2 Boltzmann equation with velocity-dependent force

The procedure of the derivation of the hydrodynamic equations from the Boltzmann equation for particle distribution function F_s of the particle s is thoroughly described in a number of textbooks. However, it is commonly assumed that the Boltzmann equation can be written in the form (we use the Einstein summation law)

$$\frac{\partial F_s}{\partial t} + \xi_{sh}\frac{\partial F_s}{\pa... ..._s}{\partial \xi_{sh}} = \left(\frac{{\rm d}F_s}{{\rm d}t}\right)_{{\rm coll}},$$ (1)

i.e. it is assumed that the force f_s is independent of velocity, which is acceptable when we consider gravitational or electrical forces. However, the latter assumption is not valid for the case of a wind driven by radiative force coming from line absorption, which is strongly dependent on a velocity gradient. Therefore, we rederive here hydrodynamic equations without an assumption of the force independent of velocity. In this case, the Boltzmann equation for the one-particle distribution function F_s of particles of type s is written as

 

$$\frac{\partial F_s}{\partial t} +\xi_{sh} \frac{\partial F_s}{\pa... ...h}}{m_s}F_s\right) = \left(\frac{{\rm d}F_s}{{\rm d}t}\right)_{{\rm coll}}\cdot$$ (2)

Here $\xi_{sh}$ (h=1,2,3) are the velocity components of individual particles of a type s with mass m_s, and f_sh are the components of an external force acting on them. The right-hand side term expresses the effect of collisions. By a definition, the integral of the distribution function over the velocity space is a number density n_s of s-particles,

 

$$n_s=\int {\rm d}\vec{\xi}_s F_s.$$ (3)

Now, the usual way to obtain hydrodynamic equations is the following. One multiplies the Boltzmann Eq. (2) by multipliers m_s, $m_s\xi_{sh}$ and $m_s\xi_{sh}\xi_{sk}$and integrates it over the velocity space. For the discussion of the role of velocity dependent force in the Boltzmann equation we confine us to the left-hand side of the Boltzmann equation. The right-hand side (i.e. the collisional term) remains unaffected by the presence of such forces, so we assume in this section that the gas is collisionless, i.e. that the right hand side of the Boltzmann equation is zero.

2.1 Continuity equation

Multiplying the Boltzmann Eq. (2) by m_s and integrating over the velocity space we obtain the continuity equation

 

$$\frac{\partial }{\partial t}\left(n_s m_s\right)+\frac{\partial }{\partial x_k}\left(n_s m_s v_{sk}\right)=0,$$ (4)

where v_sk are the components of the mean velocity of particles s, and

$$n_s m_s v_{sk}= m_s \int {\rm d}\vec{\xi}_s \xi_{sk} F_s.$$ (5)

Here, the force term disappeared through integration by parts and the collisional term is also zero. Apparently, velocity-dependent external force does not change the continuity equation.

2.2 Momentum equation

Multiplying the Boltzmann Eq. (2) by $m_s\xi_{sh}$ and integrating over the velocity space we obtain momentum equation

 

$$\frac{\partial }{\partial t}\left(n_s m_s v_{sh}\right)+ \frac{\p... ...(n_s m_s v_{sh} v_{sk}+p_{s,hk}\right)- \int {\rm d}\vec{\xi}_s f_{sh} F_s = 0,$$ (6)

where p_s,hk are the components of the momentum transfer tensor, and

$$n_s m_s v_{sh} v_{sk}+p_{s,hk} = m_s \int {\rm d}\vec{\xi}_s \xi_{sh} \xi_{sk} F_s.$$ (7)

Again, the velocity term was integrated by parts. Note, that due to the velocity dependence of the external force the term containing f_shcannot be moved in front of the integral.

