Model neutron star atmospheres with low magnetic fields. 1. Atmospheres in radiative equili

a r X i v :a s t r o -p h /9604072v 1 13 A p r 1996

A&A manuscript no.

(will be inserted by hand later)

1.Introduction

Thermal radiation from the surfaces of neutron stars (NSs)provides important information about their ther-mal histories and the properties of the high-density mat-ter in their interiors (Pethick 1992;Tsuruta 1995).The NS thermal-like radiation has been observed recently in

the soft X-ray range with ROSAT (¨Ogelman 1995;and

references therein)and EUV E (e.g.,Foster et al.1996;Halpern et al.1996)and in the UV-optical range with HST (Pavlov et al.1996).For the proper interpretation of these observations,one should take into account that NSs are not perfect blackbody emitters —properties of their radiation are determined,like in usual stars,by the radiative transfer in their atmospheres,and the emergent spectra can di?er substantially from the Planck spectrum,depending on the chemical composition,magnetic ?eld,energy ?ux and gravity in surface layers (Pavlov et al.1995,and references therein).

There are at least two factors which make NS atmo-spheres di?erent from those of normal stars:the immense gravity and huge magnetic ?elds.The ?rst factor is im-portant for all NSs;it leads to strong compression and non-ideality of the atmospheric matter,which a?ects con-siderably equation of state (EOS)and opacity.Even much greater and more complicated e?ects on the opacity and EOS can be expected from the very high magnetic ?eld which,in particular,makes the NS atmospheres essentially anisotropic.

First NS atmosphere models have been developed by Romani (1987;hereafter R87).He employed (nonmag-netic)opacities from the Los Alamos Opacity Library (LAOL)to calculate atmosphere models and emergent spectra for a few e?ective temperatures,5.5≤log T e?≤6.5,and chemical compositions (pure helium,carbon,oxy-gen,iron,and a mixture of elements with cosmic abun-dance)and showed that the spectrum could indeed be quite di?erent from the blackbody,especially for the he-

2V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds

lium models.Further work has been concentrated on NS atmospheres with“typical”(strong)NS magnetic?elds, B=1011?1013G,which drastically change opacities and EOS(Miller1992;Shibanov et al.1992;Shibanov et al. 1993;Ventura et al.1993;Pavlov et al.1994;Pavlov et al.1995;Shibanov&Zavlin1995;Shibanov et al.1995a; Zavlin et al.1995a).These papers(except that of Miller) were devoted to pure hydrogen NS atmospheres because no reliable opacities and EOS had been developed for non-ideal partly ionized gases of heavier elements.It has been demonstrated that the strong magnetic?elds not only dra-matically a?ect the emergent spectra,but they also lead to remarkable anisotropy and strong polarization of the NS thermal radiation.

Importance of the magnetic?eld depends on how large is the electron cyclotron energy,hνB=heB/m e c,in com-parison with other characteristic energies.For instance, if we consider a completely ionized(hot)atmosphere, where the main source of opacity is the free-free transitions and/or scattering on electrons,the opacity is not a?ected by the?eld for hνB?hν,k B T(see,e.g.,Kaminker et al.1983).Since the energy transfer in the atmosphere is mainly driven by photons with hν～(1?10)k B T,one can neglect the magnetic e?ects if B?(m e c/he)k B T～1010(T/106K)G.If a substantial fraction of bound atoms or ions is present,the magnetic e?ects are determined by the ratio hνB/Z2Ry～(B/109G)Z?2,where Z is a char-acteristic ion charge,and Ry=13.6eV is the Rydberg energy.When this ratio is<～1,magnetic e?ects on the bound-bound transitions between lower quantum levels, as well as on the bound-free transitions well above pho-toionization thresholds,can be neglected.Although tran-sitions which involve highly excited levels may be a?ected by much lower?elds,they usually are not important for conditions typical for NS atmospheres.

Thus,depending on atmospheric temperature and chemical composition,one can neglect the magnetic e?ects on opacities and EOS if B?108?1010G.This means that there exists at least one class of NSs,millisecond pul-sars,for which the magnetic e?ects can be expected to be small.Four millisecond pulsars have been detected in soft X-rays(Halpern1996,and references therein).The radiation from at least one of them–a nearby,bright PSR J0437?4715,with the?eld B～3×108G–has been shown to have a thermal-like component apparently originating from the NS surface(Becker&Tr¨u mper1993; Edelstein et al.1995;Halpern et al.1996).To interpret these data properly,the observed spectra and the light curves should be compared to nonmagnetic atmosphere models.One cannot exclude also that high magnetic?elds of ordinary pulsars are concentrated in small“platelets”, whereas a substantial fraction of the NS surface has much lower?elds(Page et al.1995).Finally,since the NS mag-netic?elds are subject to decay(although characteristic time of this decay remains highly uncertain—see,e.g., Bhattachrya1995),one can expect that very old NSs,most of which do not show pulsar activity,have magnetic?elds su?ciently low for nonmagnetic atmosphere models could be applied to analyze their thermal radiation.Detection of two old NS candidates has been reported recently(Stocke et al.1995;Walter et al.1996).Although identi?cation of these objects as NSs requires further con?rmation,the amount of dead pulsars observable through their thermal radiation should exceed considerably that of active pul-sars,so that one can expect more such objects to be dis-covered in near future.

Thus,investigation of the nonmagnetic NS atmosphere models can be justi?ed not only by their relative simplicity —the models can be applied to analysis of observations of real objects.Although a set of such models has been pre-sented in R87,there exist several reasons to re-examine the previous results and to continue work on the nonmag-netic models.

