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Mass Loss at the Tip of the AGB
L. A. Willson
Iowa State University
©L. A. Willson 5/2004
I will try to persuade you that
• 1. None of the mass loss formulae now
in print provide what is needed for
stellar evolution and PN formation
• 2. However, we know quite well which
stars are dying from terminal mass loss
• 3. There is a problem with standard
core mass - luminosity relations
©L. A. Willson 5/2004
The physics of mass loss
Radiative transfer in dynamical
atmospheres with periodic shocks -Non-LTE, non-RE
Molecules and grains in quenched flow
Non-equilibrium H2/H (Bowen)
Metastable eutectic condensates?
(Nuth)
Gas-grain interactions
Quenched chemistry vs. equilibrium:
What is the equilibrium state of cake in a hot oven?
©L. A. Willson 5/2004
Some things we do know
• Periodicity condition applied to Miras
– Constraints on M and R of AGB tip stars, i.e.
constraints on the evolutionary tracks
• Importance of departure from LTE and RE in
the dynamics (adiabatic shocks)
– Mass loss is enhanced by departures from RE
– Makes it difficult to use dynamic models for
radiative transfer
– Makes it difficult to study the mass loss process
observationally
©L. A. Willson 5/2004
Periodicity condition
H&W, W&H 1979
Po
must be ≤ P
NOT ∆v = gP
because ∆r/r is
not <<1
If the material does not have enough time to fall back to its initial
position, then the atmosphere expands.
Expansion => a stable periodic structure with larger scale height
and/or a wind
©L. A. Willson 5/2004
∆v/vescape depends on Q = P√(/Sun)
2
∆v=gP
2vo/vescape
Note:
Overtone
models have
both smaller Q
and smaller
vescape for a
given P
1.5
1
F-mode
Mira
models
0.5
Q = 0.01
0.1
©L. A. Willson 5/2004
1
Isothermal or adiabatic shocks?
Deep in the atmosphere, cooling times are << P
and the shocks are effectively isothermal.
LTE cooling times are fast; Non-LTE cooling (or
heating) times increase with decreasing density
At some critical density, cooling times become ~ P
For densities << critical, the shocks will be effectively
adiabatic.
©L. A. Willson 5/2004
Isothermal
/
Adiabatic
hybrid
model:
Mass loss rate
depends on the
cooling rates
(Willson & Hill 1979)
©L. A. Willson 5/2004
Shock compression -> heating -> radiative losses;
expansion between shocks -> cooling and slower radiative gains.
T/1000K
10
The level of T here
depends on details of the
model including non-LTE
cooling and mass outflow
8
6
4
2
T<TRE
2
4
6
8
©L. A. Willson 5/2004
10
12
14
Bowen model
16 R*
Shocks form and propagate outward
Vesc
2
4
6
8
10
R/R*
©L. A. Willson 5/2004
12
14
16
Bowen model
Two kinds of models
• Models developed to study physical and
chemical processes in detail
– Höfner, others:
• Dynamics with radiative transfer to fit spectra.
– Sedlmayr, others:
• Dust nucleation & growth in carbon stars
• Models developed to study the pattern of
mass loss at the tip of the AGB
– Bowen models approximate nonLTE, transfer and
dust processes for O-rich stars
©L. A. Willson 5/2004
NOTE
One cannot run LTE radiative transfer on a
Bowen model (or any other model with
approximate nLTE) because the cooling
assumes nLTE; LTE transfer gets more energy
out than was put in and detailed nLTE doesn’t
generally match schematic cooling rates
everywhere.
