Application of Gas Dynamical Friction toPlanetesimals

Evgeni Grishin & Hagai B. Perets1

Lund Observatory, Lund, Sweden

Supported by European FP7CAG grand

Exoplanets in Lund, 06.05.2015

1Technion, Israel Institute of Technology, Haifa, Israel

Gas planetesimal interaction play an important role in planetformation

few×106yr disk lifetimes

(Pfalzner., 2014)

Gas drag is important for small

planetesimals

Type I migration is important

for large planetary embryos

How does gas aect

intermediate mass

planetesimals?

Aerodynamic gas drag is eective for small planesimals

Gas Drag Formula

Fd =−12CD(Re)Aρgv

2rel

CD - Drag coecient

Re - Reynolds number

A - Cross section

vrel - relative velocity

Tightly couples small grains

Inective for large planetesimals

Keeps relative velocities low

Increases growth

Planetary migration is dominant for large protoplanets

(Masset., 1999)Planetary Migration

Exchange of angular momentum

with the gas (Lin & Papaloizou,

1979)

Spiral density wave (Goldreich &

Tremaine, 1980)

Resonant Lindblad and corotation

torques

m|Ω(r)−Ωp(r)|=±κ(r), m ∈ Z

Eective for masses of m & 1025g

(Hourigan & Ward, 1984; Takanka

& Ida, 1999)

Scaling with Planetesimal Mass

Dynamical Friction (DF) is an eective gravitational dragmechanism

DF is a Loss of momentum of a massive object in a background

medium, by creating an over-density gravitational wake

Collisionless systems (Chandrasekhar, 1943)

Gaseous medium (Ostriker, 1999)

Gravitational perturbatition on uniform gaseous medium:

Calculate the gravitational wake α(x,t) = ∆ρ(x,t)/ρ0

Calculate the eective force FGDF =∫

ρ∇Φextd3r

Dynamical Friction (DF) is an eective gravitational dragmechanism

DF is a Loss of momentum of a massive object in a background

medium, by creating an over-density gravitational wake

Collisionless systems (Chandrasekhar, 1943)

Gaseous medium (Ostriker, 1999)

Gravitational perturbatition on uniform gaseous medium:

Calculate the gravitational wake α(x,t) = ∆ρ(x,t)/ρ0

Calculate the eective force FGDF =∫

ρ∇Φextd3r

Dynamical Friction in Gaseous Medium (GDF)

Linear perturbation theory yields an outgoing pressure wave

Solving Inhomogenous wave equation with retarded potential

Point mass perturber

GDF peaks near v ∼ cs

F = F0×I (M )

where F0 = 4πG2M2ρ0

c2s

M - object mass

cs - speed of sound

I (M ) = 1M 2 ×

12ln(1+M1−M

)−M M < 1

12ln(1−M−2) + lnΛ M > 1

Approximate formula:

I (M ) =

M /3 M 1

lnΛ/M 2 M 1

Λ = rmax/rmin is called Coulomb

logarithm

Previous works considered only masses of &M⊕

GDF in the context of planet formation:

Vertically averaged, steady state GDF (Muto el. al., 2011)

GDF dominant for highly eccentric orbit

Disk planet interactions for highly inclined orbits (Rein., 2012)

Interaction of accreting planet (Lee & Stahler., 2012; Canto

et. al., 2012)

Secular interaction of self gravitating disk (Teyssandier et. al.,

2013)

All consider masses of fully evolved planets, at least &M⊕

Power law Disk Structure

Protoplanetary Disk Structure (Goldreich & Chiang., 1997)

Radial Structure:

Temperature prole: Tdisk ≈ 120(a/AU)−3/7KSound speed cs ≈ 4.7×104(a/AU)−3/14cm/sAspect ratio H0 = h(a)/a = 0.022(a/AU)2/7

Radial gas density: ρg (a) = 3×10−9(a/AU)−16/7g/cm3

Vertical structure:

Vertical Gas density:

ρg (a0,z)∼ ρg (a0,0)× exp(−z2/2h2)g/cm3

Isothermal disk

Relative velocity due to pressure gradients:

Pressure gradient: P ∼ (a/AU)−β where β = 19/7vrel = |vK −vϕ,gas | ∼ βH2

0vK cs

Power law Disk Structure

Protoplanetary Disk Structure (Goldreich & Chiang., 1997)

Radial Structure:

Temperature prole: Tdisk ≈ 120(a/AU)−3/7KSound speed cs ≈ 4.7×104(a/AU)−3/14cm/sAspect ratio H0 = h(a)/a = 0.022(a/AU)2/7

Radial gas density: ρg (a) = 3×10−9(a/AU)−16/7g/cm3

Vertical structure:

Vertical Gas density:

ρg (a0,z)∼ ρg (a0,0)× exp(−z2/2h2)g/cm3

Isothermal disk

Relative velocity due to pressure gradients:

Pressure gradient: P ∼ (a/AU)−β where β = 19/7vrel = |vK −vϕ,gas | ∼ βH2

0vK cs

GDF is stronger than Gas Drag for Radii R & 200km

Gas drag scales as ∼ R2

GDF scales as ∼ R6

Corollary

Exists a critical value

R?(G ,ρm,vrel ,Re,M ) where

both forces are equal

r? = 0.29

[CD(Re)

I (M )

]1/4vrel√Gρm

GDF is stronger than Gas Drag for Radii R & 200km

Gas drag scales as ∼ R2

GDF scales as ∼ R6

Corollary

Exists a critical value

R?(G ,ρm,vrel ,Re,M ) where

both forces are equal

r? = 0.29

[CD(Re)

I (M )

]1/4vrel√Gρm

−1 0 1 2 3 4

100

200

500

log [a/AU]

Crirical siz

e [km

]

GDFDominates

Gas Drag Dominates

−1 0 1 2 30

0.5

1

1.5

log [a/AU]

Mach N

um

ber

e=0e=0.02e=0.04e=0.1

Scaling with Planetesimal Mass

GDF is dominant for imtermediate mass planetesimals

GDF induced damping time is comparable to disk lifetime

Planetesimal with orbital parameters (a,e, I ) under disturbing force

Perturbation due to disturbing force F = Fr r +Fϕϕ +Fzz

mpda

dt= 2

a3/2√GM?(1− e2)

[Fre sin f +Fϕ (1+ e cos f )]

For circular orbit the SMA damping timescale is

τa = a/a≈ 1

4π

H2

0c3s

G 2ρgmp

= 3×(

mp

2 ·1025g

)−1Myr

Eccentricity and inclination damping is ∼ 2−3 orders of

magnitude faster

τe = e/e ∼ eτa τI = I/I ∼ 5H2

0τa

We integrate numerically 2-body problem with external force

Results: Coplanar massive bodies (m = 1025g) damp (a,e)in disk lifetimes

0 1 2 3

0.2

0.4

0.6

0.8

1

Time [Myr]

Sem

imajo

r axis

[A

U]

e=0e=0.02e=0.1e=0.3e=0.8e=0.04

−3 −2 −1 00

0.2

0.4

0.6

0.8

log (Time/Myr)

Eccentr

icity

−3 −2 −1 0

−2

−1

0

1

2

Time [Myrs]

log

(M

ach

)

Results: Inclined massive bodies (m = 1025g) damp (a,e, I )in disk lifetimes

0 0.2 0.4 0.6 0.8 1

0.8

0.85

0.9

0.95

1

Time [Myr]

Se

mim

ajo

r a

xis

[A

U]

e=0e=0.04e=0.1e=0.3e=0.8

−3 −2 −1 00

0.2

0.4

0.6

0.8

log (Time/Myr)

Ecce

ntr

icity

−4 −3 −2 −1 0

−2

−1

0

1

2

Time [Myrs]

log

(M

ach

)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

Time [Myr]

Inclin

ation

Results: Dependence on disk surface density Σg ∼ a−α

0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

1

Time [Myr]

Sem

imajo

r axis

[A

U]

α=1

α=1.5

α=2

e=0e=0.1

10−3

10−2

0

0.02

0.04

0.06

0.08

0.1

Time [Myr]