(i)New,much improved opacity codes have been de-veloped by the OPAL group,and a comprehensive set of the opacity data has been computed(Iglesias et al.1992; Rogers&Iglesias1994).These new opacities substantially exceed those from LAOL for heavy elements just at the densities and temperatures characteristic for the NS at-mospheres.Particularly interesting for the application to NS atmospheres is a set of monochromatic opacities for pure iron composition,generated at request of,and in collaboration with,Roger Romani(Romani1996,private communication).Rajagopal and Romani(1996;hereafter RR96;see also Romani et al.1996)have used the results to develop modi?ed NS atmosphere models.1

(ii)No hydrogen atmosphere models and only three helium models(for T e?=105.5,106and106.5)were presented in R87,and a few hydrogen and helium non-magnetic models were presented in our previous pa-pers(Shibanov et al.1992;Zavlin et al.1994;Pavlov et al.1995),mainly for comparison with the magnetic mod-els,without a systematic analysis.However,the low-?eld NSs,including the millisecond pulsars,are assumed to have been passed,during their long life,through an ac-cretion stage when a substantial amount of light elements have been accumulated at the NS surface.(It may even be that the surface magnetic?elds are low because of accretion-induced?eld decay—see,e.g.,Bhattacharya 1995).Due to gravitational strati?cation(Alcock&Illar-ionov1980)and/or nuclear spallation reactions which de-stroy heavy elements(Bildstein et al.1992,1993),the up-per atmospheric layers can be expected to consist mainly of lightest elements unless they have been dragged down-ward by convection or burned in nuclear reactions.Only two hydrogen atmosphere models(for T e?=105.5and 106K)were presented in RR96,which is not enough for a detailed analysis of how the atmospheric structure and emergent spectra depend on the model parameters.

V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds3

(iii)Only the(local)?ux spectra of the emergent radi-ation have been studied in both R87and RR96.However, the temperature distribution over the NS surface should be,as a rule,nonuniform due to either the presence of a hot polar cap in active pulsars(which is demonstrated by the light curve of PSR J0437?4715—Becker& Tr¨u mper1993)or anisotropy of the heat conduction in the NS crust(Shibanov&Yakovlev1996).This means that the?ux from a NS is to be determined by integra-tion of the angle-dependent speci?c intensities over the observable NS surface;this?ux may di?er substantially from the local?ux due to the frequency-dependent limb-darkening e?ect which has not been investigated for the nonmagnetic models.In addition,gravitational bending of the photon trajectories is to be taken into account to obtain correct light curves of the NS thermal radiation (Shibanov et al.1995b;Zavlin et al.1995b,c).

(iv)It would be desirable to verify the models pre-sented in R87and RR96using an alternative,more mod-ern numerical approach which has been applied by our group to modeling magnetic NS atmospheres.In partic-ular,that approach takes into account anisotropy of ra-diation in both solving the radiative transfer and calcu-lating the atmosphere temperature distribution,whereas the Lucy-Uns¨o ld method employed in R87and RR96for calculating the temperature distribution is correct only in the case of isotropic radiation?eld.In addition,Thomson scattering,treated in R87with an oversimpli?ed approach, and excluded from consideration in RR96,may be impor-tant for light element atmospheres at T e?>～106K and is to be included consistently in the atmosphere modeling.

In this paper we present a set of nonmagnetic NS at-mosphere models of di?erent chemical compositions com-puted with an advanced atmosphere modeling code com-bined with the OPAL opacities.We study the atmospheric structure,and the spectra and angular distribution of emergent radiation in a wide range of e?ective temper-atures and surface gravities.We consider the atmospheres in radiative equilibrium,which means that the total en-ergy?ux is transferred only by radiation,whereas other heat transport mechanisms(convection,electron heat con-duction)are of no importance.The models with convec-tion,which plays a role at lower T e?,will be analyzed elsewhere(see Zavlin et al.1996for preliminary results). Our estimates show that,for the models considered,con-tribution of electron heat conduction to the heat?ux never exceeds a few tenths of percent at unit optical depth.It may reach several percent only at very bottoms of heavy element atmospheres with lowest e?ective temperatures; the intensities of far Wien tails of the emergent radiation formed at these layers are too low to be of any practical importance.It should be mentioned that our estimates of the conductive opacity exceed those of RR96(making electron heat conduction less important)because RR96 used an approximate equation with the Coulomb loga-rithm omitted.On the other hand,the equations we use (Hochstim&Massel1969)also become inaccurate at high densities and low temperatures,so that the problem needs further consideration.

We describe our approach to atmosphere modeling in Section2,and present the results in Section3.Some ob-servational implications are discussed in Section4.

2.General description of the model

2.1.Basic equations

Since a characteristic thickness of the NS atmosphere,～0.01?1cm,is much smaller than the NS radius,and characteristic densities are rather high,we consider the plane-parallel atmospheres in local thermodynamic equi-librium.Following a standard approach(e.g.,Mihalas 1978),we proceed from the radiative transfer equation in the form of a second-order boundary problem:

μ2

d

kν

d

kν

d

2

(I+ν+I?ν),I+ν=Iν(y,+μ)and I?ν=Iν(y,?μ)are the speci?c spectral intensities of the outward-and inward-directed streams of radiation,y is the column density(d y=?ρd z),and Bν=Bν(T(y)) is the Planck function.The monochromatic opacity kν= kν(y)=αν(y)+σν(y)is the sum of the true absorption opacityανand scattering opacityσν.For the source func-tion Sν=Sν(y)we use

Sν=(σνJν+ανBν)k?1ν,(3) where Jν=Jν(y)= 10uνdμis the mean intensity.This form of the source function implies the isotropic scatter-ing approximation which proved to be very accurate even in the case when scattering dominates over true absorp-tion(e.g.,Mihalas1978).The boundary conditions(2) assume that there is no incident radiation at the outer boundary y=0,and the photon energy density tends to its equilibrium value at large depths.

In the present paper we consider only atmospheres in radiative equilibrium,where convection and electron heat conduction are negligible,and the energy is trans-ferred solely by radiation.In this case the radiative?ux is independent of y and coincides with the total?ux, F≡σSB T4e?,

∞

Fνdν=4π ∞0dν 10dμμ2d y=σSB T4e?,(4)

where Fν(y)is the spectral?ux,σSB is the Stefan-Boltzmann constant,and T e?is the e?ective temperature.

