2. Accretion as a Source of Energy PhD Course, University of Padua - - PowerPoint PPT Presentation

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

• L. Zampieri
• 2. Accretion as a Source of Energy

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

• L. Zampieri

High Energy and Time Resolution Astrophysics

High Energy and Time Resolution Astrophysics deals mostly with the Astrophysics of Neutron Stars (NSs) and Black Holes (BHs), and their environment. High Energy (HE) emission and rapid time variability phenomena are “boosted” when compact objects reside in binary systems and interact with their companion stars because of mass transfer.

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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Preliminaries

Motion of a test particle of mass m in the gravitational field of a compact object. Order of magnitude estimate of the energy that can be extracted:

∆E/mc2 = GMc/c2rc = 0.15(Mc/M⊙)(rc/106 cm)−1 = ⇒ compactness Mc/rc

(1) Hamilton-Jacobi equation (S = Ldt, L action of a single particle):

gij ∂S ∂xi ∂S ∂xj + m2c2 = 0

(2) In the Schwarzschild metric (ǫ = E/mc2, h = L/m, S = −Et + Lφ + Sr(r), θ = π/2):

˙ r2 c2 + Veff = ǫ2

(3)

˙ r = pr m = 1 mgrr

∂Sr

∂r

• (4)

Veff =

• 1 − rS

r 1 + h2 r2c2

• rS = 2GM/c2

(5)

Veff = h2/2r2c2 − GM/rc2 in the Newtonian case

(6)

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Maximum energy extraction: η = Veff(∞)1/2 − Veff(rISCO)1/2 = 1 − 2

√ 2 3

= 0.057

where rISCO = 3rS is the innermost stable circular orbit (ISCO) and hISCO =

√ 3 rS · c.

In the Kerr metric: η = 0.42 Accretion luminosity ( ˙

M accretion rate, mass accreted per unit time): Lacc = η ˙ Mc2 ≃ 1038η0.1 ˙ M−8 erg/s

(7) If this energy is radiated without being thermalized (without that radiation has reached thermodynamic equilibrium with the accreting matter):

GMmp/rc ∼ kT = ⇒ T ∼ 1012 K ∼ 100 MeV

(8) If it is radiated as blackbody radiation (matter optically thick):

Lacc = 4πr2

cσT 4

= ⇒ T ∼ 107 K ∼ 1 keV

(9) For a steadily radiating source and assuming spherical symmetry, the maximum luminosity above which accretion quenches can be obtained setting the radiative force (Frad = σTL/4πr2) equal to the gravi- tational force (Fg = GMmp/r2) acting on a electron-proton pair:

LEdd = 1.3 × 1038M1 erg/s Eddington luminosity

(10)

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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Wind and Roche-lobe fed binary systems

WIND ACCRETION

• Stellar wind from early type companions: vW ∼ 1000 km s−1 >> vs
• Compact object orbital velocity (Kepler’s law): vc ∼ 200M 1/3

∗,1 (1 + Mc/M∗)1/3P −1/3 day

km s−1

• Wind velocity relative to the compact object: vrel ∼ (v2

W + v2 c)1/2 ≈ vW

• Radius at which the gravitational energy equals a particle kinetic energy (accretion radius):

racc = 2GM/v2

rel ≃ 2GM/v2 W

(11)

• Fraction of stellar wind captured by the compact object:

f = πr2

acc

4πa2 ≃ G2M 2

c

v4

Wa2 = 2.1 × 10−3M 4/3 c,1 v−4 W,1000P −4/3 day

(12) where a = (G/4π2)1/3M 1/3

c

P 2/3.

• Luminosity produced by wind accretion (assuming matter has sufficient angular momentum to form

a disk):

˙ MW = f ˙ M∗ = 10−8f−3 ˙ M∗,−5 M⊙/yr

(13)

Lacc,W = 1038η0.1 ˙ MW,−8 erg/s

(14) Often ˙

M∗ < 10−5 M⊙/yr and/or matter does not form a disk, so that η < 0.1. Therefore, wind

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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accretion is inefficient. Only thanks to the high mass loss rates of early type stars, can sources powered in this way be luminous.

