Dynamical and Radiative Properties of X-Ray Pulsar ...mason.gmu.edu/~pbecker/papers/west2/west_etal_2017b.pdf · Dynamical and Radiative Properties of X-Ray Pulsar Accretion Columns:

Dynamical and Radiative Properties of X-Ray Pulsar Accretion Columns:Phase-averaged Spectra

1Brent F. West1, Kenneth D. Wolfram2, and Peter A. Becker3

Department of Electrical and Computer Engineering, United States Naval Academy, Annapolis, MD, USA; [email protected] Naval Research Laboratory (retired), Washington, DC, USA; [email protected]

3 Department of Physics and Astronomy, George Mason University, Fairfax, VA, USA; [email protected] 2016 October 14; revised 2016 November 21; accepted 2016 December 5; published 2017 January 23

AbstractThe availability of the unprecedented spectral resolution provided by modern X-ray observatories is opening upnew areas for study involving the coupled formation of the continuum emission and the cyclotron absorptionfeatures in accretion-powered X-ray pulsar spectra. Previous research focusing on the dynamics and the associatedformation of the observed spectra has largely been confined to the single-fluid model, in which the super-Eddington luminosity inside the column decelerates the flow to rest at the stellar surface, while the dynamicaleffect of gas pressure is ignored. In a companion paper, we have presented a detailed analysis of the hydrodynamicand thermodynamic structure of the accretion column obtained using a new self-consistent model that includes theeffects of both gas and radiation pressures. In this paper, we explore the formation of the associated X-ray spectrausing a rigorous photon transport equation that is consistent with the hydrodynamic and thermodynamic structureof the column. We use the new model to obtain phase-averaged spectra and partially occulted spectra for Her X-1,Cen X-3, and LMC X-4. We also use the new model to constrain the emission geometry, and compare the resultingparameters with those obtained using previously published models. Our model sheds new light on the structure ofthe column, the relationship between the ionized gas and the photons, the competition between diffusive andadvective transport, and the magnitude of the energy-averaged cyclotron scattering cross-section.

Key words: accretion, accretion disks pulsars: general radiation: dynamics radiative transfer X-rays: binaries shock waves

1. Introduction

Nearly half a century has passed since the first observations ofX-ray emission from accretion-powered X-ray pulsars. In thesesources, the gravitational potential energy of the accreting gas isefficiently converted into kinetic energy in an accretion columnthat is collimated by the strong magnetic field ( ~B 1012 G).Since the surface of the neutron star represents a nearlyimpenetrable barrier, the kinetic energy is converted into thermalenergy at the bottom of the column and escapes through thecolumn walls in the form of X-rays. Hence the emergent X-rayluminosity, LX, is essentially equal to the accretion luminosity,

* *L GM M Racc , where M* and R* denote the stellar mass

and radius, respectively, and M is the accretion rate. The X-rayemission is generated along the column and in the vicinity of oneor both of the magnetic polar caps, thereby forming hot spotson the stellar surface. The X-rays were initially thought toemerge as fan-shaped beams near the base of the accretioncolumn, but it soon became clear that a pencil-beam componentwas sometimes necessary in order to obtain adequate agreementwith the observed pulse profiles (Tsuruta & Rees 1974;Bisnovatyi-Kogan & Komberg 1976; Tsuruta 1975).

Analysis of X-ray pulsar spectra is usually performed byaveraging over many rotation periods to obtain phase-averagedprofiles. The resulting spectra are characterized by a power-lawcontinuum extending up to a high-energy exponential (thermal)cutoff at 1060 keV. The spectra also frequently displayputative cyclotron absorption features (used to estimate themagnetic field strength), and a broadened iron emission line at6 keV (e.g., White et al. 1983; Coburn et al. 2002).

Until recently, attempts to simulate the spectra of accretion-powered X-ray pulsars using first-principles physical modelsdid not yield good agreement with the observed spectra. Theearliest spectral models were based on the emission ofblackbody radiation from the hot spots, and these were unableto reproduce the nonthermal power-law continuum. Thepresence of the cyclotron absorption feature led to thedevelopment of more sophisticated models based on a staticslab geometry, in which the emitted spectrum is stronglyinfluenced by cyclotron scattering (e.g., Yahel 1980b; Nagel1981; Mszros & Nagel 1985a, 1985b). While the magnetizedslab models are able to roughly fit the shape of the observedcyclotron absorption features, they remained unable toreproduce the nonthermal power-law X-ray continuum.The lack of a physics-based model for the formation of X-ray

pulsar spectra led to the characterization of the observed spectrausing parameters derived from a variety of ad-hoc functionalforms. Coburn et al. (2002) describe in detail three analyticalfunctions commonly used to empirically model the X-raycontinuum. The first function is the power law with high-energy cutoff (PLCUT), given by

( )( )( )

( )( )

=>

-G- -

E AEE E

e E EPLCUT

1, ,

, ,1

E E Ecut

cutcut fold

where E is the X-ray energy, is the photon spectral index, andEcut and Efold denote the cutoff and folding energies,respectively. The second function (Tanaka 1986) uses thesame power-law index , combined with a FermiDirac form

The Astrophysical Journal, 835:130 (25pp), 2017 February 1 doi:10.3847/1538-4357/835/2/130 2017. The American Astronomical Society. All rights reserved.

1

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for the high-energy cutoff (FDCO), defined by

( ) ( )( )= +-G

-E AE

eFDCO

1

1. 2

E E Ecut fold

The third function (Mihara 1995) uses two power laws, G1 andG2, in combination with an exponential cutoff (NPEX), writtenas

( ) ( ) ( )= +-G +G -E AE BE eNPEX . 3E E1 2 fold

Although these functional forms have no physical basis, theyare often used to characterize the observed spectral shapes.

In later work, it eventually became clear that the observedpower-law continuum was the result of a combination of bulkand thermal Comptonization occurring inside the accretioncolumn. The first physically motivated model based on theseprinciples, that successfully described the shape of the X-raycontinuum in accretion-powered pulsars, was developed byBecker & Wolff (2007, hereafter BW07). This new modelallowed for the first time the computation of the X-rayspectrum emitted through the walls of the accretion columnbased on the solution of a fundamental radiation-transportequation. The BW07 model was recently ported to XSPECusing a model described by Wolff et al. (2016). Anothervariation of the original BW07 model was developed andutilized by Farinelli et al. (2012, 2016), which added anadditional second-order scattering term to the transportequation related to the bulk flow. It is important to note thatthe Wolff et al. (2016) and Farinelli et al. (2012, 2016)implementations were premised on the utilization of anapproximate form for the accretion velocity profile, whichwas assumed to stagnate at the stellar surface while varying inproportion to the scattering optical depth measured from thestellar surface. This approximate velocity profile stagnates atthe stellar surface as required, but it does not follow the exactexpected variation with altitude (Becker 1998).

While the BW07 and F16 models have demonstrated notablesuccess in fitting the observed X-ray spectra for higherluminosity sources, these analytical models do not include allof the fundamental hydrodynamic and thermodynamic pro-cesses occurring within the accretion column. It is thereforenatural to ask how the incorporation of additional physicswould impact the ability of the BW07-type models to fit theobserved X-ray spectra. This has motivated us to investigatethe importance of additional radiative and hydrodynamicalphysics within the context of a detailed numerical simulation.The new simulation includes the implementation of a realisticdipole geometry, rigorous physical boundary conditions, andself-consistent energy transfer between electrons, ions, andradiation. In particular, we explore the dynamical effect of thegas pressure, which was neglected by BW07 and F16, in orderto examine how the radiation and gas pressures combine todetermine the dynamical structure of the accretion column andthe associated radius-dependent and vertically integrated,partially occulted X-ray spectra.

This is the second in a series of two papers in which wedescribe the new coupled radiative-hydrodynamical model. Theintegrated approach involves an iteration between an ODE-based hydrodynamical code that determines the dynamicalstructure, and a PDE-based radiative code that computes theradiation spectrum. The iterative process converges to yield aself-consistent and unique description of the dynamicalstructure of the accretion column and the energy distribution

in the emergent radiation field. In West et al. (2017, hereafterPaper I), we focus on solving the set of five coupledhydrodynamical conservation equations that describe theunderlying gas and radiation hydrodynamics within theaccretion column. We employed Mathematica to solve the setof nonlinear hydrodynamical equations in dipole coordinates todetermine the accreting bulk fluid velocity, the electron and iontemperature profiles, and the variation of the total energy flux.In this paper (Paper II), we focus on solving the fundamental

PDE photon transport equation to describe the production ofthe observed radius-dependent and phase-averaged X-rayspectra. The bulk velocity and electron temperature profilesdeveloped using the Mathematica calculation (described inPaper I) are used as input for the analysis conducted here, inwhich we solve the PDE photon transport equation using theCOMSOL multiphysics module. The iterative dual-platformcalculation is based on the transfer of information between theCOMSOL radiation spectrum calculation and the Mathematicahydrodynamical structure calculation. Using this iterativeapproach, we are able to model realistic X-ray pulsar accretioncolumns in dipole geometry by solving the photon transportequation using self-consistent spatial distributions for theaccretion velocity profile v(r) and the electron temperatureprofile Te(r).