4V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic ?elds

It follows from Eqs.(1)and (4)

that the

radiative

equilib-

rium

condition

can be

also presented in the form

∞

αν(J ν?B ν)d ν=0.(5)

We consider the atmospheres in hydrostatic equilib-rium,which implies that the gas pressure is P =g y ,where g is the gravitational acceleration (the radiative force is in-signi?cant for the temperatures of interest,T e??107K).The above equations are to be supplemented with EOS,P =P (ρ,T ),and true absorption and scattering opac-ities,αν(ρ,T )and σν(ρ,T ).The opacities can be either extracted from opacity libraries or calculated with the use of atomic data (radiative cross sections).In the latter case,the set of equations of ionization equilibrium (EOIE)should be solved to ?nd the fractions of ions in di?erent stages of ionization as well as the electron number density N e (ρ,T ).

The input parameters of a model are the e?ective tem-perature T e?,surface gravitational acceleration g (or the NS mass and radius)and chemical composition.Solving these equations yields the atmospheric structure (T (y ),ρ(y ),N e (y ),etc.),and the radiative ?eld in the atmo-sphere (F ν(y ),J ν(y ),u ν(y,μ),including the angular and spectral distribution of the emergent radiation,I ν(0,μ)=2u ν(0,μ)).

It should be noted that the models give the radiation and the atmospheric structure for a local element of the NS surface.If the surface conditions are not uniform (e.g,the e?ective temperature may be higher at the magnetic poles),the observable photon ?ux should be obtained from integration of I ν(0,μ)over the visible NS surface,with allowance for the gravitational redshift and bending of the photon trajectories (e.g.,Shibanov et al.1995b;Zavlin et al.1995b,c).

2.2.Method of solution

Solving the system of the model equations splits into two parts,each of which can be solved with the knowledge of the other:calculation of the radiation ?eld given the structure of the atmosphere,and calculation the structure given the radiation ?eld.We start from calculating the temperature distribution in the Eddington approximation,integrating the equation d T e?

=3

T 3

(6)

with the boundary condition T (0)=0.841T e?.Here k R =k R (T,ρ)is the Rosseland mean opacity which is calculated at each step of the integration,with ρ(y )being found from EOS and hydrostatic equilibrium.The initial atmospheric structure is used to obtain a ?rst approximation for the radiation ?eld.

Computing the radiation ?eld starts from calculating the Eddington factors,f ν=

J ?1ν

1

μ2

u νd μ

and h ν=

J ?1ν

1

μu νd μ.(7)

In deep layers,where the radiation is nearly isotropic and the di?usion approximation is valid,we have f ν?1

2.The dependences of f νand h νon y account for the depth dependence of anisotropy of radiation.Integrat-ing Eqs.(1)and (2)over μyields the ordinary di?erential equation for the mean intensity,d k ν

d

k ν

d d y 1d y f νJ ν=αν(J ν?B ν)?(10)

?αν

d B ν

∞

αν′(d B ν′/d T )d ν′

,

with the boundary conditions (9)and with the Eddington factors calculated as described above.Since the spectral ?ux is F ν=

4πd y f νJ ν,

(11)

the additional (as compared to Eq.(8))integral term in Eq.(10)automatically provides the total radiative ?ux F to be constant throughout the atmosphere at any given it-eration.Moreover,this integral term turns into zero iden-tically when the radiation ?eld and atmospheric structure satisfy the radiative equilibrium condition (5).

The mean intensity found from Eq.(10)is used to cal-culate a new temperature T +δT for each atmospheric

V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds5 layer,with the correctionδT equal to the ratio of the two

integrals in the additional term in Eq.(10).Once the new

T(y)is obtained,we can complete calculation of the at-

mospheric structure(and new opacities)making use of the

hydrostatic equilibrium,EOS and EOIE,and then com-

pute the corresponding radiation?eld as described above

(starting with calculation of the Eddington factors).The

global iterations continue until the maximum relative cor-

rection|δT/T|becomes smaller than a given value(typi-

cally～1%).Note that using the modi?ed transfer equa-

tion(10)substantially accelerates the global iterations(a

reasonable accuracy is reached in2–4iterations).

We solve the model equations numerically by discretiz-

ing on the depth variable y,photon energy E=hνand

angular variableμ.Typically,we use a grid with100?150

depth points logarithmically spaced between10?8?10?6

and1?10g cm?2,100?200energy points logarithmi-

cally spaced between0.1k B T e?and60k B T e?,and50an-

gular points forμbetween0and1.A fourth-order Hermi-

tian discretization method(Auer1976)is applied to solve

Eqs.(8),(9)and(10).We also use the elimination scheme

suggested by Rybicki&Hummer(1991),which provides

a higher accuracy in comparison with the standard Gaus-

sian method.

Since the monochromatic opacities of heavy elements

(particularly iron)include many thousands of spectral

lines,their frequency dependence should be properly

binned to use them for construction of the atmosphere

models.We average the original monochromatic opacities

arithmetically in each bin of the logarithmic frequency

grid(typically(1?2)×103frequency points per bin)and

use these binned opacities to compute the atmospheric

structure and a“smoothed”spectrum as described above.

