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Date: 22-12-2015
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Sub-Eddington two-temperature accretion (ADAFs)
As in the previous section, we consider an axisymmetric steady-state disk accretion flow. However, the azimuthal velocity is no longer assumed to be given by the Keplerian velocity so that
(1.1)
Thus, the non-relativistic equation for the radial velocity now has an extra centrifugal force term:
(1.2)
For simplicity, we will deal only with the non-relativistic equations in this section. Again, vr is defined positive for inflow. In this model, we ignore the dynamical effects of the magnetic field. The shear viscosity, ν, but is now written as
(1.3)
where α is the r -independent α-viscosity parameter (Shakura and Sunyaev 1973) and now cs is the isothermal sound speed (c2s≡ P / ρ = 2kBT/mp). This viscosity causes a change in ω with r which follows from the vertically integrated form of the azimuthal component of the Euler equation:
(1.4)
Both equations assume that the flow is viscous and therefore reduce to unphysical results (ω ∝ r−2) in the limit of ν→ 0. Using the first law of thermodynamics, written as
(1.5)
where s is the specific entropy and assuming a constant γ and μ = 0.5, as
(1.6)
where again a prime (' ) denotes d/dr . Here, we ignore magnetic heating and instead assume that the heating is purely due to an anomalous shear viscosity so that
(1.7)
It is the ratio η ≡ Γ/Λ which determines the type of accretion flow: η 1 corresponds to the standard thin-disk solution, η1 corresponds to the flow discussed here, and η 1 corresponds to Bondi-type solutions. All three types may be present near Sgr A*. In this section, we are interested in solutions in which the viscous heating rate _ is much greater than the radiative cooling rate Λ; for the Keplerian model, Γ/Λ << 1 except very close to the outer edge of the flow. The η >> 1 solutions, it turns out, require sub-Eddington mass accretion rates and a two-temperature plasma with hot ions and cool electrons. These models are known as Advection Dominated Accretion Flows (ADAFs; see Narayan et al 1998) because the ions of the gas are heated by viscous dissipation and accretion occurs before the cooler electrons can radiate. Bondi–Hoyle-type solutions also advect most of their energy without radiating (thus the low radiative efficiency described earlier) so the term ADAF is slightly misleading.
If we further assume (Ichimaru 1977, Narayan and Yi 1994, Narayan et al 1998) that Λ → 0 and α is constant with r , we find a self-similar solution exists. That is, n, T , v, and ω can be given by simple power laws: ρ ∝ r−3/2, v ∝ r−1/2, T ∝ r−1, and ω ∝ r−3/2. In particular, if A ≡ ω/_ and the Keplerian velocity is
(1.8)
then (1.2), (1.4), and (1.6) become
(1.9)
(1.10)
and
(1.11)
Solving these three and taking the limit α << 1 gives:
(1.12)
(1.13)
(1.14)
and
(1.15)
These results are similar to the profiles derived for spherical Bondi-type accretion.
In most ADAF models, it is generally assumed that a perfectly tangled magnetic field exists with a magnetic pressure that is comparable to the thermal pressure. A purely tangled field results in an added specific internal energy density term of B2/8πρ and a radial pressure term of B2/24π; this pressure is assumed to be a constant fraction (1 − βB) of the total (thermal plus magnetic) pressure. Then we have
(1.16)
where Pth is the thermal component of the pressure and Ti and Te are the ion and electron temperatures (assumed equal at this point). For simplicity we assume a hydrogen mass fraction of one so that the mean molecular weights of both the ions or protons, μi, and electrons, μe are unity. The magnetic field changes the effective value of γ so that if it is assumed that the hydrodynamical adiabatic index, γad = 5/3, then
(1.17)
This treatment of the field is not fully accurate since even a tangled field will also result in some resistance to the shear viscosity. In addition, the Balbus–Hawley magneto-rotational instability (Balbus and Hawley 1991) will probably eventually result in a magnetic field with a dominant azimuthal component. In the following, βB = 0.5.
A problem with self-consistency arises at this point: do the profiles given by (1.12)-(1.15) actually result in Γ >> Λ? Substituting the self-similar results gives
(1.18)
Similarly, assuming cooling is due to thermal bremsstrahlung (Rybicki and Lightman 1979) gives (in cgs units):
(1.19)
Therefore we get:
(1.20)
We parametrize the mass accretion rate with f ≡ ˙M / ˙ME, so that ˙M = f × 1024M6 g s−1, where ˙ME is the accretion rate that, with 10% efficiency, produces the Eddington-limited luminosity, and M6 is the mass of the black hole in units of 106Mּ. Then in order to have an ADAF solution, we must have
(1.21)
where here ro is the outer boundary of the flow in units of rs. If one includes magnetic bremsstrahlung and Compton cooling, f must be even smaller than the limit given in (1.21). Therefore, it is unlikely that large-scale single temperature ADAFs exist except in rare circumstances where the accretion rate is particularly low.
The usual way around this difficulty is to assume that the cooling is not given by (1.19) because the temperature of the radiating electrons, Te, is not the same as that of the bulk of the gas. It is argued that viscous heating applies primarily to the massive ions and the Coulomb coupling between the hot ions and cool electrons is sufficiently weak to result in a so-called ‘two-temperature plasma’. Ideally, such a problem should be solved using a full two-fluid approach, but for simplicity we shall just assume that Λ → 0 for the ions and solve explicitly for the ion and electron temperatures. Thus, instead of (10.93), Ti is given by (Esin et al 1997)
(1.22)
and
(1.23)
where
(1.24)
After some manipulation, it can be shown that
(1.25)
The heating of the electrons due to Coulomb coupling with the ions, Γie, is given by (Stepney and Guilbert 1983)
(1.26)
The Ki are the modified Bessel functions, σT is the Thomson cross-section, θe ≡ kBTe/mec2, θp ≡ kBTi/mpc2, and ζ ≡ (θe + θp)/θeθp. It is assumed in (1.26) that the Coulomb logarithm is 20. Since it is assumed that Ti > Te everywhere, there is no direct feedback from the electrons to the ions; since ρc2s≡ Ptot, there is indirect feedback. The profiles for the gas radial velocity, rotational frequency, and density are still given by (1.12), (1.14), and (1.15), respectively but with T = Ti. Note that in earlier work (Narayan and Yi 1995a)
Figure 1.1. Examples of electron temperature profiles for a two-temperature ADAF. Here, βB = 0.5, Λi = 0, α = 0.1, γad = 5/3, and ro = 105rs. The curves correspond to different values of log10( f ): −4 (full), −5 (dotted), −6 (dashed), −7 (dot–dashed), and −8 (chain dot–dashed).
the electrons were assumed to be in isentropic balance so that the left-hand side of (1.23) was set to zero.
The existence of a two-temperature ADAF solution now depends on the Coulomb coupling timescale compared to the accretion timescale. A detailed calculation shows that as long as f < 0.3α2, a two-temperature ADAF solution exists. Representive profiles for Te are shown in figure 1.1 with α = 0.1. For this plot, the assumption that α << 1 has been relaxed. Also, radiation pressure and the optical depth are assumed to be negligible. The cooling function for the electrons includes magnetic bremsstrahlung, thermal bremsstrahlung (both electron-electron and electron-ion), and line cooling but not inverse Compton scattering. For large accretion rates, the last could result in significant cooling within <10 rs (Mahadevan 1997). For comparison, the ion temperature goes as r−1 and Ti