VII. Hydrodynamic theory of stellar winds
observations
winds exist everywhere in the HRD
hydrodynamic theory needed
to describe stellar atmospheres
with winds
Unified Model Atmospheres:
- based on the hydrodynamics of radiation
driven winds
- fully self-consistent hydrodynamic transition
from hydrostatic photosphere to supersonic
winds
- first developed by
R. Gabler, A. Gabler, Kudritzki, Pauldrach, Puls
1989, A&A 226, 162
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1. Hydrodynamical equations of an ideal compressible fluid
Definitions:
momentum density (momentum per unit volume)
force per unit volume,
acts on dV
momentum flux tensor
flux of momentum of component i
perpendicular to direction j
macroscopic
momentum flow
stochastic motion
p = gas pressure
eq. of continuity
Gauss theorem
Eq. 1
2
momentum equation
change of volume
momentum change
momentum with time
through external forces
momentum flow through
borders of V
Gauss theorem
definition
Eq. 2
note: for any scalar or vector
Eq. **
=0
see, Eq. 1
substantial derivative
in co-moving frame
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energy equation
e internal energy per unit mass
for an ideal gas
1st law of thermodynamics for volume
element dV with mass
work done
per unit mass
energy equation in co-moving frame
external energy
added or lost
per unit mass
Eq. 3a
Eq. 3b
external energy added or lost
i.e., radiation, sound waves,
convection, friction,
magnetic fields, etc.
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2. Stationary, spherically symmetric winds
r is dimensionless radial
coordinate in units of
stellar radius
as in previous chapters
Eq. 1
Eq. 4a, continuity
Eq. 2
gravity
Eq. 4b, momentum
radiation,
magnetic fields
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Eq. 3
Eq. 2
Eq. 4c, energy
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Eq. 4c  condition for existence of stellar winds
integration
select
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left side only > 0, if at least one of the integrals >> 0
stellar wind can only escape potential well of the star, if
or
● significant momentum input
● significant energy input
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3. “Naïve” theory without additional forces
first simple approach: isothermal wind without additional
forces driven by gas pressure only
● example: solar wind
significant
in corona
pressure driven wind
Eq. 4b
with
pure ionized H
hot solar gas
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*
starting point to derive
equation of motion for v(r)
next step: replace ρ
using Eq. 4a
Eq. 5, density/velocity
relationship
Eq. 5 can be used to replace density in eq. *
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Eq. 6, equation of motion
differential eq. for v(r) of pressure driven winds
has singularity called the critical point at
Eq. 7a
Eq. 7b
left side Eq. 6 zero at critical point
right side must also be zero
T ~ const  vs ~ const.
because v = vs at r = rs , the critical point is also
called the sonic point
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Example: solar wind of solar corona
photospheric escape velocity of sun
sonic point in outer corona
Note: if temperature of corona were as cool as photosphere,
we would still have a solar wind, but
supersonic just at 6 AU
in principle, all stars will have a (weak) wind, but
the solar wind is strong because of
high temperature in corona
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the importance of the critical point
differential eq. 6 has several families of solutions
critical point  disentangle topology of solutions
 find physically relevant solution
for simplicity,
re-write Eq. 6
with
Eq. 8a
Eq. 8b
the implicit solution Eq. 8b for v(r) together with
the conditions for the critical point Eq. 7a,b can
be used to discuss topology of solution families
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Eq. 8b
case I
Eq. 8a
v(r) exists for 1 ≤ r ≤ ∞ and is transsonic, i.e.
all monotonic solutions 8b with C = -3
What is the value of
at sonic point?
Eq. 8a
L’Hospital:
at sonic point
Eq. 9a
two solutions possible
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case I, 1
transonic, monotonic, increasing
this solution is realized in solar wind
case I, 2
transonic, monotonic, decreasing
this solution is not realized
sketch of
2 solutions
I, 1:
for
I, 2:
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also possible: case II
v(r) exists for 1 ≤ r ≤ ∞, but
2 possibilities
case II, 1: v ≥ vs for 1 ≤ r ≤ ∞, C > -3
supersonic solution
case II, 2: v ≤ vs for 1 ≤ r ≤ ∞, C ‹ -3
subsonic solution
both cases not realized
however, II,2 was for a
long time discussed as
discussed as a possible
option, until space
observations showed that
solar wind is supersonic
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also not realized is case III
non-unique solutions
in
with
or
note: discussion of solution
topologies crucial to identify
physically realized solution.
