Linear Theory of Stellar Pulsation
Mir Emad Aghili∗1
1
1
Department of Physics and Astronomy, University of Mississippi, University, MS, 38677
Introduction
The first pulsating star to be discovered was omicron Ceti, in constellation Cetus. This star was discovered
in 1638 by Johannes Holwarda when he realized that this star changes its brightness drastically in a period
of 11 months. This star later was named Mira which means wonderful. Since then, many pulsating stars
has been discovered.
Most of the stars in some stages of their life go out of hydrostatic equilibrium and start to pulsate, i.e. they
either change their size (radial pulsation), change their shape (non-radial pulsation) or both of them at the
same time. Pulsation affects the surface temperature of the star and therefore the luminosity. The pulsation
can give information about inner structure of a star, some bounds on the radius etc. If the change in the
luminosity is regular and the same way for the stars of the same type, it can be used as a standard candle1 .
An example of such objects are Cepheid stars. A relation between brightness and period of Cepheids was
established by Henrietta Swan Leavitt [1]. Based on this relation one can use the stars pulsation period to
obtain the absolute magnitude and therefore the distance to the star. This method can be calibrated by
using parallax2 of some nearby Cepheids.
Cepheids are the most common pulsating stars. Their period is between 1 to 50 days. W Virginis stars are
essentially the same as Cepheids but with metal deficiency. RR-Lyrae stars have nearly the same luminosity
with periods between 1 to 24 hours and they can be used as standard candles as well. The δ-Scuti stars
are the ones with very short period of a few hours. All of aforementioned types of pulsating stars on H-R
diagram are located on a long strip that is called the instability strip fig(1). Another group of pulsating
stars are β-Cepheids which are early B stars, they are very hot they are outside the instability strip but the
mechanism causing the pulsation is the same. On the right hand side of the H-R diagram are the pulsating
stars with periods between 100 to 700 days. These type of stars are called long period variables (LPV)
including Mira variable stars and semi regular pulsating stars [3].
The aim of this paper is to give an introduction to the mechanism deriving the pulsations and the simplest
theoretical models behind them. In section 2, there will be a discussion on the linear radial pulsation of the
stars and in section 3 there will be a general discussion of the results and the applications.
2
Radial Pulsation
Understanding of stellar pulsation has been a challenge for a long time. For some types of pulsating stars
this issue is more or less understood and for some there still exists some confusion.
The simplest mathematical model for pulsation is to naively think that the star mass is concentrated at the
center and the surface of a star is a thin shell with a mass m much smaller than the total mass M of the star
at radius R, and the volume between the center and the shell is filled with a massless gas, which provides the
∗ [email protected]
1 An
object in astrophysics that has a known luminosity and can be used for distance measurement.
is the displacement of an object with respect to background stars due to change of position of the observer
(perspective). This quantity can be used to determine the distance to the object d[pcs] = 1/π[arc sec], where π is parallax.
2 Paralax
1
Figure 1: Position of different types of pulsating stars on H-R diagram. credit: J.P. Cox, courtesy of Institute
of physics.
pressure to support the sell from falling. The force exerted on the shell as it is displaced from equilibrium is
mr̈ = 4πr2 p −
GM m
.
r2
(1)
If one assumes that the star is surrounded with vacuum, we are able to assume an adiabatic equation of
state p/ρΓ1 = const. which after linearization3 procedure simplifies to
δr̈ = − (3Γ1 − 4)
GM
δr.
R3
(2)
p
This is a simple harmonic motion with frequency ω = (3Γ1 − 4) GM/R3 . Although the result is in good
agreement with that of Cepheids and discovery of Leavitt, it does not explain the physical reason behind
the pulsation of the star.
The first proposal for pulsation mechanism was introduced by Eddington [4, 5]. He predicted that pulsation
is caused by a sort of valve mechanism. The heat that is trapped under a layer increases pressure which in
turn pushes the top layers and after expansion it cools down and contracts. This mechanism needs at least
a large enough layer to be opaque.
As we go towards the center of a star, opacity decreases due to dependence4 on density ρ and temperature
T by κ ∝ ρ/T 7/2 . Although density and temperature are both increasing functions as we move inward,
the dependence on temperature is with much higher power. The decrease in opacity does not support the
valve mechanism. It was later found that solution to this ambiguity is the partial ionization zones, such as
3 Linearization of an equation, is to find the solution to the equation when the variables are infinitesimally changed from a
known solution, this way one can safely consider only up to first order in (A → A + δA) deviation which is called a linearized
equation.
