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Black Hole Astrophysics Chapters 9.3 All figures extracted from online sources of from the textbook. Part I Equation of state Pressure and Internal energy of various types of gases (Ch 9.3.1~9.3.2) Introduction To this stage, we have presented all the conservation laws that would be needed to calculate how plasma behave in a general gravitational field. ฯc 2 + ๐๐ ๐ ฮฑฮฒ gas = ๐๐๐ฅ ๐ฆ ๐๐ ๐๐๐ง ๐ฆ ๐๐๐ฅ โ2๐๐ฃ,๐ ๐ด xx โ ๐๐ฃ,๐ ๐ฉ + ๐๐ ๐๐ โ2๐๐ฃ,๐ ๐ด xy ๐๐๐ง โ2๐๐ฃ,๐ ๐ด xz โ2๐๐ฃ,๐ ๐ด yx โ2๐๐ฃ,๐ ๐ด zx โ2๐๐ฃ,๐ ๐ด yy โ ๐๐ฃ,๐ ๐ฉ + ๐๐ โ2๐๐ฃ,๐ ๐ด zy โ2๐๐ฃ,๐ ๐ด yz โ2๐๐ฃ,๐ ๐ด zz โ ๐๐ฃ,๐ ๐ฉ + ๐๐ ๐๐๐ผ = โ๐พ๐ ๐ 2 ๐ฮฑฮฒ ๐ป๐ฝ ๐ + ๐๐๐ฝ ๐ป๐ฝ ๐ ๐ผ However, we see above that there are lots of quantities that we donโt know yet โ For gases: ๐๐ ๐๐ ๐พ๐ ๐๐ฃ,๐ ๐๐ฃ,๐ (energy density, pressure, thermal conductivity, viscosity coefficients) For radiation: ๐๐ ๐๐ ๐พ๐ ๐๐ฃ,๐ ๐๐ฃ,๐ Therefore, what we do next is to relate them to density ฯ and temperature T, and in some cases, plasma composition. Composition of gases Since the most abundant elements in the universe are Hydrogen and Helium, we usually express the composition of a gas in terms of mass fraction of the elements X Y Z for Hydrogen for Helium for anything heavier (often called โmetalsโ) Unless the gas is exotic (ex electron-positron), the mass fractions sum to 1. X+Y+Z=1 X Y Z Solar Abundance 0.71 0.27 0.02 Early Universe 0.75 0.25 4 × 10โ10 The general distribution function for Thermal gases (9.3.1) According to statistical mechanics, we can find that gases are distributed in momentum according to (the particle density per unit momentum) dn ๐๐ 4ฯp2 = dp โ3 e ๐ ๐ โ๐chem kT ± 1 ๐ ๐ = ๐2 ๐ 2 + ๐0 ๐ 4 particle energy; โ = 6.62607 × 10โ27 erg · ๐ Plankโ constant ๐chem chemical potential; ๐๐ degeneracy factor +1 is for Fermions, half-spin particles (๐ โ ๐ + ๐+ ๐ ๐๐ . . .) โ1 is for Bosons, integer spin particles (๐พ๐ ± ๐ 0 . . .) Classical Maxwellian Determining Energy and Momentum from the distribution function dn ๐ Since the distribution function dp = โ3๐ 4ฯp2 tells us e ๐ ๐ โ๐chem kT ±1 how many particles (per unit volume) are contained within a momentum interval, The total kinetic energy is simply to sum over that of each momentum interval dn๐ ๐๐ = ๐๐พ ๐ dp dp Classical Maxwellian And the pressure, being momentum flux as we discussed last week, is ๐ ๐ = 1 3 ๐ ๐ฃ dn๐ dp dp Particle flux Non-Relativistic Ideal Gas: Tenuous, Warm Fermions (Ch9.3.1.1) For fermions at not too high a density, the chemical potential is very negative. And for non-relativistic gases, ๐ โ ๐0 ๐ 2 + ๐๐พ >> kT Thus, the distribution function reduces to dn ๐๐ 4ฯp3 8๐ = 3 ๐ ๐ โ๐ โ e chem kT + 1 dp โ e โ3 ๐2 8๐ ๐ โ 2 โ 3 e chem โ๐0๐ kT ๐2 e 2๐0kT โ ๐2 โ ๐2 e 2๐0kT ๐chem โ๐0 ๐ 2 ๐๐พ kT ๐ 2 eโkT Evaluating the internal energy and pressure, we find the very familiar formulas: 3 ๐๐ = nkT 2 ๐ ๐ = nkT 5 The