Thin Discs, Thick Discs and Transition Zones

Accretion onto a compact object must occur through a disc when the material has some initial angular momentum. Thin discs and the thicker low radiative efficiency accretion flows (LRAFs) are solutions to this problem that have been widely studied and applied. This is an introduction to these accretion flows within the context of X-ray binaries and cataclysmic variables. Accretion describes the increase of the mass of an astrophysical object when matter is deposited onto it. The conversion of a fraction of the gravitational potential energy of the accreted material into radiation can dominate the emission from the system. Accretion is a major source of energy in a varied lot of astrophysical objects including interacting close binaries, active galactic nuclei and protostellar systems (the standard textbook is Frank et al. 2002). My focus is on close binaries in which a normal star transfers mass onto a black hole/neutron star (X-ray binaries, XRBs) or white dwarf (cataclysmic variables, CVs) via Roche-lobe overflow. Exhaustive reviews of their observational properties may be found in Lewin et al. (1995) and Warner (1995) respectively. In these systems, an accretion disc forms around the compact object due to angular momentum conservation. The detailed characteristics of the flow depend upon how matter dissipates its potential energy and gets rid of its angular momentum in the disc. This basically leads to two extremes: the “low radiative efficiency” accretion flows (LRAFs, which are geometrically thick) and the radiatively efficient, geometrically thin discs (hence the title). This is an introduction to these solutions. §1 groups the basic tools needed to present the solutions in §2 (thin discs) and §3 (thick discs); §4 is a brief discussion of the transition from one to the other. 1 The building blocks 1.1 Accretion efficiency, Eddington luminosity In Newtonian dynamics, the available gravitational energy from a small mass m moved from infinity to the surface R⋆ of the accretor is GM⋆m/R⋆. The energy 1 California Institute of Technology, MC 130-33, Pasadena, CA 91125 [email protected] c © EDP Sciences 2008 DOI: (will be inserted later) 2 Title : will be set by the publisher released can be a sizeable fraction η of the rest-mass energy mc when the accretor is a neutron star or black hole: assuming a thin disc and free fall at the innermost stable circular orbit then η ≈ 0.42 for a maximally rotating Kerr black hole (Novikov & Thorne 1973; see also Gammie 1999). The Eddington luminosity LE is the luminosity for which radiation pressure exactly balances the gravitational pull on the accreted material. Accretion is quenched above this luminosity (but see Begelman 2001; Shaviv 2001). For a spherical inflow of ionised hydrogen such that the radiation pressure is due to Thomson scattering on electrons and the dominant gravitational pull is on the protons, LE is: GM⋆mp R2 = LE 4πR2 σT c hence LE ≈ 10 38 (M⋆/M⊙) erg · s −1 (1.1) The luminosities of XRBs and CVs are generally lower than this limit (Lewin et al. 1995; Warner 1995). The Eddington mass accretion rate ṀE is defined as: ηṀEc 2 = LE i.e. ṀE ≈ 10 18 (M⋆/M⊙)(0.1/η) g s −1 (1.2) 1.2 Accretion flow temperature Assuming the gravitational energy is released into radiation at the surface of the compact object, the minimum temperature of the flow is that of the black body radiating the same luminosity. At the Eddington limit this gives: 4πR ⋆σT 4 bb = LE i.e. kTbb ≈ 1.5 keV (LE/10 erg s)(R⋆/10 km) −1/2 (1.3) Accreting black holes and neutron stars should radiate mainly in soft X-rays while white dwarfs (R⋆ ≈ 10 4 km) should radiate in UV. This is consistent with observations (Lewin et al. 1995; Warner 1995). On the other hand, the maximum temperature is obtained when all the gravitational potential energy is transformed into thermal energy e without radiation losses (adiabatic flow): e = 3 2 kTg μmH = GM⋆ R⋆ i.e. kTg ≈ 45 MeV (M⋆/M⊙)(R⋆/10 km) −1 (1.4) assuming ionised hydrogen (μ = 0.5). This is twice the virial temperature of bound particles in circular orbit at R⋆. Accretion onto a compact object can power the emission of gamma rays. 1.3 Disc formation Matter infalling onto the compact object will generally have some non-zero angular momentum. A particle with a ballistic trajectory will therefore have RΩ = (RΩ)t=0. At the radius of closest approach to the compact object 1/2(RΩ) 2 c = GM/Rc so that 2GMRc = (R Ω)0. Only a particle with very low angular momentum can directly hit the compact object (Rc < R⋆). This condition is not Guillaume Dubus: Thin discs, thick discs and transition zones 3 met in compact binaries where matter comes from an orbiting companion and the stream will go round the compact star and intersect itself. Subsequent shocks lead to the dispersion of energy and the stream settles onto a circular orbit with the initial angular momentum (see Lubow & Shu 1975 for details). A steady supply of matter with some specific initial angular momentum piles up in a ring at this circularisation radius. There is no accretion unless some matter can transfer its angular momentum to reach smaller orbits. Under such a process, an accretion disc forms extending down to the compact object. 