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Advances in Astronomy
Volume 2011 (2011), Article ID 412620, 10 pages
http://dx.doi.org/10.1155/2011/412620
Research Article

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

Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582, Japan

Received 22 December 2010; Accepted 19 March 2011

Academic Editor: Jerome Orosz

Copyright © 2011 Jun Fukue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [113]. 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., [1417]), or proposed the variable Eddington factor (e.g., [1824]), or performed the numerical simulations using, for example, the flux-limited diffusion (FLD) approximation (e.g., [2535]).

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., [3740]) 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 [610, 12, 13, 4244]. 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𝜕𝐼𝜈𝜕𝑡+(𝐥)𝐼=𝜈03𝜌𝑗0𝜅4𝜋0+𝜎0𝐼0+𝜎0𝑐𝐸0,4𝜋(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 𝜈𝜈0=𝛾1+𝜷𝐥0,𝜈(2)𝐥=0𝜈𝐥0+𝛾1𝛽2𝜷𝐥0𝜷𝜈+𝛾,(3)𝐼=𝜈04𝐼0,(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 (=1/1𝛽2) the Lorentz factor, 𝛽 being 𝐯/𝑐.

The zeroth and first moment equations are, respectively,𝜕𝐸+𝜕𝑡𝜕𝐹𝑘𝜕𝑥𝑘𝑗=𝜌𝛾0𝜅0𝑐𝐸0𝜅𝜌𝛾0+𝜎0𝜷𝐅0,1𝑐2𝜕𝐹𝑖+𝜕𝑡𝜕𝑃𝑖𝑘𝜕𝑥𝑘𝛽=𝜌𝛾𝑖𝑐𝑗0𝜅0𝑐𝐸0𝜅𝜌0+𝜎0𝛾1𝛽2𝛽𝑖𝑐×𝜷𝐅01𝑐𝜌𝜅0+𝜎0𝐹𝑖0,(5) 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 approximation in the comoving frame,𝑃0𝑖𝑘=𝑓𝑖𝑘𝐸0,(6) 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 radius 𝑟, the transfer equation (1) is expressed as 1𝑐𝜕𝐼𝜕𝑡+𝜇𝜕𝐼+𝜕𝑟1𝜇2𝑟𝜕𝐼=𝜈𝜕𝜇𝜈03𝜌𝑗0𝜅4𝜋0+𝜎0𝐼0+𝜎0𝑐𝐸0,4𝜋(7) where 𝜇 is the direction cosine in the inertial frame. Inserting the transformation (4) in the left-hand side, this equation (7) becomes 𝑣𝑣01𝑐𝜕𝐼0𝜕𝑡+𝜇𝜕𝐼0+𝜕𝑟1𝜇2𝑟𝜕𝐼0𝑣𝜕𝜇4𝑣20𝐼01𝑐𝜕𝑣0𝜕𝑡+𝜇𝜕𝑣0+𝜕𝑧1𝜇2𝑟𝜕𝑣0𝑗𝜕𝜇=𝜌0𝜅4𝜋0+𝜎0𝐼0+𝜎0𝑐𝐸0.4𝜋(8)

To calculate the derivatives of 𝐼0 [9], we apply the chain rules, and, after some manipulations, we have𝜕|||𝜕𝑡𝑟𝜇𝜈=𝜕|||𝜕𝑡𝑟𝜇0𝜈0+𝜕𝜇0||||𝜕𝑡𝑟𝜇0𝜈0𝜕𝜕𝜇0+𝜕𝜈0||||𝜕𝑡𝑟𝜇0𝜈0𝜕𝜕𝜈0=𝜕|||𝜕𝑡𝑟𝜇0𝜈0𝛾21𝜇20𝜕𝛽𝜕𝜕𝑡𝜕𝜇0𝛾2𝜇0𝜈0𝜕𝛽𝜕𝜕𝑡𝜕𝜈0,𝜕|||𝜕𝑟𝑡𝜇𝜈=𝜕|||𝜕𝑟𝑡𝜇0𝜈0+𝜕𝜇0||||𝜕𝑟𝑡𝜇0𝜈0𝜕𝜕𝜇0+𝜕𝜈0||||𝜕𝑟𝑡𝜇0𝜈0𝜕𝜕𝜈0=𝜕|||𝜕𝑟𝑡𝜇0𝜈0𝛾21𝜇20𝜕𝛽𝜕𝜕𝑟𝜕𝜇0𝛾2𝜇0𝜈0𝜕𝛽𝜕𝜕𝑟𝜕𝜈0,𝜕||||𝜕𝜇𝑟𝑡𝜈=𝜕𝜇0||||𝜕𝜇𝑟𝑡𝜈0𝜕𝜕𝜇0+𝜕𝜈0𝜕𝜇0||||𝑟𝑡𝜈0𝜕𝜕𝜈0=𝛾21+𝛽𝜇02𝜕𝜕𝜇0𝛾2𝛽1+𝛽𝜇0𝜈0𝜕𝜕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 as𝜈𝜈0=𝛾1+𝛽𝜇0,𝜇𝜇=0+𝛽1+𝛽𝜇0.(10)

