Rotation and Disks

- Rotating Clouds- Magnetic Braking- Centrifugal Radius- First appearance of disks- Thin disk approximation- The observable emission of a dusty disk- Molecular gas in protoplanetary disks

Stahler & Palla (2004): Sections 9.2, 10.4, 11.3, 17.3, 18.4

Equilibrium structures with rotation & pressure support

A disk is born when infalling gas begins to miss the protostellar surface. Material with sufficiently large angular momentum lands first in a disk and only later spirals to the stellar surface. We first consider rotation at the earliest stages of the core.

Coordinate system used for studying rotating clouds. The origin lies at the center of the cloud.

The z-axis points along the direction of rotation. Two fluid elements are shown (r and r’). Each fluid element has a certain steady velocity uφ.

Assume that the cloud is axisymmetric. ϖ ≡ r sin θ is the cylindrical radius.

Rotating Clouds

The specific angular momentum about the z-axis:

€

j ≡ ϖ uφThe centrifugal force per unit mass on each element is:

€

j 2 /ϖ 3

To see how rotation affects cloud structure, the centrifugal force must be incorporated into the equation for hydrostatic equilibrium:

€

−1ρ∇P −∇Φg = 0

€

P = ρ aT2Using an isothermal equation of state:

The force balance in the ϖ- and z-directions requires:

€

−aT2

ρ∂ρ∂ϖ

−∂Φg

∂ϖ= −

j 2

ϖ 3

−aT2

ρ∂ρ∂z

−∂Φg

∂z= 0

Thus, the specific angular momentum can only be a function of ϖ ! j and uφ must be constant along cylinders centered on the z-axis. In fact, differentiating [1] w.r.t z and [2] w.r.t ϖ and subtracting the two:

1

2

€

−1ϖ 3

∂j 2

∂z=aT2

ρ∂ρ∂z

∂ρ∂ϖ

−∂ρ∂ϖ

∂ρ∂z

&

' ( )

* + = 0

Poincaré-Wavre theorem in the more general case of P being an arbitrary function of density.

Illustration of the Poincaré-Wavre theorem: j is constant along cylinders that are centered on the rotation axis.

For rotational stability j must increase outward.

Using the fact that j is independent of z, the force balance can be reformulated in a more concise manner:

€

−aT2

ρ∇ρ −∇Φg −∇Φcen = 0

Φcen ≡ −j 2

ϖ 3 dϖ0

ϖ

∫

€

(lnρ +Φg /aT2 +Φcen /aT

2 ) is now a spatial constant.

Magnetic Braking

Magnetic field threading the gas core allows transfer of angular momentum.

Therefore, in the initial stages of collapse, angular momentum is not conserved.

The centrifugal radius may be very small. It is actually an active research problem to understand how protostellar disks can form (see PPVI chapter by Z.-Y. Li et al.).

Presumably as the collapse proceeds and the ionization fraction decreases, the coupling of the gas to the B-field becomes weaker and magnetic braking becomes less efficient.

Mathematical details of 10.4.1 and 10.4.2 are not examined.

Returning to the assumption of angular momentum conservation,during collapse, the initial angular rotation speed Ω0 of a fluid element originally located at ϖ0 increases to Ω0(ϖ/ ϖ0)2 once the element reaches the smaller radius ϖ (assuming every element conserves its j).

€

β ≡T rot

|W |≈Ω2L3

24GM=1×10−3 Ω

1 km s-1 pc-1

(

) *

+

, -

2L

0.1 pc(

) *

+

, -

3M

10 M!

(

) *

+

, -

−1

β will reach unity when ϖ will be decrease by 10-3 ! rotation must play a key role in the deep interior of a collapsing cloud (where the density is also large enough that the matter and the magnetic field decouple)!

The centrifugal force therefore increases as ϖ-3, faster than the ϖ-2 rise in gravitational attraction toward any fixed interior mass. If, initially:

The matter inside some volume of the central region indeed conserves angular momentum and spins up as it approaches the central protostar.

Since the centrifugal force rises faster than gravity, each fluid element inevitably veers away from the geometrical center of the cloud.

Centrifugal Radius (S&P 10.4.3)

A range of specific angular momenta is present in the infalling matter at any time.