adiabatic index ๐ค = 3 3 The polytropic index ๐ = 2 5 3 Specific heats ๐ถ๐ = 2 ๐ ๐ถ๐ฃ = 2 ๐ Pressure for different non-relativistic ideal gas compositions ๐ ๐ = nkT = ฯkT ๐ ๐ is the mean molecular weight, expressed in units of grams per mole Gas Type ๐ Neutral Hydrogen Gas 1 Fully Ionized hydrogen gas 0.5 General Composition Neutral Gas 1 ๐ + 0.25๐ + 0.06๐ 1 2๐ + 0.75๐ + 0.56๐ General Composition Fully Ionized Gas ๐(๐๐จ๐ฅ๐๐ซ) 0.61 ๐ ๐ = 1.63ฯkT Simple explanation for mean molecular weight: ๐total ๐tot 1 ๐= = = ๐๐ป + ๐He + ๐Metal ๐tot · ๐ + ๐tot · ๐ 4 + ๐metal · ๐ ๐metal ๐ + 0.25๐ + ๐ ๐metal ๐tot · ๐ = ๐๐ป = ๐๐ป · 1 ; ๐tot · ๐ = ๐He = ๐He · 4 ; ๐tot · ๐ = ๐metal = ๐metal · ๐metal Relativistic Ideal Gas: Tenuous, Hot Fermions (Ch9.3.1.2) dn ๐๐ = dp โ3 e 4ฯp3 ๐ ๐ โ๐chem 8๐ โ e kT ± 1 โ3 ๐chem โ๐0 ๐ 2 ๐๐พ kT ๐2 eโkT Since for general situations, the kinetic energy is ฮตK = m0 c 2 2 + pc This changes the distribution to ๐0 ๐ 2 2 + pc 2 โ๐0 ๐ 2 dn 8๐ ๐ 2 โ kT = 3 e chem โ๐0๐ kT ๐2 e dp โ ๐2 eโ m=100 ๐2 +๐2 โ๐ ๐ Classical Maxwellian m=1 m=0.0001 Shape of cut-off is affected by the mass 2 โ m0 c 2 The highly relativistic case When the kinetic energy is much greater than the rest mass energy, it is mainly dominated by the pc term. dn 8๐ = e dp โ3 ๐chem โ๐0 ๐ 2 ๐๐ kT 2 โ kT ๐ e Evaluating the internal energy and pressure, we find the very familiar formulas: ๐๐ = 3nkT ๐ ๐ = nkT m=100 ๐2 +๐2 โ๐ โ ๐ ๐2 e 4 The adiabatic index ๐ค = 3 The polytropic index ๐ = 3 Specific heats ๐ถ๐ = 4๐ ๐ถ๐ฃ = 3๐ m=1 m=0.0001 As we would expect, this will turn out to be very much the same as photons since photons have rest mass and their energies are only kinetic. Photon Gas: Hot Bosons (Ch 9.3.1.3) Taking the distribution for photons and using the fact that ฮตK = ฮต = pc = hฮฝ and g s = 2 for two polarization states dn ๐๐ 4๐๐2 8๐ ๐2 = = dp โ3 e ๐ ๐ โ๐chem kT โ 1 โ3 epc kT โ 1 If we look at the spectral energy distribution, we see that it should be very familiar dn 8๐hฮฝ ๐2 ๐ = 3 hฮฝ kT dฮฝ ๐ e โ1 It is simply the Plankian SED ! As for the intensity, ๐ dn 2hฮฝ ๐2 ๐ผ ๐ = ๐ = 2 hฮฝ kT 4๐ dฮฝ ๐ e โ1 = ๐ต๐ ๐ This is also the out familiar form of the Plank function that describes the intensity per unit frequency. (Black Body Distribution) Energy and Pressure for a Photon gas Evaluating the internal energy and pressure, we find: ๐๐ = 3๐ ๐ = aT 4 a = 7.56577 × 10โ15 erg · cmโ3 ๐พ โ4 This gives: 4 The adiabatic index ๐ค = 3 The polytropic index ๐ = 3 Specific heats ๐ถ๐ = 4๐ ๐ถ๐ฃ = 3๐ Which is the same as a relativistic Fermion gas. Denerate Gas: Dense Fermions (Ch9.3.1.4) Previously, we have discussed cases where the chemical potential is very negative and therefore causes the exponential term to be much larger than 1. dn ๐๐ 4ฯp2 = dp โ3 e ๐ ๐ โ๐chem kT ± 1 However, when the density of Fermions, for example, becomes so high that the Pauli Exclusion Principle canโt be neglected, then