Helioseismology: Studying Solar Oscillations and Wave Propagation, Papers of Physics

Download Helioseismology: Studying Solar Oscillations and Wave Propagation and more Papers Physics in PDF only on Docsity! Helioseismology: Probing the Interior of the Sun Steve Kaeppler Physics 325 Submitted: 21 April 2006 Abstract: This paper will focus on helioseismology, acoustic waves that propagate through the interior of a star. The equilibrium equations and conditions for stars are discussed. Wave oscillations are density and pressure perturbations onto the equilibrium state of the Sun. The properties of wave propagation is also considered, which sets up the conditions for standing waves within the interior of the Sun. 1 I. Introduction The Sun has long been one of the most fundamental objects in the study of astronomy. It was one of the first objects Galileo viewed through his telescope. Sunspots and their cycles have been recorded since the 1700s. The advent of modern astronomy in the early 1900s brought new life into astronomy by using physical techniques, such as spectroscopy, to study fundamental astronomical questions. The early 1900s also began the rise of theoretical physics, which brought some of the best minds to ponder the most fundamental questions in astrophysics. Astronomers pondered on two very fundamental questions in solar physics: what mechanisms power the sun and how is the interior of the sun structured. The question of powering mechanisms was partially answered by Hans Bethe (1939), in which he described the process of combining four hydrogen atoms into helium, called the p-p chain. This theory is considered the dominant theory on how energy is produced at the core of a low mass star. More recently, scientists have been investigating the solar neutrino problem as a means to more fully understand the solar energy system, along with fundamental particle theory. Although the power mechanism within the sun is fairly well understood, the internal structure is still an active area in research. The sun is optically thick (neutrinos were thought to be one method to bi-pass this optical thickness), as a result the photons we see from the photosphere are coming from the upper most regions of Sun. A photon emitted from the core will take nearly a million years to reach the photosphere! Light is not a useful method, so other means are needed to probe the solar interior. In the 1960’s, Robert Leighton discovered radial surface oscillations that had periods on the order of five minutes and amplitudes around one hundred meters per second (Harvey, 1995). Initially, Leighton dismissed the discovery as the local atmosphere’s response to the convective surface; however, in 1968 Edward Frazier disproved this theory, explaining instead that the oscillations came from cavities below the surface (Harvey, 1995). Three other astronomers built on Frazier’s work, later theorized that the oscillations were acoustic waves trapped in subsurface cavities. In 1975, observations by Franz Deuber demonstrated that the predictions made by the acoustic theory were correct (Harvey, 1995). With the evidence in favor of acoustic waves, the field of helioseismology was born. Helioseismology is a method that detects the many order of oscillations on the solar surface to deduce waves being transmitted in the interior. This paper intends to outline some of the fundamental physics that dictates solar surface oscillations. To begin, a brief introduction will be given on equilibrium stellar structure, including the equations of state. Next, oscillations will be explained through perturbation theory of two parameters, pressure and density. These perturbations are applied to the equations of state. Wave propogation conditions, including the development of standing waves, are also discussed. 2 III. Pressure/Density Perturbations The introduction to the equations that govern stellar interiors was mainly to demonstrate the equilibrium conditions for a star. In solar oscillations, the equations of state are manipulated by assuming that there is a small perturbation applied on two parameters, density and pressure (Christensen-Dalsgaard, 2004). Before diving into the derivation, it should be noted and is of extreme importance that this derivation is not rigorous! For a more rigorous treatment of the problem please refer to Christensen-Dalsgaard, (2004 or 1991). It is important to indicate the assumptions in applying perturbation theory. The periods oscillations are very small compared to the