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

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.

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 Sections 3 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 the mixed frame, where the variables in the inertial and comoving frames are used, is expressed as1𝑐𝜕𝜕𝐼𝑡+(𝐥⋅∇)𝐼=𝜈𝜈₀³𝜌𝑗⁰4𝜋−𝜅⁰+𝜎₀𝐼₀+𝜎₀𝑐𝐸₀4𝜋,(1) 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 𝜈𝜈₀𝐥=𝜈₀=𝛾1+𝜷⋅𝐥₀,(2)𝜈𝐥⁰+𝛾−1𝛽²𝐼=𝜈𝜷⋅𝐥₀+𝛾𝜷,(3)𝜈₀⁴𝐼₀,(4) 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,𝜕𝐸𝜕𝑡+𝜕𝜕𝐹𝑥^𝑘𝑘1=𝜌𝛾𝑗₀−𝜅₀𝑐𝐸₀−𝜌𝛾𝜅₀+𝜎₀𝜷⋅𝐅₀,𝑐²𝜕𝐹^𝑖𝜕𝑡+𝜕𝜕𝑃𝑥^𝑖𝑘𝑘=𝜌𝛾𝛽^𝑖𝑐𝑗⁰−𝜅₀𝑐𝐸₀−𝜌𝜅₀+𝜎₀𝛾−1𝛽²𝛽^𝑖×𝜷⋅𝐅₀𝑐−1𝑐𝜌𝜅⁰+𝜎₀𝐹^𝑖0,(5) where the frequency-integrated radiation energy density 𝐸, the frequency-integrated radiative flux 𝐅, and the frequency-integrated radiation stress 𝑃^𝑖𝑘 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 approximation in the comoving frame,𝑃^0𝑖𝑘=𝑓^𝑖𝑘𝐸₀,(6) where 𝑓^𝑖𝑘 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 radius 𝑟, the transfer equation (1) is expressed as 1𝑐𝜕𝜕𝐼𝑡=+𝜇𝜕𝜈𝜕𝐼𝑟+1−𝑟𝜇²𝜕𝜕𝜇𝐼𝜈₀³𝜌𝑗⁰4𝜋−𝜅⁰+𝜎₀𝐼₀+𝜎₀𝑐𝐸₀4𝜋,(7) where 𝜇 is the direction cosine in the inertial frame. Inserting the transformation (4) in the left-hand side, this equation (7) becomes 𝑣𝑣₀1𝑐−𝜕𝜕𝐼4𝑡⁰𝑣+𝜇𝜕𝜕𝐼𝑟⁰+1−𝑟𝜇²𝜕𝜕𝐼𝜇⁰𝑣²⁰𝐼₀=𝜌𝑗₀1𝑐𝜕𝜕𝑣𝑡⁰+𝜇𝜕𝜕𝑣𝑧⁰+1−𝑟𝜇²𝜕𝜕𝑣𝜇⁰4𝜋−𝜅⁰+𝜎₀𝐼₀+𝜎₀𝑐𝐸₀4𝜋.(8)

To calculate the derivatives of 𝐼₀ [9], we apply the chain rules, and, after some manipulations, we have𝜕𝜕𝑡|||_𝑟𝜇𝜈=𝜕𝜕𝑡|||_{𝑟𝜇0𝜈0}+𝜕𝜇⁰𝜕𝑡||||_{𝑟𝜇0𝜈0}𝜕𝜕𝜇₀+𝜕𝜈⁰𝜕𝑡||||_{𝑟𝜇0𝜈0}𝜕𝜕𝜈₀=𝜕𝜕𝑡|||_{𝑟𝜇0𝜈0}−𝛾²1−𝜇²⁰𝜕𝛽𝜕𝑡𝜕𝜕𝜇₀−𝛾²𝜇₀𝜈₀𝜕𝛽𝜕𝑡𝜕𝜕𝜈₀𝜕,𝜕𝑟|||_𝑡𝜇𝜈=𝜕𝜕𝑟|||_{𝑡𝜇0𝜈0}+𝜕𝜇⁰𝜕𝑟||||_{𝑡𝜇0𝜈0}𝜕𝜕𝜇₀+𝜕𝜈⁰𝜕𝑟||||_{𝑡𝜇0𝜈0}𝜕𝜕𝜈₀=𝜕𝜕𝑟|||_{𝑡𝜇0𝜈0}−𝛾²1−𝜇²⁰𝜕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,(9) where 𝜇 is the direction cosine in the comoving frame. In addition, the Doppler shift (2) and the aberration (3) are, respectively, expressed as𝜈𝜈₀𝜇==𝛾𝜇₀1+𝛽𝜇₀,+𝛽1+𝛽𝜇₀.(10)

