HELIOSEISMOLOGY Using Seismic Waves to Look Into the Sun's Interior

HELIOSEISMOLOGY

Using seismic waves to look into the Sun’s interior Challenge

Waves are the basis of the most powerful diagnostic utility known to all humanity.

—the most powerful by far! Perspective Where are we?

Frequencies of waves we use. Bright Matter

The Sun in visible light. Density of the photosphere: 3.5×10−7 g/cm3, i.e., about 2 ten-thousandths of the density of the air in this room. Mass of the solar photosphere: 1.8×1023 g, i.e., about three lunar masses. Dark Matter: Bad Bright Matter: Worse

Irony: It appears that most of the matter of which our universe cannot be seen, other than by its gravitational attraction. We characterize this ob- scurity by the term “dark”. What is ironic is how eﬀective obviously bright matter is at obscuring so much of the little that remains. The entire solar interior is totally obscured by a photosphere that is less than a ten billionth of the total solar mass. Only two possibly diagnostic agents are known to penetrate deep into the interiors of stars and re- emerge with coherent information therefrom: (1) neutrinos, and (2) seismic waves. The latter is the subject of helioseismology. The Crux of wave diagnostics is motion of the medium that conducts the waves. In the early 1960s, Robert Leighton, Robert Noyes and George Simon developed the technique to make line-of- sight Doppler maps of the motion of the solar photosphere. The Doppler Signature of Seismic Waves

Using the Doppler eﬀect to make maps of the surface motion of the Sun. Line-of-Sight Doppler Snapshot of the Sun.

Doppler snapshot of the line-of-sight surface motion of the Sun from SOHO/MDI. Note (1) the gradient due to solar rotation and (2) the Evershed eﬀect in Sunspots. The Solar Supergranulation

Doppler motion averaged over an hour shows celluar horizontal ﬂows with a scale size of 30-50 Mm, a major discovery of Leighton, Noyes and Simon. The 300-Second Oscillations

Power spectrum of Doppler oscillations observed by SOHO/MDI. Basic Properties

Peak-to-peak amplitude of the solar oscillations: ∼20 km. Characteristic period: ∼5 minutes. Leighton, Noyes and Simon quickly recognized the solar oscillations as the signature of waves propagating throughout the solar interior. This is widely regarded to be the advent of helioseismology. These waves are most generally thought to be excited by convection, mostly in a thin layer a few hundred km beneath the photosphere. Mean Seismic Energy Flux: ∼1 kWatt/m2. This is 0.0025% of the ﬂux of sunlight through the photosphere, but happens to be roughly equal to the ﬂux of sunlight at Earth. Wave Mechanics

Linear Harmonic Oscillations

A broad class of idealized harmonic oscillations are based upon restoring forces acting against inertia. Hence, if φ is an amplitude with inertia m, and is confronted with a restoring force characterized by a modulus κ, the equation of motion is

d2φ m = −κφ, (1.1) dt2 and the resulting motion takes the form

iωt φ = φ0e , (1.2) with κ ω = . (1.3) rm For linear systems involving multiple amplitudes, φi, with i ∈ {1, 2,...,n}, each with inertia mi,

d2φ n m i = − κ (φ − φ ). (1.4) i dt2 ij i j Xj=1

Problem: Show that the matrix κij is “Hermitian”, i.e., that κij = κji. Hint: Impose that momentum,

n dφ P ≡ m i (1.5) i dt Xi=1 must be conserved. Charles Hermite

Charles Hermite 1822–1901, Parisian mathematician. In Hermite’s lifetime, the question of how to “square the circle” with a straight-edge and compass was a holy grail to mathematicians from antiquity, centuries older than Jesus of Nazereth. In the early 1870s, Her- mite showed that the number e is transcendental. This was the crux to Ferdinand von Lindeman’s proof, in the early 1880s, that π is likewise transcendental. Since then, the grail of how to square the circle has enjoyed the highest honor mathematics can confer: as holy as ever—and securely non-existent. Given that κij is Hermitian, it follows directly that motion of such a system can be expressed in terms of “normal modes”, with amplitudes

N Φi = Λijφj, (1.5) Xj=1 each of which oscillates independently of the others at its own characteristic frequency, νi. The Sun shows some qualities of such a system. Line-of-Sight Doppler Shift of Integrated Sunlight

Sample of line-of-sight Doppler shift integrated over the entire solar disk. Fourier Analysis

Jean Baptiste Joseph Fourier. Line-of-sight Doppler spectrum of integrated sunlight determined by Fourier analysis. Stellar Seismology

Line-of-sight Doppler power spectrum of the Sun (top) and of a nearby star, α-Centauri A, (bottom). Acoustics

For a considerable class of idealized systems, the composition is that of a continuous medium, the amplitude, φ, is a ﬁeld over the spatial domain of the system, and each spatial element of the medium interacts only with other elements that are relatively nearby. The restoring agent is stated in terms of force densities, f, let’s say “twangs”, that act in some proportion to the the relative “curva- ture” of the ampitude proceeding in some direction through the point at which the force is to be evaluated. In a one-dimensional medium, with the spatial dimension represented by a distance z, the twang is something like the second derivative of φ with respect to z times an appropriate modulus, κ, hence ∂2φ ∂2φ ρ = κ . (2.1) ∂t2 ∂z2 For a uniform isotropic medium in three dimensions, this generalizes to

∂2φ ρ = κ∇2φ. (2.2) ∂t2 Problem: Consider an isotropic ﬂuid, the defor- mation of which is expressed by a vector displace- ment, ξξξξξ(r), as a function of location, r. Show that the linear wave equation for such a medium is

∂2ξξξξξ ρ = ∇κ∇· ξξξξξ. (2.3) ∂t2

Problem: Show that equation (2.3) is time-reversal invariant.

