Linear Theory of Stellar Pulsation Mir Emad Aghili 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 638 by Johannes Holwarda when he realized that this star changes its brightness drastically in a period of 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 candle. An example of such objects are Cepheid stars. A relation between brightness and period of Cepheids was established by Henrietta Swan Leavitt []. 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 parallax 2 of some nearby Cepheids. Cepheids are the most common pulsating stars. Their period is between to 5 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 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). 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 to 7 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 maghili@go.olemiss.edu An object in astrophysics that has a known luminosity and can be used for distance measurement. 2 Paralax 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] = /π[arc sec], where π is parallax.
Figure : 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 m r = 4πr 2 p GMm r 2. ) If one assumes that the star is surrounded with vacuum, we are able to assume an adiabatic equation of state p/ρ Γ = const. which after linearization 3 procedure simplifies to δ r = 3Γ 4) GM δr. 2) R3 This is a simple harmonic motion with frequency ω = 3Γ 4) GM/R 3. 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 dependence 4 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 pρ, T, X) and nuclear energy production rate per unit mass ɛρ, 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 dp dr = Gmρ r 2 Hydrostatic Equilibrium 3) dm dr = 4πr2 ρ Conservation of Mass 4) dt dr = 3ρ κ 6σT 3 F rad Radiative Flux 5) d r 2 F rad + F conv ) ) = r 2 ρɛ dr Thermal Equilibrium 6) F conv = F conv ρ, T, X, g; ), = p dt, T dp Convective Flux 7) where κ 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 = 2 5. If rad < ad at some radius r, then the layer will be radiative and the convective flux F conv vanishes, on the other hand if rad > ad convection starts and F conv >. The functionality of F conv 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 δ r = 4Gm r 2 4πr 2 δp ) δp = Rλp 4ζ xr x p g ζ + δp ) 8) p δρ ζ = 3 ζ r 9) ρ r δl = 4ζ δκ + 4 δt + ) δt L κ T d ln T ) r dr T ) ) δp δρ = Γ + ρ Γ 3 ) δ ɛ L ), ) t p t ρ p where ζ = δr r is the relative change in distance, R is the star radius at equilibrium, x = r /R is the relative distance, λ p = dr d ln p = p ρg is the pressure height, L is the luminosity and Γ 3 = + d ln T/d ln ρ) ad. We can use the conservation of mass equation dm = 4πrρ 2 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... r ζ = 4πr 2 ζ d dm [3Γ, 4) p ] + 2πrΓ 2, p ζ +6π 2 r 2 r 3 Γ, p ρ ζ ) 4πr 2 where Γ = d ln p/d ln ρ) ad. For small adiabatic oscillations, one can use [ ρ Γ 3, ) δ ɛ L )], 2) δp δρ = Γ, 3) p ρ 3
in momentum equation equation 8) and the oscillation equations become r ζ = ζ4πr 2 d dm [3Γ, 4) p ] + ) 6π 2 Γ, p ρ r 6 ξ. 4) r A simplified equation is called homologous motion, which means motion when ζ is independent of mass m. In such a motion dp dm = Gm/4πr4 and therefore r ζ = 3Γ, 4) Gm r 2 ζ = 3Γ, 4) 4r G ρζ, 5) 3 where ρ is the average density of the star. The solution to this equation simple harmonic motion with frequency ω = 4πGm 3Γ, 4)) /r, in agreement with primitive model that was discussed previously. This result gives stable pulsation if Γ > 4/3 and we will have instability, including both relaxation or blowing up solutions for Γ < 4/3. For a more general case a standing wave solution to this equation requires that ζr, t) = αr) exp iωt), which changes the partial differential equation to an ordinary differential equation d Γ pr 4 dα ) [ + α ω 2 ρr 4 + r 3 d ] dr dr dr {3Γ 4) p} =, 6) where 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 = at the center and on the surface we can choose different types of boundary conditions. We can require δp = 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 6), the coefficient of dζ dr is singular at the center of the star, therefore we need to require dζ dr r= =. 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/λ p ), 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/λ p in the atmosphere of real stars, we can drop the smaller terms and find d 2 α dx 2 dα λ p dx + ω2 νs 2 =, 7) where ν s = Γ p/ρ is the adiabatic speed of sound. With basic knowledge of calculus we can see that α = A exp ) 2 ) 2 ω + + B exp ) 2 ) 2 ω. 8) 2λ p 2λ p ν s 2λ p 2λ p ν s To discuss the solutions we need to consider two cases. In the first case, we consider 2λ p > ω ν s, in which the solutions are real. In this case looking at the kinetic energy of the sound wave for two solutions to the equation 8) shows us that lim x K A = and lim x K B =, which means the second one is not physical and therefore we should have A =. For the second case we have 2λ p < ω ν s 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 =. 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 4
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 [] H. S. Leavitt, E. C. Pickering, Periods of 25 Variable Stars in the Small Magellanic Cloud, Harvard College Observatory Circular. 73:, 92). [2] J. P. Cox, Theory of Stellar Pulsation, Princeton University Press, 98) [3] B. W. Carroll, D. A. Ostlie, An Introduction to Modern Astrophysics, second edition, Pearson- Addison Wesley, 27) [4] A. S. Eddington, M.N.R.A.S, 72, 2, 98) [5] A. S. Eddington, M.N.R.A.S, 72, 77, 98) [6] J. P. Cox and R. T. Guili, Principles of Stellar Structure, N.Y. Gordon and Breach, 968) 5