# Relativistic Milne-Eddington Type Solutions with a Variable Eddington Factor for Relativistic Spherical Winds

Authors: Jun Fukue
Journal: Advances in Astronomy
 (2011)

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## Abstract

Relativistic radiative transfer in a relativistic spherical flow is examined in the fully special
relativistic treatment. Under the assumption of a constant flow speed and using a variable
(prescribed) Eddington factor, we analytically solve the relativistic moment equations in the comoving
frame for several restricted cases, and obtain relativistic Milne-Eddington type solutions.
In contrast to the plane-parallel case where the solutions exhibit the exponential behavior on
the optical depth, the solutions have power-law forms. In the case of the radiative equilibrium,
for example, the radiative flux has a power-law term multiplied by the exponential term. In the
case of the local thermodynamic equilibrium with a uniform source function in the comoving
frame, the radiative flux has a power-law form on the optical depth. This is because there is an
expansion effect (curvature effect) in the spherical wind and the background density decreases
as the radius increases.


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## Body

## 1. Introduction

The research field of radiative transfer has been developed in astrophysics and atmospheric science [1–13]. Relativistic radiative transfer and relativistic radiation hydrodynamics have been also developed in astrophysics and applied to various energetic phenomena in the universe: nova outbursts, gamma-ray bursts, astrophysical jets, black-hole accretion disks, and black-hole winds. In the subrelativistic regime, some researchers adopted the diffusion approximation in the comoving frame (e.g., [14–17]), or proposed the variable Eddington factor (e.g., [18–24]), or performed the numerical simulations using, for example, the flux-limited diffusion (FLD) approximation (e.g., [25–35]).In the highly relativistic regime, however, we cannot treat the relativistic radiative transfer properly. Hence, the research and development of the radiative transfer problem in relativistically moving media is now of great importance in this field.At the present stage, the numerical approach using the FLD approximation is limited within the subrelativistic regime. In addition, even in the subrelativistic regime, the FLD method cannot reproduce the radiative force precisely in the optically thin region [36]. This is because in the optically thin region the radiative flux vector is not generally parallel to the gradient of the radiation energy density due to the effect of the distant radiation source. On the other hand, the analytical approach is very restricted in the special cases. Indeed, even for the nonrelativistic case only a few analytical solutions have been found, but for the relativistic case little is known. Recently, the relativistic radiative transfer in relativistically moving atmospheres have been investigated from the analytical view points in the plane-parallel case (e.g., [37–40]) and in the spherical case (e.g., [41]).In Fukue [40], under the assumption of a constant flow speed, the relativistic moment equations in the comoving frame were analytically solved using a variable Eddington factor for several cases, such as the radiative equilibrium (RE) or the local thermodynamic equilibrium (LTE), and the relativistic Milne-Eddington type solutions for the relativistic plane-parallel flows have been newly found. In the present study, we also consider the relativistic radiative transfer from the analytical view point. In contrast to Fukue [40], we examine the relativistic moment equations for the spherical case, solve the equations under the assumption of a constant flow speed, and obtain several new analytical solutions for the relativistic spherical flows.In the next section, we describe the radiative moment equations in the comoving frame for spherical flows. In Sections3 and 4, we show and discuss analytical solutions in the RE and LTE cases, respectively. The final section is devoted to concluding remarks.


## 2. Relativistic Radiative Transfer Equation

The radiative transfer equations are given in several standard references [6–10, 12, 13, 42–44]. The basic equations for relativistic radiation hydrodynamics are given in, for example, the Appendix E of Kato et al. [43] in general and vertical forms (see also [39, 40]).


### 2.1. General Form

In a general form, the radiative transfer equation in themixed frame, where the variables in the inertial and comoving frames are used, is expressed as(1)1c∂I∂t+(l⋅∇)I=(νν0)3ρ[j04π-(κ0+σ0)I0+σ0cE04π],
where c is the speed of light. In the left-hand side, the frequency-integrated specific intensity I and the direction cosine vector l are quantities measured in the inertial (fixed) frame. In the right-hand side, on the other hand, the mass density ρ, the frequency-integrated mass emissivity j0, the frequency-integrated mass absorption coefficient κ0, the frequency-integrated mass scattering coefficient σ0, the frequency-integrated specific intensity I0, and the frequency-integrated radiation energy density E0 are quantities measured in the comoving (fluid) frame. In this paper, instead of the weakly anisotropic Thomson scattering, we assume that the scattering is isotropic for simplicity.The Doppler effect, the aberration, and the transformation of the intensities are expressed as(2)νν0=γ(1+β⋅l0),(3)l=ν0ν[l0+(γ-1β2β⋅l0+γ)β],(4)I=(νν0)4I0,
where ν and ν0 are the frequencies measured in the inertial and comoving frames, respectively, the direction cosine l0 measured in the comoving frame, β (=v/c) the normalized velocity, v being the flow velocity, and γ (=1/1-β2) the Lorentz factor, β being v/c.The zeroth and first moment equations are, respectively,(5)∂E∂t+∂Fk∂xk=ργ(j0-κ0cE0)-ργ(κ0+σ0)β⋅F0,1c2∂Fi∂t+∂Pik∂xk=ργβic(j0-κ0cE0)-ρ(κ0+σ0)γ-1β2βic×(β⋅F0)-1cρ(κ0+σ0)F0i,
where the frequency-integrated radiation energy density E, the frequency-integrated radiative flux F, and the frequency-integrated radiation stress Pik are measured in the inertial frame, while those with the subscript 0 are measured in the comoving frame.As a closure relation, we adopt the Eddington approximationin the comoving frame,(6)P0ik=fikE0,
where fik is the Eddington tensor, which is generally a function of the optical depth and flow speed in the relativistic radiative flow.


