Models of magnetized neutron star atmospheres: thin atmospheres and partially ionized hydrogen atmospheres with vacuum polarization
aa r X i v : . [ a s t r o - ph . S R ] M a y Astronomy&Astrophysicsmanuscript no. 12121tx c (cid:13)
ESO 2018August 14, 2018
Models of magnetized neutron star atmospheres: thinatmospheres and partially ionized hydrogen atmosphereswith vacuum polarization
V. Suleimanov , , A. Y. Potekhin , , K. Werner Institut f¨ur Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics, Universit¨at T¨ubingen, Sand 1, 72076T¨ubingen, Germany Kazan State University, Kremlevskaja str., 18, Kazan 420008, Russia Io ff e Physical-Technical Institute, Polytekhnicheskaya str., 26, St. Petersburg 194021, Russia Isaac Newton Institute of Chile, St. Petersburg Branch, RussiaReceived XX Xxxxx 2009 / Accepted XX Xxxx XXXX
ABSTRACT
Context.
Observed X-ray spectra of some isolated magnetized neutron stars display absorption features, sometimes interpreted as ioncyclotron lines. Modeling the observed spectra is necessary to check this hypothesis and to evaluate neutron star parameters.
Aims.
We develop a computer code for modeling magnetized neutron star atmospheres in a wide range of magnetic fields (10 − G) and e ff ective temperatures (3 × − K). Using this code, we study the possibilities to explain the soft X-ray spectra of isolatedneutron stars by di ff erent atmosphere models. Methods.
The atmosphere is assumed to consist either of fully ionized electron-ion plasmas or of partially ionized hydrogen. Vacuumresonance and partial mode conversion are taken into account. Any inclination of the magnetic field relative to the stellar surfaceis allowed. We use modern opacities of fully or partially ionized plasmas in strong magnetic fields and solve the coupled radiativetransfer equations for the normal electromagnetic modes in the plasma.
Results.
Spectra of outgoing radiation are calculated for various atmosphere models: fully ionized semi-infinite atmosphere, thinatmosphere, partially ionized hydrogen atmosphere, or novel “sandwich” atmosphere (thin atmosphere with a hydrogen layer abovea helium layer). Possibilities of applications of these results are discussed. In particular, the outgoing spectrum using the “sandwich”model is constructed. Thin partially ionized hydrogen atmospheres with vacuum polarization are shown to be able to improve the fitto the observed spectrum of the nearby isolated neutron star RBS 1223 (RX J1308.8 + Key words. radiative transfer – methods: numerical – stars: neutron – stars: atmospheres – X-rays: stars – stars: individual: RXJ1308.8 +
1. Introduction
In the last two decades, several new classes of neutron stars(NSs) have been discovered by X-ray observatories. They in-clude X-ray dim isolated NSs (XDINSs), or Magnificent Seven(see review by Haberl 2007), central compact objects (CCOs)in supernova remnants (Pavlov et al. 2002, 2004), anomalous X-ray pulsars and soft-gamma repeaters (AXPs and SGRs; see re-views by Kaspi 2007; Mereghetti et al. 2007; Mereghetti 2008).The NSs in the last two classes have superstrong magnetic fields( B > ∼ G) and are commonly named magnetars. The XDINSscan have B ∼ a few × G, as evaluated from period changesand from absorption features in the observed spectra, if they areinterpreted as ion cyclotron lines (see reviews by Haberl 2007;van Kerkwijk & Kaplan 2007).These NSs are relatively young with ages ≤ yr and suf-ficiently hot ( T e ff ∼ − K) to be observed as soft X-raysources. The thermal spectra of these objects can be described byblackbody spectra with (color) temperatures from 40 to 700 eV(see, for example, Mereghetti et al. 2002). Some of the XDINSsand CCOs in supernova remnants have one or more absorptionfeatures in their X-ray spectrum at the energies 0.2 – 0.8 keV
Send o ff print requests to : V. Suleimanov Correspondence to : e-mail: [email protected] (Haberl 2007). The central energies of these features appearto be harmonically spaced (Sanwal et al. 2002; Schwope et al.2007; van Kerkwijk & Kaplan 2007; Haberl 2007). The opticalcounterparts of some XDINSs are also known (see review byMignani et al. 2007), and their optical / ultraviolet fluxes are a fewtimes larger than the blackbody extrapolation of the X-ray spec-tra (Burwitz et al. 2001, 2003; Kaplan et al. 2003; Motch et al.2003).The XDINs are nearby objects, and parallaxes of some ofthem can be measured (Kaplan et al. 2002a). Therefore, theygive a good possibility to measure the NS radii, yielding usefulinformation on the equation of state (EOS) for the NS inner core(Tr¨umper et al. 2004; Lattimer & Prakash 2007), which is oneof the most important problems in the NS physics. For example,the EOS is necessary for computations of templates of gravi-tational wave signals which arise during neutron stars merging(e.g. Baiotti et al. 2008).For a su ffi ciently