On the Modeling Aggregation of Dust Fractal Clusters in the Protoplanetary Laminar Disc

A.V. Kolesnichenko*, M.Ya. Marov*** M.V. Keldysh Institute of Applied Mathematics, RAS, Moscow

** V.I. Vernadsky Institute of Geochemistry and Analytical Chemistry, RAS, Moscow

The Third Moscow Solar System SymposiumSpace Research Institute, Moscow, Russia

October 8-12, 2012

The Background

The authors’ study in the field of stellar-planetary cosmogony is rooted in their research of:

• multicomponent turbulent gases;• heterogenic mechanics;• physical - chemical kinetics;• non-equilibrium thermodynamics;• magnetohydrodynamics;• chaotic and self-organization processes;• coagulation processes;• mechanisms of hydrodynamic instability.

Based on the results derived from these research methods of mathematical modeling were developed in application to disc formation and evolution involving Solar system origin.

What do we really know Primary dust particles may combine in the collisional processes under velocities <

1 m/s because mostly of Van-der-Waals ‘ force or hydrogen bond.

Amount of monomers (primary fine-dyspersated particles) is fast exhausted as a result of sticking process in the big dust aggregates.

At low speeds of collisions the cloud of dust particles evolves in fractal clusters with quasi monodispersible allocation to the dimensions.

Fractal structure is not preserved when energy of the collisions grows which results in porous structures set up.

Radial drift separates solid particles from the carrying gas phase, particles migrating inside the protoplanetary disc influencing mass distribution and chemical processes in it.

Radial drift of small (< cm) particles can be partially compensated by the turbulent diffusion.

Vertical mixing and small replenishment of monomers thanks to accretion processes are necessary for an explanation of observable disk structure.

Problems demanding the answer

What maximum size of dust formations in the protoplanetary cloud is achievable?

What are the velocity constraints and dust composition/properties to make collisional combinations possible?

What main physical parameters (for example, a velocity, angle of attack, fractality, porosity, material, form, mass, etc.), defining outcomes of collisions?

If the disc is turbulent indeed, what is coherent structures (turbulent curls, rings, etc.) time life to afford concentrations of large size solid aggregates?

How radial drift affects the disc structure and how dust formations behave in dense regions of subdisk?

How fragmentation of relatively small dust formation occur and do they acquire fluffy/fractal structures?

The goal of the study is to model hydrodynamic and coagulation processes in the accretion gas-dust protoplanetary disc based on the new developed approach of dust clusters set up and follow on evolution.

In contrast to the classic models of protoplanetary cloud based on the continuous mechanics approach when fractions of dusty medium and its fractal nature were not distinguished, our model deals with the set of disperse (fluffy) dust aggregates as a specific kind of fractal continuous medium where there are hollow regions not fulfilled with particles.

The point specifically emphasized is that fluffy structure of clusters significantly facilitates probability of integration in the collisional processes because of larger geometrical cross section and patterns of motion in the gas medium in terms of friction force change dependence.

Hydrodynamic modeling of such a medium having non-integer mass dimensionality can be performed in the framework of differential mode of the fractional-integral model with the use of fractional integrals of the order corresponding to fractal dimensionality of the disc medium.

The Goal

Time sequence of the protoplanetary accretion disc evolution including dense dust subdisk formation caused by particles sedimentation

towards equatorial plane and gravity instability development when condition of the critical density is fulfilled followed by the primary dust

clusters and planetesimals set up.Образование пылевых сгущений

0.4-0.9 млн.лет

0.1-0.5 млн.лет 0.5 -1 млн.лет

1-10 млн.летАккумуляция сгущений, образование зародышей

Образование диска

Аккреция газа и пыли через диск на Солнце

Рост частиц, оседание к средней плоскости диска, дрейф к Солнцу

Scenario of the Disc Evolution

Simultaneous protostar and gas-dust formation from a turbulent molecular cloud and continuing accretion of gas and dust onto disc and protostar (~ 5 mil. yrs.).

Viscous turbulent disc dissipation at the T-Tauri stage (~ 5-10 mil. yrs.).

Grows of dust particles from submicron to decimeter size in due course of mutual collisions incorporating electrostatic charges influence.

Dust subdisk formation and its density increase up to critical (~ 0.1-0.5 mil. yrs.).

Subdisc fragmentation into dust clusters due to gravity instability and asteroid-size bodies (10-100 km) formation (~ 0.5-1 mil. yrs.).

