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