Using these expressions, after some manipulations, we have the radiative transfer equation in the comoving frame for the spherical flow:𝛾1+𝛽𝜇₀1𝑐𝜕𝜕𝐼𝑡⁰+𝛾𝜇⁰+𝛽𝜕𝐼⁰𝜕𝑟+𝛾1+𝛽𝜇⁰1−𝑟𝜇⁰𝜕𝜕𝜇𝐼⁰₀+4𝛾𝛽1−𝜇²⁰𝑟𝐼⁰×1−𝜇₂₀−𝛾³1+𝛽𝜇₀−𝛾₃𝜕𝜕𝜇𝐼⁰₀−4𝜇₀𝐼₀1𝑐𝜕𝜕𝛽𝑡=𝜌𝑗₀𝜇₀+𝛽1−𝜇²⁰𝜕𝜕𝜇𝐼⁰₀−4𝜇₀𝐼₀𝜕𝜕𝛽𝑟4𝜋−𝜅⁰+𝜎₀𝐼₀+𝜎₀𝑐𝐸₀4𝜋.(11)

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:𝛾𝜕𝑐𝐸⁰𝑐𝜕𝑡+𝛾𝜕𝜕𝐹𝑟⁰+𝛾𝛽𝜕𝑐𝐹𝜕𝑡⁰+𝛾𝛽𝜕𝑐𝜕𝐸𝑟⁰+𝛾𝑟2𝐹⁰+𝛾₃+𝛽3𝑐𝐸₀−𝑐𝑃₀+𝛾₃2𝐹₀+𝛽𝑐𝐸₀+𝑐𝑃₀𝑐𝜕𝜕𝛽𝑡=𝜌𝑗₀2𝛽𝐹₀+𝑐𝐸₀+𝑐𝑃₀𝜕𝜕𝛽𝑟𝛾𝜕𝐹₀−𝜅₀𝑐𝐸₀,𝑐𝜕𝑡+𝛾𝜕𝜕𝑐𝑃𝑟⁰+𝛾𝛽𝜕𝑐𝑐𝜕𝑃𝑡⁰+𝛾𝛽𝜕𝜕𝐹𝑟⁰+𝛾𝑟2𝛽𝐹⁰+𝛾₃−𝑐𝐸₀+3𝑐𝑃₀+𝛾₃2𝛽𝐹₀+𝑐𝐸₀+𝑐𝑃₀𝑐𝜕𝜕𝛽𝑡=−𝜌𝜅2₀𝐹₀+𝛽𝑐𝐸₀+𝑐𝑃₀𝜕𝜕𝛽𝑟+𝜎₀𝐹₀,(12) 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) becomes𝑃₀=𝑓(𝜏,𝛽)𝐸₀,(13) 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:𝑓(𝛽)=1+3𝛽²3+𝛽²,(14) 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 called prescribed 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 𝛾𝜇₀+𝛽𝑑𝐼⁰𝑑𝑟−𝛾³𝜇₀+𝛽1−𝜇²⁰𝜕𝐼⁰𝜕𝜇₀−4𝜇₀𝐼₀+𝛾1+𝛽𝜇₀𝑑𝑑𝛽𝑟1−𝑟𝜇²⁰𝜕𝜕𝜇𝐼⁰₀+4𝛾𝛽1−𝑟𝜇²⁰𝐼⁰=𝜌𝑗⁰4𝜋−𝜅⁰+𝜎₀𝐼₀+𝜎₀𝑐𝐸₀𝛾𝑑𝐹₀4𝜋,(15)𝑑𝑟+𝛾𝛽𝑑𝑐𝑑𝐸𝑟⁰+𝛾³2𝛽𝐹₀+𝑐𝐸₀+𝑐𝑃₀𝑑𝛽+𝛾𝑑𝑟𝑟2𝐹⁰𝛾𝑑𝑐𝑃₀+𝛽3𝑐𝐸₀−𝑐𝑃₀=𝜌𝑗₀−𝜅₀𝑐𝐸₀,(16)𝑑𝑟+𝛾𝛽𝑑𝑑𝐹𝑟⁰+𝛾³2𝐹₀+𝛽𝑐𝐸₀+𝑐𝑃₀𝑑𝛽+𝛾𝑑𝑟𝑟2𝛽𝐹⁰−𝑐𝐸₀+3𝑐𝑃₀=−𝜌𝜅₀+𝜎₀𝐹₀.(17)