2. Hydrodynamical Model Review

Our self-consistent model for the hydrodynamics and theradiative transfer occurring in X-ray pulsar accretion flows isbased on a fundamental set of conservation equationsgoverning the flow velocity, v(r), the bulk fluid mass density,

( )r r , the radiation energy density, Ur(r), the ion energydensity, Ui(r), the electron energy density, Ue(r), and the totalradial energy transport rate, ( )E r , where r is the radiusmeasured from the center of the neutron star. In ourhydrodynamical model, the field-aligned flow is deceleratedby the combined pressures of the ions, the electrons, and theradiation. The total pressure is therefore given by the sum

( )= + +P P P P , 4e i rtot

where Pe, Pi, and Pr denote the electron, ion, and radiationpressures, respectively. The equations-of-state for the materialcomponents are

( )= =P n kT P n kT, . 5i i i e e e

We note that in the X-ray pulsar application, the value of Prcannot be computed using a thermal relation because, ingeneral, the radiation field is not in thermal equilibrium witheither the matter or itself. Hence the radiation pressure must becomputed using an appropriate conservation relation. Thecorresponding energy densities are related to the pressurecomponents via

( ) ( ) ( )( )

g g g= - = - = -P U P U P U1 , 1 , 1 ,6

i i i e e e r r r

where (see Paper I) we set g = 3e and g = 5 3i . The ratio ofspecific heats for the radiation field is given by g = 4 3r .As shown in Paper I, the mathematical model can be reduced

to a set of five coupled, first-order, nonlinear ordinarydifferential equations satisfied by v, E , and the ion, electron,and radiation sound speeds, ai, ae, and ar, respectively, defined

2

The Astrophysical Journal, 835:130 (25pp), 2017 February 1 West, Wolfram, & Becker

by

( )gr

gr

gr

= = =aP

aP

aP

, , . 7ii i

ee e

rr r2 2 2

The ion and electron temperatures, Ti and Te, respectively, arerelated to the corresponding sound speeds via

( )g g= =a kTm

akT

m, , 8i

i ie

e e2

tot

2

tot

where mi and me denote the ion and electron masses,respectively, and +m m me itot . We assume here that theaccreting gas is composed of pure, fully ionized hydrogen.

2.1. Conservation Equations

Here, we briefly review the hydrodynamical conservationequations and the underlying model assumptions. The funda-mental independent variables in our model are the photonenergy, , and the radius, r, measured from the center of thestar. Since our model includes only one explicit spatialdimension, the structure across the column at a given valueof r is not treated in detail. Therefore, all of the physicalquantities (velocity, temperature, pressure, flux, and density)developed here are interpreted as mean values across thecolumn (see Appendix A).

The mass continuity equation can be written in dipolegeometry as (e.g., Langer & Rappaport 1982)

( )( )

[ ( ) ( ) ( )] ( )r r

= -

r

t A r rA r r v r

1, 9

where ( )

detail in Paper I and are shown here as

E

( )

s g

g g g

= + -

+ +-

+-

+-

-

da

dr

a

r v

dv

dr

R

m c

M

A a

v a a a

r

2

3 1

2

2 1 1 1

1,

21

r r g r

r

e

e

i

i

r

r

tot

2 2 2 2

( )

( )

g g=

-+ +

da

dr

a

r v

dv

dr

R

c

A

M

U

a

1

2

3 1,

22

i i i g i i

i2 2

( )

( )

g g=

-+ +

da

dr

a

r v

dv

dr

R

c

A

M

U

a

1

2

3 1,

23

e e e g e e

e2 2

E

( ) ( ) ( )*

g g t=

- - ^

d

dr

a

c v

R

R rc

c

1min , , 24r

r r g

2

2 1

3

3

1 2

E

( )

[( ) ( ) ] ( )

s

g g g

g g

=- -

+- +

+ +-

+-

+-

-

+ - + -

dv

dr

v

v a a

a a

r r

R

m c

M

A

v a a a

r

R

c

A

MU U

3 1

2 1 1 1

1

1 1 , 25

i e

i e g

i

i

e

e

r

r

gi i e e

2 2 2

2 2

2tot

2 2 2 2

2

where the cross-sectional area A(r) of the dipole accretioncolumn is given by Equation (10). The ions do not radiateappreciably, and therefore the ion energy density is onlyaffected by adiabatic compression and Coulomb energyexchange with the electrons (see Langer & Rappaport 1982).Conversely, the electrons experience freefree, cyclotron, andCompton heating and cooling, in addition to adiabaticcompression and the Coulomb exchange of energy with theions. The radiation energy density responds to both the creationand absorption of photons via freefree and cyclotronprocesses, in addition to Compton scattering with the electrons.

The thermal coupling terms in Equations (22), (23), and (25)describe a comprehensive set of electron and ion heating andcooling processes, which are broken down as follows,

( )= + + + + +

=-

U U U U U U U

U U

,

. 26

e

i

brememit

bremabs

cycemit

cycabs

Comp ei

ei

The terms in the expression for Ue denote, respectively, thermalbremsstrahlung (freefree) emission and absorption, cyclotronemission and absorption, the Compton exchange of energybetween the electrons and photons, and electron-ion Coulombenergy exchange. In our sign convention, a heating term ispositive and a cooling term is negative. The mathematicaldescriptions of each term in Equation (26) are provided inPaper I. We note that the term UComp plays a fundamental rolein the energy exchange between photons and electrons, asdiscussed in detail below and in Section 3.

Thermal bremsstrahlung emission plays a significant role incooling the ionized gas, and in the case of luminous X-raypulsars, it also provides the majority of the seed photons thatare subsequently Compton scattered to form the emergent

X-ray spectrum. The electrons in the accretion columnexperience heating due to freefree absorption of low-frequency radiation, and they also experience heating andcooling due to the absorption and emission of thermalcyclotron radiation. These emission and absorption processescan play an important role in regulating the temperature of thegas. On average, however, cyclotron absorption does not resultin net heating of the gas, due to the subsequently rapid radiativede-excitation, and we therefore set =U 0cyc

abs in our dynamicalcalculations. Near the surface of the accretion column,however, photons scattered out of the outwardly directed beamare not replaced, and this leads to the formation of the observedcyclotron absorption feature in a process that is very analogousto the formation of absorption lines in the solar spectrum(Ventura et al. 1979). While cyclotron absorption does notresult in net heating of the gas, due to the rapid radiative de-excitation, cyclotron emission will cool the gas. The electronscan also be heated or cooled via Coulomb collisions with theprotons, depending on whether the electron temperature Teexceeds the ion temperature Ti.Compton scattering plays a fundamental role in the

formation of the emergent X-ray spectrum. It is criticallyimportant in establishing the radial variation of the electrontemperature profile through the exchange of energy betweenphotons and electrons. The Compton cooling rate, prior todimensionless unit conversion, is expressed in terms of themass density, , the electron sound speed, ae, and the radiationsound speed, ar. The complete derivation of the electroncooling rate due to Compton scattering is given in Section 3.4,and the final result is given by (see Equation (76))

( ) [ ( ) ] ( ) ( )s= -U r n cm c

kT g r U r4 1 , 27e

ee rComp 2

where the temperature ratio function, g(r), is defined by

( ) ( )( )

( )g r T rT r

. 28e

IC

The derivation of TIC is provided later in Section 3.4.Equation (27) can be stated in terms of the sound speeds as

( )( )

( )sg g g

r=-

-U

g

m ca a

4 1

1. 29

e e r rr eComp

2 2 2

The sign of UComp depends on the value of g. The electronsexperience Compton cooling if

Solving the five coupled conservation equations to determinethe radial profiles of the quantities v, E , ai, ae, and ar requiresan iterative approach, because the rate of Compton energyexchange between the photons and the electrons depends on therelationship between the electron temperature, Te, and theinverse-Compton temperature, TIC, which is determined by theshape of the radiation distribution (see Equation (77)). In orderto achieve a self-consistent solution for all of the flow variables,while taking into account the feedback loop between thedynamical calculation and the radiative transfer calculation, thesimulation must iterate through a specific sequence of steps.The steps required in a single iteration are (1) the computationof the initial estimate of the accretion column dynamicalstructure, which is obtained by solving the five conservationequations described in Paper I under the assumption that

( ) ( )=T r T r ;eIC (2) the calculation of the associated radiationdistribution function by solving the radiative transfer equationdescribed here in PaperII; (3) the computation of the inverse-Compton temperature profile ( )T rIC based on integration of theradiation energy distribution using Equation (77); and (4) there-computation of the dynamical structure using the newestimate for ( )T rIC . The iteration continues until adequateconvergence is achieved between successive temperatureprofiles for all values of r. The iterative process is discussedin detail in Paper I and also in Section 4.1 here in Paper II.

2.2. Hydrodynamic Boundary Conditions

Here we summarize the set of boundary conditions utilizedin the Mathematica integration of the set of hydrodynamicalconservation equations, required in order to determine thestructure of the accretion column. At the top of the accretioncolumn (radius =r rtop), the inflow velocity v equals the localfree-fall velocity, so that

( ) ( )* = -

v v r

GM

r

2. 31top ff top

top

1 2

We also assume that the local acceleration of the gas is equal tothe gravitational value, and therefore

( )*==

dvdr

GM

r2. 32

r r top3

1 2

top

At the upper surface of the dipole-shaped accretion funnel,the radiation escapes freely, forming a pencil-beam comp-onent in the observed radiation field. This physical principlecan be used to derive a useful boundary condition at the uppersurface of the accretion column. The radial component of theradiation energy flux, averaged over the cross-section of thecolumn at radius r, is given in the diffusion approximation by(see Equation (71))

( ) ( )s

= - +F rc

n

dU

drvU

3

4

3, 33r

e

rr

where the first term on the right-hand side represents theupward diffusion of radiation energy parallel to the magneticfield, and the second term represents the downward advectionof radiation energy toward the stellar surface (with

transport of the photons through the energy space due tostochastic electron scattering (the Kompaneets operator); thespatial diffusion of radiation; the injection of seed photons; theabsorption of radiation by the accreting gas; and the differentialwork performed on the photons due to the convergence of theaccreting gas (Gleeson & Webb 1978; Cowsik & Lee 1982). Itis important to note that Equation (38) must be supplementedby suitable radiation boundary conditions for the photondistribution function f, imposed at the column walls, the stellarsurface, and at the top of the column. This is further discussedin Section 3.1.

