Planetesimal Formation —Snow line & "No-drift" mechanism
Highlights- Various “runaway” pile-up (instability) modes of pebbles/dust are reported.
- These occur around the snow line and beyond the snow line.
- Beyond the snow line, the “no-drift” mechanism can stop pebble drift,
leading to a runaway pile-up.

Ryuki Hyodo
Ida S., Guillot T., Hyodo R., et al. (2021) A&A, 646, A13
Hyodo R., Guillot T., Ida S., et al. (2021) A&A, 646, A14
Hyodo R., Ida S., Guillot T. (2021) A&A, 645, L9

Today’s Contents- Introduction
Importance of planetesimal formation study- Consequences of pebble drift
Pile-up around the snow line
Pile-up by “no-drift” mechanism- Overall results
Dependence on the disk structure & pebble mass flux

- Discussion
Diverse disk evolutions & diverse planetesimal formations
Snow line “fossilized”- SummarypebblesInitial planetesimal formation
is a missing piece

Planet Formation

Planetesimal formation

Smooth?
2AUE-type3AU

Annulus?

©AAS Nova

6AU
C-typeO-typeAsteroids

Mercury

Venus

O

O
E

*Just example values

E

Earth
C

Mars
E

O
E

O

Gradie & Tedesco 1982

Initial planetesimal formation is a missing piece

The initial planetesimal distribution borns
smooth, or borns in an annulus?

An annulus distribution

The classical view

ALMA observations &planetesimal formation

Surface density

Smooth or Annulus?

A smooth distribution

Distance to stare.g., Weidenschilling 1977

Asteroid belt
depleted(e.g., ∼1/1000)

~μm

Today

~100km>1000kmmodified from Turrini et al. (2014)©Ryuki Hyodoe.g., Marel et al. 2019

Asteroid belt
implanted?

TodayPressure Bump
- edges of dead zones(e.g., Chatterjee & Tan 2014; Charnoz+2019)

Planetesimal Formation
A (local) elevated concentration of solids may
be a favorable condition for planetesimalformation, followed by streaming instability
and/or gravitational instability.

e.g., Johansen et al. 2009
Carrera et al. 2015
Yang et al. 2017- curved by planet gravity(e.g., Pinilla+2012; Kanagawa+ 2015, 2016)- etc.

Whipple 1972

Snow Line
Stevenson & Lunine 1988
Ciesla & Cuzzi 2006
Birnstiel et al. 2010
Saito & Sirono 2011
Ros & Johansen 2013
Morbidelli et al. 2015
Bitsch et al., 2015
Estrada et al. 2016
Ida & Guillot 2016
Okuzumi et al. 2016
Schoonenberg & Ormel 2017
Drazkowska & Alibert 2017
Hyodo et al. 2019b

OthersMarel et al. 2019

- Dust evolution (e.g., Wada et al, 2009; Okuzumi et al. 2012; Zhang et al. 2015)- Secular GI (e.g., Ward 1976, 2000; Youdin 2011; Takahashi & Inutsuka 2014)- Anti-cyclonic vortex (e.g., Barge & Sommeria 1995; Inaba & Barge 2006)- “No-Drift” mechanism (e.g., Hyodo et al. 2021b)- etc.

Fig. of Hyodo et al. 2021c

Class Ⅱ

Snow Line Evolution
up and/or evolution stages of the disk.

Times

Snow line migrates, depending on the build-

lifetime of ~few Myr

snow line
Class Ⅰ
lifetime of ~0.5 Myr—Other Parameters —· ·Pebble-to-gas mass flux (Fp/g ≡ Mp / Mg)

Distance to stare.g., Oka et al. 2011
Bitsch et al., 2015
Morbidelli et al. 2016
Lichtenberg et al. 2021

Nonuniform Turbulence
A nonuniform turbulence structure may beubiquitous.
—Parameters —αacc : α-parameter for disk gas accretion
αDr : α-parameter for radial mixing
αDz : α-parameter for vertical stirring©Philip Armitage

Things around the Snow Line — Qualitative

Hyodo et al. 2021b- Pebbles (τs ≃ 0.1) quickly drift due to gas drag (e.g., Garaud 2007; Lambrechts et al. 2014).
- Silicate dust (τs ≪ 1) well couples to the gas and drift with the gas (Brnstiel et al. 2010; Morbidelli et al. 2015)→ causing traffic-jam (e.g., Ida et al. 2016; Hyodo et al. 2019).