4 Opacity is a measure for the amount of energy from radiation that is being absorbed and can not find its way out. In general,
there are different processes that contribute to the opacity, and each one depends on temperature and density in a different
way. Some of those processes are Thompson electron scattering, free-free absorption, bound-free absorption and bound-bound
absorption. For intermediate distances from the core of the star Thompson scattering is negligible and the opacity is governed
by free-free and bound-free opacity, which depend on temperature and density as stated in the text.
2
partially ionized Hydrogen or Helium or in the case of β-Cepheids the partially ionized Iron layer. In these
layers the opacity does not decrease, because the energy that is given to these layers will be used to ionize
the elements more and therefore these layers will not let the heat out.
This mechanism is good enough assumption for description of Cepheid type variable stars dynamics. Next
step is to formulize the problem of pulsation and theoretically derive the pulsation frequency and overtones.
A star with densityρ, temperature T , hydrogen mass portion X, equation of state p(ρ, T, X) and nuclear
energy production rate per unit mass (ρ, T, X) in equilibrium should satisfy static equations. For the
simplest case of spherically symmetric stars (which is true for the many cases) we have
−Gmρ
dp
=
dr
r2
dm
= 4πr2 ρ
dr
3ρκ̄
dT
Frad
=−
dr
16σT 3
d 2
r (Frad + Fconv ) = r2 ρ
dr
Fconv = Fconv (ρ, T, X, g; ∇) ,
∇=
p dT
,
T dp
Hydrostatic Equilibrium
(3)
Conservation of Mass
(4)
Radiative Flux
(5)
Thermal Equilibrium
(6)
Convective Flux
(7)
where κ̄ is the Rossland mean opacity, σ is the Stefan-Boltzmann constant and g is the the gravitational
acceleration. The boundary between radiation dominated and convection dominated zones of a star is
determined by the degree of adiabaticity of the star at that zone. The quantity that is used to determine the
adiabaticity is ∇ad = 52 . If ∇rad < ∇ad at some radius r, then the layer will be radiative and the convective
flux Fconv vanishes, on the other hand if ∇rad > ∇ad convection starts and Fconv > 0. The functionality of
Fconv depends on the convection theory that is being used. Another point that should be mentioned here is
that the viscosity is neglected in these equations.
When a star undergoes pulsation, it goes out of equilibrium state and therefore we should modify the static
equilibrium equations. For adiabatic pulsations up to linear order in deviation from equilibrium we have
4Gm
∂ δp
δp
R
xR
2 ∂δp
δr̈ = 2 − 4πr0
⇒
ζ̈ +
=
4ζ −
(8)
r
∂m
∂x p
λp
g
p
δρ
∂ζ
= 3 − ζ − r0
(9)
ρ0
∂r0
δL
δκ
δT
1
∂
δT
= 4ζ −
+4
+ d ln T0
(10)
L0
κ0
T0
∂r
T0
0
dr0
∂ δp
∂L
ρ (Γ3 − 1)
∂ δρ
= Γ1
+
δ −
,
(11)
∂t p
∂t ρ
p
∂m
where ζ = δr
r0 is the relative change in distance, R is the star radius at equilibrium, x = r0 /R is the relative
p
distance, λp = − d dr
ln p = ρg is the pressure height, L is the luminosity and Γ3 = 1 + (d ln T /d ln ρ)ad . We
can use the conservation of mass equation dm = 4πr02 ρ0 dr to write all the derivatives in terms of mass. We
have chosen the mass to be independent variable because if we choose the correct frame of reference we can
always have the mass of the layer to be conserved however, this is not true for the radius [2].
After doing some manipulations one can combine all of the equations of motion to write one single equation
...
r0 ζ =
d
∂ ζ̇
[(3Γ1,0 − 4) p0 ] + 12πr02 Γ1,0 p0
dm
∂m
!
∂
∂ ζ̇
∂
∂L
+16π 2 r02
r03 Γ1,0 p0 ρ0
− 4πr02
ρ0 (Γ3,0 − 1) δ −
,
∂m
∂m
∂m
∂m
4πr02 ζ̇
(12)
where Γ1 = (d ln p/d ln ρ)ad . For small adiabatic oscillations, one can use
δρ
δp
= Γ1,0
p0
ρ0
3
(13)
in momentum equation equation (8) and the oscillation equations become
1 ∂
∂ξ
d
[(3Γ1,0 − 4) p0 ] +
16π 2 Γ1,0 p0 ρ0 r06
.
r0 ζ̈ = ζ4πr02
dm
r0 ∂m
∂m
(14)
A simplified equation is called homologous motion, which means motion when ζ is independent of mass m.