Equations (38) and (39) can be combined to obtain the singleequivalent equation (e.g., Becker 2003)

( ) ( )

( )

s

k

+ =

+

+ +

+ +

v

v

f

tf

n c

m cf kT

f

ff

f f

1

1

3. 40

e

ee2 2

4

prod abs

In order to solve this equation, we must adopt a specificgeometrical model. The accretion flow in an X-ray pulsar ischanneled by the strong magnetic field, and therefore we willemploy a dipolar geometry for the accretion column, in whichthe cross-sectional area of the column, A(r), is given byEquation (10). The angle-dependent operators in Equation (40)can be removed by averaging across the column cross-section,and by assuming azimuthal symmetry around the magneticfield axis. The resulting transport equation, satisfied by theisotropic photon distribution function ( )f r, can be written inthe dipole geometry as

( )( )

( )[ ( ) ]

( )

s

k

+

=

+

+

+

+ + +

f

tv

f

r

n c

m cf kT

f

A r rA r

f

r

A r

A r v

r

ff f f

1

1

1

3,

41

e

ee2 2

4

prod abs esc

where the spatial diffusion coefficient in the radial direction isgiven by

( )( )

( )

ks

=rc

n r3. 42

e

We demonstrate in Appendix A that the photon escape termfesc can be implemented using an escape-probability formalismby writing

( ) ( )( )

( ) = -f r f rt r

,,

, 43escesc

where the mean escape time at radius r, denoted by ( )t resc , isgiven by

( ) ( )( )

( )=^

t r r

w r. 44esc

esc

Here, ( ) resc denotes the perpendicular escape distance acrossthe column at radius r, computed using

( ) ( ) ( )*

= -

r

r

R, 45esc 2 1

3 2

and ( )w r is the perpendicular diffusion velocity, given by

( )( )

( ) ( ) ( ) ( )t

t s= =^^

^ ^

w r c

c

rr n r rmin , , , 46e esc

where s denotes the electron scattering cross-section forphotons propagating perpendicular to the magnetic field, and

( )t r is the corresponding perpendicular optical thickness ofthe accretion column at radius r. The parameters * q= R1 1 and

* q= R2 2 in Equation (45) represent the inner and outer arc-length surface radii at the base of the accretion column,respectively (see Equation (11)). When q= = 01 1 , thecolumn is completely filled; otherwise, the center of thecolumn is devoid of matter, which may reflect the manner inwhich the gas is entrained onto the field lines at large radii, oralternatively, it may reflect the possible presence of aquadrupole field component (see Section 6.4). Note that ift

*= =r R 10 km (the stellar surface) up to = ~r r 20top km(the top of the accretion column). The precise value for rtop isdetermined self-consistently for each source, as discussed inPaper I. In the energy dimension, the computational domainextends from a minimum photon energy = 0.01 keVmin to amaximum photon energy = 100 keVmax . The final form ofthe transport equation is found by converting Equation (47) tothe dimensionless radius and velocity variables r and v (seeEquation (19)). The result obtained is

( )

s

s

-

-

+

- +

+

= + -

rR r

c

n R

f

rcv

f

R rn c

m cf kT

f cv

R

f

r

R r f ff

t

3

1

3

3

,

48

ge g

ge

ee

g

g

2 2 3

3 3 22

2

2 3 3prod abs

esc

where we have also utilized Equation (42). Since COMSOLemploys the FEM, it is well-suited for the iterative solutionmethod required to solve the photon transport Equation (48).The associated boundary conditions are discussed in detailbelow, along with the forms utilized for the terms describingradiation injection, absorption, and escape.

3.1. Photon Transport Boundary Conditions

In order to solve the partial differential transport equation(Equation (48)) for the photon distribution function f in theCOMSOL finite-element environment, we are obliged todevelop and apply a suitable set of physical boundaryconditions in radius and energy. The upper surface of theaccretion column, located at radius =r rtop, represents the lastscattering surface for photons diffusing up from deeperlayers in the column. Hence at this radius, the photon transportmakes a transition from classical diffusion to free-streaming,as occurs in the scattering layer above the photosphere ina stellar atmosphere. In the dipole geometry considered here,the specific flux in the radial direction is given by (seeEquation (39))

( )

k -

-

Ff

r

v f

3. 49rph,

Using Equation (42) to substitute for the spatial diffusioncoefficient , we find that the transition from diffusion to free-streaming implies that (see Equation (34))

( )s

- c

n

df

drc f r r

3, , 50

etop

and therefore

( )

-

F c fv f

r r3

, . 51rph, top

This relation expresses the spatial boundary condition appliedto the photon distribution function f at the top of the column.

We must also develop and apply an appropriate boundarycondition at the stellar surface, where *r R . In our idealized,one-dimensional model, the density of the accreting gasformally diverges as it settles onto the star, and therefore no

radiation flux can penetrate the stellar surface. This concept isphysically manifested in the mirror boundary condition,which states that the radiation streaming in the radial directionmust vanish as *r R . The mathematical implementation ofthis condition is given by

( )* F r R0, . 52rph,

The mirror condition described above provides the spatialboundary condition that must be satisfied by f at the stellarsurface.The lowest and highest energies treated in our simulations

are denoted by = 0.01 keVmin and = 100 keVmax , respec-tively. The boundary conditions applied at these two energiesare determined by the requirement that the radiation transportrate along the energy axis must vanish in the limit , andalso in the limit 0. The numerical values we choose formin and max are such that we can assume the radiationtransport rate vanishes at these two energies, which implies thatno photons cross the boundaries of the computational domainat = max and = min . The high-energy boundary conditioncan be written as (see Equation (47))

( )

s +

-

= =

n c

m cf kT

f v f

r30, , 53e

ee2

2max

and likewise, the low-energy boundary condition is given by

( )

s +

-

= =

n c

m cf kT

f v f

r30, . 54e

ee2

2min

Taken together, Equations (51)(54) comprise the set of fourboundary conditions that we apply on the four edges of the( ) r, computational domain employed in the COMSOLenvironment.

3.2. Photon Sources

The injection of seed photons due to the various emissionmechanisms operating inside the accretion column is repre-sented by the term fprod in Equation (48). Following BW07, wecan express the photon production rate by writing

( ) ( )

( )=W

=+ +

Wf

Q

r r

Q Q Q

r r, 55prod

prod

2

brem cyc bb

2

where the source functions Qbrem, Qcyc, and Qbb correspond tobremsstrahlung, cyclotron, and blackbody emission, respec-tively. The source functions are normalized so that

( ) Q r d dr,2 gives the number of photons injected per unittime between radii r and r+dr, with energy between and + d . The source function is therefore related to the volumeemissivity, n , via

( ) ( ) ( ) ( ) = WQ r d dr r r n r d dr, , , 562 2

where ( ) n r d, gives the number of photons injected perunit time per unit volume in the energy range between and + d .Computation of the thermal bremsstrahlung (freefree)

source function, Qbrem, is based upon Equation (5.14b) fromRybicki & Lightman (1979), which gives for the freefreevolume emissivity (in cgs units)

( ) r= - - -n T e3.7 10 . 57e kTff 36 2 1 2 1 e

7


We can combine Equations (56) and (57) to obtain for thebremsstrahlung source function

( ) r= W - - -Q r T e3.7 10 . 58e kTbrem 36 2 2 1 2 3 e

The cyclotron volume emissivity, ncyc, gives the production

rate of cyclotron photons per unit volume per unit energy. Forthe case of pure, fully ionized hydrogen, we can employEquations (7) and (11) from Arons et al. (1987) and substituteinto Equation (56) to obtain

( )

( )

r d= -

- -

n B H kT e2.1 10 ,

59e

kTcyc 36 212

3 2 cyccycecyc

where ( ) ( ) =r B r11.57cyc 12 keV denotes the cyclotronenergy, ( ) ( ) ( )B r B r 10 G12 12 , and H(x) is a piecewisefunction defined by

( ) ( )T Te IC, and they gain energy from the photons when

luminosity, Lacc, defined by

( )*

*L

GM M

R, 78acc

which assumes that the heating of the star during the accretionprocess is negligible. Finally, the remaining seven parameters,listed in Table 6 from Paper I, are derived quantities that arefunctions of the six free parameters and the boundaryconditions, as discussed in Section 2.2.

The resulting photon distribution function ( )f r, is acentral result of this paper. All transport phenomena arecalculated based on ( )f r, , including the total radiationenergy flux inside the column, Fr(r), the radiation energydensity, Ur(r), and the photon number density, nr(r). Here inPaper II, we present both phase-averaged and radius-dependentX-ray spectra computed using our model for Her X-1, Cen X-3,and LMC X-4. We compare the resulting X-ray spectra withthe observational data for these three sources in order toexercise the model using sources spanning a wide range ofluminosities, and to gain new insight into the emissionphenomena occurring in X-ray pulsar accretion columns.

