giant planet formation - uni-jena.detheory/dips/talks/broeg.pdf · giant planet formation: episodic...

Christopher Broeg 12.10.2010

Giant Planet Formation:episodic impacts vs. gradual core growth

C. Broeg1, W. Benz1, G. Wuchterl2

1University of Berne, Institute for Space and Planetary Sciences2Thüringer Landessternwarte Tautenburg

!Japanese-German Workshop: „Dust in Planetary Systems“Universität Jena

27.9. - 1.10.2010, Jena, Deutschland

Dienstag, 12. Oktober 2010


Outline

• Motivation: core growth by giant impacts important for giant planet formation?

• Methodsimpact modeling and numeric scheme

• Validation• Results• Conclusion



• This model could be called the standard model and has been first worked out by Mizuno (1980, Progress of Theoretical Physics, 64, 544) and Bodenheimer & Pollack (1986, Icarus, 67, 391). It was refined in great detail by Pollack et al. (1996, Icarus, 124, 62) and extended recently by Alibert et al. (2004, AA, 434, 343).

• In these models a solid core accretes first. Once this core reaches a critical mass (of order 10 Mearth) the gaseous envelope is accreted in a runaway process.

the core accretion paradigm



Motivation

• The planetesimal accretion rate is an important parameter in the core accretion scenario

• For numerical convenience, formation models use gradual core growth modelled by a rate equation

• However, in the oligarchic growth regime, possibly:‣ the core growth is dominated by large impacts‣ the mass ratio is large, e.g. 0.1

Does this change the current picture of giant planet growth?



Methods



Procedure

• Replace constant dMz/dt with „impacts“• Impacts are modelled as Gaussian dMz/dt curve: width

gives timescale• Parameters:

‣ impact mass‣ impact timescale‣ (initial & background rate)

• Study thermal response on impact



impact model

8.000 8.005 8.010 8.015 8.020time [Ma]

0.000000

0.000002

0.000004

0.000006

0.000008

0.000010

0.000012

0.000014

0.000016

0.000018

dMz/

dt[M

e/yr

]

0.02 Me in 103 yr

impact vs. continuousepisodic vs. gradual

when same mass accreted: stop accretion and compare

parameters:• impact mass• impact timescale

τEW = σ√

2π

τττEW

compare here!core growth rate



core growth rate log scale

8.0 8.2 8.4 8.6 8.8 9.0time [Ma]

10−10

10−9

10−8

10−7

10−6

10−5

10−4

dMz/

dt[M

e/yr

]

1Me in 104 yr

at minimum rate until same mass accreted

comparison: continue continuous accretion

impact

initial model from continuous

accretion



Calculation

• Henyey type code with self-adaptive 1D grid• Stellar structure equations• Quasi-hydrostatic equilibrium• Impact timescale timp:• Neglect energy deposition in atmosphere• Material

‣ Saumon et al. (1995) EOS‣ Opacities: [Bodenheimer & Pollack (1986) + Alexander &

Ferguson (1994) + weiss et al. (1990)] bzw. [Ferguson et al. (2005)]

tdyn � timp � tKH



code verification


Christopher Broeg GruSi 2010 11

Verification: Jupiter formation (Pollack J1)

• Model‣ feeding zone: left and right of planet‣ give Sigma0‣ no migration

• Simplifications / differences:‣ capture radius = core radius‣ feeding zone width = 4 hill radii‣ const. grav. focussing: Fg=105

‣ outer BC: hill radius• Maximum gas accretion rate 10-4 Me/yr


10−2 10−1 100 101 102

time [Ma]

0.000000

0.000001

0.000002

0.000003

0.000004

0.000005

0.000006

0.000007

0.000008

lum

inos

ity[L⊙

]Pollack J1, 15

0 2 4 6 8time [Ma]

10−10

10−9

10−8

10−7

10−6

10−5

lum

inos

ity[L⊙

]

Pollack J1, 15


luminosity

L

log L

time [Ma]


0 1 2 3 4 5 6 7time [Ma]

10−8

10−7

10−6

10−5

10−4

dM/d

t[M

e/yr

]

Pollack J1, 15


0 2 4 6 8 10time [Ma]

0

10

20

30

40

50

60

70

80

90

M[M

e]