4
0
In such a motion dp
dm = −Gm/4πr0 and therefore
r0 ζ̈ = − (3Γ1,0 − 4)
4r0
Gm
ζ = − (3Γ1,0 − 4)
Gρ̄ζ,
2
r0
3
(15)
where ρ̄ is the p
average density of the star. The solution to this equation simple harmonic motion with
frequency ω = (4πGm (3Γ1,0 − 4)) /r0 , in agreement with primitive model that was discussed previously.
This result gives stable pulsation if Γ1 > 4/3 and we will have instability, including both relaxation or
blowing up solutions for Γ1 < 4/3.
For a more general case a standing wave solution to this equation requires that ζ(r, t) = α(r) exp (iωt), which
changes the partial differential equation to an ordinary differential equation
dα
d
d
Γ1 pr4
+ α ω 2 ρr4 + r3 {(3Γ1 − 4) p} = 0,
(16)
dr
dr
dr
where 0 subscript is dropped for simplicity. This equation is known as “Linear Adiabatic Wave Equation”
(LAWE). Like any other physical equation, we need to provide the boundary conditions to the equations to
find a unique solution. This equation is of second order, therefore two boundary conditions are required.
With the spherical symmetry we require the star to have δr = 0 at the center and on the surface we can
choose different types of boundary conditions. We can require δp = 0 at the surface of the star or set a
quantity such as p/ρ that is proportional to δT equal to zero. We should not also forget that when we
expand the first term in equation (16), the coefficient of dζ
dr is singular at the center of the star, therefore we
dζ
need to require dr |r=0 = 0.
Atmosphere of the stars are isothermal at at higher layers [6] and therefore for such an isothermal atmosphere
it is easy to see that p, ρ ∝ exp (−x/λp ), where x is the height measured from a reference point. By change
of variable x = r − R, where R is the reference point and keeping in mind that R/λp 1 in the atmosphere
of real stars, we can drop the smaller terms and find
d2 α
1 dα ω 2
−
+ 2 = 0,
2
dx
λp dx
νs
where νs =
(17)
p
Γ1 p/ρ is the adiabatic speed of sound. With basic knowledge of calculus we can see that




s
s
2 2
2 2
1
1
ω 
1
1
ω 
α = A exp 
+
−
+ B exp 
−
−
.
(18)
2λp
2λp
νs
2λp
2λp
νs
To discuss the solutions we need to consider two cases. In the first case, we consider 2λ1p > νωs , in which
the solutions are real. In this case looking at the kinetic energy of the sound wave for two solutions to the
equation (18) shows us that limx→∞ KA = ∞ and limx→∞ KB = 0, which means the second one is not
physical and therefore we should have A = 0.
For the second case we have 2λ1p < νωs which is pure imaginary and represents ingoing and outgoing waves.
We can physically have only outgoing waves so in this case we again have A = 0. The outgoing wave solution
can not represent a standing wave which means the standing wave is only possible for the first case.
A more detailed model includes more sophisticated variational methods to calculate eigenvalues (pulsation
frequencies) and eigen functions (how the wave propagates through the star).
3
Conclusion
Derivation of equations of pulsation for the star is very complicated. For simplicity this equations are usually
linearized, but a price should be paid for this simplicity. The first consequence of this simplification is that
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the amplitude of the oscillations can not be determined and therefore it is not possible to get an estimation
for the size of the star. The second consequence is that we are not able to derive non-radial pulsations that
are frequent for asymptotic giant branch stars.
In many of this equations, the equation of the state is assumed to be adiabatic, which is true for the most
of the parts in star but not everywhere. Overall the results are reliable up to a small percentage for many
Cepheid like stars but not true for more complicated cases such as AGB stars with extended atmospheres
and shock waves traveling through them coming from nonlinear effects in hydrodynamics.
Also in the present case, the convection is neglected, which could make the equation very complicated and
chaotic and it usually depends on the choice of convection theory.
References
[1] H. S. Leavitt, E. C. Pickering, Periods of 25 Variable Stars in the Small Magellanic Cloud, Harvard
College Observatory Circular. 173: 1, (1912).
[2] J. P. Cox, “Theory of Stellar Pulsation”, Princeton University Press, (1980)
[3] B. W. Carroll, D. A. Ostlie, “An Introduction to Modern Astrophysics”, second edition, PearsonAddison Wesley, (2007)
[4] A. S. Eddington, M.N.R.A.S, 72, 2, (1918)
[5] A. S. Eddington, M.N.R.A.S, 72, 177, (1918)
[6] J. P. Cox and R. T. Guili, “Principles of Stellar Structure”, N.Y. Gordon and Breach, (1968)
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