the โ1โ in the denominator becomes important. dn 8๐ = dp โ3 e ๐2 ๐ ๐ โ๐chem dn 8๐ = 3 3 ๐๐พ 2 + 2๐๐พ ๐0 ๐ 2 dฮต๐พ โ ๐ kT +1 ๐๐พ + ๐0 ๐ 2 ๐๐พ +๐0 ๐ 2 โ๐chem kT e +1 How to define โdegenerateโ? In our introduction to degenerate gases, we noted that for dense fermions, the +1 must be considered. It should then be obvious that the exponential term canโt be too large. dn 8๐ = 3 3 ๐๐พ 2 + 2๐๐พ ๐0 ๐ 2 dฮต๐พ โ ๐ ๐๐พ + ๐0 ๐ 2 ๐๐พ +๐0 ๐ 2 โ๐chem kT e To be more precise, we can define a โFermi Temperatureโ ๐๐น = ๐๐พ โ๐๐น kT The exponential then becomes e +1 ๐๐น k = ๐chem โ๐0 ๐ 2 ๐ . Now, we see that it is clear that there are two cases: 1. ๐๐พ โซ ๐๐น ๏ผThe exponential term is large, we have a non-degenerate gas. 2. ๐๐พ โช ๐๐น ๏ผThe exponential term is small. A degenerate gas. Pressure and Energy Evaluating the pressure and energy, we get: ๐= 8๐ 3 ๐0 ๐ 3 2๐ ๐ ๐ 0 โ ๐ฅ ๐= 8๐ 3 ๐0 ๐ 3 2๐ธ ๐ ๐ 0 โ ๐ฅ ๐ ๐ฅ โก ๐ ๐น๐ 2 0 With the normalized energy and pressure functions: ๐ ๐ฅ = ๐ฅ 2๐ฅ 2 โ 3 ๐ฅ 2 + 1 + 3sinhโ1 ๐ฅ ๐ธ ๐ฅ = 3๐ฅ 2๐ฅ 2 + 1 ๐ฅ 2 + 1 โ 8๐ฅ 3 โ 3sinhโ1 ๐ฅ Polytropic index Non-rel gas n=1.5 Relativistic gas n=3 Non-rel gas 5 ๐ค= 3 Relativistic gas 4 ๐ค= 3 Some handy numbers Handy expressions for the pressure for a degenerate electron gas are, for the nonrelativistic and relativistic cases, and for a degenerate neutron gas with ฯ and ฮผ in cgs units, and the standard ฮต = p/(ฮ โ 1) giving the internal energy density for each. Note the similarity between the two different degenerate gases in the relativistic cases. The boundaries between the non-relativistic and relativistic cases are approximately 1.9 × 106 ๐ · cmโ3 for the degenerate electron gas and 1.15 × 1016 ๐ · cmโ3 for degenerate neutrons. ๐๐น = 10 kT Rel, Non-Rel ๐ โ 1.9 × 106 ๐ · cmโ3 Rel, Non-Rel n 1.15 × 1016 ๐ · cmโ3 Radiation Pressure Non-Rel Degenerate neutron Non-Rel Degenerate electron Relativistic Degenerate electron Usually happens in unstable stars (Collapsing) Nonthermal gases (Ch 9.3.2) Possibly due to Fermi acceleration in the universe, many sources exhibit a powerlaw spectrum in the high energy end. The Crab Nebula is given as an example to the left. (Radio lobes, jets often also show this behavior) This is usually called non-thermal since particles that emit this radiation must have energies way higher than the thermal value โkTโ. Lorentz factors can go even up to 106 or higher. Power law spectra For such cases, it is common to assume that the particles distribute in energy as a power law shape: dn 1โ๐ฝ โ๐ฝ = ๐ 1โ๐ฝ ๐ ๐พ 1โ๐ฝ dฮต๐พ ๐ โ๐ ๐พ,Max ๐พ,min normalization Which energies ๐๐พ,min < ๐๐พ < ๐๐พ,Max If ฮฒ > 1, then the distribution function is steep and dominated by low-energy particles, perhaps even a very lowenergy thermal distribution. On the other hand, If ฮฒ < 1, then the distribution is shallow, dominated by the high-energy end, and must be cut off more steeply beyond ๐๐พ,Max . ฮฒ=0.5 ฮฒ=1 ฮฒ=2 Energy and Pressure for nonthermal particles Evaluating the energy and pressure for non-thermal particles, we find that 1โ๐ฝ ๐ = 3๐ = ๐ 2โ๐ฝ 4 2โ๐ฝ 2โ๐ฝ 1โ๐ฝ 1โ๐ฝ ๐๐พ,Max โ ๐๐พ,min ๐๐พ,Max โ ๐๐พ,min This gives a Adiabatic Index ๐ค = 3, same as for highly relativistic particles. (This should be trivial since by origin, they are highly relativistic) Part II Equation of state Conductivity and Viscosity (Ch9.3.3~9.3.7) Thermal Conductivity (Ch9.3.3) Recall From last week With the knowledge that ๐ ๐ = โ๐พ๐ ๐ป ๐ and that it corresponds to the ๐ i0 and ๐ 0j terms, we could guess that in locally flat space-time, the components would read as ๐ฆ 0 ๐๐๐ฅ ๐๐ ๐๐๐ง ๐๐๐ฅ 0 0 0 ฮฑฮฒ ๐ Conduction = ๐ฆ ๐๐ 0 0 0 ๐๐๐ง 0 0 0 However, we can see that ๐๐ is actually still a 3-vector and the above form is simply from an educated guess. Therefore we need to first rewrite ๐๐ into a 4-vector ๐๐๐ผ . We find that it can be expressed as 1 ๐๐๐ผ = โ๐พ๐ ๐ 2 ๐ฮฑฮฒ ๐ป๐ฝ ๐ + ๐๐๐ฝ ๐ป๐ฝ ๐ ๐ผ with ๐ฮฑฮฒ = ๐ 2 ๐ ๐ผ ๐๐ฝ + ๐ฮฑฮฒ Or, ๐ ๐ = โ๐พ๐ ๐2 ๐ · ๐ป ๐ + ๐ ๐ · ๐ป ๐ with ๐ = 1 ๐2 ๐โ๐+ ๐ โ1 A simple kinetic picture Consider a picture like the one on the left. If we consider that a pair of particles are exchanged, then there will be a net energy transfer from top to bottom. ฮT Therefore we can write heat flux as (particle number flux)×(energy difference) For a thermal gas, the energy that is required to heat it by ฮT is ฮ๐ธ = ๐ถ๐ ฮT. In terms of differential quantities, we can write ฮT = โ๐ 1 dT dz dT Putting it all together, we get Q โ โ 3 ๐ ๐๐ ๐ถ๐ โ๐ dz . 1 Comparing with ๐ = โ๐พ๐ ๐ป ๐, we find the diffusion coefficient ๐พ๐ โ โ 3 (๐ถ๐ ๐) ๐๐ โ๐ Q1.Why is โ๐ the mean free path? Q2.Why is it ๐ถ๐ ? http://en.wikipedia.org/wiki/Thermal_conductivity Thermal Conductivity 1 As we have just found, the thermal condutivity is equal to ๐พ๐ โ 3 (๐๐ถ๐ ) ๐๐ โ๐ For thermal conduction in a electron-ion plasma, it would be sufficient to only consider electrons since they are fast. 3 For a classical thermal gas, ๐๐ถ๐ = 2 ๐๐ ๐ The rms velocity is ๐๐ = 3kT ๐๐ The mean free path, by definition is the inverse of the density multiplied by the collision cross1 section. โ๐ = ๐ ๐ ๐ ๐ Determining the mean free path 1 ๐ ๐๐ The mean free path โ๐ = ๐ The easiest was to estimate the collision crosssection is to give it a radius, thus, ๐๐ = ฯr๐ 2 Therefore the actual problem is to find some reasonable radius to apply into the formula. (This was actually already discussed in Ch1 of plasma Astrophys.) My own idea is like this: Since the mean free path is the distance of which a particle travels before crashing into something and thereby changing direction of motion, the cross-section associated with it would be defined by some radius within which the injected particle would be deflected by a large angle. (red oval below) http://en.wikipedia.org/wiki/Coulomb_collis Determining the mean free path