Spectral energy distribution

Dipolar magnetic field:� . � 24Ç�7�� �ÉÈ� . � W �8¥M¦ � �ÊÇs7�Ëk and con-

stant temperature (�V�������

K)[�\1Ì - �"Í � j�ÏÎsÐ Í j� �=�¤  Ç Î Ç�Í � j�ÑÎsÐÉÒ Í j �*�� Î Ç Ò , - ¥M¦ � Ç Ò �=�¤  Ç Ò(Ç Ò 2 Ð Ò ) B define the plane of incidence

1e-04

0.001

0.01

0.1

1

10

100

1000

0.001 0.01 0.1 1

Fν

(arb

itrar

y un

its)

E (keV)

BBBp=1012GBp=5 1012GBp=1013GBp=5 1013G

� Spectrum is essentially featureless� At low energies, the behavior is similar for differentmagnetic field strenghts

4.3 Observed flux

Assuming the following anisotropic temperature distribu-tion: � . � 2=Çs78� � g[ . � W �8¥�¦ � � Çs7with

� g ��!�Ó���� �� G and� g � ��� � K.Ì - � § � Í�ÔÊÕ , - ¥�¦ � Ç Ò �=�¤  Ç Î Ç Î�Ч B scale factor ÖØ× k� k

1e-04

0.001

0.01

0.1

1

10

100

1000

0.001 0.01 0.1 1

Fν

(arb

itrar

y un

its)

E(keV)

BBθo=0o

θo=45o

θo=90o

Red line is the observed flux for a single BB (�V�����s�

K).� Optical band is not very much altered for different ob-servation angles (

Ç�Ù), but high energy tail is signifi-

cantly depressed.� Broadband spectrum mimics the BB, but with an over-all reduced flux.

5 ION’S EFFECT

The effect of the ions is visible in the next figure. We com-pare the observed flux from two different NS models: a sin-gle temperature BB and the solid surface model described inÚ4.3 (with and without ions). In both cases, we take into ac-

count interstellar medium absorption (b�Û B hydrogen col-

umn density).� Blackbody:���Vr©Ó���� �

K,b1Û"�p��q D Ó������=��IMÜ�()� .� Model:

� g �Ý��� � K,b1Û��Ø��q [ Ó½��� �=� I�Ü ()� , Ç Ù �ÞÃ�� À ,� g �V!©Ó���� �� G, § � ������q !

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1

Fν

(arb

itrar

y un

its)

E(keV)

BB fitθo=90o

θo=90o w/ ions

� X-ray spectrum is practically indistinguishable (noticethat § � = 11.5).� Optical flux including ion effects (green) is 4.5 timeslarger, similary to what has been observed in INS’s.� The apparent estimated value of the radius (

�µß H z ) is3.4 times smaller for a single BB.

6 CONCLUSIONSà Solid surface models show a significantly de-pressed spectrum (up to factor 10) comparedto a single temperature BB with the same ef-fective temperature.à Spectrum is almost featureless. Minor differ-ences only at energies where the interstellarmedium absorption makes difficult to distin-guish between models.à The presence of the magnetic field producesa large anisotropy in the surface temperaturedistribution.à The size of the emitter can be understimatedby a large factor if a simplified BB model isused. No need to appeal to strange stars toexplain the apparent small radius of someNS [3, 9].à Depending on the magnetic field’s strengthand including the effects of the ions, the op-tical flux is about 5 times larger than the cor-responding BB extrapolation of the best X-ray fit. This is similar to the observed spectrafrom INS.

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