4.1. Distribution Function Convergence

Our method for determining the convergence of thenumerical solution is based on the comparison of successiveiterates of the temperature profiles Te(r) and ( )T rIC . We definethe convergence ratios, R ( )re and R ( )rIC , respectively, for theelectron and inverse-Compton temperatures using

R R( ) ( )( )

( ) ( )( )

( ) ++

++

rT r

T rr

T r

T r, , 79e

n en

en

nn

n1

1

IC1 IC

1

IC

where the superscripts represent the iteration number for thecorresponding solution vectors. The solutions are deemed tohave converged when the radial vector of convergence ratiosfor both the electron and the inverse-Compton temperatureprofiles are within 1% of unity across the entire computationalgrid. The spatial grid spans the distance from the top of theaccretion column, at =r rtop (or dimensionless radius =r rtop),to the stellar surface at *=r R (or dimensionless radius =r 4.836, assuming canonical neutron star parameters). Onceconvergence is achieved, we have obtained a self-consistent set

of results for the radiation distribution function ( )f r, , andalso for the five dynamical variables ( )v r , ( )a rr , ( )a ri , ( )a re ,and E( )r .

4.2. Interstellar and Cyclotron Absorption

Soft X-ray absorption by the interstellar medium was firstdiscussed by Bell & Kingston (1967), Brown & Gould (1970),and Charles et al. (1973). In this process, the X-ray intensityfrom the source is attenuated according to

( ) ( ) ( ) ( )( ) = s-I I A A e, , 80N0 NH NH nH

where NH ( cm2) denotes the intervening column density of

neutral hydrogen atoms, and ( )sn is the net photoelectricabsorption cross-section per H nucleus. Figure 1 in Morrison &McCammon (1983) provides the net photoelectric absorptioncross-section per hydrogen atom as a function of energy fortypical elemental abundances in the interstellar medium. Theanalytic fit to the net cross-section is approximated using

( )( ) ( ) ( )

( )

s =

+ +

-

c c c

10 cm

,

81

n24 2

0 keV 1 keV keV 2 keV keV2

keV3

where keV denotes the photon energy in keV units, and thecoefficients ( )c0 , ( )c1 , and ( )c2 in Equation (81) haveconstant values within each of the 14 energy bins (spanning arange from 0.03 to 10 keV) indicated in Table 2 from Morrison& McCammon (1983). Outside the energy range of0.0310 keV, the photoelectric cross-section s = 0n ,and ( ) =A 1NH .The cyclotron resonant scattering feature (CRSF) is modeled

as a 2D Gaussian function (e.g., Orlandini et al. 1998; Heindl &Chakrabarty 1999; Soong et al. 1990), with both spatial andenergy dependences, and is superimposed directly onto thespectrum by multiplying by the attenuation function,

( )A r,cyc , defined using

( )

( )

( ) ( )

( ) ( )

s p

s p

-

s

s

- -

- -

A rd

e

de

, 12

2, 82

r

r

r r

c

cyc2

cyc

2

rcyc2 2

cyc2

cyc2

where rcyc is the cyclotron absorption imprint radius, cyc is thecyclotron absorption centroid energy, dr and dc are strengthparameters, and scyc and sr are standard deviations, respec-tively. We can combine the two strength parameters into asingle joint strength parameter, defined as =d d dcr c r, whichreduces Equation (82) to the form

( )

( )

( ) ( ) ( ) ( ) ps s

= - s s- - - -A rd

e e, 12

.

83

cr

r

r rcyc

cyc

2 2 rcyc2

cyc2

cyc2 2

The specific values used for the strength parameter, dcr, and thestandard deviations, scyc and sr , for each source are provided inSection 5, and in Section 6 we discuss the validity of usingsuch an approximation.

Table 3Auxiliary Parameters

Parameter Her X-1 Cen X-3 LMC X-4

Log ( )N10 H (cm2) 19.72 22.20 21.97cyc (keV) 44.72 31.79 32.15scyc (keV) 11.10 11.50 11.30sr (km) 9.20 5.17 4.14dcr 353 216 120K1 (keV) 6.45 6.67 5.90sK1 (keV) 0.400 0.293 0.190dK1 0.0060 0.0084 0.0007K2 (keV) 0.96 N/A N/AsK2 (keV) 0.157 N/A N/AdK2 0.028 N/A N/Asbb (km) 0.207 0.207 0.207Blackbody area (cm2) 9 1015 N/A N/ABlackbody temperature (Kelvin) 1.06 106 N/A N/A

10


4.3. Fan-beam and Pencil-beam Components

The photon transport equation is solved in COMSOL using ahigh-density finite-element 2D meshed grid, comprising oneradial dimension and one energy dimension. This yields thenumerical solution for the fundamental photon distributionfunction, ( )f r, , from which we can calculate the inverse-Compton temperature profile ( )T rIC , the radiation flux Fr(r), theradiation number density nr(r), and the radiation energy densityUr(r). Once the iterative process is completed and convergenceis achieved, we can calculate the spectrum emerging throughthe column walls in two ways. The first option is to obtain anapproximation of the phase-averaged photon count ratespectrum, ( )F , by integrating the fan-beam component ofthe escaping spectrum with respect to radius r over the entirelength of the accretion column, from =r rtop to *=r R . Thesecond option is to explore the effects of partial occultation ofthe column due to the stellar rotation, as seen by a distantobserver, which is accomplished by integrating the fancomponent from the top of the column down to a selectedradius, r, which is above the stellar surface, so that *>r R . Inaddition to the fan component, we can also compute thepencil-beam component that emanates from the upper surfaceof the accretion column, at radius =r rtop.

Bremsstrahlung, cyclotron, and blackbody emission eachmake contributions to the escaping radiation spectrum, asdiscussed in Section 3.2. These emission components arelinked due to the inherent nonlinearity of the radiation transportequation (Equation (48)). The nonlinearity reflects the fact thatthe electron temperature profile, Te(r), is influenced by theinverse-Compton temperature of the radiation field, ( )T rIC ,which in turn depends on the shape of the radiation spectrumvia Equation (77). Despite the interconnection between theComptonized bremsstrahlung, cyclotron, and blackbody com-ponents, once the iterative process has converged and theelectron temperature profile Te(r) is known, we can thencalculate the separate contributions to the observed spectrumdue to each seed photon source. This comparison is carried outin Section 5.

Following BW07, we can express the contribution to the fan-beam component of the photon number spectrum resultingfrom the escape of photons through the sides of the columnbetween radii r and r+dr using

( ) ( )( )

( ) ( ) =W

- - -N rr r

t rf r, , s cm keV , 84

2 2

esc

1 1 1

where the mean escape timescale, ( )t resc , is computed usingEquation (44). Note that the quantity ( ) N r d dr, representsthe number of photons escaping per unit time through thecolumn side walls between radii r and r+dr in the energyrange between and + d .

Once the attenuation factors ( )ANH and ( )A r,cyc havebeen computed using Equations (80) and (83), respectively, wecan calculate the fan-beam component of the observed photonnumber flux spectrum measured in the reference frame of adistant observer, ( )F , due to radiation escaping from a sub-domain of the column between radii r and rtop using the integral

( ) ( ) ( ) ( )

( )

p= - - -

FA

DA r N r dr

4, ,

s cm keV , 85r

rNH

2 cyc

1 2 1

top

where D is the distance to the source. The sub-domainintegrated spectrum given by Equation (85) can be used tocompute the phase-averaged and the partially occulted fan-beam radiation components detected by a distant observer. Forexample, the result for ( )F corresponds to the approximatephase-averaged spectrum if we integrate N over the entirecolumn length, from =r rtop to *=r R . Alternatively, we canintegrate N over a smaller radial sub-domain that extends fromthe top of the column ( =r rtop) down to any selected radius r.In this case, the result for ( )F approximates the spectrumdetected by a distant observer when the column is partiallyocculted by the stars surface due to the stellar rotation.The individual, radius-dependent fan-beam components due

to the escape of Comptonized bremsstrahlung, cyclotron, andblackbody emission are denoted by ( )N r,

brem , ( )N r,cyc ,

and ( )N r,bb , respectively. Once these functions have been

generated, the total (pre-absorption) fan-beam photon numberspectrum is then computed using the sum

( ) ( ) ( ) ( )( )

= + + - - -

N r N r N r N r, , , ,

s cm keV . 86

tot brem cyc bb

1 1 1

The corresponding fan-beam components of the observed photonspectrum, including absorption, are consequently given by

( ) ( ) ( ) ( )( )

= + + - - -

F F F F

s cm keV , 87

tot brem cyc bb

1 2 1

where each term on the right-hand side is computed byreplacing ( )N r, in Equation (85) with the appropriatesolution for each photon source, and then carrying out theintegration with respect to radius. Depending on the integrationbounds used in Equation (85), the sum over emissioncomponents in Equation (87) can provide the numericalprediction for either the phase-averaged or the partiallyocculted X-ray flux spectrum observed from an accretioncolumn. We will compare the predicted phase-averagedspectrum with previously published data for three luminoussources in Section 5. We will also investigate the contributionsdue to each seed photon source by examining the individualcomponents in Equation (87).The observed spectrum will be the sum of the fan-beam

component (Equation (87)) and the pencil-beam component,created by the escape of radiation through the free-streamingsurface at the top of the column. The pencil-beam emissionfrom the top of the column, ( )F , including stellar absorption,is computed using

( )( ) ( )

( )

( )

p

=W

- - -

FA r r c

Df r

4,

s cm keV . 88

NH top top2 2

2 top

1 2 1

In the case of the pencil-beam component, only interstellarabsorption is included, because cyclotron absorption isconcentrated at radius rcyc, which is about half the radius atthe column top, rtop (see Tables 6 and 7 from Paper I). The totalpencil-beam emission spectrum, representing the escape ofComptonized photons from the top of the column at radius=r rtop due to all three emission mechanisms, is given by the

11


sum (see Equation (87))

( ) ( ) ( ) ( )( )

= + + - - -

F F F F

s cm keV , 89

tot brem cyc bb

1 2 1

with each component on the right-hand side computed bysubstituting the corresponding solution for f into Equation (88).

