The formation of stars and planets

The formation of stars and planets Day 3, Topic 3: Irradiated protoplanetary disks Lecture by: C.P. Dullemond

This is because energy is: Spectral Energy Distributions (SEDs) Plotting normal flux makes it look as if the source emits much more infrared radiation than optical radiation:

In that case is the relevant quantity to denote energy per interval in log. NOTE: Spectral Energy Distributions (SEDs) Typically one can say: and one takes a constant (independent of ).

Now take an annulus of radius r and width dr. On the sky of the observer it covers: and flux is: Total flux observed is then: Calculating the SED from a flat disk Assume here for simplicity that disk is vertically isothermal: the disk emits therefore locally as a black radiator.

multi-color region Wien region Rayleigh-Jeans region Multi-color blackbody disk SED F 

F Slope is as Planck function:  Suppose that temperature profile of disk is: Emitting surface: Peak energy planck: Location of peak planck: Multi-color blackbody disk SED Rayleigh-Jeans region: (4q-2)/q 3 Multi-color region:

F (4q-2)/q  Rayleigh-Jeans part modified by slope of opacity. Suppose that this slope is: Then the observed intensity and flux become: Disk with finite optical depth If disk is not very optically thick, then: Multi-color part stays roughly the same, because of energy conservation 3+

According to our derived SED rule (4q-2)/q=4/3 we obtain: Higher than observed from veiling (see later) HD104237 AB Aurigae Bad fit SED of accretion disk Remember: Does this fit SEDs of Herbig Ae/Be stars?

Viscous heating or irradiation? T Tauri star

Viscous heating or irradiation? Herbig Ae star

 Irradiation flux: Cooling flux: Flat irradiated disks Similar to active accretion disk, but flux is fixed. Similar problem with at least a large fraction of HAe and T Tauri star SEDs.

flaring vertical structure irradiation heating vs cooling Flared disks • Kenyon & Hartmann 1987 • Calvet et al. 1991; Malbet & Bertout 1991 • Bell et al. 1997; • D'Alessio et al. 1998, 1999 • Chiang & Goldreich 1997, 1999; Lachaume et al. 2003

The flaring angle: Irradiation flux: Cooling flux: Express surface height in terms of pressure scale height: Flared disks: Chiang & Goldreich model

We obtain Flared disks: Chiang & Goldreich model Remember formula for pressure scale height:

We therefore have: with Flaring geometry: Flared disks: Chiang & Goldreich model Remark: in general  is not a constant (it decreases with r). The flaring is typically <9/7

The surface layer A dust grain in (above) the surface of the disk sees the direct stellar light. Is therefore much hotter than the interior of the disk.

Heating: a = radius of grain = absorption efficiency(=1 for perfect black sphere) Cooling: Thermal balance: Intermezzo: temperature of a dust grain Optically thin case:

Intermezzo: temperature of a dust grain Big grains, i.e. grey opacity: Small grains: high opacity at short wavelength, where they absorb radiation, low opacity at long wavelength where they cool.

The surface layer again... Disk therefore has a hot surface layer which absorbs all stellar radiation. Half of it is re-emitted upward (and escapes); half of it is re-emitted downward (and heats the interior of the disk).

Chiang & Goldreich: two layer model • Model has two components: • Surface layer • Interior Chiang & Goldreich (1997) ApJ 490, 368

Flared disks: detailed models Global disk model... ... consists of vertical slices, each forming a 1D problem. All slices are independent from each other.

Flared disks: detailed models A closer look at one slice: Malbet & Bertout, 1991, ApJ 383, 814 D'Alessio et al. 1998, ApJ 500, 411 Dullemond, van Zadelhoff & Natta 2002, A&A 389, 464

Dust evaporation and disk inner rim Natta et al. (2001) Dullemond, Dominik & Natta (2001)

SED of disk with inner rim

Covering fraction

Example: HD100546 Must have weak inner rim (weak near-IR flux), but must be strongly flaring (strong far-IR flux)

Example: HD 144432 Must have strong inner rim (strong near-IR flux), but either small or non-flaring outer disk (weak far-IR flux)

Measuring grain sizes in disks The 10 micron silicate feature shape depends strongly on grain size. Observations show precisely these effects. Evidence of grain growth. van Boekel et al. 2003

Grain sizes in inner disk regions Resolving inner disk region with... ...infrared interferometry R < 2 AU R > 2 AU van Boekel et al. 2004

Probing larger grains in disks At (sub-)millimeter wavelength one can measure opacity slope (remember!). But first need to make sure that the disk is optically thin. A measured flux, if F~ 3, can come from a blackbody disk surface. Measure size of disk with (sub-)millimeter interferometry. If disk larger than that, then disk must be optically thin. A slope of F~ 3 then definitely point to large (cm) sized grains! Evidence for large grains found in many sources. Example: CQ Tau (Testi et al.)

As disk gets older: part of dust converted into big grains. Disk loses opacity, falls into own shadow. Many big grains observable at (sub-)millimeter wavelengths. Disk starts as flaring disk: strong mid/far-IR flux. Few big grains produced. Probinging the shape of disks We have sources with weak mid/far-IR flux, and sources with strong mid/far-IR flux. One of the ideas is that disk can be self-shadowed to obtain weak mid/far-IR flux.

Probinging the shape of disks Acke et al. 2004 looked for such a correlation, and indeed found it: Self-shadowed(?) disks Flaring disks

The formation of stars and planets