### 2.2. Spherical Expression in the Comoving Frame

Let us suppose a relativistic spherical flow, for example, a luminous black-hole wind. In the spherical geometry with the radiusr, the transfer equation (1) is expressed as
(7)1c∂I∂t+μ∂I∂r+1-μ2r∂I∂μ=(νν0)3ρ[j04π-(κ0+σ0)I0+σ0cE04π],
where μ is the direction cosine in the inertial frame. Inserting the transformation (4) in the left-hand side, this equation (7) becomes
(8)vv0(1c∂I0∂t+μ∂I0∂r+1-μ2r∂I0∂μ)-4vv02I0(1c∂v0∂t+μ∂v0∂z+1-μ2r∂v0∂μ)=ρ[j04π-(κ0+σ0)I0+σ0cE04π].To calculate the derivatives ofI0 [9], we apply the chain rules, and, after some manipulations, we have(9)∂∂t|rμν=∂∂t|rμ0ν0+∂μ0∂t|rμ0ν0∂∂μ0+∂ν0∂t|rμ0ν0∂∂ν0=∂∂t|rμ0ν0-γ2(1-μ02)∂β∂t∂∂μ0-γ2μ0ν0∂β∂t∂∂ν0,∂∂r|tμν=∂∂r|tμ0ν0+∂μ0∂r|tμ0ν0∂∂μ0+∂ν0∂r|tμ0ν0∂∂ν0=∂∂r|tμ0ν0-γ2(1-μ02)∂β∂r∂∂μ0-γ2μ0ν0∂β∂r∂∂ν0,∂∂μ|rtν=∂μ0∂μ|rtν0∂∂μ0+∂ν0∂μ0|rtν0∂∂ν0=γ2(1+βμ0)2∂∂μ0-γ2β(1+βμ0)ν0∂∂ν0,
where μ is the direction cosine in the comoving frame. In addition, the Doppler shift (2) and the aberration (3) are, respectively, expressed as(10)νν0=γ(1+βμ0),μ=μ0+β1+βμ0.Using these expressions, after some manipulations, we have the radiative transfer equation in the comoving frame for the spherical flow:(11)γ(1+βμ0)1c∂I0∂t+γ(μ0+β)∂I0∂r+γ(1+βμ0)1-μ0r∂I0∂μ0+4γβ1-μ02rI0-γ3(1+βμ0)×[(1-μ02)∂I0∂μ0-4μ0I0]1c∂β∂t-γ3(μ0+β)[(1-μ02)∂I0∂μ0-4μ0I0]∂β∂r=ρ[j04π-(κ0+σ0)I0+σ0cE04π].Integrating the transfer equation (11) over a solid angle, we have the zeroth and first moment equations in the comoving frame for the spherical flow:(12)γ∂cE0c∂t+γ∂F0∂r+γβ∂F0c∂t+γβ∂cE0∂r+γr[2F0+β(3cE0-cP0)]+γ3[2F0+β(cE0+cP0)]∂βc∂t+γ3[2βF0+(cE0+cP0)]∂β∂r=ρ(j0-κ0cE0),γ∂F0c∂t+γ∂cP0∂r+γβ∂cP0c∂t+γβ∂F0∂r+γr[2βF0-cE0+3cP0]+γ3[2βF0+(cE0+cP0)]∂βc∂t+γ3[2F0+β(cE0+cP0)]∂β∂r=-ρ(κ0+σ0)F0,
where E0, F0, and P0 are the radiation energy density, the radiative flux, and the radiation pressure in the comoving frame, respectively.In the present spherical one-dimensional flow, if we assume the pressure isotropy in the comoving frame, the closure relation (6) becomes(13)P0=f(τ,β)E0,
where f(τ,β) is the variable (prescribed) Eddington factor, and generally a function of the optical depth, the flow speed, and the velocity gradient [45, 46]. In the plane-parallel flow [40], the following form was adopted:(14)f(β)=1+3β23+β2,
which is 1/3 for β=0 and approaches unity as β→1 [47]. In the present spherical case, we adopt alternative appropriate forms, which are shown later.It should be noted that historically Auer and Mihalas [48] first used the term a variable Eddington factor (VEF) to express an iterative solver to obtain the Eddington factor (cf. [49]). The Eddington factors which depend on the optical depth should be calledprescribed or approximated Eddington factors, although they are often called a variable Eddington factor [50, 51]. In the previous papers, in order to express the not-constant Eddington factor, we also used the variable Eddington factor, which depends on the optical depth and flow velocity. In this paper we use both terms, variable and prescribed, but both usages express the same meanings; that is, the present Eddington factor is not constant but varies as a function of the optical depth and flow velocity.


