Radiation transport through porous media

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A medium is porous if it consists of high density clumps in a low density surrounding. If the clumps are optically thick, while the surrounding medium is not the porosity changes the effective opacity of the medium, sine photons will be able to escpae from the gas by moving through the optically thin gaps between the clumps. The clumps are self-shielding, since only the surface of the clump interacts with the radiation. The net effect of the poroisty is a reduction of the coupling between radiation and gas. This means that a star, which has formally reached the Eddington limit can stay gravitationally bound. Only in the outer layers, where the clumps themselves become optically thin does the gas feel the full force of the radiation. This allows for a continuum driven stellar wind.

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Radiatively driven mass-loss

The equation of motion for a steady state stellar wind, driven by radiation is:

v \frac{{\rm d}v}{{\rm d}r}~=~-\frac{G M_{\star}}{r^2}+ g_{\rm rad}-\frac{1}{\rho}\frac{{\rm d}P}{{\rm d}r},

with v the wind velocity, r the radius, G the gravitational constant, ρ and P the gas density and pressure respectively and grad the radiative acceleration. The density and pressure of the gas are related through the isothermal sound speed a as P = a2ρ. We can use this relationship to eliminate the density, so the equation of motion takes the form:

\left(1-\frac{a^2}{v^2}\right)v \frac{{\rm d}v}{{\rm d}r}~=~-\frac{G M_{\star}}{r^2}+ g_{\rm rad}+\frac{2a^2}{r}-\frac{{\rm d}a^2}{{\rm d}r},

where the two final terms on the right hand side result from the gas pressure gradient. They play a key role in driving the transonic wind expansion of the high-temperature (˜106K) coronae that form around the sun and other cool stars. But in the line-driven winds of massive stars, the wind temperature remains comparable to the stellar effective temperature, of order 104K. In the supersonic portions of such wind, these pressure terms are relatively unimportant, since compared to gravity and other competing terms they are or of order

w_{\rm s} \equiv (a/v_{\rm esv})^2 \approx 0.001,

where v_{\rm esc}^2 \equiv 2GM_\star/R_\star^2.

For continuum-driven winds, the neglect gas pressure is less certain, as these typically occur in giant type stars such as LBV (Luminous Blue Variable) stars. Since these stars have very large radii, the escape velocity is much lower than for Main Sequence and Wolf-Rayet stars, which makes the pressure gradient term more important, though still generally small compared to the radiative driving term.

The equation of motion can be conveniently written in a dimensionless form, using a new velocity variable w\equiv v^2/v_{\rm esc}^2, which represents the ratio of wind kinetic energy (v_{\rm esc}^2/2) to gravitational binding energy (v_{\rm esc}^2/2), and defining also an inverse radius coordinate x\equiv1-R_\star/r. If the pressure expansion term (2a2 / r) is neglected, the equation of motion then becomes:

\left(1-\frac{w_{\rm s}}{w}\right)w'~=~-1+\Gamma_{\rm rad},

with \Gamma_{\rm rad}\equiv g_{\rm rad} r^2/GM_\star and the gravitationally scaled acceleration:

w'\equiv\frac{{\rm d}w}{{\rm d}x}~=~\frac{r^2 v {\rm d}v/{\rm d}r}{G M_\star}

The contribution of the pressure gradient terms to the wind velocity can be seen in the following image,
Wind velocity for a continuum driven wind
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Wind velocity for a continuum driven wind
which shows the velocity for a continuum driven wind. The red dots in the image show the result of a hydrodynamical simulation, using the ZEUS 3D code (Stone & Norman 1992) The green and blue lines show the analytical solution of the equation of motion without and with the pressure gradient terms respectively. Obviously, the contribution of the pressure gradient terms is negligeable. However, in this particular case, the luminosity of the star was assumed to be twice the Eddington Luminosity. For lower luminosity the pressure gradient terms would become relatively more important.
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Super-Eddington outflow moderated by porous opacity

For continuum driving to be effective, the star has to reach the Eddington limit, where the radiative force due to electron scattering exceeds the gravitational binding force. Unfortunately, this presents us with a problem, since the scaling laws for gravity and radiation leads us to conclude that a star which reaches the Eddington limit becomes gravitationally unbound throughout the star. This prohibits a steady outflow from the surface (See Continuum-driven mass loss from super-Eddington stars). A possible way to get around this problem is through the porosity of the material, which effectively reduces the coupling between gas and radiation (Shaviv 2000,2001).