Using these expressions, after some manipulations, we have the radiative transfer equation in the comoving frame for the spherical flow:𝛾1+𝛽𝜇01𝑐𝜕𝐼0𝜇𝜕𝑡+𝛾0+𝛽𝜕𝐼0𝜕𝑟+𝛾1+𝛽𝜇01𝜇0𝑟𝜕𝐼0𝜕𝜇0+4𝛾𝛽1𝜇20𝑟𝐼0𝛾31+𝛽𝜇0×1𝜇20𝜕𝐼0𝜕𝜇04𝜇0𝐼01𝑐𝜕𝛽𝜕𝑡𝛾3𝜇0+𝛽1𝜇20𝜕𝐼0𝜕𝜇04𝜇0𝐼0𝜕𝛽𝑗𝜕𝑟=𝜌0𝜅4𝜋0+𝜎0𝐼0+𝜎0𝑐𝐸0.4𝜋(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𝜕𝐹0𝜕𝑟+𝛾𝛽𝜕𝐹0𝑐𝜕𝑡+𝛾𝛽𝜕𝑐𝐸0+𝛾𝜕𝑟𝑟2𝐹0+𝛽3𝑐𝐸0𝑐𝑃0+𝛾32𝐹0+𝛽𝑐𝐸0+𝑐𝑃0𝜕𝛽𝑐𝜕𝑡+𝛾32𝛽𝐹0+𝑐𝐸0+𝑐𝑃0𝜕𝛽𝑗𝜕𝑟=𝜌0𝜅0𝑐𝐸0,𝛾𝜕𝐹0𝑐𝜕𝑡+𝛾𝜕𝑐𝑃0𝜕𝑟+𝛾𝛽𝜕𝑐𝑃0𝑐𝜕𝑡+𝛾𝛽𝜕𝐹0+𝛾𝜕𝑟𝑟2𝛽𝐹0𝑐𝐸0+3𝑐𝑃0+𝛾32𝛽𝐹0+𝑐𝐸0+𝑐𝑃0𝜕𝛽𝑐𝜕𝑡+𝛾32𝐹0+𝛽𝑐𝐸0+𝑐𝑃0𝜕𝛽𝜅𝜕𝑟=𝜌0+𝜎0𝐹0,(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 𝛽1 [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 𝛾𝜇0+𝛽𝑑𝐼0𝑑𝑟𝛾3𝜇0+𝛽1𝜇20𝜕𝐼0𝜕𝜇04𝜇0𝐼0𝑑𝛽𝑑𝑟+𝛾1+𝛽𝜇01𝜇20𝑟𝜕𝐼0𝜕𝜇0+4𝛾𝛽1𝜇20𝑟𝐼0𝑗=𝜌0𝜅4𝜋0+𝜎0𝐼0+𝜎0𝑐𝐸0,𝛾4𝜋(15)𝑑𝐹0𝑑𝑟+𝛾𝛽𝑑𝑐𝐸0𝑑𝑟+𝛾32𝛽𝐹0+𝑐𝐸0+𝑐𝑃0𝑑𝛽+𝛾𝑑𝑟𝑟2𝐹0+𝛽3𝑐𝐸0𝑐𝑃0𝑗=𝜌0𝜅0𝑐𝐸0,𝛾(16)𝑑𝑐𝑃0𝑑𝑟+𝛾𝛽𝑑𝐹0𝑑𝑟+𝛾32𝐹0+𝛽𝑐𝐸0+𝑐𝑃0𝑑𝛽+𝛾𝑑𝑟𝑟2𝛽𝐹0𝑐𝐸0+3𝑐𝑃0𝜅=𝜌0+𝜎0𝐹0.(17)