We will repeat this, when
dealing with the much more
complicated case of
radiation driven winds.
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Discussion of the realistic solution:
trans-sonic, monotonic, increasing (Parker, 1958)
Eq. 10
limiting cases: 1‹ r ‹‹ rs, v ‹‹ vs:
r >> rs, v >> vs:
Eq. 11a
Eq. 11b
v  ∞ if r  ∞, consequence of the unrealistic
assumption T = const., i.e. pressure can accelerate
until infinity
in reality T drops and v remains finite
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density stratification
Eq. 4a
Eq. 12
ρ1, v1 at r = 1
photosphere
for r ‹‹ rs  hydrostatic solution, for r ‹‹ rs  ρ ‹ ρstatic
mass-loss rate
if ρ1 at bottom of wind known  with v1 (eq. 11)
for ρ1 = 10-14 g/cm3 Tcorona rs
1 106K
2
3
6.9
3.5
2.3
1.6 10-14
3.7 10-12
8.2 10-11
observed value is 2 10-13
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from Lamers & Cassinelli
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4. Additional forces: radiation
Pressure driven winds can explain solar wind (note: that in
reality you need magnetic fields to explain coronal heating)
but…
can winds of hot stars be driven by pressure only?
crucial observational facts:
● cool winds, no corona: Twind ~ Teff ~ 40000K
vs ~ 20 km/s
● sonic point v = vs very close to photosphere at r ~ 1.05
● vesc ~ 500 … 1000km/s
with pressure driven wind theory
we need additional force gadd !!!
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radiative acceleration
hot stars
Bν (T) large because T large
Iν, Jν large
large amount of photon momentum
photon absorption transfers momentum to
atmospheric plasma
calculation of gadd = gextra:
absorbed energy per unit time:
momentum
:
per time & mass :
only projection into radial direction contributes
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with azimutal symmetry of radiation field and
Eq. 13
Eq. 13 exact, as long as all absorption processes included
Eq. 14
Thompson
scattering
continuous
absorption
line
absorption
Eq. 15
remember:
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corollary: approximation for τν >> 1 at all frequencies
radiation field thermalized at all
frequencies
“diffusion approximation”
“radiation pressure”
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in hot star winds the influence of
is usually small but
influence of Thomson scattering
with
q electrons provided by helium nucleus (0 …2)
and with
Eq. 16
with Eq. 4b new momentum equation
Eq. 17
Thomson scattering reduces gravity by constant factor 25
with
Eq. 18
effective escape
velocity
and eq. 6 we obtain new equation of motion
Eq. 19
does Thomson scattering alone do the job?
if we ignore
sonic point at
Eq. 19b
typical Γ-value for hot supergiants Γ = 0.5
sonic point for hot stars still at rs ~ 50…100,
but observed is rs ~ 1.1
need additional accelerating force!!!!
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5. Very simplified description of line driven winds
As first step: very simplified description of
assumptions:
● only radially streaming photons
● only lines with
>> 1
● Sobolev approximation
contribute
>>
>>
● number of strong lines independent of r
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shell in stellar wind: calculation of
line i at
total photon
momentum
>>
fraction abs. by line I
line width
luminosity
at
contribution by all lines:
Eq. 20
effective number
of strong lines
line acceleration
~ velocity gradient !!!
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new equation of motion (T = const.)
Eq. 21a
with
Eq. 21b
because
we assume
no singularity at
r.h.s. of Eq. 21b negative
smaller than
‹
old rs
‹‹
and prove later that this is correct
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multiplying eq. 21a by
and assuming
>>
Eq. 21c
combining eq. 21b and c for
>>
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integrating from rs to r
Eq. 22
β-velocity field,
β = 0.5
from observations (Abbott, 1978, 1982)
Eq. 23
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Simplified theory of line driven winds explains
●
but does not give proportionality constant
● right order of magnitude for
but predicts
whereas
is observed
improved theory needed!!!
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