We checked by varying the number of bins that～100?200

bins provide su?cient accuracy(a relative error<～1?2%)

of the atmospheric structure and the total?ux.To obtain

the spectrum with a?ne spectral resolution,we calcu-

late the intensities of the outward-and inward-directed

streams,

I+ν(y,μ)=1

μ y1y kνd y2

d y1(12)

and

I?ν(y,μ)=1

μ y y1kνd y2

d y1,(13)

with the original monochromatic opacities(at～104fre-quency points).The spectral?ux is then computed as Fν(y)=2π 10[I+ν(y,μ)?I?ν(y,μ)]μdμ.(14)

These“output”spectra can be further smoothed,if needed,with a standard spline procedure.2.3.Opacities and EOS

The true absorption opacityανis due to the free-free, bound-free and bound-bound transitions;σνis dominated by the Thomson scattering on electrons.The main distinc-tion of the opacities,EOS and EOIE in a NS atmosphere from those in usual stars is due to high non-ideality(pres-sure e?ects).For modeling hydrogen/helium atmospheres, we calculate the opacities and EOIE of non-ideal gases using the occupation probability formalism(Hummer& Mihalas1988)as described by Zavlin et al.(1994).The re-sults of these calculation agree fairly well with the OPAL data(Iglesias et al.1992)obtained with a quite di?erent approach(Rogers1986).

For modeling iron atmospheres,we use the opacities and EOS computed by the OPAL group,in collaboration with R.Romani.The opacities kνwere computed for the domain log R=?5(1)1,log T=4.5(0.25)7.5(R≡ρ/T36,ρin g cm?3,T6=T/(106K)).They are provided on a grid of104photon energies linearly spaced in the range 2×10?3k B T≤E≤20k B T.To obtain kνat given T andρ, we use the cubic spline interpolation of log kνwith respect to log R and log T.For E>20k B T we use additional tables for the same R and T and1eV≤E≤10keV. 3.Results

We have computed a set of the atmosphere models for NSs with various e?ective temperatures T e?,gravitational accelerations g and chemical compositions.In this paper we demonstrate representative models for pure hydrogen, helium and iron compositions.The hydrogen and helium models are presented for log T e?=4.7(0.3)6.5;higher ef-fective temperatures are not expected to be observed in cooling NSs,and at lower T e?the helium atmospheres be-come convectively unstable.The iron atmosphere models, where convection can develop at higher T e?,are presented for log T e?=5.3(0.3)6.5.We chose a“basic”gravita-tional accelerations g14≡g/(1014cm s?2)=2.43(it cor-responds to standard NS mass M=1.4M⊙and radius R=10km);a few examples are given for g14=0.5and 4.5.

3.1.Atmospheric structure

The atmospheric structure has not been analized thor-oughly in the previous works;however,it is not only in-teresting in itself—it also allows one to better under-stand properties of the emergent radiation.Quite di?er-ent structures of atmospheres composed of light and heavy elements are due to very di?erent opacities.An example of the energy dependence of the opacities at the depths log y=?6,?4and?2is shown in Fig.1for the hydrogen, helium and iron atmosphere models with log T e?=5.9, g14=2.43.We see that the Fe opacities,dominated by numerous spectral lines,are,on average,greater than H

6V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic

?elds

Fig.1.Monochromatic opacities vs .photon energy at the layers y =10?6,10?4and 10?2g cm ?2of the hydrogen,helium and iron atmospheres with T e?=105.9K and g =2.43×1014cm s ?2.The upper right panel shows the original Fe opacities (104energy points);the binned opacities (200points)are shown in the bottom right panel.The horizontal dashed lines demonstrate the Rosseland mean opacities k R at the corresponding layers.

and He opacities,and their mean slope at the most im-portant frequency range is almost ?at.On the contrary,H and He opacities are dominated by the free-free and bound-free transitions (k ν∝ν?3∝E ?3)at moderate and low frequencies (energies),whereas at high frequen-cies they are dominated by Thomson scattering (indepen-dent of ν).These distinctions are responsible for di?erent atmospheric structures and properties of the emergent ra-diation.

Particularly important component of the atmospheric structure is the depth dependence of temperature,T (y ).Examples of T (y )for several values of T e?and for three chemical compositions are demonstrated in Fig.2.We see that,for any T e?,the temperature at a given depth y is systematically higher for iron atmospheres.In other words,a characteristic depth,y e?,which can be de?ned by the equation T (y e?)=T e?,is smaller for iron atmo-spheres (?4.6

e? y

0k R d y ,are due to higher Rosseland mean opacity k R (cf.Fig.1).Substantially higher surface temperatures (T s [≡T (0)]=(0.78?0.81)T e?for Fe vs .T s =(0.31?0.53)T e?for H,at log T e?=5.3?6.5)can be explained by generally ?at-ter frequency dependence of the Fe opacity,which results in temperature pro?les and surface temperatures close to those for the grey atmosphere (T s ?0.811T e?).The dif-ferences between the hydrogen and helium temperature pro?les depend on the degree of ionization in the

atmo-

Fig.2.Temperature pro?les T (y )of the models with di?erent e?ective temperatures and chemical compositions.The ?lled circles mark the points where T (y )=T e?;the corresponding y =y e?can be considered as a characteristic depth of the atmosphere.

sphere,which,in turn,is determined by T e?and g .In par-ticular,at high T e?the light element plasma is completely ionized,and its opacity in deep,dense layers is mainly de-termined by the free-free transitions:k R ～Z 3A ?2ρT ?7/2,where Z is the ion charge,and A the atomic https://www.doczj.com/doc/bd17072563.html,ing this expression for k R ,Eq.(6),and EOS for the ideal gas,ρ∝A (Z +1)?1gyT ?1,we obtain T 15/2d T ∝

V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds

7 Fig. 3.Mass density pro?lesρ(y)for two values of

log T e?=5.3and6.5and three chemical compositions.The

?lled circles indicate the characteristic densitiesρe?=ρ(y e?)

