Stellar pulsation theory – Regular versus irregular variability
Encyclopedia
Stellar pulsations are caused by expansions and contractions in the outer layers as a star seeks to maintain equilibrium. These fluctuations in stellar radius causes corresponding changes in the luminosity of the star. Astronomers are able to deduce this mechanism by measuring the spectrum and observing the Doppler effect
. Many intrinsic variable stars that pulsate with large amplitude
s, such as the classical Cepheids, RR Lyrae
stars and large-amplitude Delta Scuti
stars show regular light curves. (By regular one means that Fourier analysis shows amplitudes that are constant in time.)
This regular behavior is in contrast with the variability of stars that lie parallel to and to the high-luminosity/low-temperature side of the classical
variable stars in the Hertzsprung-Russell diagram. These giant stars are observed to undergo pulsations ranging from weak irregularity, when one can still define an average cycling time or period
, (as in most RV Tauri
stars and Semiregular variables) to the near absence of
repetitiveness in the Irregular
variables. The W Virginis variables are at the interface; the short period ones are regular and the longer period ones show first relatively regular alternations in the pulsations
cycles, followed by the onset of mild irregularity as in the RV Tauri stars into which they gradually morph as their periods get longer. Stellar evolution and pulsation theories suggest that these irregular stars have a much higher luminosity to mass (L/M) ratios.
A third category of variable stars are the non-radial pulsators which typically have much smaller pulsation
amplitudes. With relative fluctuations in brightness from ~10\% down the observable limit, non-radial pulsation is very common among stars.
Here we address the mathematical and physical reasons for the difference between the regular and irregular large amplitude stars. Intuitively, a prerequisite for irregular variability is that the star be able to change
its amplitude on the time scale of a period. In other words, the coupling between pulsation and heat flow must be sufficiently large to allow such changes. This coupling is measured by the relative linear growth- or decay rate
of the amplitude of a given normal mode
in one pulsation cycle (period). For the regular variables (Cepheids, RR Lyrae, etc) numerical stellar modeling and linear stability analysis show that
is at most of the order of a couple of percent for the relevant, excited pulsation modes. On the other hand, the same
type of analysis shows that for the high L/M models
is considerably larger (30% or higher).
imply that there are two distinct time scales, namely the period of oscillation and the longer time associated with the amplitude variation. Mathematically speaking, the dynamics has a center manifold
, or more precisely a near center manifold. In addition, it has been found that the
stellar pulsations are only weakly nonlinear in the sense that one can limit oneself to low powers of the pulsation amplitudes to describe them. These two properties are very general and occur for oscillatory systems in many other fields such as population dynamics, oceanography, plasma physics, etc.
The weak nonlinearity and the long time scale of the amplitude variation can be taken advantage of to reduce the temporal description of the pulsating system to that of only the pulsation amplitudes, thus eliminating motion on the short time scale of the period. The result is a description of the system in terms of amplitude equations that are truncated to low powers of the amplitudes. Such amplitude equations have been derived by a variety of techniques, e.g. the Poincaré-Lindstedt method
of elimination of secular terms, or the multi-time asymptotic perturbation method, and more generally, normal form theory.
For example, in the case of two non-resonant modes, a situation generally encountered in RR Lyrae variables, the temporal evolution of the amplitudes A1 and A2 of the two normal modes 1 and 2 is
governed by the following set of ordinary differential equations
where the Qij are the nonresonant coupling coefficients.
These amplitude equations have been limited to the lowest order nontrivial nonlinearities. The solutions that interest us in stellar pulsation theory are the asymtotic solutions (time → infinity) because the time scale for the amplitude variations is generally very short compared to the evolution time scale of the star which is the nuclear burning time scale. The equations
above have fixed point
solutions with constant amplitudes, corresponding to single-mode
(A1
0, A2 = 0) or
(A1 = 0, A2
0) and double-mode
(A1
0, A2
0)
solutions. These correspond to singly periodic and doubly periodic pulsations of the star. It is important to emphasize that no other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients.
For resonant modes the appropriate amplitude equations have additional terms that describe the resonant coupling among the modes. The Hertzsprung progression in the light curve morphology of classical (singly periodic)
Cepheids is the result of a well-known 2:1 resonance among the fundamental pulsation mode and the second overtone mode. The amplitude equation formalism can be further extended also to nonradial stellar pulsations.
