Transcript The formation of stars and planets - uni

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:
F d  F 
Spectral Energy Distributions (SEDs)
Typically one can say:    (log  ) and one takes (log  )
a constant (independent of ).


In that case  F is the relevant quantity to denote energy per
 F   F
interval in log.
NOTE:
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.
I (r)  B (T(r))
Now take an annulus of radius r and width dr. On the sky of

the observer it covers:
2 rdr
d 
cosi
2
d
and flux is:
Total flux observed is then:
2 cosi
F 
d2

rout
rin

B (T(r)) rdr
F  I d
Multi-color blackbody disk SED
F
Wien
region
multi-color
region
RayleighJeans region

Multi-color blackbody disk SED
Rayleigh-Jeans region:
(4q-2)/q
F
Slope is as Planck function:
F   3
3
Multi-color region:

Suppose that temperature profile of disk is:
r  T 1/ q
T(r)  rq
Emitting surface:
S  rdr  r 2
 T 2 / q
  2 / q
Peak energy planck:
max(B ) T 4   4


Location of peak planck:
 T



2) / q
 F  S max( B )   2 / q 4   ( 4 q

F   (4 q2)/ q

Disk with finite optical depth
If disk is not very optically
thick, then:
F
(4q-2)/q
Multi-color part stays
roughly the same, because
of energy conservation
3+

Rayleigh-Jeans part modified by slope of opacity.
Suppose that this slope is:
   
Then the observed intensity and flux become:

I (r)  (1 e )B    B   B
 F    B   3 
SED of accretion disk
Remember:
 3 Ý 2 1/ 4
Teff  
M K   r3 / 4
8

According to our derived SED rule (4q-2)/q=4/3 we obtain:
 F   4 / 3

Does this fit SEDs of Herbig Ae/Be stars?
Ý  2 107 Msun /yr
need M


Higher than
observed from
veiling (see later)
Ý  7 107 Msun /yr
need M
HD104237
AB Aurigae

Bad fit
Viscous heating or irradiation?
T Tauri star
Viscous heating or irradiation?
Herbig Ae star


Flat irradiated disks

0.4 r*

r
Irradiation flux:
L*
Firr  
4 r 2

Cooling flux:
Fcool   T 4
0.4 r* L* 1/ 4
T  
3 
4

r


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
vertical
structure
irradiation
heating vs
cooling
●
●
●
●
●
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:
 hs 
hs
  r   
r  r 
r
Irradiation flux:
L*
Firr  
4r 2

 hs L*
T 
 4 r 3
4
Cooling flux:
Fcool   T 4


Express surface height in terms of pressure scale height:

hs   h
  1...6
Flared disks: Chiang & Goldreich model
 hs L*
T 
 4 r 3
hs   h
4
T4 

  hL*
 4 r 3

Remember formula for pressure scale height:
4
3


k Tr
k
12 4

h
h 8  
r
T



m p GM*
m p GM* 
We obtain

 k
44  hL
L
hL
12
9
* **
h 87  
r

3
m GM  
4
4
4



r
 pp
** 
Flared disks: Chiang & Goldreich model
 k
4   L
9
*
h 7  
r

m GM   4 
 p
* 
We therefore have:
h  C 1/ 7 r 9 / 7

with
 k
4   L
*
C  

m GM   4 
 p
* 
Flaring geometry:


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.
Intermezzo: temperature of a dust grain
Optically thin case:
Heating:
Q   a2  F  d
a = radius of grain
= absorption efficiency (=1
for perfect black sphere)
Cooling:
Q  4 a
2
  B (T) d
 
 a 2
m
Thermal balance:
1
2
d
F
d d
4aB(T)
 B(T)
d



a

  F

4

2
Intermezzo: temperature of a dust grain
1
 B (T) d  4
 F  d
Big grains, i.e. grey opacity:
 444 111 LL**
TTT 
F 22

444
44rr
1r*2rT*4*4 r*2T*4
T 2
TTT
2 *
2r 4 r
44r
44


Small grains: high opacity at short wavelength, where they
absorb radiation, low opacity at long wavelength where
 cool.
they


r*
T
T*
2r
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
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
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.)
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:
Self-shadowed(?) disks
Flaring disks