- Diffused water vapor outside the snow line re-condense onto icy pebbles.
(Schoonenberg & Ormel 2017; Drazkowska & Alibert 2017; Ros et al. 2019; Hyodo et al. 2019; Garate et al. 2020)- Diffused silicate dust outside the snow line can stick to icy pebbles.
- Released silicate dust would initially have a small scale height (similar to pebbles?)and would be vertically stirred up as being away from the snow line?

See also
Stevenson & Lunine 1988
Ciesla & Cuzzi 2006
Birnstiel et al. 2010
Saito & Sirono 2011
Ros & Johansen 2013
Estrada et al. 2016

AimesPrevious studies did not correctly include the back-reaction (dust-gas inertia).
Also, the dependence on the disk structures is unclear.

• Better understanding the consequences of pebble drift to the snow line.

(silicate dust or icy pebbles?, effects of back-reaction, and scale height, etc)

· ·• It’s dependence on Fp/g ≡ Mp / Mg

• It’s dependence on the turbulent structures (αacc, αDr, αDz)

A local 1D advection-diffusion simulation

Hyodo et al. 2021c, Fig.1

Models- Three distinct non-dimensional parameters;
gas accretion ∝ αacc
radial and vertical diffusions ∝ αDr, αDz (αDr = αDz)
- A realistic scale height of dust (Ida, Guillot, Hyodo et al. 2021)- The back-reaction (BR) of solids onto gas(radial drift and diffusion)- Recycling of silicate dust and water vapor, included

Parameters· ·- Pebble-to-gas mass flux Fp/g = Mp / Mg·

(Mg = 10−8M⊙yr−1)−1/2- The temperature profile Td = 150 (r/3au)- Stokes number of pebbles τs ∼ 0.1- Stokes number of silicate dust τs ≪ 0.1- Initially, 50:50 rock-to-ice ratio.

KA local 1D advection-diffusion simulation
We are looking at Zp & Zd
around the snow line.
Zp ≡ ρp /ρg
Zd ≡ ρd /ρg

Midplane spatial density

Pebble scale heightgas surface density
gas scale height

Gas: ρg = Σg /2πHg

Stokes number
pebble-to-gas ratio (Z ≡ ρp /ρg)

Turbulence-regulated Hp

coefficient for diffusion back-reaction (K=0,1,2)
gas scale heightpebble surface density
pebble scale height

Pebble: ρp = Σp /2πHp

Silicate dust: ρd = Σd /2πHd

KH-regulated Hp

Larger one usedsilicate dust scale heightsilicate dust surface density

Richardson number(0.5 here)Snow Line Critical Considerations
ρp = Σp / 2πHpρd = Σd / 2πHdρp /ρgΛ ≡ ρg /(ρg + ρp) = 1/(1 + Z )Ida&Guillot 2016

Stokes number

pebbles

Ida & Guillot 2016
gas velocity
gas pressure

Schoonenreg&Ormel 2017✔

Hyodo et al. 2019✔

✔

Ida et al. 2021,
Hyodo et al. 2021✔

✔

a coefficient to characterize
the strength of the diffusion back-reaction(K=1 or 2 as examples; Hyodo et al. 2019)

About Diff-BR

pebbles

Schoonenreg&Ormel 2017

As pile-up occurs, the diffusivity decreases.

Hyodo et al. 2019✔

✔

Ida et al. 2021,
Hyodo et al. 2021✔

✔

*As long as K ≠ 0, the results
do not qualitatively change.

dust

snow line
Ida, Guillot, Hyodo, et al. 2021

gas

dust

gas

Hd=Hp at the snow line
w. a simple diffusion model.

pebblesdust

A realistic Hd is derived
using Monte-Carlo simulations.
(via a Lagrangian method)→ Hd: a function of disk structure.

pebblesrrsnow line

pebblesr

Hyodo Ida, Charnoz 2019

dust

Ida & Guillot 2016

Hd = Hgrsnow line(see Garate et al. 2020)

As pile-up occurs, drift velocity decreases.