(cf.Fig.2).

gZ3[A(Z+1)]?1y d y,which yields the following equation

for the asymptotic behavior of the temperature of a fully

ionized atmosphere at large y:

T(y)∝ Z3A T4e? 2/17y4/17.(15)

The value of the multiplier in front of y4/17for He ex-

ceed that for H by3%only;this means that the temper-

ature pro?les are essentially the same at su?ciently large

depths,which we observe for the models with log T e?=

6.5.At log T e?=5.9,the hydrogen atmosphere is com-

pletely ionized at the most important depths,whereas the

helium atmosphere contains a few percent of once-ionized

helium,HeII(see Fig.4),which substantially increases the

bound-free opacity(Fig.1).This leads to a greater di?er-

ence between the He and H temperatures at y～y e?.At

log T e?=5.3the di?erence is even more pronounced be-

cause a larger fraction of HeII,>～10%,is available.Finally,

at log T e?=4.7,the H atmosphere contains～10%of neu-

tral atoms HI,and the He atmosphere is mainly composed

of HeII,with an admixture of HeI.In both cases opacity

is dominated by the bound-free transitions,and the tem-

perature pro?les are close to each other at y～y e?.

The temperature dependence of the He atmosphere

with log T e?=4.7shows an interesting anomaly at small

y:it slightly increases outward at log y<～?7.5.This in-

verse behavior is due to substantial growth with y of

the HeI fraction at log y<～?7(see Fig.4),accompa-

nied by increasing role of the HeI photoionization absorp-

tion at E>

～

20?25eV.To provide the radiative equilib-

rium(Eq.(5))at small y,the local blackbody intensity

Bν(T(y))is to be greater than the mean intensity Jν(y)

Fig.4.Depth dependences of ionization for models with dif-

ferent values of log T e?(numbers near the curves).Two upper

panels show pro?les of non-ionized hydrogen and helium,x0(y),

and once-ionized helium,x1(y),for pure hydrogen and helium

models.The bottom panel shows the depth dependences of the

mean ion charge Z?(y)for pure iron models.The?lled circles

show the corresponding quantities at y=y e?.

at low energies(below the main photoionization jump),

and smaller than Jν(y)at high E(above the jump).At

log y<～?8the main jump(provided by HeII)arises at

?50?55eV,whereas at?7<～log y<～?6the jump is

shifted to?20?25eV due to a greater fraction of HeI.

Therefore,for energies between approximately20?55eV,

Bν>～Jνat log y<～?8,while the opposite inequality holds

at?7<～log y<～?6.Since Jνis practically independent

of y in all these layers,the Planck function at log y<～?8

should be higher than at?7<～log y<～?6.This decrease

of Bνat these depths is possible if only the surface temper-

ature T(0)is higher than the temperature in the deeper

layers.

Figure3shows the depth dependence of the mass

densityρ(y)for the models with log T e?=5.3and6.5.

The density grows with y linearly(exponentially with the

geometrical depth z)at the upper(exponential)atmo-

sphere,where the temperature is close to T s.In deep lay-

ers,the density grows more gradually due to the steeper

temperature growth.The density can be estimated as

8V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds

ρ～m H A(Z?+1)?1gy(k B T)?1,where Z?=Z?(y)= N e Am H/ρis the mean ion charge de?ned as the ratio of the electron number density N e to the summed number densities of atoms in all stages of ionization.Since the hy-drogen and helium atmospheres at the temperatures con-sidered are strongly ionized(Z??Z),the density of a helium atmosphere at a given y exceeds that of a hydro-gen atmosphere with the same T e?and g.In particular, for deep layers of fully ionized atmospheres the curvesρ(y) has a universal slope,

ρ∝Z?6/17(Z+1)?15/17A19/17g15/17T?8/17

e?

y13/17(16) so thatρ(He)(y)/ρ(H)(y)?2.6.Iron atmospheres at these temperatures are only partially ionized,and the di?er-ent disposition of theρ(Fe)(y)with respect toρ(He)(y) at log T e?=5.3and6.5is explained by di?erent de-grees of ionization(Z??5?7and21?22,respec-tively—see Fig.4).Characteristic densities in di?erent models,ρe?=ρ(y e?),are generally lower for iron at-mospheres,ρe?～0.003?0.02g cm?3,than for hydro-gen/helium atmospheres,ρe?～0.005?1g cm?3,for the range log T e?=5.3?6.5.The ratio z e?=y e?/ρe?gives a characteristic scale height in an atmosphere;it varies from～0.01cm for Fe at log T e?=5.3to～2cm for H at log T e?=6.5.

The atmospheric opacity and,hence,the temperature, density and properties of the emergent radiation depend essentially on the ionization state at di?erent depths.Fig-ure4demonstrates the depth pro?les of the fractions of ions in di?erent stages of ionization,x j=N j/(N0+N1+ ...+N Z)(j=0,1,...,Z is the ion charge).We show the nonionized fraction x0for the hydrogen models,nonion-ized and once-ionized fractions,x0and x1,for the helium models,and the mean ion charge Z?= Z j=1jx j for the iron models.The hydrogen atmospheres are strongly ion-ized at the considered temperatures(even at log T e?=4.7 the nonionized fraction does not exceed17%);however, the contribution of the bound-free transitions into opacity is not negligible at x0as small as10?3?10?2(cf.Fig.1). At log y<～?2.5the values of x0are determined by the usual Saha equation:x0grows with y at very outer lay-ers,where the temperature is almost constant,due to increasing density;the growth decelerates in deeper lay-ers due to increasing temperature.At large depths,where ρ>～10?2?10?1g cm?3,the fraction x0decreases steeply due to pressure ionization.Similar behavior is seen for the helium fractions x0and x1,although the ionization degree is lower than for hydrogen due to higher ionization poten-tials.At moderate temperatures,the main contribution to the opacity comes from the bound-free transitions of HeII;the non-ionized HeI plays a role at lower T e?.The iron atmospheres are only partly ionized even at high T e?and very deep layers(e..g.,Z?does not exceed21?22 through the whole atmosphere with log T e?=6.5)due to very high ionization potentials and binding energies.