### 2.3. Steady Spherical Flow

Let us further suppose a time-independent steady flow in the radial direction. In this case the transfer equation and moment equations in the comoving frame become(15)γ(μ0+β)dI0dr-γ3(μ0+β)[(1-μ02)∂I0∂μ0-4μ0I0]dβdr+γ(1+βμ0)1-μ02r∂I0∂μ0+4γβ1-μ02rI0=ρ[j04π-(κ0+σ0)I0+σ0cE04π],(16)γdF0dr+γβdcE0dr+γ3[2βF0+(cE0+cP0)]dβdr+γr[2F0+β(3cE0-cP0)]=ρ(j0-κ0cE0),(17)γdcP0dr+γβdF0dr+γ3[2F0+β(cE0+cP0)]dβdr+γr[2βF0-cE0+3cP0]=-ρ(κ0+σ0)F0.Introducing the optical depth defined by(18)dτ≡-(κ0+σ0)ρdr
and the scattering albedo,(19)a≡σ0κ0+σ0,
the transfer equation (15) and the moment equations (16) and (17) are finally expressed as
(20)γ(μ0+β)dI0dτ-γ(1+βμ0)1-μ02ρ(κ0+σ0)r∂I0∂μ0-4γβ1-μ02ρ(κ0+σ0)rI0-γ3(μ0+β)[(1-μ02)∂I0∂μ0-4μ0I0]dβdτ=I0-14πj0κ0+σ0-acE04π,(21)γdF0dτ+γβdcE0dτ-γρ(κ0+σ0)r[2F0+β(3cE0-cP0)]+γ3[2βF0+(cE0+cP0)]dβdτ=-j0κ0+σ0+(1-a)cE0,(22)γdcP0dτ+γβdF0dτ-γρ(κ0+σ0)r[2βF0-cE0+3cP0]+γ3[2F0+β(cE0+cP0)]dβdτ=F0.Here, we further introduce the spherical variables by(23)L0≡4πr2F0,D0≡4πr2cE0,Q0≡4πr2cP0,
and the moment equations (21) and (22) become(24)γdL0dτ+γβdD0dτ-γβD0-Q0ρ(κ0+σ0)r+γ3(2βL0+D0+Q0)dβdτ=-4πr2κ0+σ0(j0-κ0cE0),γdQ0dτ+γβdL0dτ-γQ0-D0ρ(κ0+σ0)r+γ3[2L0+β(D0+Q0)]dβdτ=L0,
and the closure relation (13) is written as(25)Q0=f(τ,β)D0.If we assume the streaming limit ofD0=Q0 (f=1) with a constant speed in (24), we have the exponential type solutions. In this paper we consider more general cases of Q0=fD0.Using this closure relation (25), the Eddington factor being not yet determined, and the definition of the optical depth (18), the relativistic moment equation (24) is expressed as(26)γdL0dτ+γβdD0dτ+γβ1-frD0drdτ+γ3[2βL0+(1+f)D0]dβdτ=-4πr2κ0+σ0(j0-κ0cE0),γd(fD0)dτ+γβdL0dτ-γ1-frD0drdτ+γ3[2L0+β(1+f)D0]dβdτ=L0.
After several manipulations and rearrangement, the relativistic moment equation (26) in the comoving frame is finally expressed as
(27)γ(f-β2)fdL0dτ+γβ(1-f2fr-1fdfdr)D0drdτ+γ3[2β(1-1f)L0+(1+f)(1-β2f)D0]dβdτ=-4πr2κ0+σ0(j0-κ0cE0)-βfL0,γ(f-β2)dD0dτ+γ[dfdr-(1-f)(1+β2)r]D0drdτ+2γL0dβdτ=L0+β4πr2κ0+σ0(j0-κ0cE0).
After we determine the appropriate form of the variable Eddington factor, we can solve the moment equation (27) in some restricted cases.Before solving the moment equations, we derive a relation between the optical depthτ and radius r. If the flow is steady, as is assumed, the continuity equation for the spherical case is written as
(28)4πr2ργβc=Ṁ,
where Ṁ is the constant mass-outflow rate. Using this continuity equation (28), assuming the opacities are constant, and imposing the boundary condition of τ=0 at r=∞, we can integrate the optical depth (18) to give
(29)τ=Ṁ(κ0+σ0)4πγβc1r=ρ(κ0+σ0)r,
which is also written as
(30)ττc=rcr,
where the subscript c denotes some reference position (core radius). It should be noted that the optical depth at the core radius is related to the core radius by(31)τc=ṁrg2γβrc,
where ṁ (=Ṁ/ṀE) is the mass-outflow rate normalized by the critical rate ṀE (=LE/c2), LE being the Eddington luminosity of the central object, and rg (=2GM/c2) is the Schwarzshild radius of the central object. In what follows, we use these relations, if necessary.


