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Evaporation Timescales of HI Clouds
Masahiro Nagashima (Nagasaki)with
Hiroshi Koyama (Kobe), Shu-ichiro Inutsuka (Kyoto)
1. Introduction2. 1D Simulation of cloud evaporation3. Analytic Approximation Forumula4. Summary
ref) Nagashima, Koyama & Inutsuka, 2005, MNRAS, 361, L25 (astro-ph/0503137) Nagashima, Inutsuka & Koyama, submitted (astro-ph/0603259)
2006/02/06 2
Discovery of Tiny HI clouds
sub-pc (~0.01pc) clouds“tiny” HI clouds recently discovered!
Braun & Kanekar (2005)Stanimirovic & Heiles (2005)
simple interpretation: CNM cloud surrounded by WNM
Growth rate? ... Most fundamental quantity
N~1018 /cm2
2006/02/06 3
Formation of Tiny HI Cloudsthermal instability and turbulance behind shock fronts: many clumps form
... one of natural mechanisms of HI clouds formation
Once the clouds formed,how they evolve?
Koyama & Inutuka (2002)
Myers (1978)
Log[n/cc]
Log[T/K]
constantpressure
Physical State of Diffuse ISM Clouds
P /k~103
diffuse cloudsWNM/CNM
under pressure equilibrium
Physical State of Diffuse Clouds
Diffuse (n<102cc) gas in the ISM:
- Co-existence with two phases Warm Neutral Medium (WNM) Cold Neutral Medium (CNM) - pressure balance with heat-conductive interface- negligible self-gravity
~ 1G
~1.2×108 ncm−3
−1/2
yr
l~2×103 T102K
1/2
ncm−3
−1/2
pc
~ Galactic rotation
~ Galaxy size
T~104K , n~10−1/cm3
T~102K , n~101/cm3
x
T
CNM
WNM
2006/02/06 6
Heating/Cooling rates
Koyama & Inutsuka (2000)
Our results do NOT depend on the detailsof the heating/cooling processes
2006/02/06 7
Thermal instability
sequence of heating (Γ)= cooling (Λ)
stable eq.
unstable eq.Γ<Λ
Γ>Λ
WNMCNM
phase transition between WNM and CNM
Koyama & Inutsuka's cooling function used
Frontal motion between WNMand CNM corresponds toCNM domain (cloud) growth
at “saturation pressure” p_sat,Γ=Λ over the whole system
n
P/k
T
∵pressure constant
2006/02/06 8
Evaporation & Condensation
whole system cooling :condensation
whole system heating :evaporation
n
p
cooling > heating
cooling < heating
static front(in 1D)
evaporation
condensation p psat
p psat
→saturation pressure
p psat
p psat
Pressure determines whether evaporation or condensation
Describing evaporation and condensation of clouds
2006/02/06 9
Basic equations
∂∂ t
∇⋅v=0
∂ v∂ t
v⋅∇v=−1∇ p
∂e∂ t
∇⋅e v p∇⋅v=−n2−n∇⋅∇ T
e= p /−1p=n k BT ∝T 1/2
Lheat-loss function
conductivity for neutral gas
L
0 Tlike a cubic function
continuity
EOM
energy eq.
2006/02/06 10
Characteristic scales
cool≃kTn
≃104−106 yr≃3×1011−13 s
cS≃105 T102K
1/2
cm/ s
lF= Tn2
≃3×1015−1018cm
cooling time
sound velocity
Field length: cooling = thermal conductionproviding the thickness offront (transition layer)
vl F /cool≃104cm / s≪cS
ISM is quickly separated into two phasesthen the domain grows slowly
2006/02/06 11
Numerical Simulation
Numrical simulation for spherically symmetric clouds
1D (radial) 、 Lagrangean mesh (2000), 2nd-order Godunovboundary condition: a constant outside pressure (p_WNM)
example (p=p_sat=2823kB; saturation pressure)
Radius of clouds isdefined at which thedensity is half the CNM density
cloud center WNM
snapshots ofa cloud evaporating
2006/02/06 12
Evaporation rate dM/dt
Full Numerical Simulation
Quasi-steady state(QSS) numerical solution
dM/dt at surface
McKee&Cowie's dM/dt
- Numerical simulation is consistent with quasi-steady state ev.- McKee-Cowie's dM/dt overestimates by a factor of 4-5
dM/dt(r)
Case of p=p_sat
Log[r/pc]
dM/dt
dM r dt
=4 r2r v r
2006/02/06 13
Evaporation Timescale
R~0.01pc clouds evaporate in ~Myr
If tiny HI clouds exist ubiquitously, some mechanisms are neededto create the clouds continuously.
expectation from the analytic formula
condensation
Critical Size for Condensation
- When , static clouds can exist.- Static clouds are always unstable.- 0.01pc clouds must evaporate in the standard case of Γ.
p psat
2006/02/06 15
Analytic Approximation Formula
1. Isobaric approximation because of much slower motion than sound velocity Tds=dU+pdV =d(U+pV)-Vdp =dH-Vdp→Tds=dH
2. Assuming quasi-steady state (QSS)
∂∂ t
−R ∂∂r
≡−R∂r
R : speed of a front
R
R0v>0
(in case of evaporation)
−1
k B
dTdt
− dpdt
=−L ∇⋅∇ T
−1
k B
dTdt
=−L ∇⋅∇ T
2006/02/06 16
−1
k B
ud ∂rT=−L ∂r∂rTd−1r
∂rT
Analytic Approximation Formula
in case of d-dim. spherically symmetric, energy eq. becomes
replace with LHS for d=1
−1
k B
u1∂rT=−L ∂r∂rTassuming the same
−1
k B
ud ∂rT=
−1k B
u1∂rTd−1r
∂rT
ud ∂rT=u1∂rT−1
k Bd−1r
∂rTor,
2006/02/06 17
ud ∂rT=u1∂rT−1
k Bd−1r
∂rT
Analytic Approximation Formula
・non-zero dT/dr only at front ・substitution LHS in d=1 for 1st and 2nd terms in RHS
non-zero only at r=R (cloud surface)
(fluid velocity) = (fluid velocity for d=1) + (curvature term)
a non-trivial solution is obtained at r=R, so
ud R=u1R−1
k Bd−1R
RR
mean curvature
2006/02/06 18
Analytic Approximation Formula
Curvature term is very similar to McKee & Cowie (1977)'s evaporation ratewithin a factor of 4-5
M=−4CNM R2 R
∝R∝M 1/3 for R≪Rcrit
By further transformation,
Evaporation rate:
≡V p 1− RcritR ∝R2∝M 2 /3 for R≫Rcrit
substantially differentfrom MC77 for・large p_WNM・large clouds
u R
R=V p−−1
k Bd−1R
RCNM
M=16RR
5 k B
2006/02/06 19
Evaporation Timescale
R~0.01pc clouds evaporate in ~Myr
expectation from the analytic formula
condensation
2006/02/06 20
Summary- Clouds of 0.01 pc evaporate in 1Myr- for >0.1 pc depends strongly on -> large clouds can grow
- TIny clouds must be formed continuously, or, ambient pressure would be much higher than
standard value, probably because of higher heating rate.
- Study of statistical properties such as mass function is important for future analysis
evap PWNM
P /k B≃103.5
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