Those elements with the highest j-values depart from radial trajectories earliest and ultimately land farthest from the center.

The maximum impact distance in the equatorial plane is called the centrifugal radius, ϖcen, which sets the scale of the circumstellar disk.

The inside-out nature of collapse implies that fluid elements arriving at some later epoch begin falling from more distant locations within the cloud.

These regions include higher j-values, so that some new elements cross the plane even farther from the center than before ! ϖcen increases with time.

Parabolic orbit in rotating infall. A fluid element at (r,ψ), falls into the radial distance req, where it impacts the disk.

If no disk

€

ϖ cen =m0

3aTΩ02t 3

16= 0.3 AU T

10 K$

% &

'

( )

1/ 2Ω0

10−14 rad s-1

$

% &

'

( )

2 t105 yr$

% &

'

( )

3

Equating ϖcen to the protostar radius and solving for the time:

€

t0 =16R*

m03Ω0

2aT

#

$ %

&

' (

1/ 3

= 3×104 yr R*

3 R!

#

$ %

&

' (

1/ 3Ω0

10−14 rad s-1

#

$ %

&

' ( −2 / 3 aT

0.3 km s-1

#

$ %

&

' ( −1/ 3

The mass of the protostar at this time of disk formation is then:

€

M0 ≡ ˙ M t0 =16R*aT

8

G3Ω02

$

% &

'

( )

1/ 3

= 0.2 M! R*

3 R!

$

% &

'

( )

1/ 3Ω0

10−14 rad s-1

$

% &

'

( ) −2 / 3 aT

0.3 km s-1

$

% &

'

( )

8 / 3

The protostar mass at the time of disk formation increases with either a greater aT (i.e. a higher infall rate) or a lower Ω0.

Protostars do not simply accumulate the angular momentum brought in by collapsing cloud matter. Instead, protostars must experience a strong, braking torque, presumably associated with the shedding of an MHD wind.

Protostellar Disks: First Appearance (S&P 11.3)

10

The Thin-Disk Approximation

The internal force balance is between the upward thermal pressure gradient and the downward pull of gravity. In the younger disks of relatively low mass, the vertical force balance for an element located at height z is:

€

−∂P∂z

=ρGM*zϖ 3

Using the equation of state for an ideal gas, we can thus derive the scale height Δz of the disk, i.e. the distance over which P falls appreciably:

z-component of the radial acceleration from the central star

11

€

Δz ≈ aTVKep

$

% & &

'

( ) ) ϖ

VKep ≡ GM* /ϖ

Scale Height of the Disk

Keplerian orbital speed

With increasing distance from the star, the disk temperature and aT drop and the inequality aT << Vkep is always well satisfied. Thus, we can adopt the thin disk approximation Δz << ϖ.

The accretion shock now effectively broadens to include the whole region inside the expanding centrifugal radius.

Gas entering this extended front preserves the component of its momentum || to the equatorial plane.

The mass of the central object continues to grow and the disk remains a structure of very low surface density.

collisions of streamlines during rotating,

protostellar collapse

12

Evolution of various disk and star quantities as a function of time after the onset of gravitational collapse of the cloud core (Hueso & Guillot 2005, A&A, 442, 703):

stellar massaccretion rate

disk mass

In this model, the disk is formed at t ~ 0.03 Myr, causing the jump in the dotted line and the collapse phase is finished at 2×105 yr. After the infall phase, the accretion rate quickly drops to ~10-7-10-9 M" yr-1.

The optical and UV excess observed from classical T Tauri stars and Herbig Ae/Be stars confirms that this on-going accretion indeed takes place.

disk formation

stop collapse phase

13

The observable emission of a dusty disk

Dullemond et al. 2006, PPV; Chiang & Goldreich 1997, ApJ, 490, 368

•The NIR bump comes from the inner rim (typically from small radii)

•The IR dust features from the warm surface layer (outer disk)

•The underlying continuum from the deeper (cooler) disk regions (midplane of the outer disk).

A natural explanation for the strong far-IR flux of is a flaring geometry of the disk’s surface (Kenyon & Hartmann 1987, ApJ, 323, 714).

