Transcript Presentation 2: Hot Star Winds

X-ray Group Meeting
Observations of Hot Star Winds
Absorption Spectroscopy, Radiation (line-) Driving,
and X-rays
9 February 2001
David Cohen
Allison Adelman
David Conners
Eric Levy
Geneviève de Messières
Kate Penrose
How do we know that hot star winds exist?
We see evidence in ultraviolet spectra due to line
scattering in a dense outflow
Aside: The distinction between scattering, on the one hand, and
absorption and re-emission, on the other:
Scattering is non-local; it is unaffected by the physical conditions at the
place where the scattering occurs. Whereas with absorption followed by
reemission, the properties of the radiation you observe are affected by the
physical conditions (like temperature) at the location of the reemission.
Radiative excitation followed by spontaneous (radiative) emission
key
Taken together, these processes
previous position of an electron
constitute line scattering
current position of an electron
•A photon is neither created nor
destroyed
movement of an electron
g
photon
movement of an photon
e
•Atom/electron ends up in same
configuration in which it began
external electron
g
g
Scattering may not
produce or destroy
photons, and it may
not effect the
properties of atoms or
electrons
But it can change the
direction of photons
This cartoon
illustrates the
production of P Cygni
profiles as scattering
destroys blueshifted
light from the star and
adds emission at
nearly all other
wavelengths
Observed P Cygni
profiles in two hot
stars: z Pup and t
Sco
z Pup has a much
stronger wind
(deeper absorption,
higher emission)
You can read the
maximum wind
velocity right off
the spectrum -- it’s
the maximum
blueshift of the
absorption
A somewhat more
rigorous derivation of
the P Cygni profile
Consider spherical
shells in the wind.
Each annulus leads to
emission over a
narrow wavelength
range (or absorption,
if it is in front of the
star)
By summing up
the contributions
of many shells,
we can build up
the total P Cygni
profile
Ultraviolet absorption lines
•Provide evidence of a stellar wind, but also
•Drive the wind (via their mediation of the transfer of momentum
from starlight to the matter that makes up the wind)
Light has momentum -- momentum, p, is energy divided
by velocity:
E
p
v
Light doesn’t have much momentum per energy, as its velocity
is as high as possible
E
p
c
How much of this momentum can get transferred,
and at what rate?
Note: Force is the time rate of change of momentum
dp
F
dt
This momentum transfer will accelerate the stellar wind
One particle, with an effective cross-sectional area of s (cm2)
can “catch” the flux of momentum (momentum cm-2 s-1), f/c,
at the rate given by
sf
Frad 
c
This is the rate at which the particle gains momentum from
the photons, which is the radiation force on the particle
Note that another way to see this is that the force, Frad, divided by area over which it is exerted, is a pressure.
Note that the effective cross sectional area of a particle is
not really due to a physical size, but instead can be thought
of as a propensity, or likelihood, for an atom to interact
with a photon -- the more likely the interaction is, the
bigger the cross-section
Now, for gas with lots of particles, it makes sense to ask
not about the cross section of one particle, but the cross
section per unit mass, k (cm2/g), which is called the
absorption coefficient, or opacity
Substituting for s in the last equation gives the radiation
force per unit mass, or the radiative acceleration
arad
Frad kf


m
c
The opacity of the wind material, along with the brightness
of the star (i.e. its flux) determines how much momentum
gets transferred from the starlight to the wind, and therefore
how much the wind is accelerated
opacity
arad
Frad kf


