Big, Cool, and Losing mass:
Dependence of mass loss rates on L,
R, M and Z
Examples of mass loss formulae
4x10-13 L1 R1 M-1
for red giants
4x10-1210^(0.0123P)
for Miras
L. A. Willson
Iowa State
University
More modified Reimers’ formulae
•
Mdot = 1.8e-12 (MZams/8) LR/M
Volk & Kwok 1988
•
Mdot = 1.15e-13 !BVK LR/M
with !BVK = MZAMS2 - 10.6MZAMS + 10.2
Bryan, Volk & Kwok 1990
Reimers 1975
Vassiliadis & Wood 1993
2.2x10-6(L/104)1.05(Teff/3500)-6.3
for LMC high-L sources
van Loon et al. 2005
7.2x108 Teff6.81 L2.47 M-1.95
from models for carbon stars
Wachter et al. 2002
8x10-14LR/M (Teff/4000)3.5(1+gSun/4300g*)
a modified Reimers’relation
Schröder & Cuntz 2005
Compare formulae for Mdot = -dM/dt:
• Reduce to Mdot(L,M) or Mdot(L,M,Z) using
(as needed)
P(M,R), Teff(L,R), R(L,M,Z)*.
• Find the deathline Mdot/M = Ldot/L*
•
Mdot = 4e-13 (Me,o/Me) LR/M
•
MdotBi = 4.8x10-9 MZams-2.1 L2.7 MdotR
•
MdotBi =
4.8x10-9
M-2.1
Baud & Habing 1983
L2.7
Blöcker 1995
MdotR
(Blöcker gave two versions)
• Near the deathline, find dlogMdot/dlogL and
dlogMdot/dlogM
*from evolutionary models.
First iteration: Steady (mean) dlogL/dt.
Later: Shell flash modulations included.
What models and observations tell us about
Mdot
Reimers
0
VW
SC
-4
Wachter
BW
logMdot
vanLoon
-2
• 1. Some observations, most models =>
which stars lose mass,
i.e. P, L, R, M or a combination of these for
which Mdot/M ! Ldot/L or tMdot ! tnuclear
• 2. Different observations, most models
=> dlogMdot/dlogL holding M fixed and
constraining R; dlogMdot/dlogM holding L
and R fixed.
-6
-8
-10
3
3.5
logL
4
4.5
5
Where the mass-losing AGB stars are found
For the superwind phase, near Lcrit
• nearly all the action takes place in the
“Death Zone”, where
There are 3 important numbers to compare
a mass loss formula with observations
• The Lcrit(M, Z) where Mdot/M = Ldot/L for
stars evolving up an evolutionary track
R(L, M, Z)
• 0.1 < (Mdot/M)/(Ldot/L) < 10
• The slope dlogMdot/dlogL at Lcrit
• Therefore ….
• The slope dlogMdot/dlogM at Lcrit
Constraining those numbers
Lcrit for Reimers, VW and BW
VW logLcrit
BW logLcrit
RR Lcrit
• Lcrit: Relatively easy to constrain, nearly all
observational relations give this
3lawsLcritAndSlope
4.1
4
• dlogMdot/dlogL at Lc: Look to distributions
N(L), N(P), N(Mdot) to constrain this one;
equivalent = duration of mass loss epoch
(time from 0.1 to 10 x Mdotc or from
10-7 to 10-5 solar masses/yr for AGB stars)
logLcrit
3.9
3.8
solid: BW
dashed: VW
dotted: R
3.7
3.6
• dlogMdot/dlogM: Also affects the duration and
distributions
3.5
3.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Mass
Adding lower-metallicity models
The deathzone: 0.1
to 10 x Mdotcrit = M/tnuc
3lawsLcritAndSlope
4.2
3lawsLcritAndSlope
Z /Sun =
4.8
" = (dlogMdot/dlogL)-1
0.01
4.6
4
4.4
4.2
logL
LogL
3.8
3.6
VW logLcrit
VW logLcrit+!