Introducing the optical depth defined by𝜅𝑑𝜏0+𝜎0𝜌𝑑𝑟(18) and the scattering albedo,𝜎𝑎0𝜅0+𝜎0,(19) the transfer equation (15) and the moment equations (16) and (17) are finally expressed as 𝛾𝜇0+𝛽𝑑𝐼0𝑑𝜏𝛾1+𝛽𝜇01𝜇20𝜌𝜅0+𝜎0𝑟𝜕𝐼0𝜕𝜇04𝛾𝛽1𝜇20𝜌𝜅0+𝜎0𝑟𝐼0𝛾3𝜇0+𝛽1𝜇20𝜕𝐼0𝜕𝜇04𝜇0𝐼0𝑑𝛽𝑑𝜏=𝐼01𝑗4𝜋0𝜅0+𝜎0𝑎𝑐𝐸0,𝛾4𝜋(20)𝑑𝐹0𝑑𝜏+𝛾𝛽𝑑𝑐𝐸0𝛾𝑑𝜏𝜌𝜅0+𝜎0𝑟2𝐹0+𝛽3𝑐𝐸0𝑐𝑃0+𝛾32𝛽𝐹0+𝑐𝐸0+𝑐𝑃0𝑑𝛽𝑗𝑑𝜏=0𝜅0+𝜎0+(1𝑎)𝑐𝐸0,𝛾(21)𝑑𝑐𝑃0𝑑𝜏+𝛾𝛽𝑑𝐹0𝛾𝑑𝜏𝜌𝜅0+𝜎0𝑟2𝛽𝐹0𝑐𝐸0+3𝑐𝑃0+𝛾32𝐹0+𝛽𝑐𝐸0+𝑐𝑃0𝑑𝛽𝑑𝜏=𝐹0.(22)