Basically, the key problem is: What is the physical mechanism of small particles integration in dust clusters giving birth to larger size formations and eventually to planetesimals of 0.1-1 km across? Is it porous rather than firm particles?

Problems of Meter Solid Structures Formation

Collisions resulting in destructions rather than integration. Fast exhausting of sufficient matter supply owing to radial drift towards protostar and follow up evaporation of small particles. Concurrency of gravitational and brownian coagulation of dust monomers. Concurrency of the radial and vertical motions of disc particles at the accretion stage. Efficiency of evaporation depending on opacity of the inner disc regions and dissipation of the turbulent energy.

Example of Fractal Aggregate

The Refined Compression Model

Fractional dimensionality of a cluster

All primary compact particles (monomers) of the micron size range independently from their real form and material here are considered as solid spheres having the same radius 0r and mass 0 . The number 0n of

monomers, which are a part of isotropic fractal cluster (FC) and the cluster mass сlm are defined by the formulas

f)/( 0g0DrRn , f)/( 0g000

сl DrRnm , 1/ 0g rR .

Here 2/1

1i

2

g /

N

сentrei NR xx is radius gyration (the characteristic dimensionality of an isotropic

cluster), defined as the mean square radius of aggregate measured from its barycentre,

N – the number of elements of the plotting belonging to a cluster,

ix – radius-vector of i-th monomer in a cluster,

сentrex – the FC mass centre position,

)/(ln/ln 0g0f rRnD , ( DD f1 ) – fractal mass dimensionality of cluster defining its priming

quantitative characteristics in Euclidean space with dimensionality D .

Number of Monomers Entering Cluster

Initial Cluster Forms and Collisions Outcomes

Mass dimensionality of fractal aggregates in a disk

Information about fractional dimensionality, size and properties of FC has weak experimental acknowledgement and still bases on outcomes of theoretical models.

In particular, mass dimensionality fD is defined, as a rule,

using numerical modeling of cluster behavior in gravitational (or electrical) field with the help of in-situ methods of grouping process. These methods differ in various details of cluster-cluster aggregation, in particular:

the way of cluster driving (rectilinear or Brownian);

characteristics of cluster junction depending on the probability of sticking at mutual tangency;

presence or absence of full restructuring (at which clusters are bound at three points);

isotropy violation of combined clusters, bound, for example, with directed electrical dipole in exterior electric field, clusters non-sphericity etc.

Fractal Dust Clusters Generation We assume that at the early stage of the gas-dust disc evolution monomers

embedded in the gas phase coalesced in the collision events provided relative velocities were 10 ∼ cm/s.

Collisions result in both mechanical and chemical couplings and dust clusters with mono-disperse size distribution and fluffy structure are formed.

Such cluster structures have fractional fractal dimension

As clusters grow forming large fractal clusters mechanism of particle-cluster interaction changes to cluster-cluster interaction giving rise to fractal structures of larger mass dimension.

Water ice clusters following this scenario may grow under much higher velocities (up to 50-60 м/с) ∼ and their mass dimension achieves 2.5∼ while compression rate depends on collision energy.

Therefore, large compressed dust clusters may form in the collisions of fluffy fractal clusters.

Fractal medium A great number of small clusters joining uni-modal

friable aggregates (FА) are organized so that the mean density decreases according to the law

f3g00

cl / DRr ,

where 3000 4/3 r is monomer material mass

density. That means that characteristic property FА is its ability to trap the big space (for the account of building of the openwork, strongly branched out structure) using smaller amount of material in comparison with dense aggregate. As there are hollows not filled with material the disk medium dust FA cannot be described as a traditional continuous medium, but it should be considered as a fractal medium with areas not filled with its particles.

Modeling of non-integer mass dimensionality media can be performed within the frame of fractional-integral methods using fractional integrals with the order defined by fractal dimensionality fD .

Evolutionary hydrodynamic model of the formation and growth of fluffy dust aggregates (clusters) in disperse medium of the protoplanetary laminar disc is developed.

Basically, the model proceeds from the idea of fractal structure of the primary dust clusters composed originally of the gas and submicron dust particles, which eventually results in planetesimals set up.

The disc medium is considered as thermodynamic heterogeneous complex consisted of two interacting subsystems: gas phase of the solar composition (gas continuum) and polydisperse fractal dust phase.

Polydisperse phase is addressed as multi-rate heterogenic medium composed of dust fractal aggregates and pristine condensed monomers. These subsystems are assumed to fill up simultaneously every volume of the Euclid phase space.