5. Astrophysical Applications and Discussion

In this section, we use the new model to study the three X-raypulsars Her X-1, Cen X-3, and LMC X-4, which have relativelyhigh luminosities in the range of ~ -L 10 erg sX 36 38 1. Thevalues obtained for the six model free parameters, s, s , 1, 2,Mr0, and B*, are listed in Table 1, and the values for theconstrained parameters are listed in Table 2. We compare thetheoretical predictions for the emergent column-integrated spectrawith the observed phase-averaged spectral data for each of thethree pulsars in the photon energy range from 0.1 keV to100 keV.

There are a variety of emission components that add to createthe total phase-averaged X-ray spectrum. Our model providesseparate output components for the fan-beam and pencil-beamspectral components of bremsstrahlung, cyclotron, and black-body seed photons, while additional visible spectral featuresdue to iron emission and cyclotron resonant absorption areadded using Gaussian approximations. In the case of Her X-1,there is also a soft component included in the model, which isinterpreted as emission from the accretion disk. The visualmatching procedure for extracting model parameters andsolving for the phase-averaged photon count rate spectrum,

( )F , is divided into four basic steps.

1. Step 1: Once a source is selected, provisional values areassigned to the six fundamental free parameters, s, s , 1,2, Mr0, and B*. The remaining 13 constrained andderived parameters are then computed.

2. Step 2: Mathematica is used to solve the coupledhydrodynamical ODE system, comprising Equations (21)through (25), obeying all boundary conditions. Weinitially assume (the 0th iteration) that the electrontemperature, Te(r), and the inverse-Compton temperature,

( )T rIC , are equal for all values of r, so that =U 0Comp (seeEquation (76)). This is an input assumption for the firstMathematica run, which results in an output electrontemperature vector Te(r) along with an output velocityvector v(r). These two vectors are then exported fromMathematica into COMSOL in preparation for the 0th

iteration solution of the photon transport PDE, discussedbelow.

3. Step 3: COMSOL is used to solve the photon transportequation (Equation (48)) to obtain the photon distributionfunction ( )f r, (photons cm3 keV3). The inverse-Compton temperature profile ( )T rIC is then calculatedusing Equation (77). At this point, it becomes clear thatTe(r) and ( )T rIC are not necessarily equal, and thereforethe energy transfer via the UComp term can becomesubstantial. The radius-dependent fan-beam component, ( )N r, , is computed using Equation (84), and thecorresponding phase-averaged fan-beam count rate spec-trum ( )F (photons s1 cm2 keV1) is calculated usingEquation (85). We also compute the pencil-beamcomponent, ( )F , using Equation (88).

4. Step 4: The values of the convergence ratios R ( )re andR ( )rIC defined in Equations (79) are computed. The modelis considered unconverged if either R - >1 0.01e or

R - >1 0.01IC for at least one value of the grid radius r.In this case, the updated ( )T rIC vector is returned toMathematica for inclusion in the coupled ODE system, andthe iteration procedure loops back to Step 1. The model hasconverged when the ratios of the most recently calculated

( )T rIC and Te(r) profiles, compared with their previousrespective profiles, differ by less than 1% across the entireradial grid. In this case, the calculation is finished, and theresulting phase-averaged spectrum is compared with theobservational data for the source in question. If necessary,the free parameters listed in Table 1 are varied, and theiterative procedure is restarted at Step 1.

5.1. Her X-1

Our primary findings regarding the dynamical and radiativeproperties of Her X-1 are depicted in Figure 1. The bulk fluidvelocity (vbulk) and plasma temperature profiles (Ti and Te) arediscussed in greater detail in Paper I. The fluid enters the top ofthe column in a free-fall condition at an altitude of 11.19 kmabove the stellar surface, satisfying momentum conservationwith a free-fall velocity gradient. The radiation sound speed(ar) eventually rises to exceed the decelerating bulk fluidvelocity at the radiation sonic surface, which in the case of HerX-1 is at a distance of 2 km above the stellar surface. Belowthe radiation sonic surface the bulk fluid rapidly deceleratesinto an extended sinking regime (Basko & Sunyaev 1976).Eventually the bulk fluid approaches the stellar surface with aresidual stagnation velocity * = -v c0.0084 .The ions and electrons are nearly in equilibrium with each

other along the entire column, which is consistent with theshort timescale for ion and electron equilibration (see Paper I).Above the radiation sonic surface, the photons are continuouslyadding heat to the electrons via Compton scattering (i.e.,

>T TeIC ), and we observe that the average photon energyslowly decreases as the gas descends over a distance of 8 km.Below the sonic surface, the roles are reversed, and the highlycompressed bulk fluid transfers energy from electrons to thephotons via inverse-Compton scattering (

calculated using

( )

=y

t d

dt, 90thermal

esc

thermal

where repeated Compton scattering of photons with energy kTe results in photon energization with the mean rate

( ) s=ddt

n ckT

m c

4. 91e

e

ethermal2

Bulk fluid compression also results in photon energization,with a mean rate given by

( ) ( ) = - vddt 3

, 92bulk

and therefore the bulk y-parameter is found using

( )

=y

t d

dt. 93bulk

esc

bulk

In Her X-1, bulk Comptonization is the dominant mode ofphoton energization above the radiation sonic surface, whereasthermal Comptonization begins to dominate just below thesonic surface, at an altitude of 1.5 km from the stellar surface.Both y-parameters are less than unity until the fluid approachesthe sonic surface, which establishes that the defining

characteristics of the spectrum of Her X-1 are developedwithin the last 2 km of the accretion flow, where thermalComptonization dominates over bulk compression.Figure 2 depicts the radius-dependent (lower panel) and

partially occulted (upper panel) spectra for Her X-1. Theradius-dependent spectra correspond to the spectra emitted percentimeter at a precise value of the radius r, whereas in thepartially occulted case the value of r corresponds to the lowerbound of integration in Equation (85). We use four values forthe radius in Figure 2; the thermal mound surface (green),radiation sonic surface (blue), and at distances from the stellarsurface equal to one-half and three-fourths of the column length(red and orange), respectively. The lower panel depicts the fan-beam component only (the top spectrum is not included),radiated per centimeter of the column length. In addition to acontribution from the pencil-beam at the top of the column,each spectrum plotted in the upper panel includes two ironemission lines and an additional soft blackbody component thatdominates at low energies. These additional spectral contribu-tions are discussed further below. For illustration purposes,interstellar absorption and cyclotron absorption are includedonly in the vertically integrated, partially occulted spectrum inthe upper panel.The radius-dependent contributions are very uniform at

energies below ~10 keV. The maximum emitted flux, at allenergies, occurs in the vicinity of the radiation sonic location at

Figure 1. Phase-averaged dynamical and radiative properties for Her X-1. The locations of the thermal mound, the cyclotron imprint radius, and the radiation sonicpoint are indicated. See the discussion in the text.

13


1.95 km above the stellar surface. The power-law shape of thespectrum is dominated by emission escaping from the lowestregions of the column, whereas emission from the upper half ofthe column has essentially no impact on the observed spectralshape. The photon dynamics change above 10 keV, however,and all of the radius-dependent spectra above the radiationsonic surface exhibit a slightly positive bump in the spectralmagnitude, whereas there is a clear suppression in high-energyphotons at the thermal mound. The radial integration to obtainthe partially occulted spectra in the upper panel of Figure 2begins with the pencil-beam emission component as thestarting point and increases as the integration continuesdownward toward the stellar surface. The partially occultedspectra begin to exhibit the effects from cyclotron absorption atan altitude of 5.6 km above the stellar surface, which is abouthalf of the column length in this case.

The phase-averaged photon count rate spectrum for Her X-1is calculated using Equation (87) and plotted in Figure 3 overthe energy range 0.1100 keV. We note that the phase-averaged spectrum is equivalent to the partially occultedspectrum, if the lower integration bound r is set equal to thestellar radius R* in Equation (85). The upper panel in Figure 3depicts the total spectrum without the effects of interstellar

absorption and cyclotron absorption included. The lower panelincludes both absorption effects. The red dots represent theincident phase-averaged BeppoSAX data reported by Dal Fiumeet al. (1998) with the instrumental response removed. Thecumulative contribution to the total spectrum is shown as thesolid black curve that includes blackbody (brown), cyclotron(blue), and bremsstrahlung (green) emission from the side walls(solid) and the top (dashed) of the column.The spectra plotted in Figure 3 include two iron emission

lines, which are approximated using Gaussian functionscentered at energies of = 6.45 keVK1 and = 0.96 keVK2(Oosterbroek et al. 1997; Dal Fiume et al. 1998). The auxiliaryparameters listed in Table 3 document the values used for theiron emission line centroid energies, as well as the corresp-onding standard deviations (sK1 and sK2), and the line strengthparameters (dK1 and dK2). A 2D Gaussian cyclotron absorptionfeature (see Equation (83)) is also included, centered at animprint radius =r 11.74cyc km, corresponding to a cyclotroncentroid energy = 44.72 keVcyc . Implementation of the 2DGaussian cyclotron absorption feature requires the specificationof the spatial standard deviation, s = 9.2r km, the energystandard deviation, s = 1 keVc , and the strength parameter,

=d 353cr (see Table 3). A discussion of these values, and thegeneral manner in which absorption is treated in our model, is

Figure 2. Partially-occulted spectra computed using Equation (85), integratedover the indicated range of radius r (upper panel). Radius-dependent escapingnumber spectrum at the indicated value of r, computed using Equation (84)(lower panel). The upper panel also includes the pencil-beam componentcomputed using Equation (88).