thermal timescales; therefore, it is reasonable to assume that oscillations are adiabatic (Christensen-Dalsgaard, 2004). The second assumption is that the perturbations are linearized, meaning that all terms of order greater than one can be discarded (Christensen-Dalsgaard, 2004). The linearization of the equations is presented below. ),(')(),( 10 trprptrpp +=⇒ (8) ),(')(),( 10 trrtr ρρρρ +=⇒ (9) t rv ∂ ∂ = δ (10) These three equations compromise the linear terms in the perturbation. It is important to note that ' is an Eulerian derivative, meaning that its location is fixed with respect to time. Again we take into consideration four equations that sum up the fundamental properties of oscillations. p ''' 0 0 1 0 Φ∇+∇+−∇=∂ ∂ ppp t v ρ ρ ρ (11) 0)(' 01 =+∇ rδρρ (12) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∇⋅+ Γ =∇⋅+ pr pp ppr δδ ρρ ρ 00 1 0 1 1'11' , )ln( )ln( ρd pd =Γ (13) 1 2 '4' ρπG−=Φ∇ (14) Equation (11) represents the momentum equation for a density element, which summarizes the force balance in perturbation. Equation (12) is the continuity equation (mass conservation) for the density perturbation. Equation (13) represents the adabaticity and finally equation (14) is Poisson’s equation that relates to the gravitational potential and density. In this situation we are dealing with a system that is spherically symmetric, it is assumed that spherical coordinates are the most practical system of units(the del operators are not written in spherical coordinates, but assume that is what they represent). Equations (11) – (14) can be combined such that 5 the equations are in the form of a Laplacian in spherical coordinates. The standard method used to solve this type of Laplacian is spherical harmonics (Christensen-Dalsgaard, 2004). Use of spherical harmonics is also found in quantum mechanics when solving Schrodinger’s equation for the hydrogen atom. [ ]timl eYrptrp ϖφθπφθ −= ),()('Re4),,,(' (15) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ += − ti m l m l h m lr e YY rrYrr ωφ φθ θ θ ξφθξπδ ˆ sin 1ˆ)(ˆ),()(Re4 (16) r L r ll kh ≡ + = )1( (17) Equations (15) and (16) are ansatz solutions to the real part of the perturbation pressure term and velocity. The spherical harmonic has the usual order, where n represents the radial oscillation frequency, where l represents the longitudinal degree and m represents the azimuthal degree (Christensen-Dalsgaard, 2004). The variables ξ r and ξh represent amplitude functions for the solution of the equation. Plugging equations (15) and (16) into equations (12)-(14) and then solving for ξr, ' and we get (dropping the 0 and 1 terms for the perturbations): p 'Φ ''112 22 2 2 2 2 Φ+⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Γ +−= r k p k cdr dp prdr d hh r r ωωρ ξ ξ (18) ( ) dr dp dr dp p N dr dp r ''' 22 Φ−Γ+−= ρξωρ (19) ''4'1 222 2 2 Φ+⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +=⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Φ h r s kN gc pG dr dr dr d r ρξ π (20) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − Γ ≡ dr d dr dp p gN ρ ρ 12 (21) Equations (18)-(20) are the solution the density and pressure perturbations applied to the equilibrium of a star. N2 is defined as the buoyancy frequency and cs is the speed of sound (Christensen-Dalsgaard, 2004). Equations (18)-(20) represent the general solution to the density and pressure perturbations, the standard technique is to apply boundary conditions to obtain specific solutions. These equations are solved using physically relevant boundary conditions and values can be tabulated based on the initial assumptions made on parameters in the system. Two basic boundary conditions exist that must be satisfied. Matching to these boundary conditions is similar to matching boundary conditions for a wave propagating through the air. 6 Rr r l dr d ==Φ + + Φ ,0'1' (22) Rr dr dppp r ==+= ,0' ξδ (23) Equation (22) is a surface continuity equation between the surface and the vacuum outside of the star. Equation (23) is the requirement that pressure perturbations do not exist outside of the radius of the star. The reasoning is pretty obvious, for all intents the regions immediately outside of a star is a vacuum; therefore, very small perturbations would not propagate. These boundary conditions conclude the derivation of the pressure and density perturbations for the Sun. Before moving on, it is important to recall the density and pressure perturbations form the basis for acoustic waves. IV. Wave Propagation/Standing Waves Having solved the