Introducing the optical depth defined by𝑑𝜏≡−𝜅₀+𝜎₀𝜌𝑑𝑟(18) and the scattering albedo,𝑎≡𝜎⁰𝜅₀+𝜎₀,(19) the transfer equation (15) and the moment equations (16) and (17) are finally expressed as 𝛾𝜇₀+𝛽𝑑𝐼⁰𝑑𝜏−𝛾1+𝛽𝜇⁰𝜌𝜅1₀−+𝜇𝜎²⁰₀𝑟𝜕𝜕𝜇𝐼⁰⁰−4𝛾𝛽1−𝜇²⁰𝜌𝜅₀+𝜎₀𝑟𝐼⁰−𝛾³𝜇₀+𝛽1−𝜇²⁰𝜕𝐼⁰𝜕𝜇₀−4𝜇₀𝐼₀=𝐼₀𝑑𝑑𝛽𝜏−14𝜋𝜅₀𝑗+⁰𝜎₀−𝑎𝑐𝐸⁰𝛾𝑑𝐹₀4𝜋,(20)𝑑𝜏+𝛾𝛽𝑑𝑑𝑐𝐸𝜏⁰−𝜌𝜅₀+𝛾𝜎₀𝑟2𝐹⁰+𝛾₃+𝛽3𝑐𝐸₀−𝑐𝑃₀=−𝑗2₀𝛽𝐹₀+𝑐𝐸₀+𝑐𝑃₀𝑑𝑑𝛽𝜏𝜅₀+𝜎₀𝛾𝑑𝑐𝑃₀+(1−𝑎)𝑐𝐸₀,(21)𝑑𝜏+𝛾𝛽𝑑𝑑𝐹𝜏⁰−𝜌𝜅₀+𝛾𝜎₀𝑟2𝛽𝐹⁰+𝛾₃−𝑐𝐸₀+3𝑐𝑃₀=𝐹₀2𝐹₀+𝛽𝑐𝐸₀+𝑐𝑃₀𝑑𝑑𝛽𝜏.(22)

Here, we further introduce the spherical variables by𝐷𝐿₀₀≡4𝜋𝑟²𝐹₀,𝑄₀≡4𝜋𝑟²𝑐𝐸₀,≡4𝜋𝑟²𝑐𝑃₀,(23) and the moment equations (21) and (22) become𝛾𝑑𝐿⁰𝑑𝜏+𝛾𝛽𝑑𝑑𝐷𝜏⁰−𝛾𝛽𝐷⁰−𝑄₀𝜌𝜅₀+𝜎₀𝑟+𝛾³=−4𝜋𝑟₂2𝛽𝐿₀+𝐷₀+𝑄₀𝑑𝑑𝛽𝜏𝜅₀+𝜎₀𝑗₀𝛾𝑑𝑄₀−𝜅₀𝑐𝐸₀,𝑑𝜏+𝛾𝛽𝑑𝑑𝐿𝜏⁰−𝛾𝑄⁰−𝐷₀𝜌𝜅₀+𝜎₀𝑟+𝛾³=𝐿₀2𝐿₀+𝛽𝐷₀+𝑄₀𝑑𝑑𝛽𝜏,(24) and the closure relation (13) is written as𝑄₀=𝑓(𝜏,𝛽)𝐷₀.(25)