The thermal radiation from surface layers produces dust features in emission (as seen in nearly all non-edge-on T Tauri and Herbig Ae/Be stars).€

T ∝ϖ−0.5

14

Overall SED shape for non-accreting disks with stellar irradiation. The stellar spectrum is added in grey scale (Dullemond & Dominik 2004).

•Solid line is normal flaring disk with inner dust rim•Dashed line is when the rim is made higher •Dot-dashed is when the flaring is reduced; •Dotted line is when the inner rim is at 10× larger radius.

Overall SED shape for accreting disks without stellar irradiation: •Solid line: Shakura & Sunyaev (1973) “α-viscosity” models•Dotted line: model with self-irradiation included•Dashed line: self-irradiation and irradiation by the magnetosferic accretion column on the star.

15

Gas and dust temperature in a vertical cut through the T Tauri star model at 100 AU. Overplotted are the most important atomic and molecular transitions.

gas temperature

dust temperature

The physics and chemistry of the surface layers resembles that of PDR regions, where Tgas > Tdust.

Beyond a vertical column depth of AV ~ 1, gas-dust coupling takes over.

FUV induced grain photoelectric (PE) heating is often a dominant heating process for the gas in the irradiated surface layers: the FUV photon is absorbed by a dust grain or a PAH, which eject an energetic e- to the gas.

Dullemond et al. 2006, PPV

Pictograms of the structure of a flaring protoplanetary disk, in dust (left) and gas (right).

Properties and Evolution of Dust in Protoplanetary Disks

Dominik et al. 2006, PPV; Natta et al. 2006, PPV

Dust grains in the ISM are a mixture of silicates and carbons, with size distribution from ~100 Å to ~0.2-0.3 µm. PAHs are also present.

In circumstellar disks, the pristine interstellar grains grow from sub-micron size to kilometer-sized bodies (planetesimals) and eventually planets.

Grain growth to cm- and m-sizes is mainly driven by collisional aggregation, plus gravitational instabilities in overdense dusty regions. The relative velocities of 1-100 µm grains are ~10-3-10-4 m/s, low enough for sticking.

While growing in mass, particles also start to sediment, due to vertical component of the stellar gravitational field, and the relative velocities may become high enough for the compaction of aggregates to take place.

Above the compaction limit, a runaway process is expected, where a few large aggregates grow by collisions with smaller particles.

Observational TechniquesThe most severe limits to the characterization of grains in disks come from the physical structure of disks. Because of their extremely high optical depths, only grains located in a τ=1 layer can be studied. Long-λ obs. are needed.

Interferometric observations at mm and cm λ have provided the strongest evidence that most of the original solid mass has grown to cm-size by the time the central star becomes optically visible (midplane).

The Mid-IR spectral region (~3-100 µm) is rich in vibrational resonances of abundant dust species (silicates and PAHs), sensitive probes of the chemical composition, lattice structure and shape of the particles (surface layers).

IR spectroscopy roughly probes dust particles with sizes of up to a few microns, in the temperature range between 1500 and 50 K (innermost surface layers).

In the V and NIR, scattering becomes important and properties are derived from those grains in a very narrow layer on the disk surface, but over a large range of disk radii.

Millan-Gabet et al. 2006, PPV

Observational phase-space (spectral domain and angular resolution) for optical interferometers, and for complementary techniques (shaded polygons).

The most relevant phase-space regions (rectangular boxes) are the main physical phenomena associated with young stellar objects.

20

Scattering and Polarization

HST/NICMOS (NIC2) JHK near-infrared image of HH 30 (in Taurus)

HST/NICMOS (NIC2) JHK NIR image of the Class I source IRAS 04302+2247

V and NIR observations of the dark disks seen against the background scattered light, provide direct view of disks around young stars.

The properties of the scattered radiation and the fraction of polarization can be used (in principle) to measure the size of the scattering grains (see Lucas et al. 2004, MNRAS, 352, 1347).

20’’×20’’ 10’’×10’’

Mid-IR spectroscopy

ISO and Spitzer provided us with extremely important information, even if the spatial resolution is too low to resolve the emission.