Here, we further introduce the spherical variables by𝐿04𝜋𝑟2𝐹0,𝐷04𝜋𝑟2𝑐𝐸0,𝑄04𝜋𝑟2𝑐𝑃0,(23) and the moment equations (21) and (22) become𝛾𝑑𝐿0𝑑𝜏+𝛾𝛽𝑑𝐷0𝐷𝑑𝜏𝛾𝛽0𝑄0𝜌𝜅0+𝜎0𝑟+𝛾32𝛽𝐿0+𝐷0+𝑄0𝑑𝛽𝑑𝜏=4𝜋𝑟2𝜅0+𝜎0𝑗0𝜅0𝑐𝐸0,𝛾𝑑𝑄0𝑑𝜏+𝛾𝛽𝑑𝐿0𝑄𝑑𝜏𝛾0𝐷0𝜌𝜅0+𝜎0𝑟+𝛾32𝐿0𝐷+𝛽0+𝑄0𝑑𝛽𝑑𝜏=𝐿0,(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+𝛾𝛽𝑑𝐷0𝑑𝜏+𝛾𝛽1𝑓𝑟𝐷0𝑑𝑟𝑑𝜏+𝛾32𝛽𝐿0+(1+𝑓)𝐷0𝑑𝛽𝑑𝜏=4𝜋𝑟2𝜅0+𝜎0𝑗0𝜅0𝑐𝐸0,𝛾𝑑𝑓𝐷0𝑑𝜏+𝛾𝛽𝑑𝐿0𝑑𝜏𝛾1𝑓𝑟𝐷0𝑑𝑟𝑑𝜏+𝛾32𝐿0+𝛽(1+𝑓)𝐷0𝑑𝛽𝑑𝜏=𝐿0.(26) After several manipulations and rearrangement, the relativistic moment equation (26) in the comoving frame is finally expressed as 𝛾𝑓𝛽2𝑓𝑑𝐿0𝑑𝜏+𝛾𝛽1𝑓21𝑓𝑟𝑓𝑑𝑓𝐷𝑑𝑟0𝑑𝑟𝑑𝜏+𝛾312𝛽1𝑓𝐿0𝛽+(1+𝑓)12𝑓𝐷0𝑑𝛽𝑑𝜏=4𝜋𝑟2𝜅0+𝜎0𝑗0𝜅0𝑐𝐸0𝛽𝑓𝐿0,𝛾𝑓𝛽2𝑑𝐷0𝑑𝜏+𝛾𝑑𝑓𝑑𝑟(1𝑓)1+𝛽2𝑟𝐷0𝑑𝑟𝑑𝜏+2𝛾𝐿0𝑑𝛽𝑑𝜏=𝐿0+𝛽4𝜋𝑟2𝜅0+𝜎0𝑗0𝜅0𝑐𝐸0.(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 4𝜋𝑟2̇𝜌𝛾𝛽𝑐=𝑀,(28) where ̇𝑀 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 ̇𝑀𝜅𝜏=0+𝜎014𝜋𝛾𝛽𝑐𝑟𝜅=𝜌0+𝜎0𝑟,(29) which is also written as 𝜏𝜏c=𝑟c𝑟,(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 by𝜏c=̇𝑚𝑟g2𝛾𝛽𝑟c,(31) where ̇𝑚 (=̇̇𝑀𝑀/E) is the mass-outflow rate normalized by the critical rate ̇𝑀E (=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, then 𝑗0=𝜅0𝑐𝐸0, and the relativistic moment equation (27) become𝛾𝑓𝛽2𝑓𝑑𝐿0𝑑𝜏+𝛾𝛽1𝑓21𝑓𝑟𝑓𝑑𝑓𝐷𝑑𝑟0𝑑𝑟𝑑𝜏+𝛾312𝛽1𝑓𝐿0𝛽+(1+𝑓)12𝑓𝐷0𝑑𝛽𝛽𝑑𝜏=𝑓𝐿0,𝛾(32)𝑓𝛽2𝑑𝐷0𝑑𝜏+𝛾𝑑𝑓𝑑𝑟(1𝑓)1+𝛽2𝑟𝐷0𝑑𝑟𝑑𝜏+2𝛾𝐿0𝑑𝛽𝑑𝜏=𝐿0.(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 as𝑑𝑟1𝑑𝜏=𝜅0+𝜎0𝜌𝑟=𝜏𝜏2.(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 as1𝑓2𝑟𝑑𝑓𝑑𝑟=0.(35) This equation (35) is easily integrated to give𝑓=𝐶(𝛽)𝑟21𝐶(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=𝑟c,(37) and the appropriate Eddington factor requested to the present case finally becomes𝑓=2𝛾21+𝛽2̂𝑟212𝛾21+𝛽2̂𝑟2=+12𝛾21+𝛽2̂𝜏22𝛾21+𝛽2+̂𝜏2,(38) where ̂𝑟=𝑟/𝑟c and ̂𝜏=𝜏/𝜏c. This variable Eddington factor (38) satisfies the condition 𝑓1/3 when 𝑟𝑟c and 𝛽0, and 𝑓1 when 𝑟 (𝜏0) or 𝛽1. The behavior of this Eddington factor is shown in Figure 1.

412620.fig.001
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=Γ𝐿0,𝛾1(39)𝑔𝑑𝑔𝑑𝜏𝑓𝛽2𝐷0=𝐿0.(40) In these equations,𝛽Γ𝛾𝑓𝛽2(41) is a function of the flow speed and the optical depth, and it becomesΓ=𝛾𝛽1+𝛽22̂𝜏22̂𝜏2+𝛾𝛽(42) for the Eddington factor (38), while 𝑔 is the curvature factor defined byln𝑔𝜏𝜏c(1𝑓)1+𝛽2𝑓𝛽2𝑟𝑑𝑟𝑑𝜏𝑑𝜏(43) and becomes in the present case𝑔=̂𝜏32̂𝜏2.(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 𝐿s at 𝜏=0, we finally have the comoving luminosity for the RE case:𝐿0𝐿s=2̂𝜏2+̂𝜏𝑏1exp2𝛾2,𝑏̂𝜏(45) where𝑏=2𝛾𝛽1+𝛽2𝜏c.(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.

412620.fig.002
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 𝛽0, 𝑏0 and the solution reduces to𝐿0𝐿s𝑏12̂𝜏1𝛽𝜏.(47) In the extremely relativistic limit of 𝛽1, on the other hand, 𝑏𝛾𝜏c/2 and the solution reduces to𝐿0𝐿s2̂𝜏2+̂𝜏𝑏.(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) become𝛾𝑓𝛽2𝑓𝑑𝐿0𝑑𝜏+𝛾𝛽1𝑓21𝑓𝑟𝑓𝑑𝑓𝐷𝑑𝑟0𝑑𝑟𝑑𝜏+𝛾312𝛽1𝑓𝐿0𝛽+(1+𝑓)12𝑓𝐷0𝑑𝛽𝜅𝑑𝜏=0𝜅0+𝜎0𝑊0𝐷0𝛽𝑓𝐿0,𝛾𝑓𝛽2𝑑𝐷0𝑑𝜏+𝛾𝑑𝑓𝑑𝑟(1𝑓)1+𝛽2𝑟×𝐷0𝑑𝑟𝑑𝜏+2𝛾𝐿0𝑑𝛽𝑑𝜏=𝐿0𝜅+𝛽0𝜅0+𝜎0𝑊0𝐷0,(50) where𝑊016𝜋2𝑟2𝐵0(51) is the spherical source function.

