Oscillations
Reading Assignment
Chapter 13 of Stellar-Astrophysics Notes
Radial, Adiabatic Oscillations
Let's start with radial oscillations. We will use the enclosed mass as the coordinate and we can write
and write the equation of motion of a shell of fluid (hydrostatic equilibrium if no oscillation)
Substituting the perturbed expressions and keeping only the first-order terms gives
If we use the unperturbed equation and define
and
where we used the zeroth-order continuity equation
Now if we linearize this equation we get
and
We now have two equations but three unknowns and . The final ingredient is the equation of state which for an adiabatic gas gives
so now we can write the first equation just in terms of :
We can rewrite this and a Sturm-Liouville equation
This is an eigenvalue equation for the function . There are a countably infinite number of eigenvalues, all real. We will order this such that . The smallest eigenvalue will correspond to an eigenfunction without any nodes and similarly will correspond to an eigenfunction with nodes. Furthermore, these eigenfunctions will be orthogonal with the
weighting over the interval .
Let's integrate all three terms times from to for the eigenfunction to get
and
If we take the adiabatic index to be constant and use the equation of hydrostatic equilibrium, we get
Because has no nodes, has the same sign as ; therefore
means that for all of the modes are positive (all stable) and for
at least and other might be too (at least one unstable mode).
Let's write everything in terms of dimensionless variables
to get
We see that is a constant for homologous stars. For polytropes one can also get the frequencies of the overtones
where is a dimensionless eigenvalue that depends only on and .
Can we estimate the frequencies in general for modes with many nodes? Let's use a variational principle and guess the eigenfunction to be
If we substitute this into the homologous frequency equation and integrate the denominator by parts twice we will get
Radial, Non-Adiabatic Oscillations
The key idea here is that for the modes to be sustained, they must not be strictly adiabatic! We can imagine the mode is like an engine and work must be done to drive the oscillation so heat must enter the gas at high temperature. For this to work the gas must be more opaque at the high temperature, high density part of the oscillation. We have
and
so
Therefore, the opacity will be smaller at the high density if , and this is typically the case.
However, an exception is a partially ionised layer. The compression ionises the layer but the temperature does not increase. Heat is absorbed during
the compression and released during the expansion. This is called the mechanism.
H-ionization zone at 10,000-15,000 K: HI to HII and He I to He II.
He-ionization zone at 40,000 K: He II to He III
For these zone are too close to surface not much mass!
For these zone are deeper and the fundamental can be excited. There is in instability strip around effective temperatures between 5,500 and 7,500 K.
Non-Radial, Adiabatic Oscillations
Now things get more complicated, but we will make some simplifications to make our lives easier. First, the equations
Remember that the unperturbed solution is just a function of radius . We can follow the changes in the fluid in two ways, the Eulerian (I'm standing at a point) and Lagrangian (I'm moving with the fluid). We will use the prime to denote the Eulerian perturbation and to denote the Lagrangian perturbation. We can
relate them by
where is the perturbation of the fluid from its equilibrium position. We can also think of changes in time as measured by someone
in the fluid and someone standing still
We will connect the perturbation of the pressure to that of the density
We can now write perturbed Euler equation
and we can connect the density perturbation to the displacement field through the continuity equation
We would also need to calculate the perturbation to the gravitational potential through the Poisson equation, but
we will make the Cowling approximation in which we take and we write
and get equations for the three components of the displacement
where
Notice that if the unperturbed star is adiabatic.
We can solve the two angular equations by using spherical harmonics
and
where the transverse component is related to by
so we can focus on and yielding the following two equations
and
where we have defined the Brunt-Vaisala frequency , the Lamb frequency
and the transverse wavenumber as
and
The Brunt-Vaisala frequency is simply the oscillation frequency of a blob displaced vertically
and the Lamb frequency is the frequency of sound waves travelling transversely.
Unlike the equation for purely radial modes, this is not a Sturm-Liouville equation, so we have no guarantee that
the value of is bounded from below. All that we do know is that is real and that the solutions will be orthogonal.
We can talk about modes with which will be constant on lines of constant latitude (). These are called zonal modes. We can also have modes with . These are constant on lines of longitude and are called sectoral modes. The other modes are called tesseral modes.
To make some further progress we will take a WKB approximation in which the radial variation is proportional to in the limit
where . This yields a dispersion relation for the radial wavenumber
For a wave that is oscillatory in time we have . Furthermore for it to be oscillatory in the radial direction
as well, we must have ; therefore, we can have waves for values of outside the range
in between and . We have two types of waves those with greater than both
and and those with less than both
and . Usually we have . In this case we call the first modes or pressure modes
and the second we call modes or gravity modes.
If we assume that we have
and
The size of the perturbations for the two types of modes are given by
We can approximate the number of radial nodes as
and the frequencies as
and
Check out the latest Doppler images of the Sun at http://jsoc.stanford.edu/data/hmi/images/latest/
Assignment
You will use the inlist called astero_adipls for this assignment. This model evolves a 1.2 solar mass star from the ZAMS until the hydrogen is nearly exhausted in the centre. I would like you to plot the sound speed (profile column csound) and Brunt frequency (brunt_N2) as a function of radius for the initial stage (ZAMS) , halfway through the evolution and at the final stage. Identify the convectively unstable regions on the plots. Plot the Lamb frequency for l=1,2,3. Identify the turning points of p-mode with ω=3000μHz for different values of angular momentum. What about 10,000μHz?
Similarly what are the turning points of a 2000μHz g-mode?
Plot the structure of the first several modes (l=0, n=0 to 6) for the final model. The results are in the file ttt.adipls.prt.