Bowen’s models were not designed for radiative
transfer computations, but rather …
©L. A. Willson 5/2004
…to study the evolutionary pattern
• Models designed to reveal the evolution of stars
through the Mira region at the tip of the AGB
• Mass loss rates are very sensitive to L, M, R, Teff …
• R(L, M) sensitive to mixing length (and hard to
measure)
=> Hard to predict what a particular star will do,
but there is a
very robust pattern for the evolution
at the tip of the AGB
©L. A. Willson 5/2004
Some reasons for believing these
models get the pattern right
• They fit and explain the Mira P-L relation
• They fit and explain empirical correlations
of mass loss rates with stellar parameters
©L. A. Willson 5/2004
Matching models to
populations
• Evolutionary tracks => R(L, M, Z, ) and L(t)
• Mass loss models => M(R, L, M, Z)
• Together, these produce predictions of M(t)
and thus of the maximum LAGB
©L. A. Willson 5/2004
Models by Bowen (1995 grid) using Iben R(L, M, Z):
Mdot vs. L — Solar metallicity
Mdot (Msun/yr)
10 -4
M = 0.7 1.0 1.4 2.0
2.8
4.0
The dependence of
mass loss rates on
stellar parameters
along the AGB is
VERY steep (fit by
LxM-y with
10<x, y<20.
10 -5
10 -6
10 -7
10 -8
10 -9
10 -10
1000
10000
100000
L/Lsun
Note: Because R vs. L, M is given by the evolutionary track, L serves
as proxy for L, R, and Teff, and the steep dependence on L in the figure
could be all R, all L, all Teff or (most likely) a combination of these.
©L. A. Willson 5/2004
Stars evolving up the AGB lose little mass until they are close to
“the cliff” where tmassloss ~ tnuclear:
This is a “lemming diagram”
4
0.6
2.8
logM
2
0.4
Mass Loss Rates
Too Low To
Measure
logM= -10 -8
-6
-4
First surveys
Chandrasekhar
will
find
limit
mostly stars
near the cliff
1.4
0.2
1
Short
core mass
lifetime,
obscured star
0.7
0.0
-0.2
3.0
3.2
3.4
3.6
3.8
4.0
4.2
logL
Bowen and Willson 1991
©L. A. Willson 5/2004
4.4
4.6
4.8
Empirical relations result from selection effects with very steep
dependence of mass loss rates on stellar parameters.
-4
x10
slope -10
cliff stars with
M/Sun indicated
-5
log(Mdot)
Reimers’
relation is a
kind of mainsequence
for mass loss:
It tells us
which stars are
losing mass,
not how one
star will lose
mass.
4.0
2.8
2.0
1.4
-6
0.7
1.0
Reimers' formula
-7
x0.1
slope -0.1
-8
5.6
5.8
6.0
6.2
6.4
logLR/M
©L. A. Willson 5/2004
6.6
6.8
7.0
Miras have high mass loss rates, extended
atmospheres, and large visual amplitudes:
Miras markers for the “cliff”:
logM= -10 -8
-6
4
-4
0.6
2.8
logM
2
0.4
Chandrasekhar
limit
1.4
0.2
0.0
1
0.7
core mass
-0.2
3.0
4.8
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
logL
©L. A. Willson 5/2004
Bowen and Willson 1991
Observations of Miras and OH-IR stars confirm that
Miras mark the location of the cliff:
(K-L is a mass loss
Rate indicator.)
• = Miras
©L. A. Willson 5/2004
The cliff fits the
observed Mira
P-L relation from
the LMC very well.
No parameters were
adjusted to obtain this fit.
5
logL
4
5
logL
Hipparcos distances to
Miras show a lot of
scatter.
4
2.8
4
2
1.4
0.7
1
3
2
2.2
2.4
2.6
logP
©L. A. Willson 5/2004
2.8
How sensitive are these results to
uncertain parameters mixing length (=> R vs. L, M)
cooling rates (affecting mass loss rate vs.
R, L, M)
dust formation physics (affecting mass loss
rate vs. R, L, M)
etc?
©L. A. Willson 5/2004
Increase mass loss rates by a factor of 10
- what happens to the predictions?
The critical mass
.
loss rate = M (L/L)
does not depend
on the mass loss
models, but Lcliff
does.