• Does accreting matter have enough angular momentum to form a disk? The accretion stream rotates

around the compact object with angular velocity ωorb = vc/a = 2π/P :

lW ∼ racc × (ωorbracc) = 2πr2

acc/P

(15) The angular momentum is low enough that matter accretes directly on the compact object. However, the donor may be very close to filling its Roche lobe (see below) when the compact object is at periastron, allowing the formation of an accretion stream or even a transient accretion disc (e.g. Negueruela 2010). ROCHE-LOBE OVERFLOW Motion of a test particle in the gravitational potential of two massive bodies (restricted three-body problem) in circular orbit about each other. Euler equation in the reference frame corotating with the binary governed by the Roche potential:

ΦR = − GMc |r − rc| − GM∗ |r − r∗| − 1 2(ω × r)2

(16)

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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Figure 1: From Frank, King & Rayne (2002, Accretion Power in Astrophysics; left) and https://en.wikipedia.org/wiki/Roche−lobe/media/File:RochePotential−color.PNG (right). If at a certain stage of its evolution the companion star (M2 = M∗) swells up so that its surface reaches contact with its Roche lobe R2, any small perturbation will push material over the saddle point

L1 (inner Lagrange point) of ΦR where it is eventually captured by the compact object. (− ˙ M∗) M∗ = 1 5/3 − 2q

˙

R2 R2 − 2 ˙ Jtot Jtot

• (17)

where q = M∗/Mc. If ˙

Mtot = 0 and ˙ Jtot = 0 the mass transfer is conservative.

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Mass transfer from the donor takes place if ˙

M∗ < 0 (− ˙ M∗ > 0).

If q < 5/6, this condition implies one or both the following possibilities:

˙ R2 R2 > 0

(18)

˙ Jtot Jtot < 0 for e.g. gravitational radiation, magneto − hydro wind

(19) In the first case the Roche lobe radius R2 must expand. So, to sustain a stable mass transfer, also the radius of the donor r∗ must expand. This occurs when the companion is steadily burning fuel in the core (e.g. during Main Sequence or when it is burning He in the core during the giant phase):

˙ R2 R2 ∼ ˙ r∗ r∗ ≈ 1 tnuc

(20)

− ˙ M∗ = M∗ (5/3 − 2q)tnuc ∼ 10−8M∗,1t−1

nuc,50 M⊙/yr

(21) where tnuc is the hydrogen nuclear burning timescale. Faster mass transfer on a thermal timescale can

• ccur during certain evolutionary phases of the donor.

If q > 5/6 (and ˙

Mtot = 0, ˙ Jtot = 0), mass transfer takes place if ˙ R2/R2 < 0. The Roche lobe R2 shrinks down on the donor. Unless the star contracts rapidly, the overflow proceeds on a dynamical

• r thermal timescale, depending on whether the star’s envelope is convective or radiative. This may

eventually lead to a rapid and often violent evolution (unstable mass transfer) that may end up in a

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common envelope phase. Specific angular momentum of the accreting matter for Roche lobe overflow:

lR ∼ R1 × (ωorbR1) = 2πR2

1/P >> lW

(22) Radius R0 where lR is equal to the Keplerian angular momentum lK(R0) (circularization radius):

R0 ≈ 1P 2/3

dayR⊙ >> rc

= ⇒ formation of a disk

(23)

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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Figure 2: From Frank, King & Rayne (2002, Accretion Power in Astrophysics) If there is differential rotation, because of thermal or turbulent motion stresses are generated between neighbouring radii. The resulting effect is a net ’viscous’ torque (µ is the linear mass density, see below):

G = νµr2(∂Ω/∂r)

(24)

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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Molecular viscosity alone is far too weak, as are other mixing mechanisms such as convection and tidal mixing. It was concluded that turbulent mixing was needed to generate the observed degree of viscosity. However, in the absence of magnetic fields no instabilities could be found that would drive such

• turbulence. A breakthrough came with the discovery that the driving mechanism is the magnetorotational

instability in a weekly magnetized plasma (MRI; Balbus & Hawley 1991). Figure 3: From http://mri.pppl.gov/ and https://ay201b.wordpress.com/2011/04/11/the-magnetorotational-instability/

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a) Magnetically connected fluid elements have some initial displacement b) Differential rotation increases the displacement, and magnetic tension causes the inner parcel to slow down, and the outer parcel to speed up c) This transfer of angular momentum causes the inner parcel to migrate inwards and the outer parcel to be pushed outwards. d) Both parcels are not at the Keplerian radius corresponding to their velocity. The inner parcel is too slow and has then to fall inward gaining velocity. The outer parcel is too fast and has then to move

• utward losing velocity.

e) The displacement continues to increase and so does the transfer of angular momentum. If the magnetic field is too strong (i.e. the tension in the spring is too strong), the feedback cycle will not run. The tension will instead cause the displacement between fluid parcels to oscillate rather than continue to grow.