m
c
flux
Hot star winds have
so much opacity,
that much of these
stars’ flux is missing
-- its momentum has
been used to
accelerate the wind
Note that the opacity of wind material depends on wavelength
-- lines are stronger or weaker at different wavelengths
-- also, for a given line, the Doppler shift can affect how an
atom absorbs light
Consider the light from the surface of a star--its
photosphere--the spectrum of this light is constant over a
given frequency range:
Consider a scattering spectral line from a parcel of atoms
in the wind
This line’s profile, or frequency-dependent opacity, is shown in
green. This is essentially the probability that a photon of a given
frequency will be scattered, or absorbed, by the atom
After atoms of this type have absorbed the photospheric
light, the remaining light looks like this:
Absorbing the light has accelerated the parcel, however, so
the line profile is now blueshifted a bit
The red cross-hatched area shows the light that can be absorbed by
this parcel of atoms -- note that the shifting of the profile out of the
“Doppler shadow” allows for more momentum to be absorbed, and
thus more acceleration, than if the line weren’t blueshifted
The acceleration of a line-driven wind is thus
bootstrapped -- blueshifting of a line enables further
acceleration -- the physics of a line-driven outflow is
self-regulating
But the situation is non-linear...
If the parcel of atoms gets an extra little push, it will “see” more
photospheric flux, get a bigger acceleration and thus more
blueshift, and therefore receive even more flux, etc.
Line-driving has an inherent instability
We can calculate the strength of a wind and its velocity
structure by simply solving Newton’s second law
F
a
m
However, the force on a given parcel depends on how much
radiation flux it is receiving…in other words, on its redshift,
which is a function of its acceleration
Fa
This is a highly non-linear set of equations to solve!
Note that the force, as I’ve convinced you, is a function of the rate at which a line is
blueshifted out of the Doppler shadow, which depends on how the velocity changes
with distance, dv/dr, and the acceleration is proportional to this quantity through the
chain rule: a=vdv/dr
But we can solve these equations in a time-dependent manner -here is the velocity as a function of both height above the star and
time from one simulation. Note all the structure and variability.
Here is a snapshot at a single time from the same simulation.
Note the discontinuities in velocity. These are shock fronts,
compressing and heating the wind, possibly producing X-rays.
So we see that line-scattering is the means via which the
light from the photosphere of a hot star drives a massive
stellar wind.
This line-driving is unstable by its very nature, producing a
very chaotic and violent wind, which may be the
explanation for the X-rays observed on hot stars.
Furthermore, this same line-scattering also allows us to
observe and analyze these winds.
Coda: This process of momentum transfer and line driving is
fascinating and powerful. But why isn’t it the explanation
for the solar wind?
How much momentum is required?
The total rate of momentum being lost in radiation is L/c,
where L is the luminosity of the star in ergs s-1
The total rate of momentum being taken away in the wind is
just Mlossvinf, where Mloss is the mass-loss rate in solar masses
per year (Msun yr-1) and vinf is the terminal velocity of the
wind
For the sun,
L  3.9 10 33
(ergs s-1)
dp L

  1.3  10 23
dt c
(g cm s-2)
(cgs units of momentum loss per second)
For the solar wind, the momentum loss rate is:
-1)
14
(M
yr
Ý
sun
M

10
Called this M before
loss
And called this vinf
v  300
(km s-1)
dp
19
Ý
 Mv   2  10
dt
(g cm s-2)
Thus the sun’s light has more than enough momentum to drive
the solar wind…but the light doesn’t drive the wind in the sun’s
case
This is because the opacity of the wind is very low and poorly
matched to the wavelengths where the sun emits most of its
light (optical) -- the solar wind is transparent to optical light,
which streams right through it, taking its momentum away
rather than transferring it to the wind
cm2/particle
recall:
g/particle
dp sf kf


dt
c
c
F k f
a 
m
c
flux (ergs cm2 s-1)
(g cm s-2)
F
opacity (cm2 g-1)
(cm s-2)
So, the radiation force is proportional to the flux, but
also to the opacity, k
Note: force is the rate of change of momentum (check the units -- mass times
acceleration = mass times velocity per time)
Hot stars do have a lot of radiation at the wavelengths where
wind material has its opacity (the ultraviolet)
The momentum in the light of a hot star (which has a huge
luminosity) is, for the O star z Pup:
L  5 10 39

dp L
  2  1029
dt c
(ergs s-1)
(106 times solar)
(g cm s-2)
This can drive a huge wind, potentially:
It is 1010 times the momentum in the solar wind
Indeed, the wind of zeta Pup has close to this much
momentum:
(Msun yr-1)
MÝ  5  10 6
20
(g s-1)
MÝ  3  10
v  3000
v  3  108
(km s-1)
(cm s-1)
dp
29
 MÝv   10
(g cm s-2)
dt
Its wind is very efficient at absorbing light.
To quantify the efficiency at which a given wind makes
use of the momentum of light from the star, we can form a
ratio of the winds momentum to the momentum available
in the radiation:
Ý
Mvc
L
(referred to as the “momentum number” or
“performance number” of a radiation-driven wind)
Note: in a little over a million years, a star like z Pup will have lost an appreciable fraction of its mass! This has
very serious implications for stellar evolution models, supernovae rates, etc.
There are stars, called Woft-Rayet stars, which have
momentum numbers greater than unity!
How can this be?
Well, once a photon scatters in the wind and gives up its
momentum, is it gone? No, it’s just changed its direction. It
can transfer its momentum again, and again, and again...
The problem (for theoretical astrophysicists)
then becomes: How to provide enough
opacity in the wind to keep the photons
trapped so that multiple scattering can
occur?
Note: the atoms that absorb the light’s momentum also absorb
some of its energy…there isn’t an infinite supply from a given
photon (what do you suppose the ratio of momentum to energy
lost in each scattering is?)