The two-phase model

This model, first proposed by Haardt & Maraschi (1991a, 1993), is a fundamental tool for understanding the AGN X-ray emission. Essentially, if we neglect the obscuring torus and the line regions, AGN are left with the hot corona (optically thin geometrically thick), and a geometrically thin and optically thick accretion disc emitting in the optical-UV band.

The two-phase model couples these two phases: the cold phase (i.e. the disc) provides the low temperature photons for Comptonisation in the hot phase (i.e. the corona), and part of the hard, Comptonised emission from the corona contributes to the heating of the cold phase. The basic model accounts for a plane parallel geometry where the two phases are homogeneous and isothermal layers, while further updates have been proposed for the case of a non-uniform and patchy corona (e.g. Haardt et al. 1994). Let us now focus on this model from a quantitative point of view. We define PG as the total

gravitational power and f its fraction that is dissipated in order to heat the corona. The

"heating" power is then re-emitted as the result of Comptonisation, thus we can define a Compton luminosity Lheating=fPG, while what remains of the gravitational power (1-f) is dissipated in the accretion disc. Then introducing the amplification factor A due to Comptonisation, we can derive the total luminosity provided by the corona as Ls×A, with Ls the seed luminosity. Concerning the amplification factor A, it can be evaluated by energetics of the disc/corona system and geometrical considerations (see Petrucci et al. 2013a). Comptonisation conserves the number of photons, thus there must be a relationship between the luminosity of the seed photons entering and cooling a corona and the observed Comptonised luminosity. Then, by conservation, the number of observed photons nobs has to be the same as the sum of ns,0 and nC,up where these quantities account for the number of seed photons crossing the corona without being scattered and the number of photons which are Comptonised and up-scattered, respectively. Then, if the total number of soft photons is nsand the opacity of the corona is τ we can formulate:

n_s,0= n_se^−τ (3.4)

n_C,up = 1

2n_s(1 − e^−τ) . (3.5)

The factor 1/2 is used for isotropic media where half of the power is emitted upward , while the remaining half downward. Continuing our reasoning we have:

n_obs = n_s,0+ n_C,up = 1

2n_s(1 − e^−τ) , (3.6)

so we compute ns as:

n_s = 2n_obs

1 + e^−τ . (3.7)

At this point we can consider different cases in accordance with the values of τ: if τ«1 then ns ∼ nobs and only a small percentage of photons are Comptonised; if τ»1 then ns ∼ 2nobs. Consider now the total power emitted by the corona Ltot as the sum of the seed luminosity Ls and the heating power Lh, i.e.:

L_tot = L_s+ L_h . (3.8)

The two components constituting the total coronal luminosity can be viewed as the sum of the upward and downward luminosities. The sum of the Comptonised and up-scattered (down-scattered) photons and the seed photons going upward (downward) without being scattered is Ls,up (Ls,do), therefore:

L_h = L_h,up+ L_h,do= 2L_h,up (3.9) L_a= L_s,up+ L_s,do=



L_se^−τ +1

2L_s(1 − e^−τ)

 +1

2L_s(1 − e^−τ) . (3.10)

The first equation comes from the assumption of isotropy of the Comptonisation mech-anism. Now, the observed luminosity is obtained summing up the upward (downward) luminosities e.g.:

L_obs = L_s,u+ L_h,u . (3.11)

Adopting the above formulae we can rewrite the luminosity Lh in accordance with:

L_h = 2L_obs− L_s(1 − e^−τ) , (3.12) that using Eq. 3.6 becomes:

Ltot = 2Lobs− Lse^−τ . (3.13)

Again we can have two regimes: for τ«1, Ltot '2Lobs-Ls, while when τ»1 then Ltot '2Lobs. At this point, we are ready to write down the amplification factor as:

A = L_tot

L_s = L_h+ L_s

L_s . (3.14)

The radiative equilibrium of the seed source (i.e. the disc) allows us to derive a further expression for Ls. To do this, let us introduce the intrinsic disc emission Ls,intr:

L_s= L_s,intr+ L_s,do+ L_h,do= L_s,intr+ 1

2L_s(1 − e^−τ) + 1

2L_h . (3.15) Dividing this last equation by Ls we get:

1 = L_s,intr L_s +1

2− e^−τ

2 + L_h

2Ls , (3.16)

that can be re-written as:

L_h

L_s = 1 + e^−τ − 2L_s,intr

L_s . (3.17)

Finally we can derive a further expression of the amplification factor:

A = 1 + L_h Ls

= 2 + e^−τ − 2L_s,intr Ls

. (3.18)

In the case of a passive disc (i.e. Ls,intr=0) all the gravitational power is dissipated in the Comptonising medium, thus f=1. In other words, there is not any radiation from the disc that, instead, reprocesses and re-emits the Comptonised photons emerging from the corona. Typically, an optically thin corona above a passive disc is characterized by A=3, while A=2 for an optically thick corona.

We can go further in our reasoning by introducing η as the fraction of the Comp-tonised power emitted by the corona. This parameter can be seen as an estimator of the anisotropy of the Compton scatter (η=1/2 means a fully isotropic scenario). Let us

consider a further parameter, a accounting for disc albedo. Then the radiative balance between the disc and the optically thin corona leads to the following equation for A (Haardt & Maraschi 1991b):

A = 1 + f

1 − f [1 − (1 − a)η] , (3.19)

where (1 − a) accounts for the fraction of Comptonised radiation that is absorbed by the disc. In the limiting case of a=0 and f=1, and assuming an isotropic emission, we find A=3.From observations we know that AGN has typical power-law like spectral shapes:

I(E) = I(E₀) E E₀

−α

. (3.20)

If we assume that the seed luminosity is black body-like (with the corresponding peak at

∼ 3kBT), we can derive an approximated expression for Ls=I(E0)E0, while the heating Comptonised luminosity is:

L_h =

Emax

Z

Emin

I(E)dE . (3.21)

The integral is performed from Emin ∼E1 that is the energy of the photons after just one scatter to Emax ∼ 3k_BT with T is the coronal temperature. Then, assuming that A1 is the average gain in a single scattering process (E1=E0A1), we find an equation describing the heating/cooling ratio depending now on the spectral shape (Haardt &

Maraschi 1991b):

A − 1 ' 1 1 − α

 3k_BT

E₀ − (A₁)^1−α



. (3.22)

As already seen, the spectral index α is a function of both the corona opacity and temperature, and, in particular, α = −lnτ/lnA1 for τ<1.

An iterative solution of Eq. 3.20 was found byHaardt & Maraschi(1991b). According to this solution, for f=1, α always varies between 1.1 and 1.4 (corresponding to Γ between 2.1 and 2.4) for a large range of coronal temperatures and optical depths. This means that the heating/cooling ratio is kept constant by adjusting the values of the corona opacity and temperature and this leads to a constant amplification factor A that only depends on the assumed geometry. An important work which confirmed this behaviour via simulations is provided byBeloborodov(1999). The author used the Comptonisation code byCoppi(1992) and found empirical relationships between the photon index Γ and the Compton parameter and the amplification factor:

Γ ∼ 9

4y^−2/9 (3.23)

Γ ∼ 7

3A^−0.1 . (3.24)

These relationships yield similar results to the theoretical prescriptions by Eq. 2.44 . The two-phase model was further developed for different geometries of the corona. In particular, a patchy corona was considered according to the formation of magnetic loops storing the energy that, at that point, is not dissipated via reconnection. In this model, the blob-like hot component provides only a fraction of the accretion power, and the disc is no longer passive and can supply most of the optical-UV luminosity. In this new scenario the patchy corona does not illuminate uniformly the disc that reprocesses the radiation only locally, in hot-spots.

According to this model, and particularly when the X-ray and UV luminosities are similar (e.g.Haardt et al. 1994,1997), a correlation between the hard X-ray Comptonised emission and the seed reprocessed photons is expected. Therefore, variability in this band can be explained thanks to the stochastic variations in the hot spots number or via different accretion rate states.