## 2.1. General Form

In a general form, the radiative transfer equation in themixed frame, where the variables in the inertial and comoving frames are used, is expressed as(1)1c∂I∂t+(l⋅∇)I=(νν0)3ρ[j04π-(κ0+σ0)I0+σ0cE04π],
where c is the speed of light. In the left-hand side, the frequency-integrated specific intensity I and the direction cosine vector l are quantities measured in the inertial (fixed) frame. In the right-hand side, on the other hand, the mass density ρ, the frequency-integrated mass emissivity j0, the frequency-integrated mass absorption coefficient κ0, the frequency-integrated mass scattering coefficient σ0, the frequency-integrated specific intensity I0, and the frequency-integrated radiation energy density E0 are quantities measured in the comoving (fluid) frame. In this paper, instead of the weakly anisotropic Thomson scattering, we assume that the scattering is isotropic for simplicity.The Doppler effect, the aberration, and the transformation of the intensities are expressed as(2)νν0=γ(1+β⋅l0),(3)l=ν0ν[l0+(γ-1β2β⋅l0+γ)β],(4)I=(νν0)4I0,
where ν and ν0 are the frequencies measured in the inertial and comoving frames, respectively, the direction cosine l0 measured in the comoving frame, β (=v/c) the normalized velocity, v being the flow velocity, and γ (=1/1-β2) the Lorentz factor, β being v/c.The zeroth and first moment equations are, respectively,(5)∂E∂t+∂Fk∂xk=ργ(j0-κ0cE0)-ργ(κ0+σ0)β⋅F0,1c2∂Fi∂t+∂Pik∂xk=ργβic(j0-κ0cE0)-ρ(κ0+σ0)γ-1β2βic×(β⋅F0)-1cρ(κ0+σ0)F0i,
where the frequency-integrated radiation energy density E, the frequency-integrated radiative flux F, and the frequency-integrated radiation stress Pik are measured in the inertial frame, while those with the subscript 0 are measured in the comoving frame.As a closure relation, we adopt the Eddington approximationin the comoving frame,(6)P0ik=fikE0,
where fik is the Eddington tensor, which is generally a function of the optical depth and flow speed in the relativistic radiative flow.