accurate evaluation of NS radii, agood model of the NS surface radiation for the observedX-ray spectra fitting is necessary. The isolated NS surfacelayers can either be condensed or have a plasma enve-lope (Lai & Salpeter 1997; Lai 2001). In the latter case, theouter envelope layer forms an NS atmosphere. The structureand emergent spectrum of this atmosphere can be computed Suleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres by using stellar model atmosphere methods (e.g. Mihalas1978). Such modeling has been performed by many scien-tific groups for NS model atmospheres without magnetic field(Romani 1987; Zavlin et al. 1996; Rajagopal & Romani 1996;Werner & Deetjen 2000; G¨ansicke et al. 2002; Pons et al. 2002)and by several groups for models with strong ( B > ∼ G) mag-netic fields (Shibanov et al. 1992; Rajagopal et al. 1997; ¨Ozel2001; Ho & Lai 2001, 2003, 2004; van Adelsberg & Lai 2006).These model spectra were used to fit the observed isolated NSX-ray spectra (see review by Zavlin 2009).Modeling of magnetized NS star atmospheres is based onthe theory of electromagnetic wave propagation in a magne-tized plasma in two normal modes, extraordinary (X) and or-dinary (O) ones (Ginzburg 1970; M´es´zaros 1992), and on themethods of opacity calculations for these two modes (Ventura1979; Kaminker et al. 1982, 1983). Methods of fully ionizedmodel atmospheres modeling are well developed (see, e.g.,Zavlin 2009 for references). Partially ionized hydrogen atmo-spheres have been modeled (Potekhin et al. 2004; Ho & Lai2004; Ho et al. 2008), using the opacity and EOS calculations byPotekhin & Chabrier (2003, 2004), which accurately take intoaccount the motional Stark e ff ect and plasma nonideality ef-fects in quantizing magnetic fields. Mid- Z element atmospheresfor strongly magnetized NSs have been modeled by Mori & Ho(2007), who treat the motional Stark e ff ect using a perturbationtheory (Pavlov & M´esz´aros 1993) valid at relatively low T .For magnetar atmospheres, polarization of the vacuum canbe significant, which was studied by Pavlov & Gnedin (1984)and recently by Lai & Ho (2002, 2003). Model atmospheres withpartial mode conversion due to the vacuum polarization havebeen computed by Ho & Lai (2003) and van Adelsberg & Lai(2006).If the temperature is su ffi ciently low or the magneticfield is su ffi ciently strong, the thick atmosphere can be re-placed by a condensed surface (Lai & Salpeter 1997; Lai 2001;Medin & Lai 2007) as a result of the plasma phase tran-sition (cf. Potekhin et al. 1999; Potekhin & Chabrier 2004).Emission and absorption properties of such surfaces in strongmagnetic fields have been studied by Turolla et al. (2004);van Adelsberg et al. (2005); P´erez-Azor´ın et al. (2005).In recent years, evidence has appeared that some of theXDINSs may have a “thin” atmosphere above the condensedsurface. Such atmosphere could be optically thick to low-energyphotons and optically thin to high-energy photons. Motch et al.(2003) fitted the spectrum of RX J0720.4 − − § §
3. Conclusions are sum-marized in § j O- modeX-mode a qq B k yx z, n B Fig. 1.
Geometry of the radiation transfer.
2. Method of atmosphere structure calculations
We compute model atmospheres of hot, magnetized NSs subjectto the constraints of hydrostatic and radiative equilibrium assum-ing planar geometry and homogeneous magnetic field. There aretwo versions of the code. In the first one, we consider the mag-netic field B perpendicular to the surface. In this case the angle α between B and a radiation wave vector k is equal to the angle θ between k and the normal n to the surface (see Fig. 1). It is thesimplest case, because opacities depend on α , and the geome-try of radiation propagation depends on θ . In the second version,the angle θ B between B and n is arbitrary, and calculations aremore expensive. In this case the opacities depend not only onthe polar angle θ , but also on the azimuthal angle ϕ between theprojections of B and n onto the stellar surface, therefore it isnecessary to solve radiation transfer equations for a significantlylarger number of directions.The model atmosphere structure for a NS with e ff ective tem-perature T e ff , surface gravity g , magnetic field B , and givenchemical composition is described by the following set of equa-tions:1. The hydrostatic equilibrium equationd P g d m = g − g rad , (1)where g = GM NS R √ − R S / R NS , (2) g rad = c X i = Z ∞ d ν Z π d ϕ Z + − ( k i ν + σ i ν ) µ I i ν ( µ, ϕ ) d µ (3)allows for the radiation pressure, and R S = GM NS / c is theSchwarzschild radius of the NS. Here µ = cos θ , I i ν ( µ, ϕ ) isthe specific intensity in mode i , P g is the gas pressure, andthe column density m is determined asd m = − ρ d z . (4)The variable ρ denotes the gas density and z is the verticaldistance. Of course, radiation pressure is unimportant in themodels presented below, but it can be more significant athigher e ff ective temperatures. uleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres 3