Mdot vs. L — Solar metallicity
Mdot (Msun/yr)
10 -4
M = 0.7 1.0 1.4 2.0
2.8
4.0
10 -5
10 -6
10 -7
10 -8
10 -9
10 -10
1000
10000
100000
L/Lsun
©L. A. Willson 5/2004
Result:
Cliff values of L
(and associated R)
for a given M are
not very sensitive to
the mass loss law
5
logL
4
Effect on L.vs. P
of ∆logM = 1 is
no more than
∆logL ~ 0.1
5
logL
4
2.8
4
2
1.4
0.7
1
3
2
2.2
2.4
2.6
logP
©L. A. Willson 5/2004
2.8
Bowen (Theory), Reimers, Baud & Habing, and Vassiliadis & Wood (two
independent observed relations) all identify the same Lcliff(M):
What about other mass loss laws?
Reimers formula kills stars
at higher L because it is
not steep enough - hence
the introduction of  and
BH’s introduction of
1/Menvelope
From Willson 2000 ARAA (Vol 38)
©L. A. Willson 5/2004
New, improved mass loss law?
Wachter et al 2002
-4
Wachter et al 2002
Perhaps the problem is with
the Iben tracks.
What would be needed to
get stars to die at the right
L, M with the Wachter et al
mass loss law?
-8
103
L/Sun
105
Using Iben tracks and assuming zero (or small)
RGB mass loss, this law kills stars at too high L
©L. A. Willson 5/2004
Can we use their mass loss law
with different evolutionary tracks?
WachterMdotvL
-0.1
0.1 delta
1 delta
10 delta
-0.2
DeltaLogT needed
-0.3
-0.4
-0.5
Their M ~ T6.81
To get the right
death-line we need to
shift the evolutionary
tracks by
∆LogT = -0.2 to -0.9
-- more of a shift for
higher masses
-0.6
∆logT = 0.3 takes
3500K to 1750K - much
lower than indicated by
any observations
-0.7
-0.8
-0.9
=> Wachter
et
al.’s
mass
loss4 law
cannot be
1
1.5
2
2.5
3
3.5
4.5
forced into agreement
with observed
Mass
deathline for normal evolutionary tracks
0.5
©L. A. Willson 5/2004
HR diagram data
Wachter
Models by Wachter
etBowen
al. Relative
to the cliff:
10
cliff
3
mass
2
1
0.1
1000
1
10
4
10
5
Their models are all low mass & high L, and may describe
post-Mira, post-cliff carbon stars accurately,
but they do not kill stars at the right L(Minitial) and
A. Willsonevolution
5/2004
should not be used for ©L.
stellar
calculations
L
Helium Shell Flashes another complication!
3.6
An
L-Mcore
relation
fits this
part
3.5
logL
3.4
3.3
3.2
3.1
3
2.9
2.8
1.647 10 8
1.648 10 8
1.649 10 8
1.65 108
1.651 10 8
t
During a flash, ∆logL ≤ 0.4 (apart
from very short-lived minimum)
©L. A. Willson 5/2004
1.652 10 8
L and R variation => M modulation*
Post-flash He burning
Quiescent H-burning
The height of Rmax (or Lmax) is not well determined - different models
predict different contrast, from ≤1 to ≥2 x quiescent H burning L.
Where most of the mass comes off is very sensitive to this contrast,
and thus whether most CSPN are H or He burning.
©L. A. Willson 5/2004
During a shell
flash, a Mira
moves along
the P-L relation
for P < 300d but
should leave it
for P > 300d
5
logL
4
5
logL
4
2.8
4
2
1.4
0.7
1
3
2
2.2
2.4
2.6
logP
©L. A. Willson 5/2004
2.8
Initial-final mass relation
Evolution with
mass loss and
standard core
mass - luminosity
relations don’t fit.
Mass loss preAGB tip or ??
Agnes
Bischof
Kim,
MS Thesis
2003
From Weidemann V.,
2000, A&A, 363, 647
There is a deeper problem
©L. A. Willson 5/2004
P => L => Mcore for Miras
Obs.