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Local structure of thin disks (standard Shakura-Sunyaev model)

Basic Disk Equations

• Vertical integration:

h

0 A(z)dz = hA(η) ≈ hA(0/h)

• |vz| ≪ |vr|
• hydrostatic equilibrium in the vertical direction and axial symmetry

∂µ ∂t + B5 ∂(vrµ) ∂r = 0 mass cons. (integrated)

(25)

∂vr ∂t + vr ∂vr ∂r − v2

φ

r − ∂Φ ∂r + 1 ρ ∂P ∂r = 0

• rad. mom.

(26)

∂(µl) ∂t + ∂(vrµl) ∂r − ∂G ∂r = 0

• ang. mom.

(27)

B1T

∂(µs)

∂t + ∂(vrµs) ∂r

• + G∂Ω

∂r −

h

0 s0rdz = 0

• ener. cons. (integrated)

(28) where

Φ = GM (r2 + z2)1/2 gravitational potential

(29)

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Φpn = GM (r2 + z2)1/2 − rS pseudo − newtonian grav. pot. (Paczynski & Wiita 1980) (30) µ = hρr = Σr linear mass dens. (Σ = hρ surface mass dens.), l specific ang. mom. (31) ΩK = (∂Φ/∂r|z=0)/r = (GM/r3)1/2 Keplerian ang. velocity

(32)

G = νµr2(∂Ω/∂r) viscous torque, l specific entropy

(33)

s0 = ∂F/∂z − → net radiative cooling (cooling − heating)

(34) Hydrostatic equilibrium in the vertical direction:

1 ρ ∂P ∂z − ∂Φ ∂z = 1 ρ ∂P ∂z + GMz (r2 + z2)3/2 = 0

(35)

B4P = GMρh2 r3 = ⇒ B4 P ρ = GM r3 h2 = Ω2

Kh2 =

⇒ h r = (B4P/ρ)1/2 ΩKr ≈ vs vK

(36)

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High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion

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Local Structure of Newtonian Steady Thin Disks

• v = −vr −

→ positive inward motion

• stationarity −

→ ∂/∂t = 0

• small radial velocity −

→ |v| ≪ |vφ| = Ωr (radial drift)

• disk geometrically thin −

→ h ≪ r = ⇒ vs ≪ vK

For a steady thin disk, local Kepler velocity highly supersonic

• no energy advected −

→ T∂(vµs)/∂r ≪ GΩ

′

∂(vµ) ∂r = 0 = ⇒ 4πvµ = ˙ M

(37)

v∂v ∂r + (Ω2

K − Ω2)r + 1

ρ ∂P ∂r = 0 = ⇒ v2 r + (Ω2

K − Ω2)r + v2 s

r ≈ 0 = ⇒ Ω ≈ ΩK

(38)

∂(vµl) ∂r + ∂G ∂r = 0 = ⇒ ˙ M 4πl = −νµr2Ω

′ + C =

⇒ νµ = − ˙ M 4πr2Ω

′ (l − lin) (39)

−B1T ∂(vµs) ∂r + G∂Ω ∂r − rF(h) = 0 = ⇒ rF(h) = νµr2(Ω

′)2

= ⇒ 4πrF(h) = − ˙ MΩ

′(l − lin)

(40)

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If the electron plasma is in thermal equilibrium and optically thick to photon emission-absorption (τeff =

• τabs(τes + τabs) > 1):

F(z) = − c 3κRρ ∂aT 4 ∂z

(41)

κR is the Rosseland mean opacity. Integrating in the vertical direction: F(h) = B3 acT 4 3κRρh

(42) Equations (Ω = ΩK)

h2Ω2

K = B4

P ρ

(43)

4πvΣr = ˙ M

(44)

νΣ = ˙ M 6π

• 1 −

rin

r

1/2 (45)

B3 acT 4 3κRΣ = 3GM ˙ M 8πr3

• 1 −

rin

r

1/2 (46)

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Shakura-Sunyaev disks (standard or α disks; Shakura & Sunyaev 1973)