2. The radiation transfer equations for the two modes µ d I i ν d τ i ν = I i ν − S i ν (5)where S i ν = k i ν k i ν + σ i ν B ν + (6)12 π k i ν + σ i ν X j = Z π d ϕ ′ Z + − σ i j ν ( µ, ϕ ; µ ′ , ϕ ′ ) I j ν ( µ ′ , ϕ ′ ) d µ ′ is the source function, B ν is the blackbody (Planck) intensity,and the optical depth τ i ν is defined asd τ i ν = ( k i ν + σ i ν ) d m . (7)Here, the true absorption and electron scattering opacities k i ν and σ i ν depend on µ and ϕ . Specific intensity in given direc-tion and mode can be scattered in some other direction andin both modes σ i ν ( µ, ϕ ) = π X j = Z π d ϕ ′ Z + − σ i j ν ( µ, ϕ ; µ ′ , ϕ ′ ) d µ ′ (8)3. The energy balance equation12 π X i = Z ∞ d ν Z π d ϕ × Z + − (cid:16) ( k i ν + σ i ν ) I i ν ( µ, ϕ ) − η i ν ( µ, ϕ ) (cid:17) d µ = η i ν ( µ, ϕ ) = π X j = Z π d ϕ ′ Z + − σ i j ν ( µ, ϕ ; µ ′ , ϕ ′ ) I j ν ( µ ′ , ϕ ′ ) d µ ′ + k i ν B ν . (10)Equations (1) – (10) must be completed by the EOS and thecharge and particle conservation laws. In the code two di ff erentcases of these laws are considered.In the first (simplest) case, a fully ionized atmosphere arecalculated. Therefore, the EOS is the ideal gas law P g = n tot kT , (11)where n tot is the number density of all particles. Opacitiesare calculated in the same way as in the paper byvan Adelsberg & Lai (2006) (see references therein for the back-ground theory and more sophisticated approaches). The vac-uum polarization e ff ect is taken into account following the samework. Examples of opacities as functions of photon energy andangle θ for a magnetized electron-proton plasma are shown inFigs. 2 and 3.The second considered case is a partially ionized hydrogenatmosphere. In this case the EOS and the corresponding opac-ities are taken from tables calculated by Potekhin & Chabrier(2003, 2004). The normal mode polarization vectors are takenfrom the calculations by Potekhin et al. (2004). The vacuum po-larization e ff ect is also included.For solving the above equations and computing the modelatmosphere, we used a version of the computer code ATLAS(Kurucz 1970, 1993), modified to deal with strong magnetic -11 -7 -3 -11 -7 -3 -11 -7 -3 q = 90 o KRWRQ(cid:3)HQHUJ\ (cid:3) (cid:15)(cid:3)(cid:3)(cid:3)NH9 vacuum resonanceproton cyclotron line q = 45 o SD F L W \ (cid:15)(cid:3)(cid:3)(cid:3) F P (cid:21) (cid:3) J (cid:16) (cid:20) q = 5 o T=7 10 K r = 30 g cm -3 B = 10 GESFF
Fig. 2.
Dependence of the free-free and electron scattering opac-ities in two modes on the photon energy at di ff erent anglesin a fully ionized hydrogen plasma. The proton cyclotron andvacuum resonances are also shown. The plasma temperature is7 × K, the plasma density is 30 g cm − , the magnetic fieldstrength is 10 G.fields. A nonmagnetic version of this modified code was pre-viously used to model atmospheres of super-soft X-ray sources(Swartz et al. 2002; Ibragimov et al. 2003), atmospheres ofnon-magnetized NSs (Suleimanov & Werner 2007; Rauch et al.2008), and atmospheres of spreading layers on the surface of ac-creting NSs (Suleimanov & Poutanen 2006).The scheme of calculations is as follows. First of all, theinput parameters of the model atmosphere are defined: T e ff , g , B and the chemical composition. Then a starting model usinga grey temperature distribution is calculated. The calculationsare performed with a set of 90 depth points m j distributed log-arithmically in equal steps from m ≈ − g cm − to m max ≈ g cm − in the case of a semi-infinite atmosphere. It is alsopossible to calculate thin atmospheres with arbitrary values of m max . In this case the temperature at the inner boundary is con-sidered as the temperature of a condensed NS surface.In the starting model, all number densities and opacities at alldepth points and all photon energies are calculated. We use 200logarithmically equidistant energy points in our computations in Suleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres -7 -3 -6 -4 -2 E ~ E Cp ESFF SD F L W \ (cid:3) (cid:15)(cid:3)(cid:3) F P (cid:21) (cid:3) J (cid:16) (cid:20) q (cid:3)(cid:15)(cid:3)(cid:3)GHJUHH E=1 keV electron scattering (ES)free-free absorption (FF)X - modeO - mode SD F L W \ (cid:15)(cid:3)(cid:3)(cid:3) F P (cid:21) (cid:3) J (cid:16) (cid:20) Fig. 3.
Dependence of the free-free and electron scattering opac-ities in two modes on the angle between the magnetic field linesand the direction of photon propagation for two photon energies:1 keV and at the proton cyclotron energy. Plasma parameters andmagnetic field strength are the same as in Fig. 2. -6 -4 -2 -6 -4 -2 ion cyclotron linevacuum resonanceT = 10 K r = 1 g cm -3 B = 4 10 G q = 60 o X-modeO-mode O pa c i t y , c m g - Photon energy , keV
Fig. 4.
