Supporting the Obscuring Torus by Radiation Pressure

Julian Krolik

Johns Hopkins University

Typical AGN Have Toroidal Obscuration

• Spectropolarimetry of “type 2” objects• Ionization cones• warm IR spectra• X-ray absorption

The Basic Problem

N(type 2) ~ N(type 1) implies ΔΩobscured ~ ΔΩunobscured

which in turn implies hobscured ~ r

But h/r ~ Δv/vorb , with vorb ~ 100 km/s

If Δv = cs , then T ~ 105 K >> Tsubl (dust)

If 105 K is too hot, what supports the obscuring matter?

Candidate Mechanisms

• Bouncing magnetized clouds stirred by orbital shear (K. & Begelman 1988, Beckert & Duschl 2004)

Clumping avoids immediate thermal destruction But is this degree of elasticity plausible? And their

collision rate must not be >> the orbital frequency• Warped thin disk (Sanders et al. 1989)

But well-formed ionization cones are seen close to the center, IR interferometry shows a thick structure at ~1 pc in NGC 1068, and dust cannot survive closer

• Magnetic wind (Königl & Kartje 1994)

But origin of large-scale field? And mass-loss rate can be very large: 10 NH24 (h/r)(vr100)rpc Msun/yr

Another Candidate Mechanism:Radiation Pressure (Pier & K. 1992)

• If thermal continuum is created by dust reprocessing,

there must be a large radiation flux through the obscuration

• κmidIR(dust) ~ 10—30 κT (Semenov et al. 2003)

so (L/LE)eff ~ (10–30)L/LE

What is the Internal Flux?

All previous calculations of flux have guessed the density distribution:

•Pier & K. (1993): constant density, rectangular envelope

•Granato & Danese (1994): density a power-law in r, exponential in cos()

•Efstathiou & Rowan-Robinson (1996): density a power-law in r, exponential in

•Nenkova, Ivezic & Elitzur (2002): probability of a clump a power-law in r, independent of , approximate treatment of diffuse radiation

•Hoenig et al. (2006): probability of a clump a power-law in r, Gaussian in z, transfer like Nenkova et al.

Radiation Transfer and Dynamics Must Be Consistent

• Frad moves dusty gas, altering radiation transfer.

• New transfer solution changes Frad

• Additional forces (magnetic, collisions, ...) further complicate the problem

A difficult calculation!

Qualitative Character of Solution

Greater optical depth in equatorial direction than in (half-) vertical direction guarantees most flux upward; some radial component remains, decreasing outward from the inner edge

Possible Dynamical Elements

• Gravity

• Radiation pressure

• Rotational support (j(r) not necessarily = jKep(r))

• Random motions/thermal pressure

• At inner surface, a drastic phase change

flux primarily UV, not IR, so much larger opacity

temperature much higher than in torus body

rocket effect from evaporating matter

Treat this separately!

The Simplest Self-Consistent Picture

Forces:

•Gravity

•Rotational support

•Radiation pressure

Assume hydrostatic balance

The Simplest Self-Consistent Picture

Assumptions:

• 2-d, axisymmetric, time-steady

• thermally-averaged opacity independent of T, diffusion approximation (smooth density distribution)

• no sources of infrared inside the torus

• l/lKep = j(r)

Boundary Conditions

• Must specify available matter:

choose (r,z=0) = in(r/rin)-

• After finding (r,z), locate photosphere:

insist on F ~ cE by varying • Must also have E > 0

also by varying • Location of inner boundary left undetermined

Solution: Entirely Analytic

Step 1: Hydrostatic balance + diffusion equilibrium + absence of internal radiation sources leads to

r dj2/dr + 2(1- α) j2 = 3 - 2α for α = - d ln Ω/d ln r,

so that j2(r) = [jin2 + f(α)](r/rin)2(α-1) – f(α)

In other words, If Frad,z ~ Fgrav,z in a geometrically thick disk, then Frad,r ~ Fgrav,r likewiseSo the orbiting matter must have sub-Keplerian rotation if it is to remain in equilibrium; magnetic angular momentum redistribution?

A)To fix the quantity of available matter---

(r,z=0) = (*/rin)(r/rin)-

B) To match the diffusion solution to its outgoing flux---F ~ cE at the dust photosphere (which determines )

Step 2: Requiring both components of force balance to give a consistent density leads to

(∂E/∂z)/(zΩ2) – (∂E/∂r)/{rΩ2[1-j2(r)]} = 0

which is analytically solvable by characteristics:

E = constant on (almost) elliptical surfaces

Step 3: Apply boundary conditions:

Free Parameters

Q ' ¿¤M (<r in )M B H

· T·

L EL » 1¡ 10

®= ¡ dln­ =dlnr

¿¤ = · ½inrin » 10¡ 30

As Q increases, increases

Example Solution

for α = 3/2; * = 10; Q=3, so L/LE = 0.1—0.3

radiation energy density gas density

The X-ray Column Density Distribution Predicted by the Example Solution

What About Internal Heating?

Two plausible possibilities:

Compton recoil from hard X-rays

Stellar heating if there is intense star-formation

/ L4¼(r 2+z2)

(LX =L)f recoil

/ ½3=2(r;z)

Solutions with Internal Heating

(LX /L)fcomp = 0.02

L* /L = 0.05

radiation energy density gas density

Range of Q and Permitted

X = 0.01 or P = 0.005

X = 0.1 or P = 0.25

Conclusions

• Tori convert optical/UV flux to IR; there must therefore be a large IR flux through them

• Mid-IR opacity/mass ~ 10—30 Thomson, increasing the effective F/FE by that factor

• With some simplifying assumptions, a self-consistent hydrostatic equilibrium and 2-d diffusive transfer solution can be found

• The torus becomes geometrically thick when L/LE ~ 0.03—0.3 and the midplane T ~ 1