• Newtonian steady thin disks (Keplerian: Ω = ΩK)
• α-viscosity prescription: tφr = αP =

⇒ ν = αvsh

• Free-free or electron-scattering opacity:

κR = κff = 3.7 × 1022(X + Y )(1 + X)ρT −3.5gff cm2 g−1 κR = κes = 0.2(1 + X) cm2 g−1

• Gas or radiation pressure dominated: P = Pgas = ρKT/¯

µmp or P = Prad = aT 4/3

With these assumptions the equations for Newtonian steady thin disks can be made explicit. Solution expressed in terms of 3 fundamental parameters: M, ˙

M, α.

Different solutions according to the dominant pressure and opacity source Outer free-free and gas pressure dom. region: κff > κes, Pgas > Prad Intermediate electron-scattering and gas pressure dom. region: κff < κes, Pgas > Prad Inner electron-scattering and radiation pressure dom. region: κff < κes, Pgas < Prad

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Shakura-Sunyaev disk equations in the free-free and gas pressure dominated region Figure 4: From Frank, King & Rayne (2002, Accretion Power in Astrophysics)

→ Relativistic standard thin disk equations derived by Novikov & Thorne (1973) and Page & Thorne

(1974)

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Shakura-Sunyaev disks

= ⇒ = ⇒ = ⇒

Figure 5: From Frank, King & Rayne (2002, Accretion Power in Astrophysics)

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Disks with advection (slim disks; ADAFs)

Steady, vertically integrated equations of a disk that may not be geometrically thin and in which ad- vection of energy may become important.

h2Ω2

K = B4

P ρ

(47)

4πvµ = ˙ M

(48)

v∂v ∂r + (Ω2

K − Ω2)r + 1

ρ ∂P ∂r = 0

(49)

νµ = − ˙ M 4πr2Ω

′ (l − lin)

(50)

4πrF(h) = − ˙ MΩ

′(l − lin) − B1 ˙

MT ∂s ∂r F(h) = B3 acT 4 3κRρh

(51) Slim disk solutions (Abramowicz, Czerny, Lasota & Szuszkiewicz 1988)

• Inner boundary condition: no viscous torque at the horizon
• Outer boundary condition: Keplerian motion at large r

− → Treatment of the transonic part of the flow

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Self-similar solutions for advection-dominated disks (Ichimaru 1977; Narayan & Yi 1994)

f = B1 Ts

′

(l − lin)Ω

′

advected/dissipated energy

(52) For an advection-dominated flow, f ≃ 1 throughout the flow. Taking the limit α2 ≪ 1 and a polytropic

• eq. of state

h2 r2 ≃ 2 3

γ − 1

γ − 5/9

• B4

= ⇒ disk geometrically thick

(53)

Ω ≃

2

3

5/3 − γ

γ − 5/9

1/2

ΩK = ⇒ sub − Keplerian rotation

(54)

v ≃ α γ − 1 γ − 5/9

GM

r

1/2

= ⇒ significant radial velocity

(55)

• Positive Bernoulli constant −

→ a fraction of the gas can escape to infinity (winds, bipolar flows, jets)

Assuming a varying ˙

M with r − → adiabatic inflow-outflow solutions (ADIOS; Blandford & Begelman

1999)

• Entropy increases with decreasing r −

→ ADAF solutions convectively unstable (CDAFs; Igumen-

shev et al. 1996; Igumenshev & Abramowicz 1999, 2000)

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2D magneto-hydro simulations of disks with advection at high ˙

M

Figure 6: From Oshuga and Mineshige (2011, ApJ) 2D magneto-hydro simulations show an advection-dominated disc and an outflow region, with power- ful clumpy winds driven by radiation pressure (e.g. Ohsuga and Mineshige 2011, Takeuchi et al. 2013, 2014).

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Optically thick, cold disks: emitted spectrum

Gas physical conditions determine the properties of the emitted radiation. Outer region of a standard SS disk:

τabs ≈ 33α−4/5 ˙ M 1/5

16

T ≈ 4 × 105α−1/5 ˙ M 3/10

16

M 1/4

1

R−3/4

8

K Low temperature radiation in thermal equilibrium −

→ local Blackbody radiation at the effective tem-

perature

Teff = (F/σ)1/4

(56) Frequency (energy) distribution of blackbody radiation at temperature T

Bν(T) =

2hν3

c2

• 1

ehν/kT − 1

(57) If there is no advection (all viscously dissipated energy radiated locally):

4πrF(h) = − ˙ MΩ

′(l − lin)

− → Teff =

3GM ˙

M 8πr3σ

• 1 −

rin

r

1/21/4 (58)

≃ 107 ˙ M 1/4

17 M 1/4 1

R−3/4

in,6 (r/rin)−3/4K

r ≫ rin

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Local blackbody spectrum?

