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

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

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

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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. 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:

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

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

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

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

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Viscous heating or irradiation? T Tauri star

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Viscous heating or irradiation? Herbig Ae star

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

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Flared disks flaring irradiation heating vs cooling vertical structure 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

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

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

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

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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.

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

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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.

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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).

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

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

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

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

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SED of disk with inner rim

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Covering fraction

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Covering fraction of torus with height H and radius R Reprocessed luminosity can be expressed as: Observed reprocessed luminosity depends on inclination but roughly one can write (modulo inclination):

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

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

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

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

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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.)

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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. Disk starts as flaring disk: strong mid/far-IR flux. Few big grains produced. 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.

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

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