Figure 3. Phase-averaged spectrum for Her X-1 calculated using Equation (85).The lower panel includes cyclotron and interstellar absorption, and the upperpanel neglects absorption.

14


provided in Section 6. Interstellar absorption best matches theobserved Her X-1 data when we adopt for the column density

= -N 5.25 10 cmH 19 2, which agrees closely with thespectral fit parameter ( )= -N 5.10 0.7 10 cmH 19 2 obtainedby Dal Fiume et al. (1998). The parameters for the 2Dcyclotron Gaussian for all sources are listed in Table 3, inaddition to the parameters describing the hydrogen columndensity, the iron emission, and, in the case of Her X-1, thesurface area and temperature of the excess soft blackbodycomponent, which we discuss below.

As early as 1975, it was well-documented that Her X-1 hasan intense soft X-ray flux at energies below~1 keV (Catura &Acton 1975; Shulman et al. 1975). Our model also requires thisexcess, which cannot be explained as direct emission from theaccretion column. In order to visually match the observedphase-averaged spectrum, we include a blackbody componentwith temperature = T 1.06 10 K6 ( =kT 0.091 keV) andradiating surface area 9.0 10 cm15 2. These values agree withthe results of Catura & Acton (1975), who identified ablackbody component with temperature =T 106 K andradiating surface area 9.85 10 cm15 2. Note that this area isabout three orders of magnitude larger than the entire surfacearea of the neutron star, and therefore it probably originates inthe accretion disk, or, alternatively, it may represent emissiondue to Compton scattering of the primary X-rays by the coronalgas above and below the accretion disk (Jones & Forman 1976;Becker et al. 1977; Pravdo et al. 1977; Bai 1980). For example,Bai (1980) proposed that a clumpy distribution of coronal gasabove and below the accretion disk at a distance of ~10 cm8could explain the soft X-ray contribution, which agrees withour average Alfvn radius of ~ 5.3 10 cm8 (see Paper I).Oosterbroek et al. (1997) suggested that the blackbodycomponent may be reprocessed emission originating fromregions of large optical depth within the accretion disk. Endoet al. (2000) obtained a spectral fit for Her X-1 containing a0.16 keV blackbody component with an emitting surface radiusof 240 km, at a distance similar to the expected inner radius ofthe accretion disk. Ramsay et al. (2002) calculated a thermalblackbody component at 1 keV with a similar large emittingradius located between 140 km and 500 km in the accretiondisk. Ji et al. (2009) calculated a 0.114 keV thermal blackbodysource with an emitting radius between 10 and 100 km, andconcluded that the soft component likely comes from the inneredge of the accretion disk. Finally, Frst et al. (2013) calculateda radius of 250 km for an emitting blackbody at 0.14 keV.These results are consistent with the properties of the softblackbody component included in our model for Her X-1.

The X-ray emission components depicted in Figure 3include all of the individual pencil-beam and fan-beamcontributions due to the Comptonization of bremsstrahlung,cyclotron, and blackbody seed photons. The phase-averagedspectrum is clearly dominated by radiation from the wall (fanbeam), while the low-energy spectrum is dominated by theblackbody component discussed in the previous paragraph. Anumber of authors have concluded that Her X-1 is beingviewed from a direction roughly perpendicular to the magneticfield (e.g., Mszros 1978). Simplistically the emission is cup-shaped (Bisnovatyi-Kogan & Komberg 1976). A new andsignificant prediction from our model, however, is that thepencil-beam component of the spectrum is comparable to thefan-beam component at energies close to the cyclotronabsorption feature.

5.2. Cen X-3

Our primary results for the dynamical and radiative proper-ties of Cen X-3 are plotted in Figure 4. Overall, the propertiesof the bulk fluid and the shape of the radiation patterns aresimilar to those found in the case of Her X-1. The fluid entersthe top of the accretion column at a distance of 14.25 km fromthe surface, passes through a radiation sonic surface at 2.21 km,and then approaches the stellar surface with a residualstagnation velocity * = -v c0.0081 . The region in the columnabove the radiation sonic surface is again dominated byCompton scattering, as the photons transfer their energy to theelectrons, while below the sonic surface, the electrons tend tore-heat the photons. The peak magnitudes of the advective anddiffusive radiation flux components are nearly 50% larger thanthe corresponding peaks in the case of Her X-1. The imprintradius of the cyclotron absorption feature, rcyc, is located 1.2 kmbelow the radiation sonic point, and only 940 m above thestellar surface, and there is ~10% deviation between rcyc andthe radius of maximum X-ray emission, rX.The plots of the thermal and bulk Compton y-parameters in

Figure 4 indicate that bulk Comptonization dominates in thecolumn above the radiation sonic surface, and thermalComptonization dominates below it. Again, we see that theshape of the spectrum is essentially created in the lower 2 kmof the column, where both y-parameters exceed unity. Thecyclotron energy at 31.79 keV is approximately two-thirds themagnitude seen in Her X-1, which helps to explain why theaverage photon energy is also less, remaining in the range of10.814.5 keV.Figure 5 depicts the radius-dependent and partially occulted

spectra for Cen X-3, at four indicated values for the radius r. Inthe case of the radius-dependent spectra, we plot the emergentspectrum per centimeter at the precise radius r, whereas in thepartially occulted case, the value of r corresponds to the lowerbound of integration in Equation (85). A qualitative compar-ison shows that the dominant contribution to the fan-beamcomponent is emitted near the radiation sonic surface, at2.21 km above the stellar surface, which is an order ofmagnitude larger than the contribution radiated at 10.8 km(3/4 column length). In Cen X-3, we observe an extendedsinking regime below the thermal mound, where the averagephoton energy begins to rise. The average photon energyachieves its highest value at the stellar surface, which is not thecase with Her X-1. The contribution to the Cen X-3 spectrumemitted below the thermal mound is insignificant, demonstrat-ing that most of the photons injected in this region are absorbedbefore they can escape.The best visual match of the phase-averaged photon count

rate spectrum for Cen X-3 is depicted in Figure 6. In the upperpanel, we plot the raw, non-attenuated spectrum, and in thelower panel we plot the spectrum with interstellar absorptionand cyclotron absorption included. The hydrogen columndensity is set equal to = N 1.58 10H 22 cm

2, which agreeswell with the value = N 1.95 10H 22 cm

2 obtained byBurderi et al. (2000). We have simulated the effect of ironemission by adding a Gaussian emission feature centered at6.67 keV, described by the auxiliary parameters listed inTable 3. A side-by-side comparison with the Her X-1 spectrumshows that the cyclotron and Comptonized blackbody con-tributions for Cen X-3 are more broad at energies 1 keV, andthe cyclotron component is strongest at energies below0.3 keV. The cyclotron absorption feature is centered at

15


energy 31.79 keV, which is much lower than the 44.72 keVfeature in Her X-1. The spectrum is not as hard as Her X-1 atthe highest energies, which is what we expect because theaverage photon energy is lower. The overall Cen X-3 emissiongeometry is dominated by the fan-beam component, and thefan-to-pencil ratio is larger than that obtained for Her X-1.

5.3. LMC X-4

The dynamical and radiative properties of LMC X-4 aredepicted in Figure 7. The results are similar to those obtainedfor both Her X-1 and Cen X-3 in many respects, but there arealso some important differences. The fluid enters the top of theaccretion column at an altitude of 11.30 km above the stellarsurface, and passes through the radiation sonic surface at analtitude 3.21 km. At the stellar surface, the gas has a residualstagnation velocity * = -v c0.0098 . Comparing the dynamicalresults for LMC X-4, Her X-1, and Cen X-3, we note that theradius of the radiation sonic surface tends to increase as wemove to higher-luminosity sources. The extended sinkingregime likewise becomes more extended with increasingluminosity. The photon-electron energy exchange in LMCX-4 is similar to that observed for Her X-1 and Cen X-3, withall three temperatures ( ( )T rIC , Ti(r), and Te(r)) exhibiting a slowexponential increase in the sinking regime.

In contrast to Her X-1 and Cen X-3, the imprint radius of thecyclotron absorption feature in LMC X-4 is located 1.05 km

above the radiation sonic surface, at an altitude of 4.26 kmabove the stellar surface, and the cyclotron energy for LMCX-4 is found to be 32.15 keV. Figure 7 indicates that in LMCX-4, the thermal Compton parameter, ythermal, exceeds the valueof the bulk Compton parameter, ybulk, below about ~r 16 kmin the column, whereas this transition occurs closer to~r 12 km in Her X-1 and Cen X-3 (see Figures 1 and 4).

This explains the flatter slope of the power-law section of thespectrum in the plot of the phase-averaged spectrum for LMCX-4 (see Figure 8). We note that in the case of LMC X-4, thevalue of the cyclotron imprint radius, rcyc, differs from themaximum X-ray emission radius, rX, by ~10%.Figure 8 depicts the results obtained for the radius-dependent

and partially occulted spectra for LMC X-4. We again observethat the emission is concentrated in the vicinity of the radiationsonic surface. The phase-averaged photon count rate spectrumis plotted in Figure 9. The hydrogen column density is set equalto = N 9.22 10H 21 cm2, which is somewhat larger thanprevious estimates of ~ 0.57 1021 cm2 (Dickey & Lock-man 1990; La Barbera et al. 2001; Paul et al. 2002). The ironGaussian is centered at 5.90 keV (see Table 3). The LMC X-4spectrum is harder than that of Her X-1 and Cen X-3, whichexplains the larger values obtained for the average photonenergy, in the range of 16.622.7 keV. The emission geometryindicates strong contributions from both fan and pencil-beamcomponents in this source, though the fan component stilldominates in the phase-averaged spectrum.