Dependence of the true opacities in two modes on thephoton energy for fully (dashed curves) and partially ionized(solid curves) hydrogen. The vacuum polarization e ff ect is takeninto account. The plasma temperature is 10 K, the plasma den-sity is 1 g cm − , the magnetic field strength is 4 × G. Theangle between photon propagation direction and the magneticfield is 60 ◦ . the range 0.001 -20 keV with 9 additional points near each ioncyclotron resonance E ci = .
635 keV B G ZA , (12)where Z is the ionic charge and A the atomic weight in the atomicunits. If the vacuum resonance is taken into consideration, thenanother photon energy grid is used, which is constructed usingthe “equal grid” method (Ho & Lai 2003). In this method everypoint in the depth grid m j corresponds to the point in the energygrid defined by the equation E i = E V ( ρ ( m j )) (13)(if this energy point is in the considered energy range 0.001 -20keV). Here E V ( ρ ) is the energy of the vacuum resonance at given B and ρ (see van Adelsberg & Lai 2006). The opacity averagedin the energy interval ( E i − , E i + ) is used to avoid opacity over-estimation in frequency integrals like (9). This energy grid is re-calculated after every iteration. Note that the vacuum resonanceenergy E V ( ρ ) in a partially ionized plasma is shifted relative toits value in a fully ionized plasma (see Fig. 4).The radiation transfer equation (5) is solved on a set of40 polar angles θ and 6 azimuthal angles ϕ (in the case ofinclined magnetic field) by the short characteristic method(Olson & Kunasz 1987).We use the conventional condition (no external radiation) atthe outer boundary I i ν ( µ < , m = m ) = . (14)The di ff usion approximation is used as the inner boundary con-dition I i ν ( µ > , m = m max ) = B ν . (15)The code allows one to take into account the partial modeconversion according to van Adelsberg & Lai (2006). At thevacuum resonance, the intensity in the extraordinary mode par-tially converts with the probability 1 − P jump to the intensity inthe ordinary one and vice versa I , ν → P jump I , ν + (1 − P jump ) I , ν , (16)where P jump = exp (cid:20) − π E / E ad ) (cid:21) . (17)The value E ad is a function of E , B , ρ, T , µ (seevan Adelsberg & Lai 2006 for details).The solution of the radiative transfer equation (5) is checkedfor the energy balance equation (9), together with the surfaceflux condition4 π Z ∞ H ν ( m =
0) d ν = σ T ff = π H , (18)where the Eddington flux at any given depth m is defined as H ν ( m ) = π X i = Z π d ϕ Z + − µ I i ν ( µ, ϕ, m ) d µ. (19)The relative flux error ε H ( m ) = − H R ∞ H ν ( m ) d ν , (20) uleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres 5 -4 -2 Photon energy , keV H E , e r g c m - s - k e V - T eff = 5 10 Klog g = 14.3B = 5 10 Gionized H F ROXPQ(cid:3)GHQVLW\ (cid:15)(cid:3)(cid:3)
J(cid:3)FP (cid:16)(cid:21) (cid:3) L (cid:15)(cid:3)(cid:3)(cid:3) (cid:20)(cid:19) (cid:25) (cid:3) D Fig. 5.
Emergent spectra and temperature structures of fully ion-ized hydrogen model atmospheres of neutron stars with T e ff = × K and log g = .
3. The models with and without (dash-dotted curves) strong ( B = × G) magnetic field as wellas with (solid curves) and without (dashed curves) vacuum po-larization e ff ect are shown. The blackbody spectrum with thetemperature equal the e ff ective temperature is also shown in theupper panel (dotted curve).and the energy balance error as functions of depth ε Λ ( m ) = X i = Z ∞ d ν Z π d ϕ × Z + − (cid:16) ( k i ν + σ i ν ) I i ν ( µ, ϕ ) − η i ν ( µ, ϕ ) (cid:17) d µ (21)are calculated.Temperature corrections are then evaluated using three dif-ferent procedures. The first is the integral Λ -iteration method,modified for the two-mode radiation transfer, based on the en-ergy balance equation (9). In this method the temperature cor-rection for a particular depth is found from ∆ T Λ = − ε Λ ( m ) R ∞ Φ ν d ν , (22)where Φ ν = X i = h ( Λ i ν diag − / (1 − α i ν Λ i ν diag ) i k i ν (d B ν / d T ) (23) and α i ν = σ i ν / ( k i ν + σ i µ ). Averaged opacities are defined as k i ν = π Z π d ϕ Z + − k i ν ( µ, ϕ ) d µ, (24) σ i ν = π Z π d ϕ Z + − σ i ν ( µ, ϕ ) d µ. (25) Λ i ν diag is the diagonal matrix element of the Λ operator depend-ing on the mean optical depth in given mode i . The mean opticaldepth is defined by d τ i ν = ( k i ν + σ i µ ) d m (see Kurucz 1970 fordetails of Λ ν diag – τ ν dependence). This procedure is used in theupper atmospheric layers. The second procedure is the Avrett-Krook flux correction, which uses the relative flux error ε H ( m ),and it is performed in the deep layers. And the third one is thesurface correction, which is based on the emergent flux error (seeKurucz 1970 for a detailed description).The iteration procedure is repeated until the relative flux er-ror is smaller than 1% and the relative flux derivative error issmaller than 0.01%. As a result of these calculations, we obtaina self-consistent isolated NS model atmosphere, together withthe emergent spectrum of radiation.Our method of calculation has been tested by a compari-son to models for magnetized NS atmospheres (Shibanov et al.1992; Pavlov et al. 1994; Ho & Lai 2001; ¨Ozel 2001; Ho & Lai2003). Model atmospheres with partially ionized hydrogen arecompared to models computed by Ho et al. (2007). We havefound that our models are in a good agreement with these cal-culations. Our results are presented in Fig. 5, where the temper-ature structures and the emergent spectra of models with andwithout vacuum polarization are compared to a model with-out magnetic field. In Fig. 6 we present emergent spectra andtemperature structures of the semi-infinite and thin model atmo-spheres with the same parameters as used by Ho et al. (2007).