VW logLcrit-!
BW logLcrit
BW logLcrit+!
BW logLcrit-!
RR Lcrit
3.4
3.2
0.6
0.8
1
1.2
1.4
Mass
1.6
1.8
2
0.1
Z /Sun =
0.3
0.001
1
4
3.8
VW logLcrit
VW logLcrit+!
VW logLcrit-!
BW logLcrit
BW logLcrit+!
BW logLcrit-!
3.6
3.4
2.2
3.2
0.5
1
1.5
2
Mass
2.5
3
Short-P
Miras have
lower
mass,
lower Z
progenitors
We know where stars die
Reverse the axes and add tracks
BW logLcrit
BW logLcrit+!
BW logLcrit-!
Cliff or DeathLine
3lawsLcritAndSlope
log"= -10 -8
0.6
4
0.6
0.4
2 MSun
0.4
Chandrasekhar
2
logM
0.2
logM
-4
2.8
0.2
limit
1.4
1 MSun
0
1
0.0
0.7
-0.2
-0.4
core mass needed
3
3.5
-6
4
4.5
5
logL
What observations tell us
Observations of Mdot, fitted to give Mdot as a function
of L, R, M, P and/or T, generally yield the location of
the death line,
Lcrit or Rcrit or Pcrit
from setting
log(M/tnuc) = Mdot(L, R, M, P and/or Teff)
using an evolutionary track R(L, M, Z, ML)
and L(R, Teff) or P(M,R) as needed
upper limit
to core mass
-0.2
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
logL
The extent of the death zone in L
•How much does L need to change to get
Mdot to increase by 2 orders of magnitude?
•That will be roughly* from
Lcrit + 1/(dlogMdot/dlogL) to
Lcrit - 1/(dlogMdot/dlogL) using M = const*
•Or (observationally simpler) from
10-7 to 10-5 Msun/yr:
Time = 2/(dlogMdot/dlogL)*
* After Lcrit the mass will be significantly smaller than it was,
so we also need dlogMdot/dlogL to do this right.
We would really like to know how
Mdot depends on L,R,M and Z
3lawsLcritAndSlope
VW slope@Lcrit
BW slope@lcrit Z=1
17
16
solid: BW Z=solar
dashed: VW
compare
Reimers = 1.68
Blöcker= 4.38
vanLoon = 0.77
SC = 2.65
Wachter = 3.08
15
slope at Lcrit
• Locally, we may study the slopes
dlogMdot/dlogL and dlogMdot/dlogM
• The initial-final mass relation is set by what is
happening in the range
0.1 # (Mdot/M)/(Ldot/L) # 10
or
0.1 # tMdot/tnuc # 10
slope dlogMdot/dlogL at Lc
14
13
12
11
and is close to Mcore(Lcrit)
for a steep Mdot formula
10
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Mass
If the slope is steep enough it affects
only the shape of the corner in
logMdot vs time or logL, not Mi vs Mf
Duration of terminal mass loss for M = 1
formula
dlog(dM/dt)/dlogL duration,
106 yrs
Reimers
1.68
# 3.6
VW
15.75
# 0.4
vanLoon
0.77
# 7.8
SC
2.65
# 2.3
Wachter
3.08
# 2.0
BW Simple
14.3
# 0.4
Observed: ~0.2e6 years
Models for mass loss*
Conclusion, part I:
• Published mass loss laws lead to very
different dlogMdot/dlogL and
dlogMdot/dlogM
• Duration and distribution of mass-losing
stars => these slopes should be very
steep, or for Mdot = ALB along an
evolutionary track:
10 # B # 20
This eliminates most published mass loss
formulae
Pulsation
• Periodic structure
– extended atmosphere
• Shocks
– compression heating
– expansion cooling
• Multiple shocks per cycle
– when P/Pacoustic > 1
• Pulsation
– Essential
– Shocks transfer outward momentum to the gas
• Dust
– Makes a big difference in some stars
– Molecule & grain chemistry is complex!