• Not quite −

→ neglects radiative transfer effects in the non-isothermal disk atmophere (frequency-

dependence of the optical depth, line opacity, scattering)

• But it provides a reasonable approximation in many cases

Flux from a disk face-on Fν (cos iFν at inclination i)

Fν =

• Iν cos θdΩs = 2π
• Iν sin θ cos θdθ = 2π

D2

• Iνrdr

(r = D sin θ)

(59) Disk spectrum for local blackbody radiation (independent of viscosity)

Fν = 4πhν3 c2D2

rout

rin

rdr ehν/kTeff(r) − 1

(60)

ν ≪ kTeff(rout)/h kTeff(rout)/h ≪ ν ≪ kTeff,max/h ν ≫ kTeff,max/h Fν ∝ ν2 Fν ∝ ν1/3 x5/3/(ex − 1) ∝ ν1/3 Fν ∝ ν3e−hν/kTeff,max x = hν/kTeff,max(r/rin)3/4 → Relativistic disk spectrum must include effects of Doppler shift and of photon propagation in a curved

spacetime (e.g. Cunningham 1975)

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DISK BLACKBODY SPECTRUM Figure 7: Standard and relativistic disk spectra for M = 10M⊙, ˙

M/ ˙ MEdd = 0.1 and cos i = 1. The

Kerr black hole has angular momentum a = 0.9981.

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Optically thin, hot disks and coronae: emitted spectrum

Inner region of a standard SS disk:

τeff ≈ 2 × 10−3α−17/16M 31/16

3

˙ M −2

17

τes = 8α−1M3 ˙ M −1

17

(61)

T ≈ 4 × 107α−1/4M −1/4

3

K

(62) High temperature radiation in an effectively optically thin medium −

→ thermal Comptonization be-

comes important. Similar physical conditions (high temperature, effectively optical thickness < 1) occur in the so called corona, a hot cloud of plasma close to the centre of the accretion disc that is believed to account for the primary power-law, X-ray emission from luminous accreting black holes (e.g. Fabian et al. 2017). The following quantity plays a crucial role when dealing with Comptonization:

y = 4kT mec2Ns Compton parameter

(63) where Ns = max(τes, τ 2

es) = ρkesct is the mean number of scattering in a medium of optical thickness

τes (1/ρkesc is the mean time between scatterings). y ∼ 1 distinguishes between the regime of unsaturated (y < 1) and saturated (y > 1) thermal

Comptonization.

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COMPTONIZATION Thermal Comptonization is described by a non-relativistic diffusion equation for the photon number density n (Kompaneets 1956):

∂n ∂y = 1 x2 ∂ ∂xx4

• n + n2 + ∂n

∂x

• ,

(64) where x = hν/kT . This equation derives from the Boltzmann equation for n in the assumption that

hv << mec2 (soft photons), kT << mec2 (electrons are non-relativistic) and the photon energy

transfer per scattering is small (compared to the electron kinetic energy). It is a second order expansion

• f the Boltzmann equation in the photon energy transfer per scattering (Fokker-Planck diffusion equation).

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SATURATED COMPTONIZATION: SPECTRUM For y > 1, the photon number tends to an equilibrium distribution:

n = 1/(eα+x − 1) Bose − Einstein distribution ,

(65) which is the stationary solution of the Kompaneets equation. When α >> 1 (small photon occupation number), the distribution tends to a Wien tail: n ∝ e−x Figure 8: Bose-Einstein distributions for different values of α.

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UNSATURATED COMPTONIZATION: SPECTRUM For y < 1, an approximate solution of the Kompaneets equation in a finite medium assuming an input

• f soft photons (hv << kT ) is:

Iν ∝ ν3+m m = −3/2 ±

• 9/4 + 4/y

(66) Figure 9: From Rybicki & Lightman (1979, Radiative Processes in Astrophysics)

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