An original approach to the modeling of hydrodynamic and coagulation processes in such a complex is suggested. It is shown that the process of cluster-cluster coagulation and their partial integration gives rise to the progressive aggregates growing.

Dust aggregates of different scales and their internal structure influencing the follow on formation of the intermediate fluffy proto-planetesimals are specially addressed, the latter appearing as the result of the combined physical-chemical and hydrodynamic processes similar to the processes of fractal clusters grow.

The Baseline

The Model: Basic Assumptions

i) primary dust particles – unimodal on composition, firm and not distorted, spheric under the form and monodispersible;

ii) incompressibility of a stuff of monomers is supposed, const0 ;

iii) true denseness of a stuff of monomers much more than a true denseness of gas compounding disk system, g0

~ ;

iv) volume content сls of a disperse phase it is not so great, so members of order 2сl )(s

can be neglected;

v) bearing phase – compressed perfect gas;

vi) viscosity of a disperse phase can to be considered;

vii) condition of a thermal equilibrium of gas and dispersible phases is supposed;

viii) the heterogeneous continuum is considered in one- pressure approach; ix) it is supposed that the fractal medium of dust clusters in some macrovolume W has mass dimensionality fD , and dimensionality on its border W is equal d (generally dimensionality d is

not equal 2 and fD -1).

Mass Balance Equations for Gas and Dust Disk Fractions

0ggg Utd

d, sgg

~ ,

сlk

clk

Q

kj1,jkjk

Dсlk

сlk

kD,

J Nmtd

d

U , сlk

clk

сlk Nm ,

here clj

clkk,j

сlkjkJ NNm – intensity of an interchanging in masses between FC for the account

of collisions; kj2

gjgkP

k,jj,k )( wRR – intensity of an association rate and k and

j- clusters (the coagulation kernel); сlj

2jj,k

сlk

сlj)k(

сlj

1k

jk;1jj)(k,j

сlk 2

1NNNNN

Q

– velocity of

a modification of a numerical denseness of clusters сlkN of k -th kind at the expense of coagulation

processes; )(),,()(

kf

kD,

UxdDc

ttd

d generalized total derivative on time;

k2f1

3kD ),(),( UxxU dcDc generalized divergence;

),(),(),,( 2f1

3f xxx dcDcdDc .

Momentum Balance Equations for the Gas and

Dust Disc Fractions x

xFPU

3g

Q

1kgk,gg

MGps

dt

dgg

,

3

clk

Q

kj1;jkj,kjjkkg,

Dсlkk

kD,

clk )(

xFUUFU MG

Jpsdt

d

, Q)1,...,(k .

Here gxx 3/MG – vector of acceleration of gravitational force (G and M – gravitation

constant and mass of the proto-sun); gP – tensor of viscous strains;

)(k gk

kg,

clk

gBkg, UUF D

NT

– force of interaction between clusters k -th kind and a bearing gas phase (kg,D – quotient of binary

diffusion of clusters k -th kind in gas);

jkclj

clk

clkk,jkj, wF NNm

, kjjk UUw

– force of interaction of clusters k - th and j-th kinds at the expense of their collision.

AdcDcA ),(),( 2fD xx 1

3 – generalised del;

Modes of Fractal Clusters Interaction

Cluster size and physical properties depend on motion patterns of primary monomers before collision and coalescence capacity.

Two mechanisms of grow of clusters possessing fractal structure are possible, both depending on number density of monomers in the unit volume: either due to monomer attachment to cluster or cluster-cluster aggregation.

Attachment of unit nucleus in moving in the straight direction corresponds to the kinetic regime ; combination of numerous monomers in diffusion motion corresponds to the diffusion or hydrodynamic regime.

Dust Clusters Diffusion in Aero-Disperse Medium

In the case of a kinetic driving condition the diffusivity of small clusters in gas is defined by the

formula 2gkg

gskinkg,

1

28

3

Rn

c

D and resisting strength of a carrying medium is described by Epstejn

law

gkgk/2cl

kclk/2

0

20ggskin

kg, ),()(c

3

28 ff

RmNr D

D

UUF ,

where ggBgs /k Tc is isothermal sound velocity in a gas.

In a diffusion driving condition (gkR ) the diffusivity of clusters in gas is defined by the

formula )(/k4 kg,Dkg,gkgBdif

kg, ReReD CRT g , where g is the coefficient of shift viscosity.