For coulomb collisions, if the particle looses most of its initial kinetic energy to the coulomb field, then it now no longer knows which direction it came from. The radial coulomb field then changes its direction according to how close the particle is. Thus, we can approximate the radius by equating the thermal kinetic energy and the Coulomb potential energy. ๐2 ๐๐ถ = = kT = ๐๐พ ๐๐ This then give us a classical Coulomb collision radius ๐2 ๐๐ = kT Putting it all together 1 The thermal conductivity ๐พ๐ โ 3 (๐ถ๐ ๐) ๐๐ โ๐ 3 ๐ถ๐ = ๐๐ ๐ 2 ๐พ๐ โ ๐ 3 kT 2๐๐ 4 ๐๐ ๐๐ = 5 2 3kT ๐๐ 1 โ๐ = ๐๐ ๐๐ ๐๐ = ฯr๐ โ 5.38 × 1018 erg cmโ1 ๐ โ1 ๐พ โ1 ๐2 ๐๐ = kT 2 ๐ 4.0 × 109 ๐พ 5 2 Inner disk > 10keV 0.1keV accretion disk 102 103 104 105 How important is it? Letโs now estimate the importance of heat flux relative to energy flux by advection from neighboring fluid elements. (Advection is from the ๐ฃ · ๐ป term) ๐๐ Heat conduction flux ๐๐ โ๐ = โ Advection energy flux ๐๐ ๐ ๐ Becomes close ๐ is the typical length scale of system. For accreting BH, it is ~ 10 โ 100 ๐๐ . Case1: Main Sequence stars: Since MS stars are in approximately in hydrostatic equilibrium, the velocity of fluid elements V will be much smaller than ๐๐ the thermal velocity. Thus, in MS stars, heat conduction is more important. Case2: Accreting BHs: In such cases, V, the infall velocity, can reach the sound speed ๐๐ = ๐๐ ๐๐ โ kT 2 ๐๐ ๐๐ 4 ๐๐ ๐ ๐๐ = 1.0 โ1 ๐๐ 5.3×1018 cmโ3 ๐ 10๐โ โ1 ๐ 10๐๐ โ1 ๐ ๐ โ 2 ๐ 4.×109 ๐พ kT ๐๐ . ้ๆฒๅๅฎXD ๐๐ โ ๐๐ ๐๐ kT 2 ๐๐ = 1.0 ๐๐ ๐๐ 4 ๐๐ ๐ 5.3 × 1018 cmโ3 โ1 ๐ 10๐โ โ1 ๐ 10๐๐ โ1 ๐ 4.× 109 ๐พ 2 Particle Viscosity (Ch9.3.4) Recall From last week Since viscosity works to transport momentum, it should manifest itself in the momentum flux term of the tensor. Iโm not so familiar with this part so below mainly follows the textbook. ๐ ฮฑฮฒ Viscosity = โ2๐๐ฃ,๐ ๐ดฮฑฮฒ โ ๐๐ฃ,๐ ๐ฉ๐ฮฑฮฒ shear Shear viscosity coefficient bulk ๐๐ฃ,๐ = ๐๐ฃ,๐ ๐, ๐ 1 Projection tensor ๐ฮฑฮฒ = ๐ 2 ๐ ๐ผ ๐๐ฝ + ๐ฮฑฮฒ 1 1 Shear tensor ๐ด ฮฑฮฒ โก 2 [๐ฮฑฮณ ๐ป๐พ ๐๐ฝ + ๐ฮฒฮณ ๐ป๐พ ๐ ๐ผ โ 3 ๐ฉ๐ฮฑฮฒ Compression rate ๐ฉ โก ๐ป๐พ ๐ ๐พ Bulk viscosity coefficient ๐๐ฃ,๐ = ๐๐ฃ,๐ ๐, ๐ A simple kinetic picture Consider a picture like the one on the left. If we consider that a pair of particles are exchanged, then there will be a net momentum transfer from top to bottom. ฮP Therefore we can write momentum flux as (particle number flux)×(momentum difference) dV In terms of differential quantities, we can write ฮP = โ๐ฃ ๐ dz Putting it all together, we get ๐ฝ๐ โ๐ ๐๐ฃ โ๐ฃ dV . dz Comparing with ๐ ฮฑฮฒ Viscosity = โ2๐๐ฃ,๐ ๐ดฮฑฮฒ โ ๐๐ฃ,๐ ๐ฉ๐ฮฑฮฒ , we find the viscosity coefficients ๐๐ฃ,๐ โ ๐๐ฃ,๐ โ๐ ๐๐ฃ โ๐ฃ 10.1098/rstl.1866.0013 The coefficients of viscosity The coefficients of viscosity ๐๐ฃ,๐ โ ๐๐ฃ,๐ โ๐ ๐๐ฃ โ๐ฃ look very familiar to the thermal 1 conductivity ๐พ๐ โ 3 (๐ถ๐ ๐) ๐๐ โ๐ . However, in case of momentum, for an electron-proton plasma, the momentum is mainly carried by the protons. Thus, both ๐๐ฃ and โ๐ฃ have to use values for protons. โ๐ฃ = 1 ๐๐ ๐๐ = ๐๐ kT 2 ๐๐๐ 4 Typo in textbook? Thus, for ๐ โ ๐+ plasma, ๐๐ฃ,๐ โ 1 3๐๐ kT ฯe4 5 2 โ 3.