Hd = Hppebblesdust*Including back-reaction onto the gas motion
does not qualitatively change the results.

Keplerin orbital
frequency

Diffusions (Diff-BR)✔

scale heightgas orbital
frequency

dust

gas

gas
scale height

About Drift-BR

Schoonenberg&Ormel 2017

scale height

Radial drift (Drift-BR)

Scale height of silicate dust Hd

scale height

Back-reaction

snow line(i.e., αacc, αDr, αDz)

An Example Result

αacc = 1 × 10−2, αDr=αDz = 1 × 10−3, Fp/g = 0.3 with K = 0 (Hyodo et al. 2019)snow line

silicate dust-to-gas
ratio (ρd /ρg)

pebbles(rock component)

water vapor

Mid-plane

Silicate dust

snow line
Stokes number

pebble-to-gas
ratio (ρp /ρg)

pebbles(ice component)

Here, the resultant pile-up is a steady-state.

# By changing disk structures (αacc, αDr, αDz) and Fp/g,
a “runaway” pile-up occurs (a key point of this study).

Another Topic…

“Before reaching the snow line,the ‘No-drift’ mode may occur.”
Hyodo, Ida, Guillot (2021c), A&A Letters*This process does not require snow line, pressure bump, and/or pebble growth.

“No-drift” runaway pile-up —A consequence of drift back-reaction

Fig.1 from Hyodo et al. 2021b, A&A Lettersαacc: α-parameter for gas accretion
αmid: α-parameter for vertical (radial) mixing

Case example settings- Vertical layered gas disk (e.g., due to the disk wind) accretes onto the star ( ∝ αacc)- The inner disk midplane is MRI-dead (αmid = αdead). The outer disk is MRI-active (αmid = αacc).
- The scale height of pebbles (Hp) depends on the midplane turbulent stirring ( ∝ αmid).
Pebble scale height

Pebble drift velocity

Stokes number
pebble-to-gas ratio (Z ≡ ρp /ρg)

Turbulence-regulated Hp

KH-regulated Hp

coefficient for diffusion back-reaction
gas scale heightΛ ≡ ρg /(ρg + ρp)= 1/(1 + Z )Drift velocity decreases as pile-up proceeds (drift back-reaction)→ the “No-drift” mode induced

gas accretion velocity

Richardson number(0.5 here)gas pressure
gas orbital
frequency

Larger one used
Keplerin orbital
frequency

“No-drift” runaway pile-up — Numerical

Steady state
K=1, αacc=10-2, Fp/g=0.11(a)αmid/αacc“No-drift” occurs
for “given” ≤ “Analytical”“No-drift” runaway pile-up
K=1, αacc=10-2, Fp/g=0.17(b)Analytical αmid,crit/αacc

K=1, αacc=10-2, Fp/g=0.47(c)Analytical αmid,crit/αacc

K=1, αacc=10-2, Fp/g=0.17(d)Analytical αmid,crit/αacc

Analytical αmid,crit/αacc

100

Given

10-1
10

Given

Given

Analytical

Analytical

Analytical

-2

Given

No dependence on thearchitecture of the dead zone

Analytical

Analytical — A critical value
100

Below which the “No-drift” mode occurs

(a)

10

ρp/ρg

(b)

t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

1

critical(d)t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

criticalρp /ρg = 1

0.1

(c)

t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

ρp /ρg = 1

0.01
2

4

6

8

10

12

Distance to star [au]

14 2

4

6

8

10

12

Distance to star [au]

14 2

4

6

8

10

12

Distance to star [au]

14 2

4

6

8

10

12

Distance to star [au]14“No-drift” runaway pile-up — Numerical

Steady state
K=1, αacc=10-2, Fp/g=0.11(a)αmid/αacc“No-drift” occurs
for “given” ≤ “Analytical”“No-drift” runaway pile-up
K=1, αacc=10-2, Fp/g=0.17(b)Analytical αmid,crit/αacc