Suppose a medium consists of localized clumps in a transparrent surrounding medium and each clump has identical length l and mass m. If we assume that the clumps have an optical thickness \tau_{\rm cl}\approx\kappa m/l^3 \gg 1 their effective cross section to the radiation field will be \sigma_{\rm eff} \approx l^2. More generally, the fraction of the radiation field attenuated by each blob will scale as 1 − exp( − τcl) so the effective cross section becomes \sigma_{\rm eff}\approx l^2(1-exp(-\tau_{\rm cl})). Therefore we can define the effective opacity of the clumps as:

\kappa_{\rm eff} \equiv \frac{\sigma_{\rm eff}}{m} \approx \frac{l^2}{m}(1-e^{-\tau_{\rm cl}})\approx \kappa \frac{1-e^{\tau_{\rm cl}}}{\tau_{\rm cl}}

Generally speaking this means that as long as the clumps are optically thick, the opacity of the medium has been reduced by a factor:

k \equiv \frac{\kappa_{\rm eff}}{\kappa}

Of course, the assumption that all blobs are equal is highly unrealistic. If in fact the clumps vary in optical depth, the reduction of the opacity becomes:

k \equiv \frac{\kappa_{\rm eff}}{\kappa}~=~\sum_{\rm i} f_{\rm i} \frac{1-e^{-\tau_{\rm i}}}{\tau_{\rm i}}\approx \int_0^{\infty} {\rm d}\tau \frac{{\rm d}f}{{\rm d}\tau} \frac{1-e^{-\tau}}{\tau}

with df / dτ the clump distribution over optical depth. We make the assumption that this distribution can be approximated by an exponentially truncated power law:

\tau\frac{{\rm d}f}{{\rm d}\tau}~=~\frac{1}{\Gamma(\alpha)}\left(\frac{\tau}{\tau_0}\right)^\alpha e^{-\tau/\tau_0},

with τ0 the optical depth of the strongest clump and α > 0 the powerlaw index. The solution for the opacity reduction of the clumped medium is:

k(\tau_0)~=~\frac{(1+\tau_0)^{1-\alpha} - 1}{(1-\alpha)\tau_0},

for \alpha\neq 1. For fixed clump characteristics the optical depth τ0 is given by:

\tau_0 \equiv \rho \kappa h_0,

where h0 the pororsity length, which is defined as L3 / l2, with L the separation between clumps. Since k is a scaling factor for the coupling between radiation and gas, we can rewrite the equation of motion as:

w'(x)~=~\Gamma k(\tau_0) -1


For a complete derivation see Owocki, Gayley & Shaviv (2004).

Mass loss rates
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Mass loss rates
Terminal velocities
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Terminal velocities
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Numerical results

We have made a parameter study of continuum driven winds, using the porosity length formalism, covering a range of values for both Γ and α to investigate how they influence the continuum driven wind.

The mass loss rates increase with Γ, since a high Γ means that there is more luminosity available to drive the wind. However, they decrease with higher α as this decreases the coupling between matter and radiation. The wind velocity increases Γ and also increases with α, since for a give luminosity a higher α means that less mass has to be lifted against gravity. Terminal velocities are typically of the same order of magnitude as the escape velocity at the surface. Mass loss rates vary from high to extremely high, proving the efficiency of super-Eddington continuum driven winds. The basic input model for all simulations is a 50Msol, 50Rsol star with tiring parameter η = 1.0 and a surface temperature of 50,000K.

The effect of photon tiring on the wind velocity.
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The effect of photon tiring on the wind velocity.
The effect of photon tiring on terminal wind speed.
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The effect of photon tiring on terminal wind speed.
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The photon tiring limit

While the porosity length formalism discussed in the previous sections gives us a mass loss rate based on the stellar parameters, its is not certain that the star will actually be able to drive off that much material. Given a stellar luminosity L_\star, conservation of energy limits the mass loss rate to:

\dot{m}~=~2\frac{L_\star}{v_{\rm esc}^2}.

For line driven winds, this is never a problem, since line-driving will never cause such a high mas sloss rate. Continuum driving is a different story. The mass loss rates are much higher and will come close to, or even exceed the photon tiring limit. This means that the numerical simulation described in the previous section will have to be modified.

We accomplish this by calculating the work integral along the radial gridline and subtracting it from the available amount of radiation. in other words, at a given radius r the available amount of radiation equals:

L(r)~=~L_\star - \int_{R_\star}^r \dot{m}(r) g_{\rm rad} dr ~=~L_\star - \int_{R_\star}^r 4\pi r^2 \rho(r)v(r) g_{\rm rad} dr

Note that for a negative velocity this means that the radiation field will actually be increased. Numerical results show the effect of photon tiring quite clearly. The mass loss rate remains the same, since it is dictated by the input parameters of the porosity length formalism. The wind velocity decreases in order to satisfy conservation of energy leading to lower terminal velocities at high values of Γ.

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Alternaive Applications

Stars are not the only objects that can exceed the Eddington limit. Other examples are Novae and certain accretion disks around black holes, which are thought to be at the heart of Ultra-Luminous X-Ray sources (ULXs). Since such objects also lose mass in winds, they too can serve as test cases for our porosity model


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See Also

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