If we assume the streaming limit of 𝐷₀=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 as𝛾𝑑𝐿⁰𝑑𝜏+𝛾𝛽𝑑𝑑𝐷𝜏⁰+𝛾𝛽1−𝑟𝑓𝐷⁰𝑑𝑟𝑑𝜏+𝛾³2𝛽𝐿₀+(1+𝑓)𝐷₀𝑑𝛽=−4𝜋𝑟₂𝑑𝜏𝜅₀+𝜎₀𝑗₀𝛾𝑑𝑓𝐷₀−𝜅₀𝑐𝐸₀,𝑑𝜏+𝛾𝛽𝑑𝑑𝐿𝜏⁰−𝛾1−𝑟𝑓𝐷⁰𝑑𝑟+𝛾₃𝑑𝜏=𝐿₀2𝐿₀+𝛽(1+𝑓)𝐷₀𝑑𝑑𝛽𝜏.(26) After several manipulations and rearrangement, the relativistic moment equation (26) in the comoving frame is finally expressed as 𝛾𝑓−𝛽²𝑓𝑑𝑑𝐿𝜏⁰+𝛾𝛽1−𝑓𝑟𝑓²−𝑓1𝑑𝑑𝑓𝑟𝐷⁰𝑑𝑟+𝛾₃𝑑𝜏2𝛽1−𝑓1𝐿⁰+(1+𝑓)1−𝛽²=−4𝜋𝑟₂𝑓𝐷⁰𝑑𝑑𝛽𝜏𝜅₀+𝜎₀𝑗₀−𝜅₀𝑐𝐸₀−𝛽𝑓𝐿⁰𝛾𝑓−𝛽₂,𝑑𝑑𝐷𝜏⁰+𝛾𝑑𝑑𝑓𝑟−(1−𝑓)1+𝛽²𝑟𝐷⁰𝑑𝑟+2𝛾𝐿₀𝑑𝜏𝑑𝛽=𝐿₀𝑑𝜏+𝛽4𝜋𝑟²𝜅₀+𝜎₀𝑗₀−𝜅₀𝑐𝐸₀.(27) 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 depth 𝜏 and radius 𝑟. If the flow is steady, as is assumed, the continuity equation for the spherical case is written as 4𝜋𝑟²𝜌𝛾𝛽𝑐=𝑀̇,(28) 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 𝜏=𝑀̇𝜅⁰+𝜎₀4𝜋𝛾𝛽𝑐1𝑟=𝜌𝜅⁰+𝜎₀𝑟,(29) which is also written as 𝜏𝜏_c=𝑟^c𝑟,(30) 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 by𝜏_c=𝑚̇𝑟^g2𝛾𝛽𝑟_c,(31) where 𝑚̇ (=𝑀̇/𝑀̇_E) is the mass-outflow rate normalized by the critical rate 𝑀̇_E (=𝐿_E/𝑐²), 𝐿_E being the Eddington luminosity of the central object, and 𝑟_g (=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, then 𝑗₀=𝜅₀𝑐𝐸₀, and the relativistic moment equation (27) become𝛾𝑓−𝛽²𝑓𝑑𝑑𝐿𝜏⁰+𝛾𝛽1−𝑓𝑟𝑓²−𝑓1𝑑𝑑𝑓𝑟𝐷⁰𝑑𝑟+𝛾₃𝑑𝜏2𝛽1−𝑓1𝐿⁰+(1+𝑓)1−𝛽²=−𝛽𝑓𝐷⁰𝑑𝑑𝛽𝜏𝑓𝐿⁰𝛾𝑓−𝛽₂,(32)𝑑𝑑𝐷𝜏⁰+𝛾𝑑𝑑𝑓𝑟−(1−𝑓)1+𝛽²𝑟𝐷⁰𝑑𝑟+2𝛾𝐿₀𝑑𝜏𝑑𝛽=𝐿₀𝑑𝜏.(33)

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 as𝑑𝑟𝑑𝜏=−𝜅₀+1𝜎₀𝜌=−𝜏𝑟∝−𝜏⁻².(34)