× 109 erg · ๐ · cmโ3 ๐ 4.× 109 ๐พ 5 2 How important is it? Comparing the contribution of viscosity to pressure, we get the following equation ๐ ฮฑฮฒ Visc ๐ ๐๐ ๐๐ โ kT 2 ๐๐ ๐๐ 4 ๐๐ ๐ ๐๐ โ ๐๐ฃ,๐ ๐๐ฃ ๐ nkT โ kT 2 โ 3 ๐ ๐ฃ = 3 ฯe4nR = 1.0 โ1 ๐ 3.7×1017 cmโ3 ๐ 10๐โ = 1.0 โ1 ๐๐ 5.3×1018 cmโ3 ๐ 10๐โ โ1 โ1 ๐ 10๐๐ ๐ 10๐๐ โ1 2 ๐ 4.×109 ๐พ โ1 2 ๐ 4.×109 ๐พ There is an interesting thing here, when we compare with the ratio of heat flux to advection energy transfer, we find that for viscosity to dominate pressure requires even lower pressure than for heat conduction to dominate advection! Nevertheless, in low accretion rate situations, it is possible that particle viscosity may be important as well. Turbulent Viscosity (Ch9.3.5) ๅตๅตๅตโฆ ็ไธๆๅฆXD Turbulence is the random, chaotic, motion that often occurs in in ๏ฌuids on scales much smaller than the overall system size, but much larger than the distance between independent ๏ฌuid particles or even between ๏ฌuid elements. The phenomenon usually develops in ๏ฌuids undergoing shear ๏ฌow, unless the microscopic viscosity is strong enough to damp out any growing chaotic motions and turn them into heat. It also can occur in ๏ฌuids that have a weak, but non-zero, magnetic ๏ฌeld. Turbulence actually is produced by ๏ฌuid motions and is not really a separate physical microscopic process. However, the motions are so complex, compared to regular laminar ๏ฌow that statistical methods have been developed to treat chaotic ๏ฌuid motions, just as such methods were developed to handle the mechanics of multiple particles in motion in the ๏ฌrst place. If the size scale of the turbulent eddies โt is much smaller than the overall size of the system, then one can use the turbulent diffusion approximation to de๏ฌne a turbulent viscosity ฮทv,t โ ฮถv,t โ ฯ Vt โt where Vt is the RMS velocity of the chaotic motions in the eddies. ๅตๅตๅตโฆ ็ไธๆๅฆXD In the early 1970s the nature of turbulent ๏ฌow in black hole accretion ๏ฌows was largely unknown. So early investigators [351, 352] assumed that the RMS turbulent velocity was a fraction of the sound speed ๐๐ก โ ฮฑc๐ where the free parameter ฮฑ โค 1. This โalpha modelโ of turbulence was quite successful in the early days of black hole accretion studies. See Chapter 12. In addition to the obvious issues associated with choosing a diffusion approximation, treating turbulence as a viscous process has some other assumptions associated with it. Recall (Section 9.2.1) that having a viscosity implies that viscous dissipation of the shear exists (viscous heating). This is because the viscous part of the stress-energy tensor does not have its own energy density (ฮต t is missing). In