Pollack J1, 15

accretion rates

masses

coretotal

env

coreenv


Christopher Broeg GruSi 2010

Pollack J1 case

72 POLLACK ET AL.

FIG. 1. (a) Planet’s mass as a function of time for our baseline model, case J1. In this case, the planet is located at 5.2 AU, the initial surfacedensity of the protoplanetary disk is 10 g/cm2, and planetesimals that dissolve during their journey through the planet’s envelope are allowed tosink to the planet’s core; other parameters are listed in Table III. The solid line represents accumulated solid mass, the dotted line accumulatedgas mass, and the dot–dashed line the planet’s total mass. The planet’s growth occurs in three fairly well-defined stages: During the first �5 � 105

years, the planet accumulates solids by rapid runaway accretion; this ‘‘phase 1’’ ends when the planet has severely depleted its feeding zone ofplanetesimals. The accretion rates of gas and solids are nearly constant with MXY � 2–3MZ during most of the �7 � 106 years’ duration of phase2. The planet’s growth accelerates toward the end of phase 2, and runaway accumulation of gas (and, to a lesser extent, solids) characterizes phase3. The simulation is stopped when accretion becomes so rapid that our model breaks down. The endpoint is thus an artifact of our technique andshould not be interpreted as an estimate of the planet’s final mass. (b) Logarithm of the mass accretion rates of planetesimals (solid line) and gas(dotted line) for case J1. Note that the initial accretion rate of gas is extremely slow, but that its value increases rapidly during phase 1 and earlyphase 2. The small-scale structure which is particularly prominent during phase 2 is an artifact produced by our method of computation of theadded gas mass from the solar nebula. (c) Luminosity of the protoplanet as a function of time for case J1. Note the strong correlation betweenluminosity and accretion rate (cf. b). (d) Surface density of planetesimals in the feeding zone as a function of time for case J1. Planetesimals becomesubstantially depleted within the planet’s accretion zone during the latter part of phase 1, and the local surface density of planetesimals remainssmall throughout phase 2. (e) Four measures of the radius of the growing planetary embryo in case J1. The solid curve shows the radius of theplanet’s core, Rcore , assuming all accreted planetesimals settle down to this core. The dashed curve represents the effective capture radius forplanetesimals 100 km in radius, Rc . The dotted line shows the outer boundary of the gaseous envelope at the ‘‘end’’ of a timestep, Rp . The long-and short-dashed curve represents the planet’s accretion radius, Ra .

when the protoplanet has virtually emptied its feeding zone volve interacting embryos for accretion to reach the desiredculmination point (Lissauer 1987, Lissauer and Stewartof planetesimals.

If this simulation had been done in a gas-free environ- 1993). However, it is possible to carry our simulations ofthe formation of the giant planets to a reasonable endpointment, as might be appropriate for the formation of the

terrestrial planets, then the next phase would have to in- without involving interacting embryos, because of the im-

0

50

100

150

200

250

300

350

mas

s[M

earth

]

Jupiter formation, case J1

Mc

Menv

Menv,unlim.

M

10−7

10−6

10−5

10−4

10−3

dM/d

t[Mea

rth/y

r]

0 2 4 6 8 10time [Ma]

10−8

10−7

10−6

10−5

10−4

10−3

L[L ⊙

]

comparison with Pollack (1996)

pollack

broeg


Christopher Broeg GruSi 2010

Verification summary

• Good agreement with Pollack• L_max = 10-5 Lsun

(10-3 when limiting accretion to 0.01 instead of 10-4 Me/yr)• Jupiter values at 4.5 Gyr:

‣ Mass: 1.008 Mjup (by construction)‣ Radius (4.5 Ga) = 1.03 RJup‣ Mz = 34 Mearth‣ L = 0.76 L_jup_internal

• Mach number of inflow: -0.4• Further tests:

‣ static (Mizuno 1980), ‣ CoRoT-9b, ‣ HD209458b (all verification successful)

15



Results: impact vs gradual growth

• 1 example case: 1 Me impact on 10 Me target coreenvelope mass, gas accretion rate, luminosity

• all targets for 1 Me impact



Scenario

• Growing proto-planet core at 3 AU in MMSN, solar host star

• Nominal core accretion rate: 10-6 Earth masses / yr• At desired impact core mass:

‣ impact followed by no solid accretion‣ compare to gradually growing case

• Parameter study:‣ different impact masses 0.02, 0.1, 0.5, and 1 Earth

masses‣ different target masses Mc=1,2,3,…,15 Earth masses



envelope mass (impact 1 on 10 Me)

sequence:

1. gas ejection

2. fast accretion

3. gas replenished after 0.055 Myr

4. gas accretion slows down

! net more gas accreted

0.0 0.2 0.4 0.6 0.8 1.0time [Myr]

10−1

100

enve

lope

mas

s[M

⊕]

1 / 10

gradual growth

mas

s ga

in



gas accretion rate

0.00 0.01 0.02 0.03 0.04 0.05 0.06time [Myr]

−700

−600

−500

−400

−300

−200

−100

0

gas

accr

etio

nra

tedM

env/

dt[M

⊕/M

yr]

1 / 10



0.00 0.01 0.02 0.03 0.04 0.05 0.06time [Myr]

−700

−600

−500

−400

−300

−200

−100

0

gas

accretion

rate

dM

env/

dt

[M

⊕/M

yr]

gas accretion rate

0.0 0.2 0.4 0.6 0.8time [Myr]

10−1

100

101

102

gas

accr

etio

nra

tedM

env/

dt[M

⊕/M

yr]

1 / 10

• larger accretion rate at end• caused by lack of solid

accretion• all solid accretion finished

with impact



luminosity

0.0 0.2 0.4 0.6 0.8 1.0time [Myr]

1019

1020

1021

lum

inos

ity[W

]

negative

1 / 10

impact energy

removedadditional

energy from gas accretion



10 Me target, 1 Me impact

L

T

P

M

R



envelope mass during 1 Me impact

0.00 0.01 0.02 0.03 0.04 0.05 0.06time / Myr

10−3

10−2

10−1

100

101

Men

v/M⊕

1

2

3

4

5

6

78

910

1112

1314

15



envelope mass during 0.1 Me impact

0.00 0.01 0.02 0.03 0.04 0.05 0.06time / Myr

10−3

10−2

10−1

100

101

Men

v/M⊕

1

2

3

4

5

6

78

910

1112

1314

15



envelope mass after 1 Me impact

2 4 6 8 10 12 14time [Myr]

10−2

10−1

100

101

102

enve

lope

mas

s[M

⊕]



luminosity evolution 1Me impacts

0.015 0.020 0.025 0.030 0.035 0.040time [Myr]

0

5

10

15

20

25lu

min

osity

[kL j

up]

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

core luminosity luminosity



ejected envelope mass as a function of target size for 4 different impact sizes

10−2 10−1 100 101 102 103 104

energy impact / total binding energy

10−2

10−1

100

enve

lope

mas

sej

ecte

d/e

nvel

ope

mas

s E imp 1.0E imp 0.5E imp 0.1E imp 0.02

insufficient energy to remove envelope

more than enough energy to remove

envelope

100%

effi

cienc

y

smaller coreslarger cores



envelope accretion rate: ratio episodic vs continuous

0 2 4 6 8 10 12 14 16target core mass [Me]

100

101

102

103

ratio

Men

v

triggered runaway accretion

insufficient energy

E imp 1.0E imp 0.5E imp 0.1E imp 0.02

ratio

gas

acc

retio

n ra

te

(corrected y-axis label from talk)



Discussion

• Results show that the impact scenario yields more massive envelopes compared to gradual core growth

• Most of the energy can be transported at very high luminosity immediately after envelope ejection

• The Kelvin-Helmholtz timescale becomes very small during the impact and the energy from solid accretion can be shed quickly (For a 10 Me core: before: 0.2 Myr; during: 200 yr; after: 1.6 Myr)

• The subsequent phase without solid accretion quickly accumulates a large envelope



comparison with stopped core accretion



impact accretion vs. no accretion

10−2 10−1 100

time [Myr]

10−1

100

101

102

enve

lope

mas

s[M

⊕]

8

8

9

9

10

10

11

11

12

12

core accretion stop

gradual accretion

impact

beyond tcomp

see Ikoma et al. 2000, ApJ



Summary & Conclusion

• We were able to calculate episodic large impacts in the quasihydrostatic approximation

• Results show that the impact scenario yields more massive envelopes compared to the gradual core growth

• The impact itself leads to a very rapid loss of the deposited energy

• Gas accretion as fast as the shut-off case with the larger (post-impact) core

• In the oligarchic growth regime, this effect can be very important

• With this method, formerly sub-critical cores can accrete large amounts of gas

Broeg & Benz 2010, in prep.


giant planet formation - uni-jena.detheory/dips/talks/broeg.pdf · giant planet formation: episodic...

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