Instead, the second term on the left-hand side of (33) can be dropped, if we impose the restricted condition on the Eddington factor as1−𝑓²𝑟−𝑑𝑑𝑓𝑟=0.(35) This equation (35) is easily integrated to give𝑓=𝐶(𝛽)𝑟²−1𝐶(𝛽)𝑟²+1,(36) 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 as𝑓=1+3𝛽²3+𝛽²at𝑟=𝑟_c,(37) and the appropriate Eddington factor requested to the present case finally becomes𝑓=2𝛾²1+𝛽²𝑟²̂−12𝛾²1+𝛽²𝑟²̂+1=2𝛾²1+𝛽²−𝜏̂²2𝛾²1+𝛽²+𝜏̂²,(38) where 𝑟̂=𝑟/𝑟_c and 𝜏̂=𝜏/𝜏_c. This variable Eddington factor (38) satisfies the condition 𝑓→1/3 when 𝑟→𝑟_c 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 factor 𝑓 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) become𝑑𝐿₀𝑑𝜏=−Γ𝐿⁰𝛾1,(39)𝑔𝑑𝑑𝜏𝑔𝑓−𝛽²𝐷₀=𝐿₀.(40) In these equations,Γ≡𝛽𝛾𝑓−𝛽²(41) is a function of the flow speed and the optical depth, and it becomesΓ=𝛾𝛽1+𝛽²2𝜏̂²2−𝜏̂²+𝛾𝛽(42) for the Eddington factor (38), while 𝑔 is the curvature factor defined byln𝑔≡−^𝜏𝜏_c(1−𝑓)1+𝛽²𝑓−𝛽²𝑟𝑑𝑑𝜏𝑟𝑑𝜏′(43) and becomes in the present case𝑔=𝜏̂³2−𝜏̂².(44) 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 𝐿_s at 𝜏=0, we finally have the comoving luminosity for the RE case:𝐿₀𝐿_s=√2−𝜏̂√2+𝜏̂^𝑏exp1√2𝛾²𝑏𝜏̂,(45) where𝑏=√2𝛾𝛽1+𝛽²𝜏_c.(46)

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 of 𝛽 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 to𝐿₀𝐿_s∼1−𝑏√2𝜏̂∼1−𝛽𝜏.(47) In the extremely relativistic limit of 𝛽→1, on the other hand, 𝑏→𝛾𝜏_c/√2 and the solution reduces to𝐿₀𝐿_s∼√2−𝜏̂√2+𝜏̂^𝑏.(48)

In contrast to this comoving luminosity, it is still difficult to obtain analytical solutions of the spherical radiation energy density 𝐷₀. 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,𝑗₀4𝜋=𝜅⁰𝐵₀,(49) where 𝐵₀ (=𝜎𝑇⁴⁰/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) become𝛾𝑓−𝛽²𝑓𝑑𝑑𝐿𝜏⁰+𝛾𝛽1−𝑓𝑟𝑓²−𝑓1𝑑𝑑𝑓𝑟𝐷⁰𝑑𝑟+𝛾₃𝑑𝜏2𝛽1−𝑓1𝐿⁰+(1+𝑓)1−𝛽²=−𝜅₀𝑓𝐷⁰𝑑𝑑𝛽𝜏𝜅₀+𝜎₀𝑊₀−𝐷₀−𝛽𝑓𝐿⁰𝛾𝑓−𝛽₂,𝑑𝑑𝐷𝜏⁰+𝛾𝑑𝑑𝑓𝑟−(1−𝑓)1+𝛽²×𝐷₀𝑟𝑑𝑟𝑑𝜏+2𝛾𝐿⁰𝑑𝛽=𝐿₀𝑑𝜏+𝛽𝜅⁰𝜅₀+𝜎₀𝑊₀−𝐷₀,(50) where𝑊₀≡16𝜋²𝑟²𝐵₀(51) is the spherical source function.

