Download Lecture 21

ASTR 400/700:
Stellar Astrophysics
Stephen Kane
Stellar Models:
The complete set of differential equations describing the interiors
of stars is therefore:
Equation of Continuity:
Hydrostatic Equilibrium:
Energy Generation:
Temperature Gradient:
dM ( r )
= 4π r 2 ρ
dr
dP − G M ( r ) ρ
=
dr
r2
dL
= 4π r 2 ρ ε
dr
− 3 κ ρ Lr
 dT 

 =
3
2
dr
4
ac
T
4
π
r

 rad
− 1 GM ( r )
 dT 

 =
2
dr
C
r

 ad
P
Stellar Pulsation
Chapter 14.3, 14.4, 14.5
A simple pulsation cycle
• At one point in the pulsation
cycle, a layer of stellar material
loses support against the star’s
gravity and falls inwards.
• This inward motion tends to
compress the layer, which heats up
and becomes more opaque to
radiation.
• Since radiation diffuses more slowly
through the layer (as a consequence
of its increased opacity), heat builds
up beneath it.
N.B.
N.B. These
These diagrams
diagrams are
are
definitely
definitely not
not to
to scale!!
scale!!
• The pressure rises below the
layer, pushing it outwards.
• As it moves outwards, the layer
expands, cools, and becomes more
transparent to radiation.
• Energy can now escape from
below the layer, and pressure
beneath the layer drops.
• The layer falls inwards and the
cycle repeats.
Eddington’s Thermodynamic
Heat Engine
But this does not work for most stellar material! Why?
ρ
opacityµ 3.5
T
The opacity is more sensitive to the temperature than to the density,
so the opacity usually decreases with compression (heat leaks out).
But in a partial ionization zone, the energy of compression ionizes
the stellar material rather than raising its temperature!
In a partial ionization zone, the opacity usually increases with
compression!
Partial ionization zones are the direct cause of stellar pulsation.
Partial ionisation zones
•In most stars there are two main partial ionisation zones.
•The hydrogen partial ionisation zone is a broad region with a
characteristic temperature of 1 to 1.5 × 104 K, in which the
following cyclical ionisations occur:
•The helium II partial ionisation zone is a region deeper in the
stellar interior with a characteristic temperature of 4 × 104 K,
where further ionisation of helium takes place:
Pulsating Variables: The Valve Mechanism
Partial He ionization zone is opaque and
absorbs more energy than necessary to
balance the weight from higher layers.
=> Expansion
Upon expansion,
partial He ionization
zone becomes more
transparent, absorbs
less energy => weight
from higher layers
pushes it back inward.
=> Contraction.
Upon compression, partial He ionization zone
becomes more opaque again, absorbs more
energy than needed for equilibrium => Expansion
• The pulsation properties of a star depend primarily on where its
partial ionisation zones are found within the stellar interior.
-9
H
-8
-7
Surface
-10
He
-6
-5
Centre
log (1-Mrr/Mstar
star)
• The location of the partial
ionisation zones is determined
by the star’s temperature.
• For stars hotter than Teff ~
7500K, the partial ionisation
zones are located too close to
the star’s surface, where there
is not enough mass to drive the
oscillations effectively.
-4
-3
Teff ~ 7500K
-9
-8
-7
-6
-5
H
-4
He
-3
Teff ~ 5500K
Centre
log (1-Mrr/Mstar
star)
• For stars cooler than Teff ~ 5500K, on
the other hand, the partial ionisation
zones are deep in the stellar interior.
• However at low temperatures, energy
transport via convection becomes quite
efficient in the stellar interior, preventing
the build-up of heat and pressure beneath
the driving pulsation layer.
Surface
-10
The Instability
Strip
• Majority of pulsating
stars lie in the instability
strip on the H-R diagram
• As stars evolve along
these tracks they begin
to pulsate as they enter
the instability strip and
cease oscillations once
they leave it.
∆T ~ 600 – 1100 K
Modelling pulsations
• Computer modelling of stellar pulsation suggests that it is
primarily the helium II ionisation zone which is responsible for
the observed oscillations of stars on the instability strip.
R/Rmin
Teff
V
Observed properties of a classical
Cepheid
νr
• The hydrogen ionisation
zone, however, still plays an
important role, producing
an observable phase lag
between the star’s
maximum brightness and its
minimum radius.
Time
Time (days)
(days)
Modeling Stellar Pulsation
Consider the adiabatic, radial pulsation of a gas-filled shell.
Linearize the equation of motion
2
dR
GMm
2
m
= 4π R P − 2
dt
R
by setting
R = R0 + δ R
2
to get
P = P0 + δ P
d
2GMm
2
m δ R = 8π R0 P0δ R+ 4π R0 δ P +
δR
2
dt
R0
Nonradial Oscillations
Pulsational corrections δf to equilibrium model scalar
quantities f0 go as (the real part of)
l = 0 radial
m > 0 retrograde
m < 0 prograde
m = 0 standing
Two Types of Nonradial Modes
p modes
a surface
gravity
wave
g modes
SOHO
(Solar and Heliospheric
Observatory)
Michelson Doppler Interferometer
(MDI)
- measures vertical motion of
photosphere at one million points
-can measure vertical velocity
as small as 1 mm/s