Disk-planet interaction
andgap formationinduced by a giant planet inprotoplanetary disks
Kazuhiro Kanagawa (ILTS, Hokkaido University)
T. Muto (Kogakuin Univ.), H. Tanaka (ILTS, Hokkaido Univ.),

T. Tanigawa (UOEH), T. Takeuchi (Tokyo Tech)
Kanagawa et al. (2015), MNRAS
Kanagawa et al, (2015), ApJL in press(arXiv: 1501.05422)
3rd DTA Symposium@ NAOJ (Mitaka) 6/2/2015

Planet induced structure in a disk(A) planet(s) embedded in a disk perturbs
the disk due to gravitation interaction.
PlanetGiant planet can create adensity gap with the planetaryorbit
Spiral waves and Gaps are critical to the planetary
migration.

Density waves
excited by a planet1

Observations of protoplanetary disks
Gaps and spirals are discovered inprotoplanetary disks.

SpiralGaps

HL tau(ALMA partnership +15)

HD169142
Momose+ in press

Grady+ 2014

These structures may indicate existance of planet
embedded in the disks.
How are gaps and spirals related to planets?
2

Gap opening process

Planetary torque to open the gap
vsViscous diffusion to close the gap

Gap structure depends on:• Planet’s mass (Mp)

• Disk aspect ratio (h/R), or T
• Disk viscosity (α)

Important parameterfor gap shape3

Angular momentum conservation
A planet gives the angular momentum thesurrounding gas due to the disk-planet
interaction
Angular momentum
conservation
Gap edge

Gap bottom(planet location)

Planetary torqueThis relation always has to be satisfied
Angular momentum
flux on viscous disk
advection

viscosity4

Assumption: A planet exerts only on the gapbottom (Σ ~ Σp)
If the gap is sufficiently wide, thisassumption is valid.
In this case, the planetary torque isgiven as

Torque density

Surface density

Wide-limit gap model

It is proportional to the surface
density of gap bottom Σp(R-Rp)/hp5

Relation between planet mass
and gap depth
Angular momentum conservation (wide-limit gap):

AMF at Gap edge

AMF at Gap bottom

Planet torqueGap depth (wide-limit gap model)This relation is very similar to empirical formula given by high-resolution hydrodynamic
simulations (Duffell&MacFardyen 2013, Fung et al. 2014)6

vs. Hydrodynamic simulations
Gap depth obtained by the widelimit gap

wide-limit

This relation is in a goodagreement with previoushydrodynamic simulations !!

Assumptions of wide-limit gap- Gap is in steady state- Planetary torque is distributed
only within the gap bottom
7Torque distribution (small K)
Where does the planet exert the gravitational torque ?
Hydrodynamic simulation using FARGO.
Mp /M*=10-4, h/R=0.05, α = 10-3 (K=32)

Snapshot at 104 orbit

Torque density

Planet

The torque density mainly distributes within the gap.
Wide-limit gap would be valid in this case.
8

Torque distribution (large K)
Where does the planet exert the gravitational torque ?
Hydrodynamic simulation using FARGO.
Mp /M*=10-3, h/R=0.05, α = 10-3 (K=3200)

Snap shot at 104 orbit

Torque density

Planet

The torque density distributes not only the gap bottom , but also the gap edge.
Wide-limit gap is valid ?
9Planetary torque
Planetary torque (exerted on R>Rp ) given by hydrodynamic
simulations with various parameters
For large K, the onesided torque is only 2-3times larger than that ofwide-limit gap !

The wide-limit gap model seems to good model for the gap formation.
10

Relation between gap depth
and planet mass
Wide-limit gap can be used to estimate the planet
mass in observed gap.
If the gap depth, disk aspect ratio(temperature) are observed, theplanet mass can be obtained.

11

Example: HL Tau
HL Tau (ALMA band 6)

Stellar mass: ~ 1Msun
Distance from the sun: ~ 140 pc
There are three observed gaps:
1st gap: ~ 15 AU
2nd gap: ~ 30 AU
3rd gap : ~ 90 AU
Using data of band6(230GHz) andband7(345GHz), we can obtain an
optical depth and disk temperature.

observables
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Optical thickness and gas temperature
Optical depth and temperature around the 2nd gap

Independentof β

Gap depth is estimated as ~1/3– 1/7.
(if gas and dust are well-mixed)

Temperature can be estimated as 55 K
(h/R ~ 0.07)
13Dust filtration
The dust and gas distribution becomes different due to the differenceof their rotation speed .
Result of hydrodynamic
simulation done by Zhu et al.
(2012)The distribution of only large size dust is much different
from that of gas.
(small grains have almost same distribution of gaseousmatter)
If the dust filtration is critical, the dust particles arepilling up at the gap edge.
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HL tau’s case
Simulation (Zhu et al. 2012)

No decreasing ofdensity

HL tau’s data

No pilling up
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Estimated planet mass of HL tau
The planet mass in 2nd gap of HL tau’s disk is about 0.3 MJ if exists.
(we assume that gas and dust are well-mixed)

Σmin ~ 0.3 Σ0
T=55K (h/R=0.07)

16Summary We derive the analytic formula between the planet mass,disk parameters, and the observed gap depth.
 This relation can reproduce the gap depth given bynumerical simulations.
 The planet mass in the observed gap can be estimated bythis relation between the planet mass and the gap depth.
 We estimated the planet mass in 2nd gap of HL Tau’s disk

as about sub Jupiter mass, if the planet exists.

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