These equations (50) and (51) can be rearranged as𝛾𝑓𝛽2𝑓𝑑𝐿0+𝑑𝜏𝛾𝛽𝑓1𝑓2𝑟𝑑𝑓𝑑𝑟𝑑𝑟𝜅𝑑𝜏0𝜅0+𝜎0𝐷0+𝛾312𝛽1𝑓𝐿0𝛽+(1+𝑓)12𝑓𝐷0𝑑𝛽𝜅𝑑𝜏=0𝜅0+𝜎0𝑊0𝛽𝑓𝐿0,𝛾𝑓𝛽2𝑑𝐷0+𝛾𝑑𝜏𝑑𝑓𝑑𝑟(1𝑓)1+𝛽2𝑟𝑑𝑟𝜅𝑑𝜏+𝛽0𝜅0+𝜎0×𝐷0+2𝛾𝐿0𝑑𝛽𝑑𝜏=𝐿0𝜅+𝛽0𝜅0+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:𝛾𝛽𝑓1𝑓2𝑟𝑑𝑓𝑑𝑟𝑑𝑟𝜅𝑑𝜏0𝜅0+𝜎0𝛾=0,𝑑𝑓𝑑𝑟(1𝑓)1+𝛽2𝑟𝑑𝑟𝜅𝑑𝜏+𝛽0𝜅0+𝜎0=0.(53) Eliminating 𝜅0/(𝜅0+𝜎0) from (53), we obtain the differential equation for the variable Eddington factor 𝑓,𝑑𝑓+𝑑𝑟𝑓1𝑟=0,(54) as long as 𝑑𝑟/𝑑𝜏0.

This equation (54) is easily integrated to give𝑓=1𝐶(𝛽),̂𝑟(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=𝑟c,(56) and the appropriate Eddington factor requested to the present case finally becomes2𝑓=11𝛽23+𝛽212̂𝑟=11𝛽23+𝛽2̂𝜏,(57) where ̂𝑟=𝑟/𝑟c and ̂𝜏=𝜏/𝜏c. This variable Eddington factor (57) satisfies the condition: 𝑓1/3 when 𝑟𝑟c and 𝛽0, and 𝑓1 when 𝑟 (𝜏0) or 𝛽1. The behavior of this Eddington factor is shown in Figure 3.

412620.fig.003
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=Γ𝐿0𝜅Δ0𝜅0+𝜎0𝑊0,(58)𝑑𝐷0=Γ𝑑𝜏𝛽𝐿0𝜅+Γ0𝜅0+𝜎0𝑊0,(59) where𝛽Γ𝛾𝑓𝛽2=𝛾𝛽3+𝛽23+𝛽2,𝑓2̂𝜏Δ𝛾𝑓𝛽2=𝛾3+𝛽221𝛽2̂𝜏3+𝛽2,2̂𝜏(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 𝐿0𝐿s=(1𝑝̂𝜏)𝑞,(61) where2𝑝3+𝛽2,𝑞𝛾𝛽3+𝛽22𝜏𝑐.(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 as𝐿0𝐿s=(1𝑝̂𝜏)𝑞+𝛾𝜏c1𝑞3+𝛽22𝛽𝛾𝜏ĉ𝜏𝛾2𝜅0𝜅0+𝜎0𝑊0𝐿s.(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.

412620.fig.004
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 𝛽0, 𝑝2/3, and 𝑞3𝛽𝜏c/2, and the solution becomes a linear function of 𝜏. In the extremely relativistic limit of 𝛽1, on the other hand, 𝑝1/2 and 𝑞2𝛾𝜏c, the solution reduces to 𝐿0𝐿s(1𝑝̂𝜏)𝑞𝜅0𝜅0+𝜎0𝑊0𝐿s.(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 𝜏0 or 𝛽1 (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).

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