## 2.2. Spherical Expression in the Comoving Frame

Let us suppose a relativistic spherical flow, for example, a luminous black-hole wind. In the spherical geometry with the radiusr, the transfer equation (1) is expressed as
(7)1c∂I∂t+μ∂I∂r+1-μ2r∂I∂μ=(νν0)3ρ[j04π-(κ0+σ0)I0+σ0cE04π],
where μ is the direction cosine in the inertial frame. Inserting the transformation (4) in the left-hand side, this equation (7) becomes
(8)vv0(1c∂I0∂t+μ∂I0∂r+1-μ2r∂I0∂μ)-4vv02I0(1c∂v0∂t+μ∂v0∂z+1-μ2r∂v0∂μ)=ρ[j04π-(κ0+σ0)I0+σ0cE04π].To calculate the derivatives ofI0 [9], we apply the chain rules, and, after some manipulations, we have(9)∂∂t|rμν=∂∂t|rμ0ν0+∂μ0∂t|rμ0ν0∂∂μ0+∂ν0∂t|rμ0ν0∂∂ν0=∂∂t|rμ0ν0-γ2(1-μ02)∂β∂t∂∂μ0-γ2μ0ν0∂β∂t∂∂ν0,∂∂r|tμν=∂∂r|tμ0ν0+∂μ0∂r|tμ0ν0∂∂μ0+∂ν0∂r|tμ0ν0∂∂ν0=∂∂r|tμ0ν0-γ2(1-μ02)∂β∂r∂∂μ0-γ2μ0ν0∂β∂r∂∂ν0,∂∂μ|rtν=∂μ0∂μ|rtν0∂∂μ0+∂ν0∂μ0|rtν0∂∂ν0=γ2(1+βμ0)2∂∂μ0-γ2β(1+βμ0)ν0∂∂ν0,
where μ is the direction cosine in the comoving frame. In addition, the Doppler shift (2) and the aberration (3) are, respectively, expressed as(10)νν0=γ(1+βμ0),μ=μ0+β1+βμ0.Using these expressions, after some manipulations, we have the radiative transfer equation in the comoving frame for the spherical flow:(11)γ(1+βμ0)1c∂I0∂t+γ(μ0+β)∂I0∂r+γ(1+βμ0)1-μ0r∂I0∂μ0+4γβ1-μ02rI0-γ3(1+βμ0)×[(1-μ02)∂I0∂μ0-4μ0I0]1c∂β∂t-γ3(μ0+β)[(1-μ02)∂I0∂μ0-4μ0I0]∂β∂r=ρ[j04π-(κ0+σ0)I0+σ0cE04π].Integrating the transfer equation (11) over a solid angle, we have the zeroth and first moment equations in the comoving frame for the spherical flow:(12)γ∂cE0c∂t+γ∂F0∂r+γβ∂F0c∂t+γβ∂cE0∂r+γr[2F0+β(3cE0-cP0)]+γ3[2F0+β(cE0+cP0)]∂βc∂t+γ3[2βF0+(cE0+cP0)]∂β∂r=ρ(j0-κ0cE0),γ∂F0c∂t+γ∂cP0∂r+γβ∂cP0c∂t+γβ∂F0∂r+γr[2βF0-cE0+3cP0]+γ3[2βF0+(cE0+cP0)]∂βc∂t+γ3[2F0+β(cE0+cP0)]∂β∂r=-ρ(κ0+σ0)F0,
where E0, F0, and P0 are the radiation energy density, the radiative flux, and the radiation pressure in the comoving frame, respectively.In the present spherical one-dimensional flow, if we assume the pressure isotropy in the comoving frame, the closure relation (6) becomes(13)P0=f(τ,β)E0,
where f(τ,β) is the variable (prescribed) Eddington factor, and generally a function of the optical depth, the flow speed, and the velocity gradient [45, 46]. In the plane-parallel flow [40], the following form was adopted:(14)f(β)=1+3β23+β2,
which is 1/3 for β=0 and approaches unity as β→1 [47]. In the present spherical case, we adopt alternative appropriate forms, which are shown later.It should be noted that historically Auer and Mihalas [48] first used the term a variable Eddington factor (VEF) to express an iterative solver to obtain the Eddington factor (cf. [49]). The Eddington factors which depend on the optical depth should be calledprescribed or approximated Eddington factors, although they are often called a variable Eddington factor [50, 51]. In the previous papers, in order to express the not-constant Eddington factor, we also used the variable Eddington factor, which depends on the optical depth and flow velocity. In this paper we use both terms, variable and prescribed, but both usages express the same meanings; that is, the present Eddington factor is not constant but varies as a function of the optical depth and flow velocity.