2.3 Energy equation

Multiplying the Boltzmann Eq. (2) by $m_s\xi_{sk}\xi_{sk}$and integrating over the velocity space we obtain energy equation

 

$$\frac{\partial }{\partial t}\left(\frac{1}{2}n_s m_s v_s^2+\frac{... ...s,ki}+\frac{1}{2}p_{s,kki}\right)- \int {\rm d}\vec{\xi}_s \xi_{sk}f_{sk}F_s=0,$$ (8)

where p_s=1/3 p_s,kk is the scalar hydrostatic pressure and the last term was simplified using integration by parts. The equation for the temperature can be derived by subtracting the momentum Eq. (6) dot-multiplied by v_sh from the Eq. (8),

 

$$\frac{\partial }{\partial t}\left(\frac{3}{2} p_s\right)+\frac{\p... ...int {\rm d}\vec{\xi}_s f_{sh} F_s-\int {\rm d}\vec{\xi}_s \xi_{sh}f_{sh}F_s =0.$$ (9)

Note that the last two terms cancel if the force does not depend on the velocity. For this "standard'' case the external force does not give any contribution to net heating.

2.3.1 Gayley-Owocki (Doppler) heating

Let us explore now the last term on the left hand side of Eq. (9) for the case of the force caused by absorption of radiation in spectral lines. This force depends on particle velocity through the velocity dependence of the line absorption coefficient owing to the Doppler effect. Therefore, Gayley & Owocki (1994, hereafter GO) termed the heating effect by Doppler heating, but terming it Gayley-Owocki heating (or GO heating in the abbreviated form) might be more appropriate. Let us denote the heating term in the comoving fluid-frame as

$${Q}_{{\rm i}}^{{\rm GO}} = \int {\rm d}{\vec{\xi}}_{{\rm i}} \xi_{{\rm i},h}f_{{\rm i},h}{F}_{{\rm i}}.$$ (10)

Here the index ${\rm i}$stands for "absorbing ions''. Although this term is written in the comoving fluid-frame, the same expression holds in the non-relativistic case also in the observer frame. Let us assume complete redistribution and an angle independent opacity and emissivity in the atomic frame. However, the emissivity in the comoving fluid-frame is generally angle dependent due to the Doppler effect. Therefore, the radiative force acting on an atom in the comoving fluid-frame is

 

$$f_{{\rm i},h} = \frac{m_{\rm i}}{c} \int_0^\infty {\rm d} \nu \oi... ...ga \left[\kappa(\vec{n},\nu) I(\vec{n},\nu) - \epsilon(\vec{n},\nu)\right] n_h,$$ (11)

where $\kappa({\vec n},\nu)$ and $\epsilon({\vec n},\nu)$ are the absorption and emission coefficients per unit mass, which can be expressed as

$$\kappa({\vec n},\nu) = \kappa\, \varphi\left(\nu-\frac{\nu_0}{c}{\vec{\xi}}_{{\rm i}} \!\cdot\!{\vec n}\right),$$ (12a)

$$\epsilon({\vec n},\nu) = \epsilon\, \varphi\left(\nu-\frac{\nu_0}{c}{\vec{\xi}}_{{\rm i}} \!\cdot\!{\vec n}\right),$$ (12b)

where $\varphi(\nu)$ is the absorption (emission) profile in the atomic frame. After some rearrangement we obtain

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{m}_{{\rm i}}}{c} \int_{0}^{\inf... ...\varphi\left(\nu-\frac{\nu_0}{c}{\vec{\xi}}_{{\rm i}} \!\cdot\!{\vec n}\right).$$ (13)

If for the calculation of the last integral we choose one of the velocity axes parallel to the direction of n, then the integrals over other two axes vanishes (we integrate odd function) and the Gayley-Owocki heating formula becomes

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{m}_{{\rm i}}}{c} \int_{0}^{\inf... ...m i}}({\xi}_{{\rm i}}) \varphi \left(\nu-\frac{\nu_0}{c}{\xi}_{{\rm i}}\right).$$ (14)

where ${\xi}_{{\rm i}}$is a velocity component in the direction of ${\vec n}$. If we rewrite photon-line-of-sight velocity component as ${\xi}_{{\rm i}}=wv_{{\rm th,i}}$and use frequency displacement from the line center in Doppler units $x=\left(\nu-\nu_0\right)/\Delta\nu_{\rm D}$, then the GO heating takes the form