The atmospheric structure depends also on the gravi-tational acceleration g.Our computations show that vary-ing g in the range expected for the surfaces of NSs a?ects the temperature pro?le only slightly:T∝g2/17in very deep layers(see Eq.(15)),and the e?ect on the surface temperature is even smaller.The characteristic depth and density depend on g much stronger;in particular,ρ∝g in surface layers,andρ∝g15/17in deep layers.The ioniza-tion degree is a?ected accordingly:generally,the ionized fractions grow with g,particularly at the surface layers.

3.2.Spectral?ux

In Fig.5we show the spectral?uxes of the emergent radi-ation for selected e?ective temperatures and three chemi-cal compositions.The blackbody?uxesπBν(T e?)are also shown for comparison.

We see that the model spectra are substantially di?er-ent from the blackbody spectra with the same T e?.The most striking distinction is much harder high-energy tails of the spectra emergent from H and He atmospheres(see also R87,Pavlov et al.1995,and RR96).The reason for such behavior is that the H and He free-free and bound-free opacities rapidly decrease with increasing photon en-ergy(∝E?3in a wide energy range—see Fig.1),so that the radiation of higher energies is formed in deeper and hotter layers.The maxima of the spectra of strongly ionized atmospheres(at log T e?>～5.0and5.5for H and He,respectively)lie at E=(4.6?5.0)k B T,i.e.,they are shifted by a factor of1.6?1.8from the correspond-ing blackbody maxima towards higher energies.Thom-son scattering dominates the opacities at high energies (Fig.1),which must be taken into account for calculating the high-energy tails(i.e.,at E>～1keV for the models with log T e?=5.9).

At lower T e?,the light element atmospheres are less ionized,and photoionization edges and spectral lines are seen in their spectra.In particular,the Lyman jump and Lyman-αline of HeII become perceptible in the helium atmosphere spectra at log T e?<～6,when the fraction of the once-ionized helium is as small as a few percent(see Fig.4).The same fraction of neutral hydrogen is needed to form the hydrogen Lyman features which become well pronounced at log T e?<～5.

The low-energy tails(E?k B T)of the model spectra follow the Rayleigh-Jeans law(∝E2)corresponding to the surface temperature T s(

1.This occurs because the opacities grow with decreasing

E so that the low-energy radiation emerges from the very surface layers.The low-energy?ux is most strongly sup-pressed(by a factor of up to?3.2)in hot H and He atmospheres because of smallness of the ratio T s/T e?(see Fig.2).

The spectra emitted from the iron atmospheres are much more complicated,mainly due to numerous spectral lines and photoionization edges produced by ions in vari-

V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds

9 Fig.5.Spectral?uxes of emergent radiation for hydrogen,helium and iron atmospheres with di?erent values of log T e?(numbers near the curves).Dotted curves show the corresponding blackbody?uxesπBν(T e?

).

Fig.6.Spectral?uxes emergent from helium atmospheres with

T e?=105K and di?erent surface gravities g=g14×1014cm

s?2:g14=0.5(dashed curve),g14=2.43(solid)and g14=4.5

(dot-dashed).

ous stages of ionization.Generally,they are closer to the

blackbody spectra,and their high-energy tails are,on av-

erage,softer than those emitted from light element atmo-

spheres;their low-energy tails lie only slightly below the

blackbody(Rayleigh-Jeans)spectra.The reason is that

the energy dependence of the iron opacities,in spite of

its complexity,is,on average,?atter than that for H and

He.This makes the emergent spectrum to be closer to the

grey atmosphere spectrum which,in turn,only slightly

di?ers from the blackbody spectrum.However,the local

deviations from the blackbody spectra are quite substan-

tial,especially in the regions of the photoionization edges,

and strongly dependent on the e?ective temperature.

Figure6demonstrates how the gravitational accelera-

tion a?ects the spectral?ux emitted by the helium atmo-

sphere with log T e?=5.0.The e?ect is seen most vividly

in the Lyman continuum of HeII(E>54eV):the Ly-

man discontinuity becomes progressively smoothed,and

the trough at E～0.1keV becomes shallower,with in-

creasing g.This occurs because the density grows with

g and the pressure e?ects smooth out the photoioniza-

tion jump in the opacity spectra(Zavlin et al.1994).The

gravity e?ect(pressure broadening)is also seen in the pro-

?les of the absorption lines,particularly of the20eV line

of HeI.At higher temperatures,when the light element

atmospheres are strongly ionized and the spectra are con-

tinuous,the gravity e?ect becomes insigni?cant.In the

case of iron composition the e?ect of surface gravity can

be seen at higher temperatures.

10V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds

Fig.7.Spectra of the speci?c intensity of emergent radiation for the hydrogen and iron atmospheres with T e?=105.9K at di?erent values ofμ=cosθ(numbers near the curves);θis the angle between the surface normal and direction of emission. 3.3.Speci?c intensities:spectra and angular distributions

Figure7demonstrates spectra of the speci?c intensities of

the emergent radiation at di?erent anglesθ=arccosμbe-

tween the direction of emission and the surface normal for

the models with hydrogen and iron compositions and the

e?ective temperature of105.9K.We see that the inten-

sity is rather anisotropic—it decreases with increasing

θ(limb-darkening e?ect).This is explained by the fact

that radiation emitted at largerθescapes from shallower

atmospheric layers where the temperature is lower.The

anisotropy depends on photon energy(the intensity spec-

trum varies withθ)and on chemical composition of the

atmosphere.Note that the maximum of the intensity spec-

trum shifts towards lower energies with increasingθ.

The energy-dependent degree of anisotropy can be de-

scribed by the ratio

Iν(μ=1)?Iν(μ=0)

aν=

V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic ?elds

11

Fig.9.Polar diagrams of the speci?c spectral intensities (in units of 1019erg cm ?2s ?1keV ?1ster ?1)emitted from hydro-gen,helium and iron atmospheres (dash-dotted,dashed and solid lines,respectively)with T e?=105.9K,g =2.43×1014cm s ?2at three values of photon energy depicted at the up-per-right corners of the panels.The atmosphere normal is directed upward.The dotted lines show the corresponding (isotropic)blackbody intensities.