Figure 4. Same as Figure 1, except here we treat Cen X-3.

16


6. Discussion

The coupled radiative-hydrodynamical simulation developedhere, and the associated calculation of the phase-averagedphoton count rate spectrum, represents the first self-consistentmodel for the hydrodynamical and radiative properties ofaccretion-powered X-ray pulsars. The model is cast in arealistic dipole geometry, and includes rigorous calculations ofthe radial profiles of the flow velocity v(r) and the electrontemperature Te(r). We have iteratively solved an ODE-basedhydrodynamical code that determines the dynamical structureof the accretion column, coupled with a PDE-based radiativecode that computes the radiation spectrum as a function ofradius and energy by employing the FEM to solve a second-order, elliptical, nonlinear PDE. The inverse-Compton temp-erature profile, ( )T rIC , provides the link between the hydro-dynamics developed in Paper I and the solution for theradiation distribution function developed here in Paper II. Theiterative process converges to yield a self-consistent descriptionof (1) the dynamical structure over the full length of theaccretion column and (2) the energy distribution in theemergent radiation field.

Our model employs a mirror boundary condition for theradiation field at the stellar surface, which means that we striveto obtain zero net radiation flux there. This condition is satisfiedin Cen X-3 and LMC X-4, but in the case of Her X-1, there is aslightly positive residual radiation energy flux at the stellar

surface. The average photon energy is relatively constant in allthree sources and is comparable to the cyclotron energy nearthe cyclotron imprint radius.Our analysis of the y-parameters show that bulk Comp-

tonization dominates thermal Comptonization in the top 80%of the column for both Her X-1 and Cen X-3, but in LMC X-4,which is also the brightest source of the three pulsars, bulkComptonization dominates only in the top 50%. The shape ofthe spectrum, however, is altered only in the region dominatedby thermal Comptonization, when the y-parameters exceedunity, and we conclude that the spectrum shape is significantlydependent on thermal Comptonization. This takes place withinapproximately 2 km of the stellar surface in the case of Her X-1and Cen X-3, while for LMC X-4 the region extends to nearly5 km above the surface.The magnitude of the radiated fan-beam component reaches

its maximum value near the radiation sonic surface for eachsource. In the case of Her X-1, the cyclotron absorption imprintradius matches the maximum luminosity location. However, inthe cases of Cen X-3 and LMC X-4, the imprint radius isslightly below and above it, respectively. The reason for thisoffset is not yet known. It is reasonable to argue that theabsorption imprint radius should correspond to the radius ofmaximum wall emission. One possible explanation is that ourassumed values of stellar mass and radius for Cen X-3 and

Figure 5. Same as Figure 2, except here we treat Cen X-3.Figure 6. Same as Figure 3, except here we treat Cen X-3.

17


LMC X-4 require modification, which is planned infuture work.

6.1. Cyclotron Absorption Feature

The CRSF is a complex shape and often non-Gaussian(Schnherr et al. 2007). A correct treatment of the line profileshould include modeling the physical causes for the naturalline width and Doppler broadening (Trmper et al. 1977;Harding & Lai 2006). This is currently a noted limitation in thepresent state of our model. The cyclotron absorption featurewas approximated using a 2D Gaussian function (seeEquation (83)), which mathematically describes an ad hocmultiplication of two 1D Gaussian functions. We introducedstandard deviations in both the energy and spatial dimensions,given by s and sr , respectively, and we also introduced a jointstrength parameter, given by dcr.

The model values obtained for the spatial standard deviation,sr , are 9.20 km (Her X-1), 5.17 km (Cen X-3), and 4.14 km(LMC X-4). We expect that in the case of Her X-1, the relativewidth of the CRSF line, D , can be approximated by D < 0.35cyc (e.g., Staubert et al. 2014). Combining this

with the dipole variation of the magnetic field, -B r 3, weshould observe a relative radial width on the order ofD

our model, s is a fitting parameter, so it is interesting tocompare the values we obtained for this parameter with thosecomputed using Equation (94). This comparison is carried outin Table 4. We note that in the case of Cen X-3, the two valuesagree to within 14% while the error for LMC X-4 is 13%.In the case of Her X-1, the error is larger, but the two valuesstill agree to within a factor of approximately three.

6.3. Partially Occulted Spectra

In Figure 10, we plot the radius-dependent (left column) andpartially occulted (right column) spectra for all three sources atfour radii: the thermal mound surface (green), the radiationsonic surface (blue), and at distances from the stellar surfaceequal to one-half and three-fourths of the column length (redand orange), respectively. For illustration purposes, interstellarabsorption and cyclotron absorption are included only in theplots for the partially occulted spectra. The partially occultedspectra are interpreted as resulting from the rotation of thepulsar, which causes various portions of the accretion columnto disappear behind the neutron star. Although the partiallyocculted spectra provide clues as to how the rotation of the starwould affect the observed spectrum, we cannot make anydefinitive conclusions in the absence of a complete model forthe stellar rotation and the emission geometry, including theangle between the spin axis and the line of sight to the

observer, and also the angle between the spin axis and theradiating magnetic pole (or poles). Such a model should alsoinclude the effects of general relativistic light bending, and alsothe effect of the directional dependence of the cyclotronabsorption and emission processes occurring in the outer sheathof the column surface. A calculation of this type is beyond thescope of the present paper, but we plan to pursue thedevelopment of a complete geometrical picture in future work.The results obtained here for the partially occulted spectra

establish that the largest spectral contributions occur near theradiation sonic surface. We find that the observed spectra areessentially generated in the last few kilometers above the stellarsurface, where the y-parameters exceed unity and thermalComptonizaton dominates. The sequence of partially occultedspectra plotted for each source demonstrate that the observedspectra are due to emission escaping from altitudes between the

Figure 8. Same as Figure 2, except here we treat LMC X-4.Figure 9. Same as Figure 3, except here we treat LMC X-4.

Table 4Parallel Scattering Cross-section

Source s sT s sT RatioThis Paper Becker et al. (2012)

Her X-1 1.02103 3.44104 2.96Cen X-3 7.51104 8.78104 0.86LMC X-4 4.18104 4.78104 0.87

19


column top and the thermal mound. The escape of photos fromaltitudes below the thermal mound makes a negligiblecontribution to the total spectra, due to the dense atmospherein the extended sinking regime where the parallel absorption

optical depth exceeds unity, and most of the photons areabsorbed.The two-dimensional approximations employed in the Wang

& Frank (1981) model suggest that 90% of the observed

Figure 10. Radius-dependent (left column) and partially-occulted spectra (right column) computed using Equations (84) and (85), respectively.

20


radiation escapes in the fan-beam component, with only 10%contributed by the pencil beam. As these authors note, theprincipal weakness of their model is the lack of anyimplementation of energy exchange between the photons andthe plasma. Our model results establish that the fan-beamcontribution generally dominates over the pencil beam, thoughthis can be violated in the vicinity of the cyclotron absorptionenergy. We also note that the new model provides additionalradius-dependent and energy-dependent information basedupon realistic physics within the column, which is unavailablein the model of Wang & Frank (1981).

Langer & Rappaport (1982) developed a hydrodynamicalmodel for accretion flows in lower luminosity X-ray pulsars( -L 1.9 10 erg sX 36 1). Their model includes a partialimplementation of the energy exchange processes linking thephotons with the plasma, but it does not treat the radiationtransfer problem in detail. A major goal of our work has been toproperly model the interactions between the gas and theradiation due to thermal emission and absorption, cyclotronemission and absorption, and Compton scattering. Hence ourmodel provides a self-consistent description of the hydro-dynamics, the thermodynamics, and the radiative transferoccurring in the accretion column. Our results suggest that,even for low-luminosity sources, the coupling of the radiationwith the matter via Compton energy transfer can have aprofound effect on the electron temperature profile. Bykov &Krassilchtchikov (2004) also computed the hydrodynamics of a1D, two-fluid model in a dipole magnetic field geometry forlow-luminosity sources. While they go one step further thanLanger & Rappaport (1982) by adopting a cyclotron diffusionapproximation to treat the radiation transport, they do notaccount for the fundamental effects of bulk and thermalComptonization, nor do they include bremsstrahlung orblackbody source photons.

We can gain further insight into the relative importance ofthe pencil- and fan-beam components in forming the observedspectra of the X-ray pulsars treated here. In Figure 11, we plotemission ratio curves, formed by dividing the emitted pencil-beam photon spectrum by the column-integrated (phase-averaged) fan-beam photon spectrum for Her X-1 (blue), CenX-3 (black), and LMC X-4 (green). In general, for all three

sources, the emission from the wall (fan) component exceedsthat escaping from the column top (pencil) component, exceptwithin a narrow energy range centered on the cyclotronabsorption feature. The total observed photon energy andnumber fluxes associated with the pencil and fan componentscan be calculated by integrating the respective spectra over theenergy band from min to .max The observed total number fluxemitted in the fan-beam component, emanating through thecolumn walls, is calculated using

F ( ) ( )

= - -F d s cm , 95phtot tot 1 2min

max

where ( )F tot is given by Equation (87). The correspondingobserved total energy flux is given by

F ( ) ( )

= - -F d keV s cm . 96entot tot 1 2min

max

Likewise, the total observed number flux due to the pencil-beam emission escaping from the top of the column is given by

( ) ( )

= - -F F d s cm , 97phtot tot 1 2

min

max

where ( )Ftot

is evaluated using Equation (89). The corresp-onding total energy flux is computed using

( ) ( )

= - -F F d keV s cm . 98entot tot 1 2

min

max

The four observed fluxes, calculated using Equations (95)(98),are listed for each source in Table 5, separated into thecontributions due to the pencil-beam and fan-beam compo-nents. In general, the majority of the energy and number fluxesis contributed by the fan beam, though in the case of LMC X-4,the pencil-beam number flux is comparable to the fan-beamcontribution.