At small Reynolds numbers gkgggkkg, /2 wRRe cluster effective coefficient of aerodynamic

resistance is kg,D /24 ReC diffusivity is equal to ff

/1clk

0

/10Bdif

kg, )(6

k D

g

Dg mr

T

D an medium

resistance of is set by the Stokes law )()(6

gk/1cl

kclk/1

0

0difkg,

ff

UUF

D

Dg mN

r

, gkR .

Generalized Set of the Equations of Motion The following set of the equations of motion underlies the new approach to the

modeling of protoplanetary disc evolution.

,)(3

1)()(

3gg2

Q

1kkg

clk

clk

ggg

g xx

UUUUUUU M

GG

mps

t ggg

.)()()(),,(3

Q

kj1;jjk

cljjk,

clkgk

clkcl

k

Dсlk

kkfk x

xUUUUUUx

U MKG

Gmm

psdDс

t

g

Q).1,...,(k

where clj

clkjk,jk, / mmK ;

j,k –intensity of an association rate and k and j-clusters

(the coagulation kernel);

,,)()(3

24

;,)(3

28

)(

gk1)/1(сl

k/10

0dif

gk/)2(cl

k/20

gs20

clk

ff

fff

Rmсr

Rmcr

m

DD

sg

DDD

G

G

G

kin

Kinetics of Clusters Formation in the Fractal Medium

Coagulation Cores (Kernels) Two groups of models of cluster formations in the disc fractal medium were considered: adhering monomers to cluster; two clusters association. Monomer-cluster coagulation occurs in the rarefied aero-disperse medium due to drag of monomers when colliding with a cluster or due to diffuse sticking of monomer to a cluster surface.

Monomeasures - cluster coagulation

ff

2/DclkD/2

0

20

0

1Bkink1, )(

k8m

rT

− for the rarefied aerodispersible medium when drag forms as a

result of single-valued interferences of primary monomers with FC;

mgs

2/3

gkg1dif

k1, 22

34 cR difD − for diffusion character of adherence of monomers to a cluster

surface (here − m length run of monomers in a bearing gaseous fluid).

Cluster - cluster coagulation 21

clj

1

clkcl

jclk

clj

clk

D/20

20gB

jk,ff

f)()(

k8

DD mmmm

mmrT

− for free driving FC in a disk;

f

ff

/1сlm

сlk

2/1сlm

/1сlk

g

gBpmk,

)(

)()

3

2kD

DD

mm

mmT

− for Brownian coagulation of clusters;

f

11

ffff/1

)()()()(6

clj

11

clk

21

clj

1

clk

0g

02K

pjk,

D

Dmmmm

rz DDD

− for gravitational coagulation FC.

Two Problems in Explanation of Disk Dust Aggregate Growth up to 1m Size Range

First problem (not considered in the yielded work) is bound to large solids fragmentation due to destructive collisions.

Second difficulty is related to a fast loss of a "building material” due to particle radial drift. The drift velocity strongly depends on the particle size (for example, a velocity of drift of solids of the meter range on 1 а.е. in a disk compounds ~ 50 m / s, and the velocity of drift of solids of ten meter range is more narrow in 10 times less). Modeling of this appearance with use of fractal representations is one of the cores to those of the research undertaken by us.

We plan to realize two-dimensional model with fractal clusters moving both vertically and radially. As a first step, we confined the review of Brownian and gravitational coagulation of dust monomers and FC in kinetic and diffusion conditions of motion. In our judgment, that will allow to calculate accurately enough the process of erect subsidence and to define their maximum dimensionalities neglecting radial drift and a fragmentation. The second stage must include radial drift of dust "authorized" to particles moving to intrinsic areas of a disk and disappearing in an evaporation region. This will allow revealing the parameters of disk, at which the dust particles can break a drift barrier. Besides, particles, closest to the radial barrier, are to the greatest degree acquisitive to a condition of driving of bearing gas. For example, the presence of magneto-rotational turbulence a particle disk can be trapped in very percolated curls of gas that can retard radial drift twice.