In the global analysis of pulsating stars, the amplitude equations make possible to map out the bifurcation diagram (see also bifurcation theory
) between the possible pulsational states, such as the various single- and double-mode states. In this picture, the boundaries of the instability strip
where pulsation sets in during the star's evolution correspond to a Hopf bifurcation
.
The existence of a center manifold eliminates the possibility of chaotic (irregular) pulsations on the time scale of the period. Although resonant amplitude equations are sufficiently complex to also allow for chaotic solutions, this is a very different chaos because it is in the temporal variation of the amplitudes and occurs on a long time scale.
One sees that while long term irregular behavior in the temporal variations of the pulsation amplitudes is possible when amplitude equations apply, this is not the general situation. Indeed, for the majority of the
observations and modeling, the pulsations of these stars occur with constant Fourier amplitudes, leading to regular pulsations that can be periodic or multi-periodic (quasi-periodic in the mathematical literature).
, and consequently there exist no amplitude equations to help us understand these pulsations. The large 
are a prerequisite for chaos, although not a sufficient condition (see for example the Shilnikov theorem http://www.scholarpedia.org/article/Shilnikov_bifurcation). Completely different techniques are required for understanding chaotic behavior (see Chaos theory
). There are both observational and numerical hydrodynamical results strongly suggesting that, at least in some well-studied cases, the variability is due to a low dimensional chaotic dynamics. The observational evidence is particularly strong for the star R Scuti (see Low dimensional chaos in stellar pulsations
).
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
. Many intrinsic variable stars that pulsate with large amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
s, such as the classical Cepheids, RR Lyrae
RR Lyrae
RR Lyrae is a variable star in the Lyra constellation. It is the prototype of the RR Lyrae variable class of stars. It has a period of about 13 hours, and oscillates between apparent magnitudes 7 and 8. Its variable nature was discovered by the Scottish astronomer Williamina Fleming at Harvard...
stars and large-amplitude Delta Scuti
Delta Scuti
Delta Scuti is a white, F-type giant star in the constellation Scutum. It is approximately 187 light years from Earth. Delta Scuti is the prototype of the Delta Scuti type variable stars. It is a high-amplitude δ Scuti type pulsator with light variations of about 0.15 minutes...
stars show regular light curves. (By regular one means that Fourier analysis shows amplitudes that are constant in time.)
This regular behavior is in contrast with the variability of stars that lie parallel to and to the high-luminosity/low-temperature side of the classical
variable stars in the Hertzsprung-Russell diagram. These giant stars are observed to undergo pulsations ranging from weak irregularity, when one can still define an average cycling time or period
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
, (as in most RV Tauri
RV Tauri
RV Tauri is a star in the constellation Taurus. It is a yellow supergiant and is the prototype of a class of pulsating variables known as RV Tauri variables.RV Tau gives a better idea of the lives and deaths of stars like our Sun...
stars and Semiregular variables) to the near absence of
repetitiveness in the Irregular
Irregular
Something that is irregular does not follow the expected pattern. The term is used in many different fields, with quite different meanings.-Canon Law:...
variables. The W Virginis variables are at the interface; the short period ones are regular and the longer period ones show first relatively regular alternations in the pulsations
cycles, followed by the onset of mild irregularity as in the RV Tauri stars into which they gradually morph as their periods get longer. Stellar evolution and pulsation theories suggest that these irregular stars have a much higher luminosity to mass (L/M) ratios.
A third category of variable stars are the non-radial pulsators which typically have much smaller pulsation
amplitudes. With relative fluctuations in brightness from ~10\% down the observable limit, non-radial pulsation is very common among stars.
Here we address the mathematical and physical reasons for the difference between the regular and irregular large amplitude stars. Intuitively, a prerequisite for irregular variability is that the star be able to change
its amplitude on the time scale of a period. In other words, the coupling between pulsation and heat flow must be sufficiently large to allow such changes. This coupling is measured by the relative linear growth- or decay rate
of the amplitude of a given normal modeNormal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
in one pulsation cycle (period). For the regular variables (Cepheids, RR Lyrae, etc) numerical stellar modeling and linear stability analysis show that
is at most of the order of a couple of percent for the relevant, excited pulsation modes. On the other hand, the sametype of analysis shows that for the high L/M models
is considerably larger (30% or higher).Regular Variables
For the regular variables the small relative growth rates imply that there are two distinct time scales, namely the period of oscillation and the longer time associated with the amplitude variation. Mathematically speaking, the dynamics has a center manifoldCenter manifold
In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium...