3. Results
In this work we use the developed code mainly for studies of thinatmospheres above a condensed NS surface. In all calculationsbelow we use the same surface gravity, log g = ff ect. In the semi-infinite atmosphere, awide proton cyclotron line forms in agreement with the resultsof Ho & Lai (2001); Zane et al. (2001). However, the absorp-tion feature disappears with decreasing the atmosphere surfacedensity Σ . The thin atmosphere is transparent to the continuumand absorption line wings, therefore the emergent spectrum ap-proaches the spectrum of the condensed surface everywhere ex-cept for a narrow energy band at the center of the cyclotron line.The width of this absorption depends on the atmosphere thick-ness.Some XDINSs and CCOs show one or two absorption fea-tures (Haberl 2007; Sanwal et al. 2002; Schwope et al. 2007).Various hypothesis (so far inconclusive) were considered for Suleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres -7 -5 -3 -1 -7 -5 -3 -1 -7 -5 -3 -1 T , K column density , g cm -2 T eff = 5.25 10 KT infty = 4.3 10 KB = 4 10 Gz = 0.22 F l , e r g c m - s - A - Wavelength , A
Fig. 6.
Top panel:
Emergent spectra of the partially ionized hy-drogen semi-infinite (dashed curve) and thin (
Σ = . gcm − ,solid curve) model atmospheres ( T e ff = . × K, B = × G) together with the corresponding blackbody spectrum (dot-ted curve). The spectra are calculated with gravitational redshift( z = Bottom panel:
Temperaturestructures of the models from the top panel together with thetemperature structure of a fully ionized hydrogen model (dottedcurve).an explanation (see Mori & Ho 2007 and references therein).Here we suggest another one, which we name “sandwich at-mosphere”. A thin, chemically layered atmosphere above a con-densed NS surface can arise from accretion of interstellar gaswith cosmic chemical composition. In this case, hydrogen andhelium quickly separate due to the high gravity (according toBrown et al. 2002, the He / H stratification timescale can be es-timated as ∼ ρ . T − . g − s, where ρ = ρ/
10 g cm − , T = T / K, and g = g / cm s − ). In this “sandwich atmo-sphere” a layer of hydrogen is located above a helium slab, andthe emergent spectrum has two absorption features, correspond-ing to proton and α -particle cyclotron energies. In Fig. 8 theemergent spectrum for one of such models is shown. The emis-sion feature at the helium absorption line arises due to a localtemperature bump at the boundary between the helium and hy-drogen layers. This bump arises due to sharp changing of theplasma density and the opacity between helium and hydrogenlayers. Clearly, some transition zone with mixed H / He chemicalcomposition must exist between layers, and this rapid tempera- T eff = 10 KB = 10 G H E , e r g c m - s - k e V - Photon energy , keV
Fig. 7.
Emergent spectra of the thin fully ionized hydrogen at-mospheres above a solid surface with T e ff = K and B = G in comparison to the semi-infinite atmosphere (dashed curve).The spectra of atmospheres with surface densities Σ= − (dash-dotted curve) together with the cor-responding blackbody (dotted curve) are shown. Vacuum polar-ization e ff ect is not included.ture change can be reduced. We plan to calculate models withthis kind of transition zone in future work.For comparison to observations, it is necessary to integratethe local model spectra over the NS surface. The e ff ective tem-perature and magnetic field strengths are not uniform over theNS surface, and generally the magnetic field is not perpendicularto the surface (see Ho et al. 2008). Therefore, it is necessary tocompute model atmospheres with inclined magnetic field. Thispossibility is included in our code. For example, Fig. 9 showsspectra and temperature structures of model atmospheres withmagnetic field perpendicular and parallel to the NS surface.Most of the XDINSs have magnetic fields B ≥ G andcolor temperatures ≈ K (Haberl 2007). Hydrogen model at-mospheres are partially ionized under these conditions and thevacuum polarization e ff ect is also significant. Here we presentfirst results of modeling of partially ionized hydrogen atmo-spheres using our radiative transfer code. In Fig. 10 we com-pare spectra and temperature structures of the partially ionizedhydrogen model atmospheres with and without the partial modeconversion e ff ect. When the X-mode (having smaller opacity)partially converts to the O-mode in the surface layers of the at-mosphere, the energy absorbed by the O-mode heats these upperlayers. As a result, the emergent spectra are closer to the black-body.For some of the XDINSs, optical counterparts have beenfound (Mignani et al. 2007). The observed optical fluxes are afew times larger compared to the blackbody extrapolation fromX-rays to the optical range (see top panel of Fig. 12 for illus-tration). Ho et al. (2007) demonstrated, that a single partiallyionized thin hydrogen atmosphere can explain this problem inthe case of brightest isolated NS RX J1856.4 − − ≈ . uleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres 7 -5 -3 -1 -5 -3 -1 T eff = 10 KB = 10 G H E , e r g c m - s - k e V - Photon energy , keV T , K column density, g cm -2 Fig. 8.
