5 Proton illumination of disks with moderate optical depth

Figure 4: A "warm disk'', of optical depth 1.5, externally heated by ion illumination. Upper left panel: emergent model spectrum; upper right panel: heating rates from proton heating (solid line) and combined radiative cooling rates from Comptonization, bremsstrahlung and pair processes (squares); lower left panel: electron temperature $T_{\rm e}$; lower right panel: proton number density (solid line) of the layer and pair number density (dashed line).

Figure 5: Evolution from a cool disk to a warm disk by exposure to virialized protons at r=10. Initial temperature $T_{\rm e}=1$ keV The numbers at the lines denote the sequential stages, referred to in the text. The final stage [6] shows an equilibrium at $T_{\rm e} \approx 400$ keV. The dashed line shows the temperature profile after one dynamical time-scale.

Figure 6: Equilibrium temperature profiles of warm disks exposed to virialized protons at r=10 (solid line) and r=15 (dashed line) with different initial optical depths $\tau _{1/2}=2,1.5,1.$ for r=10 and $\tau _{1/2}=1.5,1$ for r=15.

The temporal evolution of an initially cool ( $T_{\rm e}=1$ keV), thin disk with an optical depth $\tau_{1/2}=1.5$ (measured from the surface to the mid-plane) at r=10 for the galactic black hole candidate (BHC) case is shown in Fig. 5. The energy flux from the virialized protons is $F_{\rm p} = 8.1\times 10^{21}$ erg cm^-2 s^-1. In the beginning the temperature of the top layers increases due to the impinging hot protons, whereas the mid-plane region cools due to bremsstrahlung (stage 0-2, Fig. 5). As the top layers are heated, the stopping power of the plasma decreases and the protons penetrate deeper into the disk. Eventually hot protons reach the mid-plane region and proton heating overcomes bremsstrahlung cooling even there [3]. At this stage no effective cooling mechanism is present and the temperature continues to rise everywhere in the disk [4-5]. At temperatures $kT \gtrsim 200$ keV pair production becomes more and more important. The extra electrons serve as additional scattering partners for the Coulomb collisions with the protons and the photons of the radiation field. Thus pair production limits the maximum attainable temperatures, and the disk adjusts to a new equilibrium state [6] at a temperatures $T_{\rm e}\simeq 400$ keV. The dashed line in Fig. 5 shows the temperature profile after one dynamical time-scale at that radius, $t_{\rm d}=1/\Omega_{\rm K}\simeq 3.5\times10^{-3}$ s.

Figure 4 shows an overview of that solution. The pair number density z at the top of slab reaches $z\approx50$% of the proton number density and drops off at the mid-plane to $z\approx 1$%. The spectrum of such a thin proton heated disk peaks at $\approx$1000 keV. We refer to those disks as warm disks. They are still considerably cooler than the local virial temperature.

Figure 6 shows the dependence of the equilibrium temperature profile on the initial optical depth of the layer and the distance from the central object. With increasing distance the proton energy flux as well as the proton penetration depth decreases. A cool disk can therefore be transferred into the warm state only within a certain distance from the BH. If the hot protons do not reach the mid-plane anymore, a cool interior can be maintained which looses its energy very efficiently via bremsstrahlung, as the hot layers above are optically thin. Our model computations show that at r=15 a thin disk can maintain a cool interior for $\tau_{1/2}=1.5$ whereas at r=10 a disk with $\tau_{1/2}=1.5$ switches into the warm state.

The temperature of the warm state also depends on both the distance and the overall optical depth of the layer. For $\tau_{1/2}=1$ and r=10 our model predicts a temperature of $\approx$1 MeV. At such temperatures our classical proton heating formalism starts deviating from the correct relativistic expression. The classical treatment underestimates the proton-electron heating rates at high temperatures (Deufel et al. 2001). But further pair processes and radiative cooling terms should also be included (see below), which again limits the maximum temperatures.

The transition from a cool disk to the warm state also takes place in AGNs. For the above AGN parameters (see Sect. 4.1) we find the transition to occur for $\tau_{1/2} \lesssim 0.2$ at $r \lesssim 10$. The temperatures of the warm state are in the MeV range. At such temperatures ($\theta>1$) our treatment of the pair processes and the radiation field needs more scrutiny. Further pair production processes should be included ( $\gamma {\rm e} \rightarrow {\rm e} {\rm e}^{+}{\rm e}^{-}$, ${\rm ep} \rightarrow {\rm ep} {\rm e}^{+}{\rm e}^{-}$) as well as additional radiative cooling terms (bremsstrahlung from ${\rm e}^{+}{\rm e}^{-}$ and ${\rm e}^{\pm} {\rm e}^{\pm}$ collisions). These will further limit the maximum attainable temperatures. Therefore we do not think that temperatures of several MeV found in our simulation of the warm state in AGNs are realistic. The important result of our investigation is that the transition is not only restricted to the BHCs but also takes place in AGNs. But compared to the BHCs the transition in the AGNs occurs in a more narrow zone around the BH (in terms of Schwarzschild radii) and the vertical extend (in terms of Thomson optical depths) is smaller.