Figure 9shows representative angular distributions of the speci?c intensity emergent from hydrogen,helium and iron atmospheres at di?erent photon energies.Although these distributions do not show the prominent features characteristic of radiation emitted from strongly magne-tized NS atmospheres (see Fig.4in Shibanov &Zavlin 1995),this picture clearly illustrates strong dependences of the distributions on both the energy and chemical com-position.4.Discussion

The presented examples of the atmosphere models show that the atmospheric structure and the properties of emergent radiation depend substantially on the chemi-cal composition of radiating layers and on the value of the e?ective temperature T e?.For the light element at-mospheres with a relatively high e?ective temperatures

(log T e?>～5.3for hydrogen,log T e?>～6for helium),the

atmospheric matter is strongly ionized,the (continuous)

spectrum is much harder at X-ray energies than the black-body spectrum,and the ?ux is lower than the blackbody ?ux at low (UV/optical)energies.For lower T e?,spec-tral features arise in the emergent spectra,particularly in the UV/EUV domain.These distinctions mean that ?tting thermal spectra observed from NSs with a black-body model may be quite misleading.In particular,the blackbody ?t of the high-energy (X-ray)tail of an ob-served spectrum would yield systematically overestimated e?ective temperature and underestimated A/d 2ratio (A is the area of the radiating surface,and d is the distance).The error introduced depends on the energy range and response function of the speci?c detector;e.g.,for the ROSAT PSPC the temperature may be overestimated by a factor of up to ?3,and A/d 2underestimated by more than an order of magnitude,at detectable NS tempera-tures.On the contrary,at low (UV-optical)energies T e?is underestimated or A/d 2overestimated ,typically by a factor of 2?3.To obtain reliable estimates of these quan-tities,one should ?t the results with various model atmo-sphere spectra.It may happen that several models (e.g.,with di?erent chemical compositions)would equally well ?t the data in a narrow spectral range.To avoid the ambi-guity,multiwavelength (e.g.,optical through X-ray)ob-servations should be carried out,whenever possible,and ?tted with the atmosphere models.

Spectra of heavy element atmospheres are much more complicated —mainly because of numerous spectral lines,as we see from the examples of iron atmospheres.Note that although the shapes of our model spectra are qualita-tively similar to those obtained in RR96,the quantitative di?erences are signi?cant,especially in domains around the strong photoionization features.The di?erence may arise from both the di?erent procedures for opacity bin-ning and di?erent numerical approaches used for the mod-eling (e.g.,due to the inaccuracy of the Lucy-Uns¨o ld tem-perature correction procedure used in RR96).Although the overall shape of the iron atmosphere spectra is not so drastically di?erent from the blackbody,using black-body models for ?tting in narrow energy intervals may be even more misleading than for light element atmospheres because even the smoothed shape of the atmosphere spec-trum depends very sharply on T e?.For instance,we see from Fig.5that increasing T e?by a factor of 2completely changes the shape of the iron atmosphere spectrum,and ?tting the spectra with the blackbody in a range of,e.g.,0.2?0.6keV would lead to quite di?erent (and quite ir-relevant)results.

We do not know a priori what is the chemical compo-sition of the NS surfaces.We can only expect that if there are light elements,they should concentrate at the very surface,but we cannot exclude that the light elements are virtually absent,and the surface layers are composed of heavier elements.To investigate the chemical composition,observations with higher spectral resolution in the soft X-

12V.E.Zavlin,G.G.Pavlov and Yu.A.Shibanov:Model neutron star atmospheres with low magnetic?elds ray range would be very useful.Our examples of the iron

models show that the main spectral features(groups of

lines and/or photoionization edges)can be resolved with

a quite moderate resolution,～3?30,which can be eas-

ily achieved with modern CCD detectors.Thus,soft X-ray

observations of NSs with future missions(AXAF,XMM,

AST RO-E)and their interpretation with the NS atmo-

sphere models will be able to determine directly which

elements reside at the NS surface.

Our results also show a very important property of the

atmospheric radiation—it is anisotropic,with anisotropy

depending on frequency.This dependence is di?erent for

di?erent chemical compositions and e?ective tempera-

tures.This means that if the temperature is not uniform

over the NS surface(which is very plausible for real NSs),

one not only should take the anisotropy into account to

explain the light curves,but also one cannot?t properly

the spectra without allowance for anisotropy.To calculate

the?ux as measured by a distant observer,one should

integrate the speci?c?uxes,Iνcosθ,over the visible NS

surface with allowance for the gravitational redshift and

bending the photon trajectories.For instance,if the radi-

ation originates from a small hot spot(polar cap)with a

uniform T e?and chemical composition,the?ux as mea-

sured by a distant observer is given by the equation(Zavlin

et al.1995c)

F obs ν=

A a

(1+z)3

Iν

(μc),(18)

where A a is the apparent area of the hot spot,z=[1?(2GM/c2R)]?1/2?1is the gravitational redshift,ν0= (1+z)ν,μc is the cosine of the angleθc between the radius-vector of the spot center and wave vector of the photon which travels to the observer along the bended trajectory. For realistic NS radii,R>～5.6(M/M⊙)km(z<～0.45),the apparent area can be estimated as A a?πR2γ2μc,where γis the angular size of the spot.The important feature of Eq.(18)is that the observed?ux is proportional not to the local?ux,but to the speci?c intensity whose spectrum (see Figs.5and7)is substantially di?erent from that of the redshifted local?ux used in RR96(their Eq.(7)).