These equations (50) and (51) can be rearranged as𝛾𝑓−𝛽²𝑓𝑑𝑑𝐿𝜏⁰+𝛾𝑓𝛽1−𝑟𝑓²−𝑑𝑑𝑓𝑟𝑑𝑑𝜏𝑟−𝜅₀𝜅+⁰𝜎₀𝐷⁰+𝛾³2𝛽1−𝑓1𝐿⁰+(1+𝑓)1−𝛽²=−𝜅₀𝑓𝐷⁰𝑑𝑑𝛽𝜏𝜅₀+𝜎₀𝑊₀−𝛽𝑓𝐿⁰𝛾𝑓−𝛽₂,+𝑑𝛾𝑑𝐷𝜏⁰𝑑𝑓𝑑𝑟−(1−𝑓)1+𝛽²×𝐷₀𝑟𝑑𝑑𝜏𝑟+𝛽𝜅₀𝜅+⁰𝜎₀+2𝛾𝐿₀𝑑𝛽=𝐿₀𝑑𝜏+𝛽𝜅⁰𝜅₀+𝜎₀𝑊₀.(52)

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:𝛾𝛽𝑓1−𝑟𝑓²−𝑑𝑑𝑓𝑟𝑑𝑑𝜏𝑟−𝜅₀𝜅+⁰𝜎₀𝛾𝑑𝑓=0,𝑑𝑟−(1−𝑓)1+𝛽²𝑟𝑑𝑑𝜏𝑟+𝛽𝜅₀𝜅+⁰𝜎₀=0.(53) Eliminating 𝜅₀/(𝜅₀+𝜎₀) from (53), we obtain the differential equation for the variable Eddington factor 𝑓,𝑑𝑓𝑑𝑟+𝑓−𝑟1=0,(54) as long as 𝑑𝑟/𝑑𝜏≠0.

This equation (54) is easily integrated to give𝑓=1−𝐶(𝛽)𝑟̂,(55) 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 as𝑓=1+3𝛽²3+𝛽²at𝑟=𝑟_c,(56) and the appropriate Eddington factor requested to the present case finally becomes𝑓=1−21−𝛽²3+𝛽²1𝑟̂=1−21−𝛽²3+𝛽²𝜏̂,(57) where 𝑟̂=𝑟/𝑟_c and 𝜏̂=𝜏/𝜏_c. This variable Eddington factor (57) satisfies the condition: 𝑓→1/3 when 𝑟→𝑟_c 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 factor 𝑓 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) becomes𝑑𝐿₀𝑑𝜏=−Γ𝐿⁰−Δ𝜅⁰𝜅₀+𝜎₀𝑑𝐷₀𝑊₀,(58)𝑑𝜏=Γ𝛽𝐿⁰+Γ𝜅⁰𝜅₀+𝜎₀𝑊₀,(59) whereΓ≡𝛽𝛾𝑓−𝛽²=𝛾𝛽3+𝛽²Δ≡𝑓3+𝛽²−2𝜏̂,𝛾𝑓−𝛽²=𝛾3+𝛽²−21−𝛽²𝜏̂3+𝛽²−2𝜏̂,(60) respectively, in the present case.

Since the index Γ 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 𝐿₀𝐿_s=(1−𝑝𝜏̂)^𝑞,(61) where𝑝≡23+𝛽²𝑞≡𝛾𝛽3+𝛽₂,2𝜏^𝑐.(62) 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 as𝐿₀𝐿_s=(1−𝑝𝜏̂)^𝑞+𝛾𝜏^c1−𝑞3+2𝛽²−𝛾𝛽𝜏_c−𝛾𝜏²̂𝜅₀𝜅+⁰𝜎₀𝑊₀𝐿_s.(63)

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 of 𝛽 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→3𝛽𝜏_c/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→2𝛾𝜏_c, the solution reduces to 𝐿₀𝐿_s∼(1−𝑝𝜏̂)^𝑞−𝜅⁰𝜅₀+𝜎₀𝑊₀𝐿_s.(64) 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 factor 𝑓(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 the essential 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 of 𝜏→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.

Acknowledgment

The author would like to thank an anonymous referee for valuable comments. This work has been supported in part by the Grant-in-Aid for Scientific Research (C) of the Ministry of Education, Culture, Sports, Science and Technology (22540251 JF).