The 10 µm silicate feature

Natta et al. 2006, PPV

amorphoussilicate grains

crystallinesilicate grains

Evidences for radial changes of dust composition in the circumstellar disks of young stars.

Infrared spectra of the inner ~1-2 AU are shown in the left panels, and of the total disk (~1-20 AU) in the right panels.

In the inner disk (probed by VLTI/MIDI), the fraction of crystalline particles is higher than in the outer disk .

Interstellar particles are mainly small amorphous Olivine grains: Fe2-2xMg2xSiO4with some admixture of Pyroxene grains: Fe1-xMgxSiO3

In our planetary system, comets show the features of crystalline Forsterite (an Mg-rich Olivine).

Millan-Gabet et al. 2006, PPVTesti et al. 2014, PPVI

The crystallization in young circumstellar disks can be a very efficient process in the central 1-2 AU, whereas the outer 2-20 AU disk possesses a markedly lower crystalline fraction, but still much higher than in the ISM.

Combined with the very young evolutionary status of these systems (as young as 1 Myr), this implies that protoplanetary disks are highly crystalline before the onset of planet formation.

The presence of crystalline dust in relatively cold disk regions (or in Solar System comets) is surprising, given that temperatures in these regions are well below the glass temperature of ~ 1000 K.

They can be explained by chemical models of proto-planetary disks that include the effects of radial mixing and local processes in the outer disks.

25

Growth to cm-size grains

At (sub-)millimeter λ and beyond, the dust emission in disks begins to be moderately thin. It is then possible to probe the bulk of the dust properties, which are concentrated in the disk midplane.

The variation of the dust opacity coefficient per unit mass with frequency, in the mm wave range, can be approximated by:

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κ ~ λ−β

The value of β is directly related to the dust properties, in particular to the grain size distribution and upper size cutoff.

In the limits of dust composed only of grains much larger than the observing λ, the opacity becomes grey and β=0.Natta et al. 2006, PPVTesti et al. 2014, PPVI €

n(a)∝ a−q between amin and amax

About 60% of the observed objects have β ≤ 1 and amax ≥ 1 cm.

Dust Emission

The specific flux (flux density):

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Fλ = IλµdΩ∫In a disk, the solid angle depends on the disk’s projected surface area:

€

Fλ =2πD2 Iλ cosθϖ dϖ

ϖ 0

ϖ D∫

To evaluate Iν, consider the solution to the radiative transfer equation and apply Kirkhoff’s law :

€

Iλ = Bλ(T)[1− exp(−Δτλ)]

In the OPTICALLY THICK regime (λ ≤ 100 µm), Iλ ~ Bλ so that:

€

Fλ =4π cosθD2

hc 2

λ5dϖ ϖ

exp(hc /λkBT) −1ϖ 0

ϖ D∫

Assuming a certain power law for the radial temperature distribution:

€

T(ϖ) = T0(ϖ /ϖ0)−q

#

€

(λFλ)disk ∝λ(2−4q ) / q

q=0.65

evidence for inner hole

In the OPTICALLY THIN regime (sub-millimeter and beyond):

€

Iλ = Bλ(T)ΔτλUsing the Rayleigh-Jeans form for the Planck function (appropriate at these long wavelengths):

€

Bλ ≈2ckBTλ4

€

Δτλ =κλΣcosθ

Substituting this and the expression of the optical depth in the expression for Fλ:

€

Fλ =4πckBκλ

D2λ4T(ϖ)Σ(ϖ)dϖ

ϖ 0

ϖ D∫

€

(λFλ)disk ∝λ−3−β

#

€

κλ ~ λ−β

Molecular Gas in Protoplanetary Disks

Dutrey et al. 2006, PPV

$Large disks (CO outer radius ~200-1000 AU)$Intrinsic line widths are small (non-thermal component of ~0.1-0.2 km/s)$All disks are in Keplerian rotation (one exception)$The kinetic temperature at the CO disk surface T(r) ∝ r-0.6 (as in flared disks)$Vertical temperature gradient (disk mid-plane cooler than CO surfaces)$Disks around Herbig Ae stars are hotter than those around T Tauri stars$The degree of CO depletion decreases with Teff

Chemical structure of protoplanetary disks

Bergin et al. 2006, PPV