reality, however, turbulence does have an energy density, a pressure also, and heat ๏ฌow as well, not just viscous-like properties. All of this physics is missing in this treatment, along with a model for how to convert turbulent energy into heat. Instead, the simple viscous approximation assumes that all mechanical energy lost due to viscosity immediately is converted into heat (equation (9.16)). While this works rather well in accretion models, it still should be remembered that turbulence can be much more complex than a simple ad hoc viscosity. Radiative Opacity (Ch9.3.6) Recall from last week In many situations that we will study in the next few chapters, the fluid will be optically thick to radiation and both will be in thermodynamic equilibrium at the same temperature Tr = Tg โก T. In this case the photon gas will contribute to the fluid plasma pressure, energy density, heat conduction, and viscosity and will add stress-energy terms similar to those discussed previously for fluids. ฯc 2 + ๐๐ ๐ ฮฑฮฒ gas = ๐๐๐ฅ ๐ฆ ๐๐๐ฅ โ2๐๐ฃ,๐ ๐ด xx โ ๐๐ฃ,๐ ๐ฉ + ๐๐ ๐๐ โ2๐๐ฃ,๐ ๐ด xy ๐๐๐ง โ2๐๐ฃ,๐ ๐ด xz โ2๐๐ฃ,๐ ๐ด yx โ2๐๐ฃ,๐ ๐ด zx โ2๐๐ฃ,๐ ๐ด yy โ ๐๐ฃ,๐ ๐ฉ + ๐๐ โ2๐๐ฃ,๐ ๐ด zy โ2๐๐ฃ,๐ ๐ด yz โ2๐๐ฃ,๐ ๐ด zz โ ๐๐ฃ,๐ ๐ฉ + ๐๐ ๐ฆ ๐๐ ๐๐๐ง ๐ = ๐๐ Total density of fluid (photons donโt contribute to this) ๐ = ๐๐ + ๐๐ Total pressure ๐ = ๐๐ + ๐๐ Total energy density ๐๐ผ = ๐๐ ๐ผ + ๐๐ ๐ผ Total heat conduction vector ๐๐ฃ = ๐๐ฃ,๐ + ๐๐ฃ,๐ Total coefficient of shear viscosity ๐๐ฃ = ๐๐ฃ,๐ + ๐๐ฃ,๐ Total coefficient of bulk viscosity Conduction by radiation Last week, we mentioned that for radiation, we can basically copy the whole set of stress-energy tensor, therefore, for the heat conduction term, ๐๐ผ = ๐๐ ๐ผ + ๐๐ ๐ผ . Thus, we can determine the total conductivity ๐พ = ๐พ๐ + ๐พ๐ . 1 Then, by analogy of ๐พ๐ โ 3 (๐ถ๐ ๐) ๐๐ โ๐ , 3 1 1 1 4acT _ ๐พ๐ = nC๐ฃ ๐๐ โ๐ = 4aT 3 ๐โ๐ = 3 3 3 ๐๐ ๐ 1 _ ๐๐ ๐ 1 = ๐ผ = โ๐ , ๐ผ is the absorption coefficient. From illustration below, we can see that it should be inverse proportional to the mean free path of photons. Output Intensity ๐ผ๐ ๐ฅ = ๐ โฮฑL Incident Intensity ๐ผ๐ 0 L For details, please see Radiative Processes in Astrophysics by Rybicki & Lightman Frequency dependent Opacity 1 Using the relations โ๐ = ๐ผ = ๐๐ _ 1 ๐ 1 ๐ = ๐๐ ๐ ๐ We can rewrite the opacity in terms of scattering/absorption coefficient ๐๐๐ ๐ ๐๐ด ๐ ๐ ๐ = = ๐๐ ๐ ๐ ๐ _ There are many ways to scatter/absorb photons. Therefore in the following we will consider a. Electron scattering b. Free-Free and Bound-Free Absorption Electron scattering (Ch9.3.6.1) General considerations Considering scattering between photons and electrons, we recall from high school that the most general case for scattering is Compton scattering. In such a case, the cross section we need to consider is the KleinNishina cross-section ๐KN . _ ๐๐๐ ๐ ๐ ๐๐ด (1+๐) 2 Using the relation ๐ ๐ ๐ = _ find ๐ ๐ ๐ = ๐๐พ๐ ๐ ๐KN ๐๐ด ๐ and