Protoplanetary Disc Stationary Model Generalised equation Smoluhovsky:

).,.....2,1k(),,(),(2

1),(),(

U),()2/(2

сlj)k(

сlj

1k

1jj)(k,j

сlj

1jj,k

сlk

kzсlk

3f2

ff

QtzNtzNtzNtzN

tzNz

zD

Q

DD

Consequence of the equations of driving:

g2g2

clkk V

1

2V

1

1)(V

L

L

L

m , g2g2

clkk V

)1(2

2V

)1(2

1)(V

LL

m ,

K

222g

)1(

2V

YX

X , K 222g

)1(

)1(V

YX

Y, 0Vg z ,

.)(6)(

1)(V f

f/1

/11сlk

g0

02K

gclk

2K

сlkk

Dz m

rz

mzm

D

G

Here )(

Ω)(

clkg

Kclk

mm

GL

,

g cl

k

k21

L

LX ,

g cl

k

k21

1

L

Y ,

gp

g2KU2

1

Generalized Smoluchowski Equation in the Cylindrical Coordinate System

In a basis of research of processes of coagulation of dust particles and the formation of fractal clusters lying generalized Smoluchowski equation.

To solve the generalized Smoluchowski equation are applied the scheme of numerical modeling of Monte-Carlo using the method of variable weight factors.

Essence of imitating model in weighted schemas consists that the considered system of dust condensations in volume V is replaced by system from comparative small number of “model particles”.

Field of flows breaks into a series of the cells which dimensions are small in comparison with characteristic scales of change of hydrodynamic parameter's of medium. The width on time ∆t is small in comparison with mean time between collisions.

Random process is constructed as sequence of collisions of "model particles", played out according to the scheme of Monte- Carlo.

Algorithm "sustains" in cells constant number of “model particles” irrespective of intensity of the collisions, both uniform, and on variable meshes. Algorithm ensures functioning with adaptable grids and calculations 2D and 3D of flows.

Imitative Model of Clusters Formation

Comparison of the Distribution Function (by Mass) of Compact and Fractal Dust Clusters

Relative velocities of collisiones between monomers of radiuses 0r

and 3/0r on distance in 1 а.е. for the minimum mass of the protoplanetary disk

On drawing top it is supposed that the denseness of dust particles is much less, than a gas denseness 1/ gd .

Are shown radial (solid line) and azimuth (dotted line) of a component of relative velocities. Shaped curves show erect relative velocities of particles which depend on height over an equatorial plane and are shown for various values hz / . As for a laminar disk dust particles organise dense beds in this area of more realistic the situation, when 1/ gd or even ≫ 1 is in a

neighbourhood of an equatorial plane.

On drawing bottom are shown relative radial and angular velocities for three various values of magnitudes 0

gd / .1 and 10. For these highly loaded masses,

magnitude should hz / be small, thus, erect velocities less than radial velocities.

Пример расчёта изменения массовой плотности пылевой фазы субдиска. Плотность увеличивается в процессе вертикального и радиального сжатия субдиска. Широкая диагональная полоса - критическая плотность, при которой субдиск становится гравитационно- неустойчивым и распадается на пылевые сгущения. Кривые 1 - 6 соответствуют моментам времени: 0, 1103, 5103, 2104, 5104 и 1105 лет от начала образования субдиска (начальный радиус rd0 = 100 а.е., диаметры частиц d = 10 см и d = 1 см).

0 0 1 1 0 1 0 0

1

2

3

2

456

0 .0 1 0 .11 0

-1 7

1 0-1 5

1 0-1 3

1 0-11

1 0- 9

1 0- 7

1 0- 5

1 0- 3

r , a .e .

d = 1 0 c ì

34

5

d

6

, ã/ñì 3~

0 0 1 1 0 1 0 0

1

2

3

4

56

0 .0 1 0 .11 0

-1 7

1 0-1 5

1 0-1 3

1 0-11

1 0- 9

1 0- 7

1 0- 5

1 0- 3

r , a .e .

d = 1 c ì

3

5

d

6

, ã/ñì 3

24

~

Распад субдискана пылевые сгущения

Критическая плотность

Условие гравитационной неустойчивости субдиска

t~ 5104 лет t~ 6104 лет

Future Priorities  Research role in evolution of a protoplanetary cloud of relative velocities of

collisions of dust aggregates in the areas of a disk “overloaded mass” (for example, in a subdisk, in large-scale rotational formations, in stagnant areas).

Problem of origin of turbulence in a disk at very high values of a Reynolds

number.

Modeling formation of dust fractal aggregates in turbulent aerodispersible disk medium.

Research optical properties of dust clusters including of fractal aggregates. Development evolutionary macroscopic model of the disk considering fractality

of dust aggregates and their collisions with arbitrary velocities.

Building of global evolutionary model of a protoplanetary disk including of turbulization mediums, electromagnetic effects, a fragmentation of dust particles, their erect and radial drift, and also turbulent agitating.

Thanks for attention