• Non-LTE gas reheating, cooling
– Also essential, particularly in low Z stars
*In case you missed Don’s talk, and a little ancient history
Periodic Structure for one shock per cycle
• Set r(tS+P) = r(tS)
where tS = time of shock passage
• During most of the cycle the motion is ballistic
• For very short periods (Q < 0.01d) this gives shock
"v # gP
• For most pulsating stars, 0.02 < Q < 0.2 =>
periodicity condition
vescP/2r = 38.3 Qd
! (#/(1-#2))+ (1- #2)-3/2 arcsin(#)
where #=vout!"v/2
From Hill & Willson 1978 and Willson & Hill 1978; also Willson 2000
Periodicity constraint on "v
Periodic Pulsation Model
Photosphere
is the red line
2.0
6
!v=gP
Two shocks
per cycle
when
P>Pacoustic
No periodic solution
!v/v
esc
1.5
1.0
Allowed
Refrigerated
region
0.5
5
R/R*
4
3
2
1
0.0
.0001
.001
.01
.1
Time ->
1
Q(r), days
Part I: (Some) Models match
observations: Are we done?
• Non-LTE cooling/heating of
compressed/expanding gas is important in
determining atmospheric structure
• Means that dynamics and radiative transfer
should be treated together
• May be approximately described in terms of
critical density:
Q/QLTE = 1/(1+$x/$) Non-LTE
• Detailed calculations so far only for some of
the constituents and some conditions
Temperature versus
radius from two
models that are
identical, except that
in the top panel the
critical density $x
=10-10 gm cm3, while
in the bottom panel
$x = 10-18 gm cm3.
Lower critical
density results in a
more extended
region where T ~TRE
between shocks.
(Figure courtesy of
GH Bowen.)
From Willson 2000
Part I: (Some) Models match
observations: Are we done?
• Can non-grey silicate grains
drive/enhance mass loss?
– They are too transparent in the optical/IR
Insert here
• 1. What is the problem? Contrasting
carbon grains and silicates
• 2. Why might non-LTE chemistry solve
the problem?
– Add iron for opacity: they get too hot
Noted independently by S. Höfner,
P. Woitke, 2006
Possible solution (Höfner & Anderson 2007):
Non-LTE chemistry => both silicate and
carbon grains form in oxygen-rich stars.
For carbon grains
Carbon grains
Silicate
grain
materials
High opacity
near 1
micron is
needed to
drive mass
loss, but
adding Fe
also => hot
grains
illustration from Woitke 2006
Radiative acceleration of a grain
1
# kdust F" (r)d"
! dust (r) = c
GM (r)
r2
Non-LTE chemistry
• When conditions change on a time
scale (dlogX/dt)-1 < destruction time
scales, then the abundances of various
species depends more on the rate of
formation than on the equilibrium value.
Thus one may not have all the C or all
the O in CO, but some of each left to
form grains.
Radiative equilibrium for an opaque grain or a planet
"*
%
$ + A() )Lstar () )d ) '
xarea
1
"
%"
%$ 0
'
! Teq4 = $
'
$
2' *
# rarea & # 4( d & $
'
A(
)
)L
(
)
)d
)
grain
$+
'
#0
&
May also be relevant
• In a range of density where collisional
excitation is followed by radiative deexcitation, under typical conditions the
excitation T is < LTE Texcitation
• kT is ~ 0.1 eV and the excitation
energies of molecules and atoms
include 0.1 - 1 eV.
• Under-excited atoms are more likely to
stick to a growing grain.
Wish List
• Successful models for mass loss from
silicate-rich stars with non-gray grains.
• An appropriate Q/QLTE to use, or an
efficient method for treating the coupling
between the gas and the radiation field.
• Better observational constraints on the
slopes (dlogMdot/dlogL and
dlogMdot/dlogM).
Questions?