Emergent spectrum (top panel) and temperature struc-ture (bottom panel) of the “sandwich” model atmosphere abovea solid surface with T e ff = K and B = G (solid curve)in comparison with the thin fully ionized hydrogen atmosphere(dashed curve) with the same parameters. The surface densitiesof both model atmospheres are 100 g cm − , in the “sandwich”model the H slab has 25 g cm − surface density and the He slabhas 75 g cm − . The corresponding blackbody spectrum (dottedcurve) is also shown in the top panel.cess optical flux can be explained by the radiation from coolsurface parts (Schwope et al. 2005).We now investigate properties of partially ionized hydro-gen modes, which can be applied to the RBS 1223 atmosphere.The color temperature of this star, found from X-ray spectra fit-ting, is close to 10 K, with magnetic field B ≈ × G(Schwope et al. 2007). In particular, we investigate the opticalflux excess in comparison to the X-ray fitted blackbody flux inthis kind of models. For this aim we have calculated two sets ofmodels with vacuum polarization and partial mode conversion.The models in the first set have e ff ective temperatures T e ff = K and the models of second one have e ff ective temperatures T e ff = . × K. In both sets B = × G, and modelswith surface densities
Σ =
1, 3, 10, 30, 100 and 10 (semi-infinitemodel) g cm − are computed. In Fig. 11 we show emergent spec-tra and temperature structures for some models from the secondset. Clearly, the X-ray spectra of the models with Σ ≤
10 g cm − are close to a blackbody and, therefore, better fit the observedX-ray spectrum. -6 -4 -2 -6 -4 -2 T eff = 1.5 10 KB = 4 10 G column density , g cm -2 T , K H E , e r g c m - s - k e V - Photon energy, keV
Fig. 9.
Emergent spectra and temperature structures of partiallyionized hydrogen model atmospheres with T e ff = . × Kwith inclinations of the magnetic field ( B = × G) to the sur-face normal θ B equal 0 ◦ (solid curves) and 90 ◦ (dashed curves).The vacuum polarization e ff ect is not included. The correspond-ing blackbody spectrum is also shown in the upper panel (dottedcurve).In Fig. 12 (bottom panel) we show the ratio of the model at-mosphere flux to the X-ray fitted blackbody flux in the opticalband depending on Σ for both sets. The observed ratio is about5 (Kaplan et al. 2002b), in agreement with the semi-infinite at-mosphere models. However, the observed blackbody like X-rayspectrum agrees with the thin atmosphere models, for whichthis ratio is close to 1. Therefore, the observed optical excesscannot be explained by the thin atmosphere model alone; in-stead, it can arise due to a nonuniform surface temperature dis-tribution, in agreement with the RBS 1223 light curve modeling(Schwope et al. 2005).
4. Conclusions
In this paper a new code for the computation of magnetized NSmodel atmosphere is presented. It can model fully ionized andpartially ionized hydrogen atmospheres in a wide range of e ff ec-tive temperatures (3 × – 10 K) and magnetic fields (10 –10 G), with any inclination of the magnetic field to the stellarsurface. The vacuum polarization e ff ect with partial mode con-version is taken into consideration. Calculated emergent spectra Suleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres -6 -4 -2 -6 -4 -2 T , K column density , g cm -2 T eff = 1.5 10 KB = 4 10 G H E , e r g c m - s - k e V - Photon energy, keV
Fig. 10.
Emergent spectra and temperature structures of the par-tially ionized hydrogen model atmospheres with T e ff = . × K with (solid curves) and without (dashed curves) vacuum polar-ization e ff ect (with the partial mode conversion). The magneticfield strength is B = × G. The corresponding blackbodyspectrum is also shown in the upper panel (dotted curve).and temperature structures of the model atmospheres agree withpreviously published ones.We presented new results obtained using this code. We havestudied the properties of thin atmospheres above condensed NSsurfaces. We demonstrated that the proton cyclotron absorptionline disappears in the thin hydrogen model atmospheres. A newthin “sandwich” model atmosphere (hydrogen layer above he-lium layer) is proposed to explain the occurrence of two absorp-tion features in the observed X-ray spectra of some isolated NSs.We analyzed the optical excess (relative to the X-ray fittedblackbody flux) in the model spectra of partially ionized hy-drogen atmospheres with vacuum polarization and partial modeconversion.A set of model atmospheres with parameters (e ff ective tem-perature and the magnetic field strength) close to the probableparameters of the isolated NS RBS 1223 were calculated. Wefound the optical flux excess ≈ Σ of the atmosphere. Spectra of thin model atmospheres fitthe observed RBS 1223 X-ray spectrum better, therefore we con-clude that the observed optical excess should be explained bynonuniform surface temperature distribution. -6 -4 -2 -6 -4 -2 -6 -4 -2 H E , e r g c m - s - k e V - Photon energy, keV T eff = 1.2 10 KB = 4 10 G T , K column density , g cm -2 Fig. 11.
