The formation of stars and planets

Day 3, Topic 3:

Irradiated protoplanetary disks

Lecture by: C.P. Dullemond

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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Fν dν = Fν Δν

Spectral Energy Distributions (SEDs)Typically one can say: and one takes a constant (independent of ).

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Δ = Δ(logν )

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Δ(logν )

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FνIn that case is the relevant quantity to denote energy per interval in log. NOTE:

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Fν ≡ λ Fλ

Calculating the SED from a flat disk

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Iν (r) = Bν (T(r))

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:

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dΩ =2π rdr

d2cosi

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Fν = Iν dΩand flux is:

Total flux observed is then:

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Fν =2π cosi

d2Bν (T(r)) rdr

rin

rout∫

Multi-color blackbody disk SED

Wien region

multi-color region

Rayleigh-Jeans region

F

F

3

(4q-2)/q

Multi-color blackbody disk SEDRayleigh-Jeans region:

Slope is as Planck function:

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Fν ∝ν 3

Multi-color region:

Suppose that temperature profile of disk is:

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T(r)∝ r−q

Emitting surface:

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S ∝ rdr ∝ r2

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max(νBν )∝T 4Peak energy planck:

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∝TLocation of peak planck:€

r∝T−1/ q

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∝−2 / q

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∝ 4

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∝T−2 / q

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Fν ∝ Smax(νBν )

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∝−2 / qν 4 = ν (4 q−2)/ q

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Fν ∝ν (4 q−2)/ q

(4q-2)/qF

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:

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κ ∝

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Iν (r) = (1− eτν )Bν ≈ τ ν Bν ∝κ ν Bν

Then the observed intensity and flux become:

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Fν ∝κ ν ν Bν ∝ν 3+β

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need ˙ M = 7 ×10−7 Msun /yr

AB Aurigae

SED of accretion disk

Remember:

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Teff =3

8πσ˙ M ΩK

2 ⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 4

∝ r−3 / 4

According to our derived SED rule (4q-2)/q=4/3 we obtain:

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Fν ∝ν 4 / 3

Does this fit SEDs of Herbig Ae/Be stars?

HD104237

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need ˙ M = 2 ×10−7 Msun /yr

Bad fit

Higher than observed from

veiling (see later)

Viscous heating or irradiation?

T Tauri star

Viscous heating or irradiation?

Herbig Ae star

Flat irradiated disks

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≅0.4 r*

rIrradiation flux:

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Firr = αL*

4π r2

Cooling flux:

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Fcool = σ T 4

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T =0.4 r* L*

4πσ r3

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 4

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T ∝ r−3 / 4

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

Flared disks

flaring

irradiation

heating vs cooling

verticalstructure

● 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

Flared disks: Chiang & Goldreich model

The flaring angle:

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=r∂

∂r

hs

r

⎛

⎝ ⎜

⎞

⎠ ⎟→ ξ

hs

r

Irradiation flux:

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Firr = αL*

4πr2

Cooling flux:

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Fcool = σ T 4

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T 4 =ξ

σ

hs L*

4π r3

Express surface height in terms of pressure scale height:

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hs = χ h

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χ =1...6

Flared disks: Chiang & Goldreich model

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T 4 =ξ

σ

hs L*

4π r3

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hs = χ h

Remember formula for pressure scale height:

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h =k Tr3

μmpGM*

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T 4 =ξ

σ

χ hL*

4π r3

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h8 =k

μmpGM*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

r12 T 4

We obtain

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h8 =k

μmpGM*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

r12 ξ

σ

χ hL*

4π r3

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h8 =k

μmpGM*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

r9 ξ

σ

χ hL*

4π

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h7 =k

μmpGM*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

r9 ξ

σ

χ L*

4π

Flared disks: Chiang & Goldreich model

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h7 =k

μmpGM*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

r9 ξ

σ

χ L*

4π

We therefore have:

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h = C 1/ 7r9 / 7

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C =k

μmpGM*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

4

ξ

σ

χ L*

4πwith

Flaring geometry:

Remark: in general χ 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.

Intermezzo: temperature of a dust grain

Heating:

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Q+ = π a2 Fν εν∫ dν

a = radius of grain

= absorption efficiency (=1 for perfect black sphere)

Cooling:

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Q− = 4π a2 π Bν (T)εν∫ dν

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κ =π a2εν

mThermal balance:

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4π a2 π Bν (T)εν∫ dν = π a2 Fν εν∫ dν

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Bν (T)κ ν∫ dν =1

4πFν κ ν∫ dν

Optically thin case:

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σπ

T 4 =1

4πF

Intermezzo: temperature of a dust grain

Big grains, i.e. grey opacity:

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Bν (T)κ ν∫ dν =1

4πFν κ ν∫ dν

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σπ

T 4 =1

4π

L*

4π r2

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T 4 =1

4σ

L*

4π r2

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T 4 =1

4σ

4π r*2σT*

4

4π r2

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T 4 =r*

2T*4

4r2

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T =r*

2rT*

Small grains: high opacity at short wavelength, where they absorb radiation, low opacity at long wavelength where they cool.

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T >r*

2rT*

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

Chiang & Goldreich (1997) ApJ 490, 368

Model has two components:

• Surface layer

• Interior

Flared disks: detailed models

Global disk model...

... consists of vertical slices, each forming a 1D problem. All slices are independent fromeach other.

Flared disks: detailed models

Malbet & Bertout, 1991, ApJ 383, 814D'Alessio et al. 1998, ApJ 500, 411 Dullemond, van Zadelhoff & Natta 2002, A&A 389, 464

A closer look at one slice:

Dust evaporation and disk inner rim

Natta et al. (2001) Dullemond, Dominik & Natta (2001)

SED of disk with inner rim

Covering fraction

Covering fraction

Covering fraction

Covering fraction

Covering fraction

Covering fraction

Covering fraction

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

van Boekel et al. 2003

The 10 micron silicate feature shape depends strongly on grain size. Observations show precisely these effects. Evidence of grain growth.

Grain sizes in inner disk regions

R < 2 AU R > 2 AU

...infrared interferometry

Resolving inner disk

region with...

van Boekel et al. 2004

Probing larger grains in disksAt (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.)

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.

Probinging the shape of disks

Acke et al. 2004 looked for such a correlation, and indeed found it:

Flaring disks

Self-shadowed(?) disks