applying it to electron scattering, we 3 2 1 + ๐๐ 1 + ๐๐ โn 1 + 2๐๐ ๐ = ๐๐ [ โ 4 1 + 2๐๐ 2๐๐ ๐๐2 hฮฝ 2 ๐๐ ๐๐ โก ๐ ๐๐ = = ๐๐ ๐ 8๐ 3 โn 1 + 2๐๐ 1 + 3๐๐ + โ 2๐๐ 1 + 2๐๐ 2 ๐ โ 5.9×109๐พ is the energy of the photon in electron rest mass units ๐2 ๐๐ ๐ 2 2 = 6.65246 × 10โ25 cm2 is the Thomson cross-section The Klein-Nishina Cross-Section ๐KN ๐ = 3 2 1 + ๐๐ 1 + ๐๐ โn 1 + 2๐๐ ๐๐ [ โ 4 1 + 2๐๐ 2๐๐ ๐๐2 In this region, the cross-section essentially is the classical one since photons have low energy. ๐ es = 0.2 1 + ๐ cm2 ๐โ1 โ 0.34cm2 ๐โ1 for solar abundance 104 < ๐ < 107 Compton Scattering region. + โn 1 + 2๐๐ 1 + 3๐๐ โ 2๐๐ 1 + 2๐๐ 2 The cross-section is very small, other forms of opacity dominate, On the other hand, if the electrons are very energetic, inverse Compton can also happen. 107 < ๐ < 1011 T โซ 1011 Absorption processes (Ch9.3.6.2) Free-Free Absorption โ an Introduction If a photon and an unbound electron collide near a positively charged ion, it is possible for the photon to be absorbed, rather than simply scattered. This process is called freeโfree absorption. The electronโs kinetic energy increases, and, when it eventually collides with another electron or ion, that extra energy will heat the plasma. Only much later may the inverse process (Bremsstrahlung emission), Section 9.4.1, or some other process, emit a photon again and convert that absorbed energy back into radiation. Photo-absorption and photo-emission, therefore, are treated as separate heating and cooling processes, rather than two parts of a single scattering. Bound-Free Absorption โ an Introduction A similar effect occurs if the electron is bound to a nucleus, but the incoming photon has enough energy to eject the electron from that nucleus and ionize it. The photon again is absorbed in the event, so this process is called boundโfree absorption. The inverse process, recombination emission, also occurs separately from boundโfree absorption, and need not involve the electron and ion that participated in the original ionization. The opacities Because free-free and bound-free are very important in stellar structure, their Rosseland means have been worked out and well known. โ โ โ 2 ๐โ bf ๐ ๐ ,ff = 7.36 × 1022 cm2 ๐โ1 ๐ + ๐ ๐๐ โ7 2 ๐ff ๐๐ Mainly dominated by H, and He ๐ ๐ ,bf = 8.68 × 1025 cm2 ๐โ1 ๐ ๐ ๐๐๐ โ7 ๐๐ Mainly dominated by heavy elements โ โ ๐ff and ๐bf are called the Gaunt factors which are generally a factor of unity. f(T) is the fraction of heavy elements that are not ionized ๐ ๐ โ 0 ๐๐ ๐ โ โ Total Opacity Again, I am too lazy to type these equationsโฆ Radiative Heat Transport v.s. Thermal Conduction (Ch9.3.7) Now that we have discussed both thermal conduction and radiative heat transport, it would be interesting to see in what cases which dominate. By taking the ratio ๐๐ ๐๐ = ๐พ๐ ๐พ๐ = 4cT3 ๐๐ ๐๐ โ ๐๐ โ๐ ๐ ๐ ๐ Radiative heat transport dominated Thermal Conduction Dominated