K=1, αacc=10-2, Fp/g=0.47(c)Analytical αmid,crit/αacc

K=1, αacc=10-2, Fp/g=0.17(d)Analytical αmid,crit/αacc

Analytical αmid,crit/αacc

100

Given

10-1
10

Given

Given

Analytical

Analytical

Analytical

-2

Given

No dependence on thearchitecture of the dead zone

Analytical

Analytical — A critical value
100

Below which the “No-drift” mode occurs

(a)

10

ρp/ρg

(b)

t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

1

critical(d)t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

Once the “no-drift” takes place
at a specific heliocentric distance,
runaway pile-up propagates outward!

criticalρp /ρg = 1

0.1

(c)

t=3x104 yrs
t=1x105 yrs
t=3x105 yrs

ρp /ρg = 1

0.01
2

4

6

8

10

12

Distance to star [au]

14 2

4

6

8

10

12

Distance to star [au]

14 2

4

6

8

10

12

Distance to star [au]

14 2

4

6

8

10

12

Distance to star [au]14“No-drift” runaway pile-up — Parameter map

Hyodo et al. 2021b, A&A Lettersαacc=10-2

boundary2

Analytical — Critical boundary1

Fp/g

Above which the “No-drift” mode occurs

No-Drift100

boundary1

ρp /ρg = 1

0.1

1

boundary2

Steady-State0.1(ρp /ρg < 1)

0.01

10-4

10-3

10-2

αmid/αacc

10-1Analytical — Critical boundary2

10boundary1

ρp/ρg

1

αacc=10-3

100

10-4

10-3

10-2

αmid/αacc

10-1

100

0.01

Above which the “No-drift” mode occurs

Resultant pile-up mode —Dependence on αacc, αDz(αDr), and Fp/g
Silicate dust
runaway pile-up

Midplane solid-to-gas ratio(ρp /ρg , ρd /ρg)rock fraction

Fp/g

dust

pebbles1

Pebblerunaway pile-up

Pebblerunaway pile-up
-2

−2
α
=
10
∆xsubl=variable
acc with αacc=10

−3
-3
α
=
10
∆xsubl=variable
with
α
=10
acc
acc

●ND

●ND

●ND

●ND

●Peb-RP
50%

●ND

●ND

●ND

●Peb-RP50%●(0.66,0.22)
51%

0.1

●Sil-RP●Peb-RP50%●Sil-RP●(0.41,0.15) ●(0.10,0.08) ●(0.10,0.06)
51%
62%
43%

●(0.18,0.12) ●(0.06,0.04) ●(0.02,0.02)
51%
50%
54%

●Peb-RP50%●Peb-RP50%●(0.30,0.79) ●(0.17,0.27) ●(0.52,0.22)
54%
79%
64%

-4

10

-3

10

-2

αDz/αacc

10

-1

10

0

10

-4

101
0

●(0.09,0.75) ●(0.08,0.17) ●(0.03,0.08) ●(0.06,0.05)
53%
51%
68%
48%

10

●(0.03,0.15)
52%

10-1

0.01
10

102

ρd/ρg

“No-drift”
runaway pile-up

10

-3

10

-2

10

-1

10

0

αDz/αacc- For αDz /αacc ≪ 1, a pile-up of silicate dust preferred.
- For αDz /αacc > 0.1 with Fp/g > 0.3 a pile-up of pebbles preferred.
−2
For
α
=
10
, the “No-drift” mode widely appears.
acc

10-2
Hyodo Guillot, Ida et al. 2021
see also Ida, Guillot, Hyodo, et al. 2021

Discussion Ⅰ —Evolving Protoplanetary Disks
A case example

Silicate dust
runaway pile-up

Pebblerunaway pile-up

αacc = 10−3

A diverse distribution of initial planetesimals could beoriginated from diverse paths of the disk evolution.
(i.e., snow line evolution & αacc, αDz(αDr), and Fp/g)

Fig.6 from Hyodo et al. 2021c, A&A
See also Ida et al. 2021, A&A

Discussion Ⅱ—
Snow line “fossilized ”

The water ice line can be "fossilized" by the rapid growth of Jupiter’s core —The inward flow of icy pebbles halted.

snow linepressurebump

Icy pebbles
pebbles blocked
by Jupiter# Depending on the size, pebbles can pass through Jupiter.