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{m}_{{\rm i}}}{c} \int_{-\infty}... ...nfty v_{{\rm th,i}}^2\,{\rm d}w\,w\, {F}_{{\rm i}}(v_{{\rm th,i}}w)\,\psi(w-x),$$ (15)

where we introduced the intensity $\tilde{I}({\vec n},x)$ as $\tilde{I}({\vec n},x)\, {\rm d}x = I({\vec n},\nu)\, {\rm d}\nu$ and the function

$$\psi(x)=\varphi(x\Delta\nu_{\rm D})$$ (16)

is normalized according to Castor (1974) as

$$\int_{-\infty}^{\infty} \psi(x) {\rm d}x=1.$$ (17)

Note that the thermal speed $v_{{\rm th,i}}$ is really ionic because it comes from the velocity distribution of absorbing ions (shall not be interchanged with $v_{{\rm th}}$, which comes from normalization of force multipliers). We neglect absorption in the resonance wings of the profile and approximated $\psi(w-x)\approx\delta(w-x)$. Finally, we assume that the velocity distribution is given by the Maxwellian velocity distribution,

$${F}_{{\rm i}}(v_{{\rm th,i}}w) =\frac{{n}_{{\rm i}}}{v_{{\rm th,i}}\sqrt\pi}\,{\rm e}^{-w^2}.$$

The latter assumption was made purely due to simplicity reasons. Relaxing it could lead to interesting effects especially if the number of collisions is not sufficient to maintain an equilibrium (cf., e.g., Scudder 1994; Cranmer 1998). However, we postpone the analysis of the non-Maxwellian effects to a future paper.

Thus, the GO heating formula takes the form of

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{m}_{{\rm i}} {n}_{{\rm i}} v_{{... ...\,x \phi(x) \left[\kappa \tilde I({\vec n},x)-\Delta\nu_{\rm D}\epsilon\right],$$ (18)

where

 

$$\phi(x)=\frac{1}{\sqrt\pi}{\rm e}^{-x^2}.$$ (19)

Due to the symmetry of the absorption profile (19) the product $x \phi(x)$ is an odd function and, thus, after integration over x, vanishes. This means that in the case of complete redistribution the process of emission gives no direct contribution to GO heating, i.e.

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{m}_{{\rm i}} {n}_{{\rm i}} v_{{... ...ega\, \int_{-\infty}^{\infty} {\rm d}x \,x \phi(x) \kappa \tilde I({\vec n},x).$$ (20)

In the static medium the product $x \phi(x) \tilde I({\vec n},x)$ is an odd function of x, thus, there is no GO heating effect in static stellar atmospheres. In the particular case of a spherically-symmetric stellar wind the expression for the Gayley-Owocki heating takes the form of

 

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{2\pi\kappa{\rho}_{{\rm i}}v_{{\r... ..._{-1}^1{\rm d}\mu \int_{-\infty}^{\infty} {\rm d}x\, x \phi(x) \tilde I(\mu,x),$$ (21)

where $\mu=\cos\theta$, which was actually used by GO.

2.3.2 Formula for Gayley-Owocki heating in the stellar wind domain

In the case of a two-level atom without continuum the solution of the transfer equation in the Sobolev approximation is (Rybicki & Hummer 1978; Owocki & Rybicki 1985, GO)

 

$$\tilde I(\mu,x)=\tilde I_{\rm c} \left\{\frac{\beta_{\rm c}}{\beta}+\left[D(\mu)- \frac{\beta_c}{\beta}\right] {\rm e}^{-\tau_\mu\Phi(x)}\right\},$$ (22)

where $\tilde I_{\rm c}$ is the core intensity, $D(\mu)$ is unity for $\mu>\mu_*$ and zero otherwise ( $\mu_*=\left(1-R_*^2/r^2\right)^{1/2}$), core penetration and escape probabilities are given by

