What influences the pebbleisolation mass?

Bertram Bitsch

Bertram Bitsch What influences the pebble isolation mass? 1 / 11

Giant planet formation

0.001

0.01

0.1

1

10

100

1000

0.1 1 10

MP [

ME]

r [AU]

r0=15 AUr0=25 AUr0=30 AU

⇒ When does solid accretion stop?

Bertram Bitsch What influences the pebble isolation mass? 2 / 11

Remember radial drift

P

v η(1− )Kep

FFG

Disc is hotter and denser close to the star: radial pressure gradient

Radial pressure gradient force mimics decreased gravity⇒ gas orbits slower than Keplerian:

vgas = vKep −∆v = vKep − ηvKep, η = −1

2

(H

r

)2∂ ln(P)

∂ ln(r)

Particles do not feel the pressure gradient force and want to orbit Keplerian

Headwind from sub-Keplerian gas drains angular momentum from particles, sothey spiral in through the disc

Bertram Bitsch What influences the pebble isolation mass? 3 / 11

Pebble accretion

Most planetesimals aresimply scattered by theprotoplanet

Pebbles spiral in towardsthe protoplanet due to gasfriction

⇒ Pebbles are accretedefficiently

Accretion rate by pebbleaccretion in Hill regime

Mc = 2( τf

0.1

)2/3

r2HΩKΣPeb

(Ormel & Klahr 2010; Lambrechts & Johansen,

2012, 2014; Johansen & Lambrechts, 2017)

Bertram Bitsch What influences the pebble isolation mass? 4 / 11

Gaps in discs: Pebble isolation mass

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

1 1.1 1.2 1.3 1.4 1.5 1.6

η

r [rp]

10 ME15 ME20 ME25 ME30 ME50 ME

Pebble isolation mass(vgas = (1− η)vkep):

⇒ Pebble accretion self-terminates:no accretion of solids any more!

(Lambrechts et al., 2014; Bitsch et al. (2018))

Pebble isolation mass: What does it depend on?

⇒ H/r , ν, dlnP/dln r and τf !

Bertram Bitsch What influences the pebble isolation mass? 5 / 11

Gaps in discs: Pebble isolation mass

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

1 1.1 1.2 1.3 1.4 1.5 1.6

η

r [rp]

10 ME15 ME20 ME25 ME30 ME50 ME

Pebble isolation mass(vgas = (1− η)vkep):

⇒ Pebble accretion self-terminates:no accretion of solids any more!

(Lambrechts et al., 2014; Bitsch et al. (2018))

Pebble isolation mass: What does it depend on?

⇒ H/r , ν, dlnP/dln r and τf !

Bertram Bitsch What influences the pebble isolation mass? 5 / 11

Pebble isolation mass

1

10

100

0.0001 0.001 0.01

Mis

o [

ME]

α

H/r=0.03H/r=0.05H/r=0.07

fit

25

18

20

22

24

26

28

30

32

-4 -3.5 -3 -2.5 -2 -1.5 -1

Mis

o [

ME]

dln P/dln r

Misofit

(Bitsch et al. 2018)

The pebble isolation mass without diffusion M†iso is given by

M†iso = 25ffitME , (1)

ffit =

(H/r

0.05

)3(

0.34

(log(α3)

log(α)

)4

+ 0.66

)1−d ln(P)d ln(r) + 2.5

6

, (2)

with α3 = 0.001.Bertram Bitsch What influences the pebble isolation mass? 6 / 11

Diffusion of dust particles

radial advection vs. turbulentdiffusion:

vr,pρp − Dρgdε

dr= 0

vr,p = −2τf∆v , D = αcsHg

Maximum loading:Hd ∼ Hb ∼ Hg

(Bitsch et al. 2018)

Estimating Hb:D

2τf∆v∼ Hg ⇒ τf ∼

α

2Π

where Π = ∆v/cs

⇒ Pebbles with τf > α/(2Π) are trapped in the pressure bump!

Bertram Bitsch What influences the pebble isolation mass? 7 / 11

Implications for pebble isolation mass

10-4

10-3

10-2

10-1 1

|∆v / cs|

10-4

10-3

10-2

α

0.0001

0.001

0.01

0.1

1

10

Sto

ke

s n

um

be

r

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

0.0010.010.11.0

503025

5030

503020

5035

50

(Bitsch et al. 2018)

Pebble isolation mass with diffusion:

Miso = M†iso +

Πcrit

λME (3)

Πcrit = α2τf

and λ ≈ 0.00476/ffit

Bertram Bitsch What influences the pebble isolation mass? 8 / 11

Implications for the solar system formation

(Kruijer et al. 2017, see also Gerber et al. 2017)

Bertram Bitsch What influences the pebble isolation mass? 9 / 11

Planet blocking the flux

(Pinilla et al. 2016)

⇒ How much mass can drift through?

Bertram Bitsch What influences the pebble isolation mass? 10 / 11

Summary

The pebble isolation mass depends on: H/r , ν, and d lnP/d ln r

⇒ Larger core masses in discs with larger viscosity

Pebbles can diffuse through pressure bump if τf < α/(2∆v/cs)

⇒ Increase of pebble isolation mass in high viscosity discs

⇒ Blocking very small pebbles is difficult!

Jupiter’s core can explain the difference in formation between NC andCC iron meteorites

⇒ exact mass of the core depends on disc structure and τf !

Bertram Bitsch What influences the pebble isolation mass? 11 / 11