Eccentr

icity

2 4 6 8 10

x 10−3

0.986

0.988

0.99

0.992

0.994

0.996

0.998

1

Time [Myr]

Sem

imajo

r axis

[A

U]

10−2

100

10−2

10−1

100

Time [Myr]

Ma

ch

nu

mb

er

Scaling with planetesimall mass τ ∼m−1p

Merger timescale for binary planetesimals is shorter

For circular binary the merger timescale is2

τa = abin/abin≈3c3s

8πG 2mbinρg∼H2

0τa∼ 0.73

(mbin

4 ·1023g

)−1Myr

for eccentric orbit around the sun, the torque is reversed

2EG & Perets., 2015 (in prep.)

Binary planetesimals of mass m ∼ 1023g merge within disklifetimes

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

Time [Myr]

Bin

ary

separa

tion [R

hill

]

ep=0, ebin=0ep=0, ebin=0.5ep=0.1, ebin=0ep=0.3, ebin=0

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

Time [Myr]

ebin

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

0.3

Time [Myr]

Orb

ital e

0 0.2 0.4 0.6 0.8

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

Time [Myrs]

log (

Mach)

Implications on Planet formation theory

Planetesimal disk evolution (Goldreich et. al., 2004)

Additional cooling term

Natural mechanism for eccentricity damping

Considerable merger rate of BPs

enhance the rate of binary hardening,

catalize encounter rate

additional heat source of the planetesimal disk

Super-Earth formation (Hansen & Murray., 2012)

In situ formation of super Earths is challenging

possible if initial rocky material enhanced by factor of & 20

Radial drift by GDF is a natural source preplanetary rocky

material

Summary

Observational constrains imply fast growth and considerable

migration

Dierent mass ranges are dominated by dierent gas

planetesimal interactions

GDF is important for dynamical evolution of intermediate

mass planetesimals

GDF keeps planetesimal disks cool with low random velocity

GDF assists in merging BPs, increases binary hardening rate

and adds heato to the disk

GDF may assist in bridging between planet formation theory

and exoplanet observations

GDF dominates of type I migration for most range of Machnumbers

Migration torque (Tanaka el. al,

2002)

TI ∼ ΣgΩ2a4 (Mp/M?)2H−20

scales as M2p , independent of M

GDF formula applicable only for

Mp . 1026g

Limitations:

non-linear regime

accretion

shear

J3 - 3D GDF (OStriker 1999)

J2 - 2D vertically averaged

GDF (Muto et. al, 2011)

Dynamical Friction in Gaseous Medium (GDF)

Governing equations (Ostriker, 1999)

Continuity equation: ∂tρ + ∇ · (ρv) = 0

Momentum equation: ∂tv+ (v ·∇)v =−∇p/ρ−∇Φext

Applying linear perturbation anylisis yields inhomogenuous

wave equation:

∇2α(x, t)− 1

c2s∂ttα(x, t) =−∇

2Φext(x, t)/c2s

The density wake propogate as a pressure wave with speed cs

Origin of Retative Velocity betweem gas and planetesimals

The gaseous disk is sub-Keplerian due to pressure gradients

v2ϕ,gas = GM?/r + rρ

dPdr

Setting P ∼ r−α we get vϕ,gas = vK (1−3 ·H20 )1/2

The relative velocity of a planetesimal in circular orbit

vrel/vK = |vK − vϕ,gas |/vK ∼ H20

The ow is subsonic vrel cs

Eccentric and inclined orbits with e, I & H0 are supersonic -

vrel & cs

Tubulence

Kolmogorov - the ow consists of self-similar eddies

Energy cascades from the largest eddy to the smallest one

Typical dimentions l0 ∼ h, v0 ∼ cs , t0 ∼ l0/v0 ∼ 1/Ω

After a few t0, the gas is well mixed - scaled of cst ∼ h are

destroyed

Small scales are intact

tη ∼ (l/l0)2/3t0 and tl/t = (Ωt)−1/3.

For t 1/Ω, the perturbation is not aected by the eddy

current,

For t ∼ 1/Ω the turbulent current of the largest eddy destroys

the wake