1.4 Angular momentum transport One process by which particles may exchange angular momentum is viscosity (see e.g. Terquem 2001 for an introduction). In gas kinetic theory, molecular viscosity arises from the exchange of momentum across the surface of a fluid element. The resulting force is proportional to νmol ∼ λu where λ is the mean free path between collisions and u is the mean thermal speed of the particles. Assuming particles in a plasma interacting only via Coulomb forces leads to: λ ∼ 1 cm (T/10 K) (ρ/10 g cm) u ∼ 10 cm s (T/10 K) using typical accretion disc temperatures and densities (Frank et al. 2002). The viscosity coefficient is therefore νmol ∼ 10 6 cm s. Anticipating a little bit on result from the thin disc model, we can get an order-of-magnitude observational estimate for viscosity in accretion discs. If the eruptions of dwarf novae, a sub-class of CVs (see §2.8), are due to the accretion of matter coming from the outer regions of a disc and transported by viscosity then νdisc ∼ Rdvr where Rd is the outer disc radius and vr is the radial velocity of the infalling matter. vr ∼ Rd/τe where τe is the timescale of the eruption. The disc radius can be estimated using e.g. the circularisation radius (§1.3) and typically will be ∼ 10 cm. For an eruption lasting a day, νdisc ∼ 10 15 cm s i.e. νdisc ≫ νmol. Molecular viscosity is much too weak to account for accretion in discs. The process by which angular momentum is transported has been the subject of intense research with turbulent transport the prime suspect. Turbulent viscosity is often modelled using the Navier-Stokes formalism but with a coefficient νturb ∼ λturbuturb where the scale and speed are those of the turbulent eddies. These are dynamic properties of the fluid, not intrinsic as with molecular viscosity. In a disc, the size and speed of the eddies are likely to be limited by the scale-height H and sound speed cs: ν = αcsH (1.5) where α is a parameter < 1. This is the famous α parameterisation of disc viscosity first described in a landmark paper by Shakura & Sunyaev (1973). In the following I will use this parameterisation. 4 Title : will be set by the publisher One would rather want to derive ν from first principles. At present, only turbulence arising from the magneto-rotational instability (MRI) has succeeded in predicting any significant viscosity (see Balbus & Hawley 1998 for a review). Fortunately, there is some sense in modelling the angular momentum transport and energy dissipation of MHD turbulence using the α parameterisation (Balbus & Papaloizou 1999). However, the mechanism may not work in weakly ionised flows and this is a problem for models with a cold accretion disc (Stepinski et al. 1993; Gammie & Menou 1998). Putative global hydrodynamical instabilities might take over in these conditions. There are other possibilities for angular momentum transport including instabilities in a self-gravitating disc (e.g. Balbus & Papaloizou 1999 and references therein), spiral waves excited by tidal torques (Spruit 1987), hydromagnetic winds launched from the disc (Blandford & Payne 1982), radiative viscosity (e.g. Loeb & Laor 1992) etc. However, these only apply in specific conditions. 1.5 Vertically integrated disc equations The disc equations derive from the equations of fluid dynamics combined with a model for viscosity (which contains the turbulent magnetic field contributions in the MRI case), an equation of state for the gas and a description of the radiative processes. A standard set of assumptions to start with is: • Axisymetry so that ∂/∂φ = 0 in cylindrical coordinates. • The only non-zero component of the stress tensor is the azimuthal shear τrφ. • The gas is perfect so the internal energy per unit mass is e = cvT and the gas pressure P = ρe(γ − 1) with γ = cp/cv. Radiation pressure is neglected here for simplicity. In addition I suppose the disc self-gravity is negligible (this assumption must be dropped for protostellar discs and AGNs) and that relativistic corrections are negligible (which is fine when more than a few gravitational radii away from the compact object). Assuming hydrostatic balance so that vz = 0, the vertical momentum conservation is (P = ρcs): ∂P ∂z = −ρgz i.e. ∂ lnP ∂ ln z = − Ω2K cs z [ 1 + z R2 − 3 2 (1.6) If the height H from the midplane of the disc is ∼< R high order terms in z/R can be neglected. This equation can be integrated analytically for a perfect gas yielding P (z) and ρ(z). Averaging P over z gives a relationship between H and the mid-plane sound speed which, to a factor of order unity, is : H = cs/ΩK (1.7) The scale-height cs/ΩK appears in Eq. 1.6. The detailed vertical balance can be bypassed by assuming the disc height is given by this relationship. The ShakuraSunyaev prescription for the viscosity becomes ν = αc2s/ΩK (1.8) Guillaume Dubus: Thin discs, thick discs and transition zones 5 Note that ν is effectively integrated over z. With a Navier-Stokes formulation of the viscosity, the integrated stress is: τrφ = νΣR ∂Ω ∂R where Σ = 2ρoH is the column density and ρo is the mean density. The radial evolution equations are then integrated and decoupled from z, resulting in a set of time-dependent 1D equations: ∂Σ ∂t + 1 R ∂ ∂R (ΣRvr) = 0 (1.9) ∂vr ∂t + vr ∂vr ∂R = RΩ −RΩ2K − 1 ρo ∂P ∂R (1.10) ∂RΩ ∂t + vr ∂RΩ ∂R = 1 ΣR ∂ ∂R ( Rτrφ )