Emergent spectra and temperature structures of the par-tially ionized hydrogen model atmospheres with T e ff = . × K with vacuum polarization e ff ect the partial mode conversionshown for various surface densities Σ (solid curves - 1 g cm − ,dashed curves - 10 g cm − , dash-dotted curves - semi-infinite at-mosphere). The magnetic field strength is B = × G. Thecorresponding blackbody spectrum is also shown in the upperpanel (dotted curve).The accuracy of the thin and sandwich model atmospheresis currently limited by the inner boundary condition for the ra-diation transfer equation. We used blackbody radiation as thiscondition, but a higher accuracy can be achieved by replacing itby the condensed surface condition (van Adelsberg et al. 2005).This will be done in a future work. We are also planning to in-clude the e ff ect of magnetic field and temperature distributionsover the stellar surface to compute an integral emergent spec-trum from isolated NSs. Acknowledgements.
VS thanks DFG for financial support (grant We 1312 / / Transregio 7 ”Gravitational Wave Astronomy”) and thePresident’s programme for support of leading science schools (grant NSh-4224.2008.2). The work of AYP is supported by RFBR grants 05-02-2203and 08-02-00837 and the President’s programme for support of leading scienceschools (grant NSh-2600.2008.2).
References
Baiotti, L., Giacomazzo, B. & Rezzolla, L. 2008, Phys. Rev. D, 78, 084033Brown, E. F., Bildsten, L,, & Chang, P. 2002, ApJ, 574, 920Burwitz, V., Zavlin, V.E., Neuh¨auser, R. et al. 2001, A&A, 379, L35 uleimanov, Potekhin & Werner: Models of magnetized neutron star atmospheres 9
10 100 1000 1000010 T eff = 10 KB = 4 10 GF BB F atm H l , e r g c m - s - A - Wavelength , A F a t m / F BB Atmosphere surface density S , g cm -2 T eff = 1.2 10 K T eff = 10 K B = 4 10 G Fig. 12.
Top panel:
Emergent spectrum of the partially ionizedhydrogen model atmosphere of neutron stars with T e ff = K with vacuum polarization e ff ect, partial mode conversion andmagnetic field strength B = × G. The blackbody spectrumfitted to the maximum flux of the spectral distribution is alsoshown (dashed curve). At the optical band the model atmosphereflux in a few times larger than the blackbody flux.
Bottom panel:
Ratios of the model atmosphere flux to the blackbody (X-ray fit-ted) flux at the optical band depending on the model atmospherethickness (surface density Σ ) for the models with T e ff = Kand T e ff = . × K. Burwitz, V., Haberl, F., Neuh¨auser, R. et al. 2003, A&A, 399, 1109G¨ansicke, B. T., Braje, T. M., & Romani, R. W. 2002, A&A, 386, 1001Ginzburg, V.L. 1970, The Propagation of Electromagnetic Waves in Plasmas,(2nd ed. Oxford: Pergamon)Haberl, F., Motch, C., Zavlin, V.E. et al. 2004, A&A, 424, 635Haberl, F. 2007, A&SS, 308, 181Ho, W. C. G. & Lai, D. 2001, MNRAS, 327, 1081Ho, W. C. G. & Lai, D. 2003, MNRAS, 338, 233Ho, W. C. G. & Lai, D. 2004, ApJ, 607, 420Ho, W. C. G., Kaplan, D.L., Chang, P., van Adelsberg, M. & Potekhin, A.Y.2007, MNRAS, 375, 821Ho, W. C. G., Potekhin, A.Y. & Chabrier, G. 2008, ApJS, 178, 102Ibragimov, A. A., Suleimanov, V. F., Vikhlinin, A., & Sakhibullin, N. A. 2003,Astronomy Reports, 47, 186Kaminker, A.D., Pavlov, G.G. & Shibanov Yu.A. 1982, Ap&SS, 86, 249Kaminker, A.D., Pavlov, G.G. & Shibanov Yu.A. 1983, Ap&SS, 91, 167Kaplan, D.L., van Kerkwijk, M.H. Anderson, J. 2002a, ApJ, 571, 447Kaplan, D.L., Kulkarni, S.R. & van Kerkwijk, M.H. 2002b, ApJ, 579, L29Kaplan, D.L., van Kerkwijk, M.H., Marshall, H.L., et al. 2003, ApJ, 590, 1008Kaspi, V.M. 2007, A&SS, 308, 