## 2.3. Steady Spherical Flow

Let us further suppose a time-independent steady flow in the radial direction. In this case the transfer equation and moment equations in the comoving frame become(15)γ(μ0+β)dI0dr-γ3(μ0+β)[(1-μ02)∂I0∂μ0-4μ0I0]dβdr+γ(1+βμ0)1-μ02r∂I0∂μ0+4γβ1-μ02rI0=ρ[j04π-(κ0+σ0)I0+σ0cE04π],(16)γdF0dr+γβdcE0dr+γ3[2βF0+(cE0+cP0)]dβdr+γr[2F0+β(3cE0-cP0)]=ρ(j0-κ0cE0),(17)γdcP0dr+γβdF0dr+γ3[2F0+β(cE0+cP0)]dβdr+γr[2βF0-cE0+3cP0]=-ρ(κ0+σ0)F0.Introducing the optical depth defined by(18)dτ≡-(κ0+σ0)ρdr
and the scattering albedo,(19)a≡σ0κ0+σ0,
the transfer equation (15) and the moment equations (16) and (17) are finally expressed as
(20)γ(μ0+β)dI0dτ-γ(1+βμ0)1-μ02ρ(κ0+σ0)r∂I0∂μ0-4γβ1-μ02ρ(κ0+σ0)rI0-γ3(μ0+β)[(1-μ02)∂I0∂μ0-4μ0I0]dβdτ=I0-14πj0κ0+σ0-acE04π,(21)γdF0dτ+γβdcE0dτ-γρ(κ0+σ0)r[2F0+β(3cE0-cP0)]+γ3[2βF0+(cE0+cP0)]dβdτ=-j0κ0+σ0+(1-a)cE0,(22)γdcP0dτ+γβdF0dτ-γρ(κ0+σ0)r[2βF0-cE0+3cP0]+γ3[2F0+β(cE0+cP0)]dβdτ=F0.Here, we further introduce the spherical variables by(23)L0≡4πr2F0,D0≡4πr2cE0,Q0≡4πr2cP0,
and the moment equations (21) and (22) become(24)γdL0dτ+γβdD0dτ-γβD0-Q0ρ(κ0+σ0)r+γ3(2βL0+D0+Q0)dβdτ=-4πr2κ0+σ0(j0-κ0cE0),γdQ0dτ+γβdL0dτ-γQ0-D0ρ(κ0+σ0)r+γ3[2L0+β(D0+Q0)]dβdτ=L0,
and the closure relation (13) is written as(25)Q0=f(τ,β)D0.If we assume the streaming limit ofD0=Q0 (f=1) with a constant speed in (24), we have the exponential type solutions. In this paper we consider more general cases of Q0=fD0.Using this closure relation (25), the Eddington factor being not yet determined, and the definition of the optical depth (18), the relativistic moment equation (24) is expressed as(26)γdL0dτ+γβdD0dτ+γβ1-frD0drdτ+γ3[2βL0+(1+f)D0]dβdτ=-4πr2κ0+σ0(j0-κ0cE0),γd(fD0)dτ+γβdL0dτ-γ1-frD0drdτ+γ3[2L0+β(1+f)D0]dβdτ=L0.
After several manipulations and rearrangement, the relativistic moment equation (26) in the comoving frame is finally expressed as
(27)γ(f-β2)fdL0dτ+γβ(1-f2fr-1fdfdr)D0drdτ+γ3[2β(1-1f)L0+(1+f)(1-β2f)D0]dβdτ=-4πr2κ0+σ0(j0-κ0cE0)-βfL0,γ(f-β2)dD0dτ+γ[dfdr-(1-f)(1+β2)r]D0drdτ+2γL0dβdτ=L0+β4πr2κ0+σ0(j0-κ0cE0).
After we determine the appropriate form of the variable Eddington factor, we can solve the moment equation (27) in some restricted cases.Before solving the moment equations, we derive a relation between the optical depthτ and radius r. If the flow is steady, as is assumed, the continuity equation for the spherical case is written as
(28)4πr2ργβc=Ṁ,
where Ṁ is the constant mass-outflow rate. Using this continuity equation (28), assuming the opacities are constant, and imposing the boundary condition of τ=0 at r=∞, we can integrate the optical depth (18) to give
(29)τ=Ṁ(κ0+σ0)4πγβc1r=ρ(κ0+σ0)r,
which is also written as
(30)ττc=rcr,
where the subscript c denotes some reference position (core radius). It should be noted that the optical depth at the core radius is related to the core radius by(31)τc=ṁrg2γβrc,
where ṁ (=Ṁ/ṀE) is the mass-outflow rate normalized by the critical rate ṀE (=LE/c2), LE being the Eddington luminosity of the central object, and rg (=2GM/c2) is the Schwarzshild radius of the central object. In what follows, we use these relations, if necessary.


## 3. Radiative Equilibrium

We first consider the case of the radiative equilibrium (RE) without heating and cooling. If the radiative equilibrium holds in the whole flow, and there is no heating or cooling, thenj0=κ0cE0, and the relativistic moment equation (27) become(32)γ(f-β2)fdL0dτ+γβ(1-f2fr-1fdfdr)D0drdτ+γ3[2β(1-1f)L0+(1+f)(1-β2f)D0]dβdτ=-βfL0,(33)γ(f-β2)dD0dτ+γ[dfdr-(1-f)(1+β2)r]D0drdτ+2γL0dβdτ=L0.These equations (32) and (33) are rather complicated yet, since they include the velocity gradient term and the derivative of the radius, which are connected with the hydrodynamical equations. Of these, except for the central accelerating region, the wind speed weakly depends on the optical depth and is almost constant in the terminal stage. Hence, as already stated, the flow speed β is assumed to be constant in this paper. On the other hand, the radius-derivative term depends on the optical depth. Indeed, it is expressed as(34)drdτ=-1(κ0+σ0)ρ=-rτ∝-τ-2.Instead, the second term on the left-hand side of (33) can be dropped, if we impose the restricted condition on the Eddington factor as(35)1-f2r-dfdr=0.
This equation (35) is easily integrated to give(36)f=C(β)r2-1C(β)r2+1,
where C(β) is an integration constant, and generally a function of the constant flow speed β. We impose the boundary condition at the core radius such as(37)f=1+3β23+β2atr=rc,
and the appropriate Eddington factor requested to the present case finally becomes(38)f=2γ2(1+β2)r̂2-12γ2(1+β2)r̂2+1=2γ2(1+β2)-τ̂22γ2(1+β2)+τ̂2,
where r̂=r/rc and τ̂=τ/τc. This variable Eddington factor (38) satisfies the condition f→1/3 when r→rc and β→0, and f→1 when r→∞ (τ→0) or β→1. The behavior of this Eddington factor is shown in Figure 1.Figure 1
Eddington factorf appropriate for the RE case as a function of the optical depth τ for the various values of the flow speed β. The values of β are 0 to 0.9 in steps of from 0.1 from bottom to top.Under these restrictive conditions, after several manipulations, (32) and (33) become(39)dL0dτ=-ΓL0,(40)γ1gddτ[g(f-β2)D0]=L0.
In these equations,(41)Γ≡βγ(f-β2)
is a function of the flow speed and the optical depth, and it becomes(42)Γ=γβ1+β22τ̂22-τ̂2+γβ
for the Eddington factor (38), while g is the curvature factor defined by(43)ln⁡g≡-∫τcτ(1-f)(1+β2)(f-β2)rdrdτdτ′
and becomes in the present case(44)g=τ̂32-τ̂2.
Since the index Γ is analyticalls expressed by the optical depth, the differential equation (39) can analytically integrate to give the comoving luminosity L0. Imposing the boundary condition of Ls at τ=0, we finally have the comoving luminosity for the RE case:(45)L0Ls=(2-τ̂2+τ̂)bexp⁡(12γ2bτ̂),
where(46)b=2γβ1+β2τc.The analytical solutions of the comoving luminosity (45) are shown in Figure 2 as a function of the optical depth for several values of the flow speed. The values of β are from 0 to 0.9 in steps of 0.1.Figure 2
Comoving luminosity for relativistic spherical flows in the RE case without heating and cooling. The values ofβ are from 0 to 0.9 in steps of 0.1 from top to bottom. The optical depth τc at the core radius is set to be 10.Although the comoving luminosity (45) has an exponential term, the power-law behavior is dominant in this case. In the nonrelativistic limit of β→0, b→0 and the solution reduces to(47)L0Ls~1-b2τ̂~1-βτ.
In the extremely relativistic limit of β→1, on the other hand, b→γτc/2 and the solution reduces to(48)L0Ls~(2-τ̂2+τ̂)b.In contrast to this comoving luminosity, it is still difficult to obtain analytical solutions of the spherical radiation energy densityD0. Even in the extremely relativistic limit, we cannot obtain the analytical solution for D0.


