1. Proto-Planetary Disk and Planetary Formation
Takayuki Tanigawa

2. Outline What are proto-planetary disks?
Basic property of the proto-planetary disk.
Disk shape
Rotation velocity
Radial density distribution
Planetary formation in the disk
Dust (~mm) motion
Planetesimal (~km) motion
Planet (~103km) motion

3. What are Proto-Planetary Disks?
Disks around young stars.
Naturally form when stars are forming.
Dissipate within years.
Planets can be formed in the disk.
Still hard to resolve the planet forming region
Fukagawa et al. 2004

4. Basic property of the disks
How the gas behave in a gravity field.
How does the disk shape determine?
Rotation velocity of the disks
Density distribution of the disks

5. Gas motion around a star
Particles around a star can rotate with Keplerian motion
Rotate on a plane including the star
Gas around a star CANNOT rotate with Keplerian motion
because of gas pressure

6. Vertical structure of the disks
Hydrostatic equilibrium
z component of star gravitational force
Equation of state
exp(-x2)
Density profile
1/e
Disk scale height (thickness)

7. Shape of the disks Disk aspect ratio The condition of disk flaring
Keplerian angular velocity
Sound speed
The condition of disk flaring
(Not depend on ρ)
For typical disks,
When
when
In general cases (like galactic disks)
Flat rotation case
Disk shape does NOT depend on density, only on the temperature.

8. Rotation velocity of the gas
Radial force in balance
2D pressure
Angular velocity of the gas: v
(η≪１)
Centrifugal force
Sound speed
~ 0.05
Keplerian velocity F
Rotation velocity of the gas is slightly slower than Keplerian motion.

9. Radial density distribution
Equation of viscous evolution of the disk (a kind of diffusion equation)
where
(α viscous coefficient)
If steady state is assumed (∂Σ/∂t = 0),
Steady accretion solution:
(q=1/2)
Early stage of the disk evolution
No accretion solution:
Late stage of the disk evolution
This radial density distribution have not been confirmed well by observations.

10. Viscosity in the disks α viscosity (Shakura and Sunyaev 1973)
speed of vortex × disk scale height
(from an analogy of the molecular viscosity coefficient)
Non-dimensional parameter α depends on physical condition in the disk,
if turbulence, α～10-4 – 10-3
if gravitational instability, α～ 1
Ordinal molecular viscosity：
random velocity × mean free path
Reynolds number
Inertial force
Viscous force ≫1
Negligible in most cases for astrophysical problems

11. Summary of the basic disk property
Disk shape
Typical disk:
Flaring
Rotation velocity v ～
Centrifugal force
Slightly slower than Keplerian rotation F
Radial density distribution

12. Planetary formation in the disks
4. Solid planets formation
1. Disk formation
2. Dust sedimentation
5. Gaseous planets formation
3. Planetesimal formation
6. Disk dissipation

13. Importance of solid particles for planetary formation
Terrestrial planets are made from solid.
Jovian planets have solid cores which are musts for the formation.
Even though solid material is minor component in the disks, solid particles play an critical role for the planetary formation.

14. Motion of small particles (Dusts)
Drag law in Epstein regime:
Vertical component of gravity of the star
Balance between the drag and gravity
Vertical density distribution
We have the terminal velocity
Dust particles settles down to the central plane.

15. Planetesimal formation through gravitational instability of the dust layer
自分自身が良く理解した上で、分散関係式などを説明すべし。
Typical size of created planetesimal

16. Difficulty for the planetesimal formation

17. Planetesimal motion
Motion is disturbed by mutual gravitational interaction
Increase of random velocity by energy exchange
> 0
Gravitational scattering
Low relative velocity case
Increasing rate decreases with the evolution
Random velocity evolution
stronger interaction
High relative velocity case
weaker interaction

18. Terrestrial-planet formation
Planetesimals grows up to be terrestrial planets through the mutual collision
Collision cross section
Gravitational focusing
Gravitational focusing factor
Geometrical cross section
Growth rate of planets
Growth time scale yr

19. Migration of the planets
Gravitational interaction with the gas become effective.
Planets lose angular momentum through the gravitational interaction with the disks.
(Tanaka et al. 2002)
The velocity of this migration increase with the mass.
Planets migrate inward faster than the growth
Significant problem of the present theory.

20. Gaseous planet formation
When the mass of a solid planet reaches 10 Earth masses, the planet starts to capture the disk gas by their strong gravity.
Because the quantity of gas material in the disk is much larger than that of solid material, gas planets can generally grow much larger than solid planets.
This is why the large planets in extra-solar planets are considered as “gaseous” planets.

21. Gap formation
If planets become large enough, the planets can create a gap in the disk and the growth stop
Planet growth is terminated by themselves through the gap formation.
The planet in the gap have to move with the disk viscous evolution.

22. Summary of the planetary formation
Planetary systems are formed in “proto-planetary disks”. .
Dust → Planetesimals
Settle down to the mid-plane.
Gravitational instability of the dust layer.
Planetesimal → Solid planets
Mutual collision and coalescence.
Solid planets → Gaseous planets
Gravitational collapse of the atmosphere by the strong gravity of the planets
There are still some problems to be addressed.
Dust is hard to settle down enough to occur the instability
Growth time scale v.s. Migration time scale
Dust → planetesimal → solid planet → gaseous planet
ここが固まっていない。しっかり考えるべし。