6.4. Future Work

Paper I includes a discussion of future enhancements to thehydrodynamical model, which could lead to an improveddescription of the structure of the accretion column. However,there are also numerous modifications that would facilitate amore detailed investigation of the partially occulted and phase-averaged spectra and associated spectral properties. Forexample, as mentioned earlier, our preliminary results forCen X-3 and LMC X-4 suggest that modification of the stellarmass and radius may improve the agreement between thecyclotron absorption imprint radius, rcyc, and the radius atwhich the luminosity per unit length (Lr) is maximized,

Figure 11. Ratio of the pencil-beam photon spectrum divided by the fan-beamphoton spectrum, plotted as a function of the photon energy . See thediscussion in the text.

Table 5Total Observed Photon Number and Energy Fluxes

Source Pencil (Top) Fan (Wall) Ratio

Number Flux ( cm2 s1) Fphtot

Fphtot Top/Wall

Her X-1 0.15 0.57 0.27Cen X-3 0.40 3.07 0.13LMC X-4 0.05 0.06 0.85

Energy Flux ( keV cm2 s1) Fentot

Fentot Top/Wall

Her X-1 1.14 2.99 0.38Cen X-3 1.13 7.77 0.15LMC X-4 0.38 0.31 1.20

21


denoted by rX. In principle, we would expect these two radii toagree.

Changes in the CRSF line energy and X-ray continuum canbe studied by varying the accretion rate, M , over a specificrange, which may provide new insights into the observedcorrelation between the cyclotron line energy and the X-rayluminosity (Mihara et al. 1995; Staubert et al. 2007; Beckeret al. 2012; Rothschild et al. 2017). To obtain a fully self-consistent and detailed theoretical study of the line width anddepth, which can be compared with recent observationalstudies, such as that by Rothschild et al. (2017), we would needto replace the 2D cyclotron Gaussian with a true energy-dependent cyclotron absorption term in both the hydrodynamicand photon transport equations. This should facilitate calcul-ation of the X-ray power-law continuum for any accretion rate,resulting in correlative studies of the hardness ratio withvariations of the source luminosity (Postnov et al. 2015).

We have assumed in our model that the dipole magnetic fieldcomponent dominates over the full length of the accretioncolumn. However, the possible presence of a magneticquadrupole component may significantly influence the overallfield geometry (Shakura et al. 1991; Panchenko & Post-nov 1994), which could substantially modify the locations andshapes of the primary emission regions, and the accompanyingemission characteristics (Gregory et al. 2006; Jardine et al.2006; Donati et al. 2007b; Long et al. 2007, 2008). Forexample, if the quadrupole component is aligned with thedipole, then the accreting matter may impact the surface as aring rather than a disk (Postnov et al. 2013). This particularpossibility is included implicitly in our model, at leastqualitatively, since we allow for a partially filled column, inwhich case the matter impacts the surface in a ringconfiguration. However, if the quadrupole component ismisaligned with the dipole component, then the accretionstructure is likely to be much more complex and asymmetric.The treatment of that type of asymmetric structure wouldrequire the development of a two-dimensional spatial model,which is beyond the scope of the work presented here, but itcould possibly be developed in the future.

The results we have obtained for the partially occultedspectra plotted in Figures 2, 5, and 8 provide motivation forfuture studies employing a more realistic source geometry,including the possibility of a quadrupole magnetic fieldcomponent, which may alter the pulse profiles, as discussedin the context of Her X-1 by Postnov et al. (2013). Here, wehave simply explored the consequences of varying the lowerintegration bound in Equation (85) for the observed fan-beamflux component ( )F as a means to roughly approximate theeffect of the stellar rotation, which would cause varyingfractions of the lower portion of the column to disappear fromview behind the star. The results we have obtained suggest thata careful consideration of the source geometry, relativisticeffects, and the directional dependence of the cyclotronscattering cross-section should be implemented in futurestudies. The authors would like to acknowledge several usefulcomments from the anonymous referee that helped tosignificantly improve the presentation of the material.

Appendix APhoton Escape Formalism

In general, the transport equation governing the evolution ofthe isotropic (angle-averaged) radiation distribution f inside the

accretion column can be written in the vector form (seeEquation (40))

( )

( ) ( )

s

k

+ =

+

+

+

+ +

v

v

f

tf

n c

m cf kT

f

f

ff f

1

1

3. 99

e

ee2 2

4

prod abs

Assuming azimuthal symmetry around the magnetic field axis,and averaging over the cross-sectional area of the accretioncolumn at radius r, we obtain (see Equation (41))

( )( )

( )[ ( ) ]

( )

s

k

+

=

+

+

+

+ + +

f

tv

f

r

n c

m cf kT

f

A r rA r

f

r

A r

A r v

r

f

f f f

1

1

1

3

, 100

e

ee2 2

4

prod abs esc

where the area of the accretion column, A(r), is given byEquation (10), and the term fesc represents the diffusive escapeof radiation through the walls of the column.Our goal here is to derive a suitable mathematical form for

fesc so that it properly accounts for the free-streaming escape ofradiation through the sides of the accretion column. Inparticular, we wish to derive an expression for the meanescape timescale, tesc, such that the escape term inEquation (100) can be written in the form

( )= -f ft

. 101escesc

In order to proceed, we return to Equation (99) and expand thevector components of the spatial diffusion term, which is theterm of interest here. For our purposes, it is sufficient to adoptcylindrical coordinates, so that we obtain

( )

( )

kr r

r kr

r jkr j

k

=

+

+

f

tf

f

f

z

f

z

1

1.

102

diff

Since we are interested in treating the escape of radiationthrough the column walls, we will focus on the radial (r)component of the photon flux. Furthermore, we will assumesymmetry with respect to the azimuthal angle j, so that

j =f 0. In this case, the radial component of the diffusiontransport in Equation (102) reduces to

( )r r

r kr

=

r

f

t

f1. 103

diff,

If we suppose for simplicity that the seed radiation is injectedat the center of the column (r = 0) and diffuses radiallyoutward to the column surface at radius r r= 0, then the radialtransport rate of the photons is conserved. In this case, we can

22


rewrite Equation (103) as

( )r kr

-

= =f

C constant. 1040

The cross-sectional variation of the spatial diffusion coefficient,, is not known with any certainty, but for our purposes here, itis sufficient to assume that it is given by the power-law form

( ) ( )k r k rr

=a

, 1050 0

where is a constant and k0 denotes the surface value of .Integration of Equation (104) with respect to then yields for

( )rf the solution

( ) ( )ra a

rr

= - +a-

f f 1

1 1, 1060

0

where f0 is the surface value of ( )rf , given by

( )k

=fC

. 10700

0

The diffusion approximation employed inside the columnmust transition to radiation free streaming at the columnsurface, and therefore the surface boundary condition forEquation (104) can be written as

( )kr

-

=r

fcf . 1080

0

By comparing Equations (104) and (108), we find that theconstant C0 is given by

( )r=C cf , 1090 0 0

which can be combined with Equation (107) to show that

( )k r= c. 1100 0The average rate of change of the distribution function f due

to the diffusive escape of radiation through the column walls isgiven by the volume-weighted integral

( )

prpr r

r rr k

rr

rr k

r

=

=

=

r

r

r

r

r

f

t

f

td

fd

f

12

2

2. 111

diff, 02 0 diff,

02 0

02

0

0

0

0

Combining the final relation with the surface boundarycondition given by Equation (108), we find that the averageescape rate is given by

( )r

= -r

f

t

cf2. 112

diff,

0

0

We note that this is nearly identical to the form ofEquation (101) for fesc.

In order to make the correspondence complete and obtain anexpression for the mean escape timescale, tesc, we write

( )

= -

r

f

t

f

t, 113

diff, esc

where f denotes the volume-weighted average of f, given bythe integral

( ) ( )pr pr r r =r

f f d1

2 . 11402 0

0

Using Equation (106) to substitute for ( )rf in Equation (114)and carrying out the integration yields

( )aa

=--

f f

3

2, 1150

which can be combined with Equations (110) and (112) toobtain the result

( )kr

aa

= ---

r

f

tf

2 2

3. 116

diff,

0

02

Comparing this result with Equation (113), we find that themean escape timescale, tesc, is given by

( )r

kaa

=--

t 2

3

2. 117esc

02

0

We can substitute for k0 using the standard relation

( )k = c 3

, 11800

where 0 is the scattering mean-free path at the column surface.The result obtained is

( )r a

a=

--

t c

3

2

3

2. 119esc

02

0

The diffusion velocity for photons propagating perpendicular tothe axis, w , can be approximated by writing

( )r

~wc

, 120

0 0

and therefore we find that

( )r a

a~

--^

t w

3

2

3

2. 121esc

0

In the context of our dipole model for the accretion column, itfollows that we can obtain an adequate approximation for theescape timescale by writing (see Equation (44))

( ) ( )( )

( )=^

t r r

w r, 122esc

esc

where ( ) resc is the width of the (either hollow or filled) columnat radius r, and (

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