## 4. Local Thermodynamic Equilibrium

Next, we consider the case of the local thermodynamic equilibrium (LTE) with a uniform source function. If the local thermodynamic equilibrium (LTE) holds in the comoving frame,(49)j04π=κ0B0,
where B0 (=σT04/π) is the frequency-integrated blackbody intensity in the comoving frame, T0 being the blackbody temperature and generally a function of the height r or the optical depth τ, but assumed to be constant in what follows.In this case the relativistic moment equation (27) become(50)γ(f-β2)fdL0dτ+γβ(1-f2fr-1fdfdr)D0drdτ+γ3[2β(1-1f)L0+(1+f)(1-β2f)D0]dβdτ=-κ0κ0+σ0(W0-D0)-βfL0,γ(f-β2)dD0dτ+γ[dfdr-(1-f)(1+β2)r]×D0drdτ+2γL0dβdτ=L0+βκ0κ0+σ0(W0-D0),
where(51)W0≡16π2r2B0
is the spherical source function.These equations (50) and (51) can be rearranged as(52)γ(f-β2)fdL0dτ+[γβf(1-f2r-dfdr)drdτ-κ0κ0+σ0]D0+γ3[2β(1-1f)L0+(1+f)(1-β2f)D0]dβdτ=-κ0κ0+σ0W0-βfL0,γ(f-β2)dD0dτ+{γ[dfdr-(1-f)(1+β2)r]drdτ+βκ0κ0+σ0}×D0+2γL0dβdτ=L0+βκ0κ0+σ0W0.Equation (52) is yet too complicated to solve analytically.Hence, in order to simplify these equations by dropping the second terms on the left-hand sides of equation (52), we impose the following two conditions:(53)γβf(1-f2r-dfdr)drdτ-κ0κ0+σ0=0,γ[dfdr-(1-f)(1+β2)r]drdτ+βκ0κ0+σ0=0.
Eliminating κ0/(κ0+σ0) from (53), we obtain the differential equation for the variable Eddington factor f,(54)dfdr+f-1r=0,
as long as dr/dτ≠0.This equation (54) is easily integrated to give(55)f=1-C(β)r̂,
where C(β) is an integration constant and generally a function of the constant flow speed β. We impose the boundary condition at the core radius such as(56)f=1+3β23+β2atr=rc,
and the appropriate Eddington factor requested to the present case finally becomes(57)f=1-2(1-β2)3+β21r̂=1-2(1-β2)3+β2τ̂,
where r̂=r/rc and τ̂=τ/τc. This variable Eddington factor (57) satisfies the condition: f→1/3 when r→rc and β→0, and f→1 when r→∞ (τ→0) or β→1. The behavior of this Eddington factor is shown in Figure 3.Figure 3
Eddington factorf appropriate for the LTE case as a function of the optical depth τ for the various values of the flow speed β. The values of β are from 0 to 0.9 in steps of 0.1 from bottom to top.Under these restrictive conditions, after several manipulations, equation (52) becomes(58)dL0dτ=-ΓL0-Δκ0κ0+σ0W0,(59)dD0dτ=ΓβL0+Γκ0κ0+σ0W0,
where(60)Γ≡βγ(f-β2)=γβ(3+β2)3+β2-2τ̂,Δ≡fγ(f-β2)=γ3+β2-2(1-β2)τ̂3+β2-2τ̂,
respectively, in the present case.Since the indexΓ is analytically expressed by the optical depth, the solution of the homegeneous part of (58), where W0 is set to be 0, is analytically obtained as
(61)L0Ls=(1-pτ̂)q,
where(62)p≡23+β2,q≡γβ(3+β2)2τc.
When the spherical source function W0 is uniform and κ0/(κ0+σ0) is also constant, the analytical solution of (58) can be obtained after some manipulations as(63)L0Ls=(1-pτ̂)q+γτc1-q[(3+β22-βγτc)-τ̂γ2]κ0κ0+σ0W0Ls.The analytical solutions of the comoving luminosity (63) are shown in Figure 4 as a function of the optical depth for several values of the flow speed. The values of β are from 0 to 0.9 in steps of 0.1.Figure 4
Comoving luminosity for relativistic spherical flows in the LTE case with a uniform source function. The values ofβ are from 0 to 0.9 in steps of 0.1 from top to bottom. The optical depth τc at the core radius is set to be 10, and the value of the source function [κ0/(κ0+σ0)]W0/Ls is set to be unity. The solid curves represent the present analytical solutions, while the dashed ones mean the solutions of the homogeneous part.In the LTE case the comoving luminosity (63) has the power-law form. In the nonrelativistic limit of β→0, p→2/3, and q→3βτc/2, and the solution becomes a linear function of τ. In the extremely relativistic limit of β→1, on the other hand, p→1/2 and q→2γτc, the solution reduces to (64)L0Ls~(1-pτ̂)q-κ0κ0+σ0W0Ls.
In contrast to the RE case, we can obtain the analytical solutions of the spherical radiation energy density D0. However, it is rather complicated, and we omit the expression for D0.


