2012/07/23

On the non-steady-state
nucleation rate
Takaya Nozawa
Kavli Institute for the Physics andMathematics of the Universe (Kavli IPMU),TODIAS, University of Tokyo1-1. Introduction
cosmic dust universally exits in space
Milky Way (optical)

Milky Way (infrared)

Dust grains absorb UV/optical lights and reemit
it by their thermal radiation at IR wavelengths!

where did these dust grains come from?

1-2. Formation sites of dust
・ mass-loss winds from AGB stars
・ expanding ejecta of supernovae (SNe)－ Huge amounts of dust grains (>108 Msun)
are detected in quasars at redshift z > 5➔ 0.1 Msun of dust per SN is needed toexplain such massive dust at high-z

theoretical works predict that 0.11.0 Msun of dust can form in SNe(e.g., Nozawa et al. 2003)

N/MIR observations detect only
< 10-3 Msun of dust in nearby SNe(e.g., Sakon et al. 2009)

z=6.4, at 350 μm

1-3. Summary of dust mass in CC-SNecool dust in SN1987A
theorymissing cold dust?

young SNRs

swept-up IS dust?
young SNeby courtesy of M. Tanaka

Far-IR to sub-mm observations are essential for revealing
the mass of dust grains produced in the ejecta of SNe2-1. Supersaturation ratio

Tdust is assumed
to equal to T=Tgas

・ supersaturation ratio, S➔ ratio of partial pressure P1 to equilibrium
partial pressure P01
gaseous atomc1
For condensation of dust,
S = P1 / P01 > 1➔ lnS = ln(P1 / P01) > 0P01

P1

solid (bulk) material
where –ΔG0/kT = A/T – B(A=8.64x104, B=19 for C grain)

lnS is higher for lower T and higher c1

2-2. Concept of nucleation theoryc1c2J2c3J3

αn-1c1cn-1 cn
Jn

βncn

・ master equations

2-3. Steady-state nucleation rate
c1c2J2c3J3αn-1c1cn-1 cn
Jn

βncn

・ current density Jn

・ “steady” ➔ Js = J2 = J3 = ・・・ = J∞

2-4. Critical number of atom for nucleation

critical number: nc

μ = 4πa02σ / kT
(σ = 1400 erg cm-2 for bulk Cmaterials)

nc = 2nc

2-5. Non-steady-state nucleation
・ steady-state nucleation rate: Js➔ Js = J2 = J3 = ・・・ = J∞・ non-steady-state nucleationn＊= 100

2-6. Steady and non-steady・ steady-state nucleation rate: Js
gas atoms

nc (= 1-104)

bulk dust

・ non-steady-state nucleation rate: J＊
gas atoms

clusters

n＊= 100

bulk dust

3-1. Basic equations for dust formation
・ Equation of mass conservation(cluster)(bulk dust)

・ Equation of grain growth

Growth rate is independent of grain radius

3-2. Evolution of gas temperature and density
・ Time evolution of c1 and T

τexp = 100 day
τcool = 100 day

c10 = 108, 107,106, 105 cm-3

s = 1.0

4-1. Steady vs. Non-steady (1)c10 = 108 cm-3

c10 = 107 cm-3・ dashed line : steady-state nucleation
・ dotted line : non-steady-state nucleation

The difference between steady and non-steady
nucleation is small for higher initial densities

4-2. Steady vs. Non-steady (2)c10 = 106 cm-3

c10 = 105 cm-3・ dashed line : steady-state nucleation
・ dotted line : non-steady-state nucleation

The difference between steady and non-steadyseems significant for lower densities

4-3. Relaxation time・ relaxation time, τrelax (e.g., Gail et al. 1984)➔ τrelax = τcoll [ (lnS)2 / μ ]τcoll = [ sn 4πa02 <v> c1 ]-1
lnS = 0lnS = μ0.5

τrelax / τcoll =[ (lnS)2 / μ ] < 1➔ steadyτrelax / τcoll =[ (lnS)2 / μ ] > 1➔ non-steady5-1. Steady vs. Non-steady: size distribution (1)c10 = 107 cm-3

The size distributionof bulk grains fromnon-steady-stateis consistent with thatfrom steady-state5-2. Steady vs. Non-steady: size distribution (2)nc ~ 1-2

c10 = 105 cm-3
n＊ = 100

Even for c10 = 105 cm-3,

the size distribution isquite similar between
steady-state and nonsteady-state5-3. Dependence on cluster maximum size

・ solid line : non-steady-state, n＊ = 1000
・ dotted line : non-steady-state, n＊ = 100

The size distribution does not depend on n＊
How many atoms are needed to be defined asbulk dust grains?

5-4. Dependence on sticking probability

The result for c10 = 107
cm-3 and s = 0.1 issimilar to that for c10 =106 cm-3 and s = 1, but
not completely same

6. Summary
・ The difference between steady and non-steadystate nucleation is not significant－ steady-state nucleation rate is a good approximate－ what is the definition of bulk dust grains?

・ In the future work
－ to extend to multiple-element grains like silicate－ to calculate the temperature of dust (clusters)➔ Dust (cluster) temperature is assumed to be equivalent
to gas temperature－ unknown quantities: sticking probability, shapeand surface energy of small clusters