The low-?eld atmosphere models can be directly ap-plied to the nearby(d～140pc)millisecond pulsar PSR J0437?4715(P=5.75ms)detected with the ROSAT PSPC(Becker&Tr¨u mper1993)and EUV E DS(Edel-stein et al.1995;Halpern et al.1996).Fitting the ROSAT spectrum of this low-?eld(B～3×108G)pulsar with the single-component blackbody model proved to be un-satisfactory—the best-?t blackbody?ux(correspond-ing to T=1.5×106K)is lower than the observed?ux in high-energy PSPC channels(above?1keV).How-ever,the ROSAT spectrum can be satisfactorily?tted with double-component models.For the“power-law+ blackbody”?t,Becker&Tr¨u mper(1993)obtained T= 1.7×106K and the radiating area A=0.05km2,while Halpern et al.(1996)obtained T=(1.0?3.3)×106K and A=0.008?1.1km2.In this model,the power law spectrum is of a nonthermal origin,whereas the blackbody spectrum is produced by the hot polar cap of the pulsar.It remains unclear with this interpretation why there is no a phase shift between the pulsations at low energies(power law component)and high energies(thermal component)—being generated at di?erent distances from the NS sur-face,these components are expected to be phase-shifted and to have di?erent shapes.The values of the emitting area are also uncomfortably low in comparison with that expected from standard models for the radio pulsar po-lar cap,A=2π2R3/(cP)～10km2(Arons1981).Edel-stein et al.(1995)ruled out that this two-component?t is compatible with the EUV E data,whereas Halpern et al. (1996)came to the opposite conclusion and explained the EUV E?ux with the power law component.

It is natural to expect that?tting of the same data with the model atmosphere spectra and light curves would yield quite di?erent results.Indeed,our preliminary anal-ysis of both the ROSAT and EUV E data indicates that a single-component model,in which the soft X-ray radiation originates from two polar caps of areas A?2?3km2cov-ered with hydrogen or helium with T e??(0.9?1.0)×106 K,?ts satisfactorily both the spectra and the light curves (Becker et al.1996).These results di?er substantially from those of RR96who?tted the J0437-4715spectrum with the local?ux spectra,without allowance for the limb dark-ening and gravitational lensing.It worths noting that the atmosphere models can be used to predict the?ux from the NS in the far-UV range(λ<～2000?3000?A),where it should overtake the?ux of the cold(T?4000K)white dwarf companion.Measuring this?ux(e.g.,with HST) would be a critical test to distinguish between di?erent models of the radiating region.Another important test of the models would be to observe the pulsar radiation in the soft X-ray range with a spectral resolution su?cient to resolve spectral features;observations at moderate X-ray energies(～1?10keV)would be useful to constrain the power law component.Such observations would be quite feasible with future X-ray missions(SAX,AXAF, XMM,AST RO-E).

Acknowledgements.We thank F.J.Rogers and C.A.Iglesias for providing the OPAL library data,J.Tr¨u mper for stimulat-ing discussions.We are grateful to the referee,R.W.Romani, whose remarks helped us to improve this paper.This work was partly supported by RFFI grant RBRF96-02-16870A,INTAS grant94-3834and NASA grant NAG5-2807.V.E.Zavlin ac-knowledges the Max-Planck fellowship.

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star法撰写成就故事范例

1. 由于是浙江选考生，在高中第一次选考中化学只取得了91分的成绩，只剩下一次选考机会，成绩说好不算好，说坏不算坏的情况下，提升有些困难。不知道该放弃还是继续参加考试。如果不能达到97或100的成绩，可能会被目标大学相同的同学在高考上拉开差距。晚上也睡不好觉，经过一段时间的调整心态，说服自己还是奋力一搏。于是到选考前我每天利用下课时间翻看化学书和整理笔记，努力记住容易混淆的基础知识点，每天晚自习刷一张化学模拟卷，自批订正纠错，找到了自己很多弱点和盲点，并针对的做专题练习。终于在四月选考中达到了97的目标。虽然很累，但是为了实现离考上理想大学的目标更进一步，还是值得的。 专业知识技能——基础化学知识 可迁移技能——记忆、纠错（找到自己的弱点）、反省、整理、权衡利弊 自我管理技能——自我控制、抗压、调整心态、坚持不懈 2. 上个学期选专业时想要进入生物科学专业，但是看到报名的人数有点多，招收人数与报名人数之比大约是1/2，要经过面试淘汰一部分人。而且在看到身边的同学都非常优秀的情况下，竞争压力有些大，我心中也有些底气不足。但我还是迫使自己鼓起勇气报了名。报名之后，我上网了解记住了一些生物科学的知识，回忆了我高中时期学习生物的一些知识和经历，找到一些我可能在面试中能够用到的素材。并在脑海中模拟了几个面试中可能会提到的问题，并想了想如何回答。此外也找到我们寝室作为生命科学学院教授的新生之友询问了一些情况。到了面试那天，我准备的其中一些素材和回答派上了用场，顺利地应对面试教授提出的问题，通过了面试，成功进入生物科学专业，提升了我的信心。 专业知识技能——生物科学部分知识 可迁移技能——记忆、模拟（假想）、询问求助、面试交流、做好充分准备 自我管理技能——鼓起勇气（自我调节心态）、有信心的、自我控制 3. 小时候老师说我钢笔字写的挺好的，在小学的钢笔字比赛中获了奖，我可能

万科物业与中海物业的分析比较

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目录管理工作流程图 1．文件控制流程图 2．记录控制流程图 3．人员和培训管理流程图 4．采购管理流程图 5．物业服务管理流程图 6．顾客满意管理流程图 7．不合格品（服务）管理流程图 8．业主投诉处理流程图 安保工作流程图 1．安保管理流程图 2．物业管理部工作流程图 3．安保主管工作流程图 4．班长日检查工作流程图 5．样板房安保员岗位工作流程图 6．侧门岗安保员工作流程图 7．巡楼安保员操作流程图 8．业主搬迁操作流程图 9．外来人员出入管理流程图 10．消防应急方案出来流程图 11．突发事件处理流程图 12．安保工作重大事项处置流程图 13．电梯困人处理流程图 意外停电处理流程图 意外停水处理流程图

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