2012/01/16

Approach to Nucleation in
Astronomical Environments
Takaya Nozawa （野沢 貴也）
Institute for the Physics and Mathematics ofthe Universe (IPMU), University of Tokyo1-1. Introduction
Cosmic dust universally exits in space
when, where, and how is dust formed?
Properties of dust evolves through the processing in
the ISM (sputtering, shattering, coagulation, growth ..)composition, size, and amount of newly formed dust
extinction curve
IR spectral featuredepletion of elements

1-2. Formation sites of dust
when, where, and how is dust formed?
in cooling gas outflows
・ abundant metals (metal: atomic number > 5)・ low gas temperature (T < ~2000 K)・ high gas density (n > ~106 cm-3)
possible formation sites of dust
－ expanding ejecta of supernovae－ mass-loss winds of AGB stars－ molecular clouds (grain growth only)
what kinds of, what size of, how much dust is formed

1-3. Missing-dust problem in CCSNe
by courtesy of M. Tanaka
cool dust in SN1987A
theorycold dust ??

young SNRs

young SNe

There is a big difference in the mass of dust between
observational estimates and theoretical predictions!!

1-3. Missing-dust problem in CCSNe
M. Tanaka
Matsuura, by..., courtesy
TN, et al. of2011,
Sciencecool dust in SN1987A
theorycold dust ??

young SNRs

young SNe

Herschel detects cool (20K) dustofin0.4-0.7
Msunoftoward
SN 1987A!
There is a big difference
the mass
dust between
observational estimates and theoretical predictions!!

2-1. How do dust grains form?
chemical approachgaseous atoms nucleation approach

molecules
formation ofseed nuclei???
large molecules
dust grains

2-2. Supersaturation ratio

Tdust is assumed① to equal to T=Tgas

・ supersaturation ratio, S➔ ratio of real partial pressure to equilibrium
partial pressure
gaseous atom

c1Pe,1For condensation of dust,
S = P1 / Pe,1 > 1➔ lnS = ln(P1 / Pe,1) > 0

P1

solid (bulk) materiallnS = A / T – B + ln(c1kT / P0)
where –ΔG0/kT = A/T – B (A=8.64x104, B=19 for C grains)lnS is higher for lower T and higher c1

2-3. Behavior of supersaturation ratio

increasing lnSlnS = A / T – B + ln(c1kT / P0)
lnS is higher for lower T and higher c1
・ number density of carbon atom to be lnS > 0－ c1 >~ 108 cm-3 at 2000 K－ c1 >~ 106 cm-3 at 1800 K－ c1 >~ 3x103 cm-3 at 1600 K

3-1. Concept of nucleation
c1c2

J1

c3

J2

αnc1cnJncn+1βncn+1・ Master equations
sn=1 is assumed
dcn/dt = Jn-1 – Jn (n ≥ 2) ③ for every n
Jn = αn c1 cn – βn cn+1 (unit of Jn : cm-3 s-1)αn = sn Sn <v> = sn 4πa02 n2/3 <v>
βn = αn ce,1 (ce,n / ce,n+1)
coagulation between② clusters is neglected

small clusters are④ assumed to be spheres

3-2. Steady-state nucleation rate
c1

c2

J1

c3

J2

αnc1cnJn

cn+1

βncn+1

Jn = αnc1cn[1 – (ce,1/c1)(ce,n/cn)(cn+1/ce,n+1)]“steady” ➔ Js = J1 = J2 = ・・・ = Jn ⑤
Js = {1/(α1c1) + ∑ (ce,1/c1)n 1/(αnc1ce,n+1)}-1(ce,1/c1)n (1/ce,n+1) = (1/c1) exp [g(n)]where g(n) = μ(n-1)2/3 – (n-1)lnS➔ Js = {1/(α1c1) + ∑ 1/(αnc12) exp[g(n)]}-1

3-3. Critical radius
Js = {1/(α1c1) + ∑ 1/(αnc12) exp[g(n)] }-1g(n) = μ(n-1)2/3 –(n-1)lnS
μ = 4πa02σ / kT (σ = 1400 erg cm-3 for C grains)
The surface energyof small clusters is⑥ assumed to be thesame as that of bulk

critical radius

3-4. Critical radius of nucleation
critical radius

n＊ = 2⑦ n＊ = n＊ – 1
assuming n＊ >> 1

Js = {1/(α1c1) + ∑ 1/(αnc12) exp[g(n)] }-1

3-5. Comparison of steady-state nucleation⑦small n＊・ solid line : approximation formula for nucleation rate

・ dashed line ： summation form for nucleation rate

Js = {1/(α1c1) + ∑ 1/(αnc12) exp[g(n)] }-1

3-5. Comparison of steady-state nucleation
n＊ = n＊ – 1⑦ assuming n＊ >> 1

reasonable !!
small n＊・ solid line : approximation formula for nucleation rate

・ dashed line ： summation form for nucleation rate

Js = {1/(α1c1) + ∑ 1/(αnc12) exp[g(n)] }-1

4-1. Non-steady nucleation rate
Js = {1/(α1c1) + ∑ 1/(αnc12) exp[g(n)] }-1“steady” ➔ Js = J1 = J2 = ・・・ = Jn ⑤
Jn = αnc1cn[1 – (ce,1/c1)(ce,n/cn)(cn+1/ce,n+1)]➔ Jn = αnc1 {cn – cn+1 exp[g’(n)]}g’(n) = μ [(n-1)2/3 – (n-2)2/3] – lnS
・ Master equations
dcn/dt = Jn-1 – Jn (2 ≤ n ≤ 100)
T = T0 (t / t0)-3/4 (T0 = 3000 K, t0 = 150 day)c1 = c10 (t / t0)-3 (c10 = 105 – 108 cm-3)

4-2. Basic equations of dust formation
Equation of conservation for atoms

Equation of grain growth4-3. Steady vs. Non-steady (1)c10 = 108 cm-3

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

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

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

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

The difference between steady and non-steady
nucleation is significant for lower densities

5-1. Relaxation time・ relaxation time, τrelax➔ τrelax = τcoll [ (lnS)2 / μ ]τcoll = [ sn 4πa02 <v> c1 ]-1
τrelax / τcoll = [ (lnS)2 / μ ] < 1 ➔ steadyτrelax / τcoll = [ (lnS)2 / μ ] > 1 ➔ non-steady5-2. Trajectories on T-c1 plot for dust formation

・ higher c1 ➔ dust condenses at lower lnS ➔ steady・ lower c1 ➔ dust condenses at higher lnS ➔ non-steady6. Current problems in dust formation theory① Dust temperature is assumed to be equivalent togas temperature② The coagulation between clusters is neglected③ The sticking probability is assumed to be one④ Small clusters are assumed to be spherical⑤ Js = J1 = J2 = ・・・ = Jn＊ for steady-state nucleation⑥ The surface energy of bulk material is used as thatof small clusters⑦ n＊ >> 1 is assumed for classical nucleation rate
➔ this assumption is valid