  

$$\beta_{\rm c}=\frac{1}{2}\int_{\mu_*}^{1}{\rm d}\mu\frac{1-{\rm e}^{-\tau_\mu}}{\tau_\mu},$$ (23)

$$\beta=\frac{1}{2}\int_{-1}^{1}{\rm d}\mu\frac{1-{\rm e}^{-\tau_\mu}}{\tau_\mu},$$ (24)

respectively, and

$$\Phi(x)=\int_{x}^{\infty}{\rm d}x'\phi(x').$$ (25)

The Sobolev optical depth $\tau_\mu$ is given by (Castor 1974; Rybicki & Hummer 1978)

 

$$\tau_\mu=\frac{{\rho}_{{\rm i}}\kappa v_{{\rm th}}r}{{\mbox{\eufont Y}}_{{\rm i}}{{v_{\rm r}}}_{{\rm i}} \left(1+\sigma\mu^2\right)},$$ (26)

where the variable $\sigma$ was introduced by Castor (1974)

 

$$\sigma=\frac{{\rm d}\ln {{v_{\rm r}}}_{{\rm i}} }{{\rm d}\ln r} -1.$$ (27)

Inserting the solution of the transfer Eq. (22) into the expression for the GO heating Eq. (21) we obtain

 

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{2\pi\kappa{\rho}_{{\rm i}}v_{{\r... ...right] \int_{-\infty}^{\infty} {\rm d}x\, x \phi(x) {\rm e}^{-\tau_\mu\Phi(x)}.$$ (28)

The effect of line ensemble is usually described using the concept of a line-strength distribution function (CAK, Abbott 1982; Puls et al. 2000)

 

$${\rm d}N(\kappa)=-N_0 \left(\frac{{\rho}_{{\rm e}}/\left(W {m}_{{... ...cm}^{-3}}\right)^{\delta} \kappa^{\alpha-2}{\rm d}\kappa\frac{{\rm d}\nu}{\nu},$$ (29)

where normalization constant N₀ is taken in the form of

 

$$N_0=\frac{c k}{v_{{\rm th}}}\left(1-\alpha\right)\alpha\, \sigma_{\rm e}^{1-\alpha}.$$ (30)

The GO heating formula for this line ensemble can be obtained by the integration of the heating term for one line (Eq. (28)) over the CAK distribution function Eq. (29). In this case it takes the form of

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{\rho}_{{\rm i}}v_{{\rm th}}v_{{... ...right] \int_{-\infty}^{\infty} {\rm d}x\, x \phi(x) {\rm e}^{-\tau_\mu\Phi(x)},$$ (31)

where $\tilde I_{\rm c} =\Delta\nu_{\rm D}L/4\pi^2 R_*^2$ was used. Finally, applying substitution $y=\kappa{\rho}_{{\rm i}}v_{{\rm th}}r/{\mbox{\eufont Y}}_{{\rm i}}{{v_{\rm r}}}_{{\rm i}}$preceding equation becomes

 

$${Q}_{{\rm i}}^{{\rm GO}} = \frac{{\rho}_{{\rm i}}v_{{\rm th,i}}k ... ...}{\sigma_{\rm e}{\rho}_{{\rm i}}v_{{\rm th}}r}\right)^{\alpha} G(\sigma,\mu_*),$$ (32)

where the function $G(\sigma,\mu_*)$ is given by the triple integration

 

$$G(\sigma,\mu_*)=\int_0^{\infty}{\rm d}y\,y^{\alpha-1}\int_{-1}^1{... ...\infty} {\rm d}x\, x \phi(x) \exp\left(-\frac{y\Phi(x)}{{1+\sigma\mu^2}}\right)$$ (33)

and in the integrals for $\beta_{\rm c}$ and $\beta$ Eqs. (23), (24) the Sobolev depth shall be computed using

$$\tau_\mu=\frac{y}{1+\sigma\mu^2}$$ (34)

instead of Eq. (26). Contrary to the radiative force formula (Castor 1974) the GO heating formula depends on the absorption profile. For the determination of the GO heating term we selected Gaussian profile (which comes from the Maxwellian velocity distribution).