## 5. Concluding Remarks

In this paper, we have examined the relativistic radiative transfer in the relativistic spherical flows in the fully special relativistic treatment. Under the assumption of a constant flow speed and using a variable Eddington factorf(τ,β), we have analytically solved the relativistic moment equations written in the comoving frame for RE and LTE cases and found new analytical solutions for several restricted situations. In both RE and LTE cases, the radiative flux decreases with the optical depth in the power-law manner, while the radiative flux has the exponential behavior in the plane-parallel case [40].We here clarify theessential difference for the relativistic radiative transfer between the plane-parallel and spherical cases; the former is the exponential type, and the latter is the power-law manner. Since the original transfer equation is the linear differential equation, the exponential behavior is natural, but there arised two different types. This essential difference is roughly understood as follows.In the relativistic plane-parallel flow [40], where we have assumed the constant flow speed, the density is also constant; there is no expansion effect. The index Γ is also constant. In this case, the natural exponential behavior emerges and the analytical solutions exhibit the exponential behavior on the optical depth. In the relativistic spherical flow in the present case, where we have also assumed the constant flow speed, the density decreases as the radius increases due to the geometrical effect; there is an expansion effect. The index Γ is no longer constant but varies as a function of r (or τ). As a result, the natural exponential behavior is lost, and the analytical solutions exhibit the power-law behavior.From the view point of the background density variation, this difference is somewhat similar to the growth of the density fluctuation of the gravitational instability in the static interstellar space and the expanding universe. Namely, in the static background, where the background density is constant, the density fluctuation increases exponentially [52], whereas it increases in a power-law manner in the expanding universe, where the background density decreases with time [53]. Hence, we can guess that, even in the plane-parallel case, there may be power-law type solutions if the flow is accelerated and the density decreases as the optical depth decreases.In order to research the physical problem, the analytical approach has several advantages. First, the analytical solutions can often reveal the essential properties of the radiative transfer problem. In the present case, we can clarify the exponential versus power-law type behavior and its causes. Secondly, they can clarify the restrictions of the assumptions and/or crucial problems inherent in the formalism. In the present case, in order to avoid the critical point in the basic equations with a traditional constant Eddington factor, we use a variable Eddington factor, which approaches unity in the limit ofτ→0 or β→1 (cf. [24, 54]). Finally, they can help us to check the precision and the validity of the numerical code for the radiative transfer problem. Particularly, in the recent research of the radiation transfer problem on the black-hole accretion using the ART code [36], the FLD approximation often adopted in the radiation hydrodynamical simulations cannot reproduce the radiative force in the optically thin region. Hence, the new numerical method should be developed for the multidimensional radiation hydrodynamical simulations, and the analytical solutions like the present case would be useful for such codes in the future.



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*Source: 412620-2011-06-21.xml*