Morbidelli+2016
see also Bitsch+ 2021

The water ice line can be "fossilized" by the “no-drift” mechanism—
The inward flow of icy pebbles suppressed.
snow line

Dead zone

αmid(r) = αdead“no-drift”
front

αmid(r) = αacc

Hp
Icy pebbles# A fraction of pebbles leaks inward by diffusion.

pebbles stopped
by the “no-drift” front

Hyodo et al. 2021c

Summary
Silicate dust
runaway pile-up

Fp/g

1

Pebblerunaway pile-up

Pebblerunaway pile-up
-2

−2
α
=
10
∆xsubl=variable
acc with αacc=10

−3
-3
α
=
10
∆xsubl=variable
with
α
=10
acc
acc

●ND

●ND

●ND

●ND

●Peb-RP
50%

●ND

●ND

●ND

●Peb-RP50%●(0.66,0.22)
51%

0.1

●Sil-RP●Peb-RP50%●(0.18,0.12) ●(0.06,0.04) ●(0.02,0.02)
51%
50%
54%

●Peb-RP50%●(0.30,0.79) ●(0.17,0.27) ●(0.52,0.22)
54%
79%
64%

10

1

●(0.09,0.75) ●(0.08,0.17) ●(0.03,0.08) ●(0.06,0.05)
53%
51%
68%
48%

10

0

●(0.03,0.15)
52%

10-1

●Sil-RP●(0.41,0.15) ●(0.10,0.08) ●(0.10,0.06)
51%
62%
43%

●Peb-RP
50%

0.01
10

-4

10

-3

10

-2

αDz/αacc

10

-1

10

0

10

-4

102

10

-3

10

-2

10

-1

10

0

10

ρd/ρg

“No-drift”
runaway pile-up-2

αDz/αacc

Pile-up around the snow line and the “No-drift“ mode

Hyodo, Ida, Charnoz (2019b), A&A
Ida, Guillot, Hyodo, et al. (2021), A&A
Hyodo, Guillot, Ida, et al. (2021b), A&A
Hyodo, Ida, Guillot (2021c), A&A Letters

via runaway pile-ups
that would lead to SI or/and GI.

- Rocky/icy planetesimal formation just inside/outside the snow line, possible.
- The resultant modes (dust/pebbles/No-Drift) depend on Fp/g & disk structure (αacc, αDr, αDz).
- Diverse planetesimal formation can be originated from diverse paths of disk evolution.

A proper description of Hd is very important

Here “pile up” means “Runaway pile-up”
Dust can piles up
inside the snow line

Dust does not pile up
inside the snow line

Diff-BR: NOT included (K=0)
Drift-BR for dust: Included
Always αDz = αacc

gas

Hd = Hp

Schoonenberg&Ormel 2017pebblesdust

gas
scale heightscale height

Ida&Guillot 2016

dust
pebblesrslightly overestimated at SL

Hyodo Ida, Charnoz 2019

Dust can piles up
inside the SL dependingon the αDz /αacc value

Ida, Guillot, Hyodo, et al. 2021
Diff-BR included (K=1 or 2)
Drift-BR for dust included

Hd=Hp at the snow linewith a simple diffusion model.

pebblesgas
scale heightgas

Diff-BR included (K=1 or 2)
Drift-BR for dust: included

dust
pebblesrrsnow line

Dependence of silicate dust pile-up upon Hdjust inside the snow line

snow line
Dust can piles up
inside the SL dependingon the αDz /αacc value

scale height

Hd = Hg

Diff-BR NOT included (K=0)
Drift-BR for dust: NOT included
Always αDz = αaccrsnow line

dust

Ida, Guillot, Hyodo, et al. 2021

snow line

What about pebbles
outside the SL?
→ Hyodo et al.2021

Hyodo Guillot, Ida et al. 2021
see also Ida, Guillot, Hyodo, et al. 2021