1Kurucz, R. L. 1970, SAO Special Report, 309 Kurucz, R. 1993, Atomic data for opacity calculations. Kurucz CD-ROMs,Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993, 1Lai, D. 2001, Reviews of Modern Physics, 73, 629Lai, D. & Salpeter, E. E. 1997, ApJ, 491, 270Lai, D. & Ho. W.C.G. 2002, ApJ, 566, 373Lai, D. & Ho. W.C.G. 2003, ApJ, 588, 962Lattimer, J.M. & Prakash, M. 2007, Phys. Rep., 442, 109Medin, Z. & Lai, D. 2007, MNRAS, 382, 1833Mereghetti, S. 2008, A&A Rev., 15, 225Mereghetti, S., Tiengo, A., & Israel, G. L. 2002, ApJ, 569, 275Mereghetti, S., Esposito, P. & Tiengo, A. 2007, A&SS, 308, 13M´es´zaros, P. 1992, High-Energy Radiation from Magnetized Neutron Stars(Chicago, Univ. Chicago Press)Mignani, R.P, Bagnulo, S., De Luca, A. et al. 2007, A&SS, 308, 203Mihalas, D. 1978, Stellar atmospheres, 2nd edition (San Francisco,W. H. Freeman and Co.)Mori, K. & Ho, W.C.G. 2007, MNRAS, 377, 905Motch, C., Zavlin, V.E. & Haberl, F. 2003, A&A, 408, 323Olson, G.L. & Kunasz, P.B. 1987, JQSRT, 38, 325¨Ozel, F. 2001, ApJ, 563, 276Pavlov, G. G., & Gnedin, Yu. N. 1984, Astrophys. Space Phys. Rev., 3, 197Pavlov, G. G., & M´esz´aros, P. 1993, ApJ, 416, 752Pavlov, G.G., Shibanov, Yu.A., Ventura, J. & Zavlin, V.E. 1994, A&A, 289, 837Pavlov, G. G., Sanwal, D., Garmire, G.P., 2002, In: Slane, P.O., Gaensler, B.M.(eds) Neutron Stars in Supernova Remnants. ASP Conf. Ser. 271, 247Pavlov, G. G., Sanwal, D., Teter, M.A., 2004, in Young Neutron Stars and TheirEnviroments (Proceedings of the IAU Symp. 218), ed. F. Camilo & B. M.Gaensler (ASP, San Francisco), 239P´erez-Azor´ın, J. F., Miralles, J. A., & Pons, J. A. 2005, A&A, 433, 275Pons, J. A., Walter, F. M., Lattimer, J., et al. 2002, ApJ, 564, 981Potekhin, A.Y. & Chabrier G. 2003, ApJ, 585, 955Potekhin, A.Y. & Chabrier G. 2004, ApJ, 600, 317Potekhin, A.Y., Chabrier G.& Shibanov, Yu. A. 1999, Phys. Rev. E, 60, 2193Potekhin, A.Y., Lai, D., Chabrier G., & Ho, W. C. G. 2004, ApJ, 612, 1034Rajagopal, M. & Romani, R. W. 1996, ApJ, 461, 327Rajagopal, M., Romani, R. W., & Miller, M. C. 1997, ApJ, 479, 347Rauch, T., Suleimanov, V. & Werner, K. 2008, A&A, 490, 1127Romani, R. W. 1987, ApJ, 313, 718Sanwal, D., Pavlov, G.G., Zavlin, V.E. et. al. 2002, ApJ, 574, L61Schwope, A.D, Hambaryan, V., Haberl, F. et al. 2005, A&A, 441, 597Schwope, A.D, Hambaryan, V., Haberl, F. & Motch, C. 2007, A&SS, 308, 619Shibanov, I. A., Zavlin, V. E., Pavlov, G. G., & Ventura, J. 1992, A&A, 266, 313Suleimanov, V. & Poutanen, J. 2006, MNRAS, 369, 2036Suleimanov, V. & Werner, K. 2007, A&A, 466, 661Swartz, D. A., Ghosh, K. K., Suleimanov, V., Tennant, A. F., & Wu, K. 2002,ApJ, 574, 382Tiengo, A. & Mereghetti, S. 2007, ApJ, 657, L101Tr¨umper, J.E., Burwitz, V., Haberl, F. & Zavlin, V.E. 2004, Nuclear Phys. B Proc.Suppl., 132, 560Turolla, R., Zane, S., & Drake, J. J. 2004, ApJ, 603, 265van Adelsberg, M. & Lai, D. 2006, MNRAS, 373, 1495van Adelsberg, M., Lai, D., Potekhin, A.Y. & Arras, P. 2005, ApJ, 628, 902van Kerkwijk, M. H., & Kaplan, D. L. 2007, A&SS, 308, 191Ventura, J. 1979, Phys. Rev. D, 19, 1684Werner, K., & Deetjen, J. 2000, in Pulsar Astronomy – 2000 and Beyond, ed.M. Kramer, N. Wex, & R. Wielebinski, ASP Conf. Ser., 202, 623Zane, S., Turolla, R., Stella, L., & Treves, A. 2001, ApJ, 560, 384Zavlin, V. E. 2009, Theory of radiative transfer in neutron star atmospheres andits applications, in Neutron Stars and Pulsars (Proceedings of the 363. WE-Heraeus Seminar), ed. W. Becker (Springer, New York), 181Zavlin, V. E., Pavlov, G. G., & Shibanov, I. A. 1996, A&A, 315, 141
List of Objects ‘RX J1856.4 −−