, or more precisely a near center manifold. In addition, it has been found that the
stellar pulsations are only weakly nonlinear in the sense that one can limit oneself to low powers of the pulsation amplitudes to describe them. These two properties are very general and occur for oscillatory systems in many other fields such as population dynamics, oceanography, plasma physics, etc.
The weak nonlinearity and the long time scale of the amplitude variation can be taken advantage of to reduce the temporal description of the pulsating system to that of only the pulsation amplitudes, thus eliminating motion on the short time scale of the period. The result is a description of the system in terms of amplitude equations that are truncated to low powers of the amplitudes. Such amplitude equations have been derived by a variety of techniques, e.g. the Poincaré-Lindstedt method
Poincaré-Lindstedt method
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail...
of elimination of secular terms, or the multi-time asymptotic perturbation method, and more generally, normal form theory.
For example, in the case of two non-resonant modes, a situation generally encountered in RR Lyrae variables, the temporal evolution of the amplitudes A1 and A2 of the two normal modes 1 and 2 is
governed by the following set of ordinary differential equations
where the Qij are the nonresonant coupling coefficients.
These amplitude equations have been limited to the lowest order nontrivial nonlinearities. The solutions that interest us in stellar pulsation theory are the asymtotic solutions (time → infinity) because the time scale for the amplitude variations is generally very short compared to the evolution time scale of the star which is the nuclear burning time scale. The equations
above have fixed point
Fixed point
"Fixed point" has many meanings in science, most of them mathematical.* Fixed point * Fixed-point combinator* Fixed-point arithmetic, a manner of doing arithmetic on computers* Benchmark , fixed points used by geodesists...
solutions with constant amplitudes, corresponding to single-mode
(A1
0, A2 = 0) or(A1 = 0, A2
0) and double-mode(A1
0, A20)solutions. These correspond to singly periodic and doubly periodic pulsations of the star. It is important to emphasize that no other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients.
For resonant modes the appropriate amplitude equations have additional terms that describe the resonant coupling among the modes. The Hertzsprung progression in the light curve morphology of classical (singly periodic)
Cepheids is the result of a well-known 2:1 resonance among the fundamental pulsation mode and the second overtone mode. The amplitude equation formalism can be further extended also to nonradial stellar pulsations.
In the global analysis of pulsating stars, the amplitude equations make possible to map out the bifurcation diagram (see also bifurcation theory
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
) between the possible pulsational states, such as the various single- and double-mode states. In this picture, the boundaries of the instability strip
Instability strip
The Instability strip is a nearly vertical region in the Hertzsprung–Russell diagram which is occupied by pulsating variable stars .The instability strip intersects the main sequence in the region of A...
where pulsation sets in during the star's evolution correspond to a Hopf bifurcation
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...
.
The existence of a center manifold eliminates the possibility of chaotic (irregular) pulsations on the time scale of the period. Although resonant amplitude equations are sufficiently complex to also allow for chaotic solutions, this is a very different chaos because it is in the temporal variation of the amplitudes and occurs on a long time scale.
One sees that while long term irregular behavior in the temporal variations of the pulsation amplitudes is possible when amplitude equations apply, this is not the general situation. Indeed, for the majority of the
observations and modeling, the pulsations of these stars occur with constant Fourier amplitudes, leading to regular pulsations that can be periodic or multi-periodic (quasi-periodic in the mathematical literature).
Irregular Pulsations
For high L/M stars no center manifold exists because of their large relative growth rates, and consequently there exist no amplitude equations to help us understand these pulsations. The large are a prerequisite for chaos, although not a sufficient condition (see for example the Shilnikov theorem http://www.scholarpedia.org/article/Shilnikov_bifurcation). Completely different techniques are required for understanding chaotic behavior (see Chaos theoryChaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
). There are both observational and numerical hydrodynamical results strongly suggesting that, at least in some well-studied cases, the variability is due to a low dimensional chaotic dynamics. The observational evidence is particularly strong for the star R Scuti (see Low dimensional chaos in stellar pulsations
Low dimensional chaos in stellar pulsations
Low dimensional chaos in stellar pulsations is the current interpretation of an established phenomenon. The light curves of intrinsic variable stars with large amplitudes have been known for centuries to exhibit behavior that goes from extreme regularity,...
).