equations that create waves within the sun, it is now time to examine how the waves propagate through the star. The topic of wave propagation is one of great depth, utilizing many techniques to deduce the properties of waves. One of the simpler approaches will be discussed, called the asymptotic method (Christensen-Dalsgaard, 2004). This method involves applying a limiting case in order to deduce fundamental wave properties. Two examples of asymptotic limiting is described in equation (24) and equation (25). These conditions would be applied to the solutions from equations (18)- (20). Equation (24) deals with the asymptotic case of p-mode oscillations, while equation (25) examines the asymptotic nature of g-modes (Christensen-Dalsgaard, 1991). 12 2 << ω N (24) 12 2 << ω hk (25) The asymptotic approximation allows us to assume that the solutions to the perturbations act like plane waves (Christensen-Dalsgaard, 1991). Already defined is the kh term, which corresponds to the “horizontal” propagation number. A similar quantity called the radial propagation number is also defined, called kr. These quantities correspond to how the pressure perturbation will propagate through the Sun. They are related to each other through the typical dispersion relation. θ̂ˆ hr krkk += r (26) [ ]222222 hrss kkckc +==ω (27) The sound wave (corresponding to the pressure perturbation) will propagate as a combination of the horizontal and radial component as illustrated in equation (27). However, there will come a location 7 and m corresponds to the azimuthal degree (Harvey, 1995). To draw the analogy to quantum mechanics, n represented the principle quantum number, l represented the angular momentum state and m represented the spin angular momentum. P-modes correspond to the radial oscillation number n > 0, f- modes correspond to n = 0 and g-modes correspond to n < 0 (Christensen-Dalsgaard, 2004). The radial oscillation number explains the order of expected oscillations, it makes sense that one would expect to see surface oscillations at n = 0. Models of f, g, and p modes are created through tabulation at different values of l. The results are then plotted onto a l-v chart, which plots the surface oscillation frequency as a function of l number (Christensen-Dalsgaard, 2004). Figure below illustrates the different modes of oscillation on a l-v plot (Christensen-Dalsgaard, 2004). 10 V. Conclusions This paper has outlined, in a simplistic fashion the basic equations that dictate equilibrium within a star. The most fundamental are the hydrostatic equilibrium and temperature equations. There was then a treatment of pressure and density as perturbations onto the equilibrium state of the star. These equations further manifested themselves by the use of spherical harmonics. Equations (18)-(20) were derived that constitute the solution to the pressure and density perturbations. The applications of physically consistent boundary conditions completed the analysis. Using asymptotic analysis and dispersion relations, the dispersion relationship for the waves, it was shown how waves propagate between two end points. The conditions on both of the endpoints were also discussed based on the physical properties inside the sun. The conditions for standing waves were then described, making the analysis of waves complete. Lastly, three standing wave modes were described, the f,p, and g mode oscillations. The physical mechanisms that cause these modes and how they can be applied to models were described. This is a summary of the basic equations the govern wave propagation within the star. Helioseismology is an incredibly rich, unsolved problem. This discussion has neglected much of the science that has been deduced from studies of these wave oscillations. The reader should refer to Jack Harvey’s article in Science magazine, which provides a good introduction to the scientific discoveries in helioseismology. Moreover, the field of oscillation detection in itself is just as exciting as the observations! New techniques in optical interferometry will open new worlds of resolution, perhaps making it possible to understand the elusive g-mode. VI. References Christensen-Dalsgaard, J. Equation-of-State and Phase Tradition in Models of Ordinary Astrophysical Matter. American Institute of Physics, 2004. Christensen-Dalsgaard, J. Solar Interiors and Atmosphere. Tucson: U. of Arizona Press, 1991. Harvey, J. Physics Today ,October, 38 (1995). "Helioseismology." 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