0.56
theory
0.60 0.64 0.72
0.85
dn
dlogP
0.7
1
200
1.4
2
400
©L. A. Willson 5/2004
2.8
Nearly all Miras
have L such that
we’d expect
Mcore > 0.6 solar
masses.
600 days
Their fate is to be white dwarf stars
Nearly all WD have
masses < 0.6 solar
masses. Observed
©L. A. Willson 5/2004
With or without overshoot
Shell flash peak
(not H-burning
luminosity)
From Herwig,
2000
Although this is 3 solar masses
and shows a limiting core mass >0.6
this is what has to happen for M ≥ 1
to keep the Mira core masses low
to match white dwarf masses
©L. A. Willson 5/2004
Another problem: ∆MRGB
• Mass loss on the RGB may be
– By reaching the Death Zone (cliff region)
– As a result of an ejection during the core flash
The character of the Death Zone is that it is hard to
go there and come out alive Most stars lose everything, or nothing
Losing a lot -> Blue HB and no Miras
Therefore, those that ascend the AGB probably have
∆MRGB mainly from the core flash event
©L. A. Willson 5/2004
masses of Miras on the cliff
0.7
1.0
1.4
2
2.8
This is
consistent with
little or no
mass loss
before the Mira
stage.
200
400
©L. A. Willson 5/2004
600 days
Can we predict Lfinal vs. Z, t?
whose position depends on mass and metallicity.
0.6
0.4
logM
0.2
Chandrasekhar
limit
0.01
0.1
0.3
0.0
Z/Z = 1.0
Again, it’s Lfinal not Mcore
this analysis tells us
core mass
-0.2
3.0
0.001
3.4
3.8
4.2
4.6
logL
These models have dlogLf/dlogZ ~ -0.1 to -0.2; even if
details are wrong, this should be a good estimate, as the
dominant effect (the only effect outside the green patch) is
©L.tracks
A. Willson 5/2004
the shift in evolutionary
dlogT/dlogZ at constant M.
Robust Results
Mass loss rates increase precipitously
stars die very soon after reaching dlogM/dlogL = -1
observations of mass loss rates and/or location of the Mira
variables tells you which stars are now dying.
Mass loss rates are sensitive to a combination of R, L, and M
such that low metallicity stars, smaller at a given L, M, reach
higher L before dying.
The generation of dust and oxygen- or carbon-rich molecules
further enhances mass loss rates for high Z stars.
Core masses do not grow as large as standard Mc-L relations
predict they should
©L. A. Willson 5/2004
Is this the usual development?
©L. A. Willson 5/2004
Symmetrical stars -> bipolar PNe
All stars pass through the low-envelope-mass zone
logM= -10 -8
-6
4
-4
0.6
2.8
logM
Miras
2
0.4
1.4
0.2
0.0
OH-IR
stars
Chandrasekhar
limit
1
Where bipolarity
arises for most PNe
0.7
core mass
-0.2
3.0
4.8
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
logL
To spin up the envelope with a companion, need m/Menvelope > ~ 0.1
Other reservoirs of angular momentum also
=> low envelope mass is necessary
get5/2004
bipolar symmetry
©L. A.to
Willson
Conclusions: What we don’t know
• We can’t yet derive a remnant mass from an initial mass
• We can’t yet predict the mass loss rate for a given star
accurately
• We don’t know whether AGB stars lose mass mostly
near the He shell flash peak or mostly during quiescent
H shell burning
• The models that fit the aggregate properties of the
populations can’t be used for radiative transfer
• Models used for radiative transfer and/or studies of dust
nucleation do not yet include all the physics needed to map
the mass loss accurately
©L. A. Willson 5/2004
Conclusions: What we do know
• We do know reasonably well where these stars die
- that is, the location of the “cliff”or death-line
both from empirical studies and from theory
• We also know that lower Z stars will reach the
same mass loss rate at a higher L for a given M,
mostly because they are smaller but also
because they make fewer molecules and grains
• We also know that the standard core mass luminosity relations overestimate Mcore for the bulk
of the Miras.
©L. A. Willson 5/2004