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Pulsating stars II: Solutions at constant density
Michel Bonnardeau
24 Feb 2006
Revised 13 July 2012
Abstract
The differential equation for a radially pulsating star is solved
for a constant density.
Introduction
Part I the differential equation of a radially
pulsating star was derived.
Let us now make the simplification that the density of matter is constant
throught the star (a "liquid" star). The differential equation (15)
of Part I is then easily solved. Although this may look grossly oversimplified,
we will see that the results are meaningful.
The differential equation for a "liquid" star
With the density rho0 constant:

(1) 

(2) 
This gives from equations (1)
and (7) of Part I:
Let us introduce, where R0 is the radius of star:

(3) 
The differential equation (15) of Part I may
then be written:
with
or
Fundamental mode solution
An obvious solution to equation (19) is X=constant and alpha=0. The
star oscillation is then:

(6) 

(7) 
For the oscillation to be stable, one needs gamma>4/3. If gamma is smaller
the star is instable: a small perturbation makes it collapse or blow up.
(gamma=4/3 is for a radiation pressure dominated gaz).
With an adiabatic gaz, gamma=5/3 and the oscillation is stable.
The pulsation period is:
Let us consider the Sun as being of constant density. With a mass of
2*10^33g, a radius of 700,000km, the density is then 1.39g/cm3 and the
computed pulsation period is 168mn. The Sun is observed with many oscillations,
from a 160mn period down to a 5mn period. The computed period happens
to be very close to the 160mn period.
More realistic models of the Sun lead to smaller periods. For example
the polytropic n=3 model gives a period of 55mn for
the fundamental mode.
So the liquid star model gives only the right order of magnitude.
The observed, weak (21cm/s), 160mn period oscillation of the Sun is NOT
the fundamental oscillation mode, and is not that easily explained.
With a Cepheid of 5 solar masses, and a radius of 60 solar radii, the
computed pulsation period is 24 days. This is a typical observed pulsation
period for this kind of star.
First overtone solution
Let search a solution to (19) of the form
the differential equation (19) becomes
this gives
The oscillations are then of the form

(9) 
with the period given by:
Second overtone solution
With a solution of the form:
one has:
with the period given by:
Oscillation amplitudes
The amplitudes of the oscillations as a function of the distance from
the star center:
Red: fundamental mode, Blue: 1st overtone, Magenta: 2nd overtone.
More
Part III: solutions to the differential equation
of the oscillations from a more realistic, polytropic model of the star
structure are presented.