Problem: Consider acoustics as prescribed by equation (2.3) for a uniform medium, meaning simply that κ and ρ are constants, independent of time and location in the medium. Then equation (2.3) becomes ∂2ξξξξξ = c2∇∇ · ξξξξξ. (2.4) ∂t2 where κ c2 ≡ . (2.5) ρ Show that equation (2.5) is satisﬁed by “longitudi- nal plane-waves”, i.e., waves of the form

ˆ i(k·r − ωt) ξξξξξ = ξ0ke (2.6) where kˆ is unit vector in the direction, k, along which the wave propagates, and the dispersion relation is ω = |k|c ≡ kc. (2.7) Nodes

Normal modes in a perfectly elastic medium ap- pear to have points at which the amplitude is null. This is easy to see in a 1-dimensional medium. What is the character of nodes in a medium of 2 or more dimensions?

Problem: Consider a ﬁnitely bounded continuous linear medium that obeys time-reversal invariance. Consider further a set among the normal modes of this system, each with a characteristic frequency νi with i ∈ {1, 2,...,n}. Show that all of the modes that apply to these frequencies can be expressed in terms of ﬁelds, ζζζζζi(r), that are compartmentalized into regions separated by nodal surfaces, i.e., surfaces on which ζζζζζi is null. (This does not preclude the possibility of a mode lacking a nodal surface, if this mode encompasses no more than a single com- partment spanning the entire spatial domain of the system.) Spherical harmonics.

Adriene-Marie Legendre Charicature by Julien-Leopold Boilly Exampes of spherical harmonics. Gravity

∂2ξξξξξ ρ = ∇κ∇· ξξξξξ − ρg∇· ξξξξξ + ρ∇g · ξξξξξ. (3.1) ∂t2 The ﬁrst term on the right side of equation (3.1) can be called the “resiliency”. The second is com- monly call “buoyancy”. I propose to call the third term the “slosh”. It introduces a phenomenon called the “gravity wave”.

Problem: Consider a fluid stratified by gravity so that its modulus, κ, is constant along any isobar, and the pressure, p, in the undisturbed medium obeys the law ∇p = ρg. (3.2) Show that the wave equation for such a fluid becomes ∂2ξξξξξ ρ = c2∇∇·ξξξξξ + (Γ − 1)g∇·ξξξξξ + ∇g·ξξξξξ, (3.3) ∂t2 where κ c2 ≡ , (3.4) ρ and dκ Γ ≡ . (3.5) dp Compression Waves

Consider the simple case of waves propagating vertically in a medium with constant sound speed and Γ. The wave equation for this instance becomes ∂2 ∂2 ∂ ξ = c2 ξ + Γ ξ. (4.1) ∂t2 ∂z2 ∂z Problem: Consider the field ζ defined by − ξ ≡ e βzζ, (4.2) where γg β ≡ . (4.3) 2c2 Show that ∂2ζ ∂2ζ = c2 + ω2ζ, (4.4) ∂t2 ∂z2 a where γg ω = . (4.5) a 2c Problem: What is the dispersion relation for equation (4.4)? Problem: How is the behavior of vertically propagating compression waves in a gravitationally stratified medium similar to the propagation of electro- magnetic waves in a uniform cold plasma? Gravity Waves

A fluid is convectively stable if it’s density de- creases rapidly with height, meaning that the bottom layers are much more dense than the top layers. If we disturbe such a fluid, it sloshes back to- wards its equilibrium configuration, generally over- shooting, hence oscillating. Some perspective into gravity oscillations, such as these, can be gained by considering an idealized liquid, i.e., a “Boussinesq fluid”, in which the modulus, κ approaches infinity. Then the first term on the right of equation (3.1) becomes an unknown Lagrange multiplier of the form ∇J, and we apply the condition

∇· ξξξξξ =0 (5.1) to supply the constraint needed to resolve it. In such a case, the buoyancy term, ρ0g∇· ξξξξξ, vanishes, and the wave equation for the vertical component of ξξξξξ becomes ∂2 2 N 2 2 ξ , . 2 ∇ − ∇h z = 0 (5 2) ∂t where ∂ N 2 ≡ g ln ρ . (5.3) ∂z 0 The constant N has units of frequency, and is called the Brunt-Väiäsäla frequency. Problem: Show that the dispersion relationship for equation (2.8) is k ω = N h . (5.4) k Problem: Show that the phase speed for gravity waves is N v = cos α , (5.5) φ k where α is the inclination of the k-vector from horizontal. Problem: Show that the group velocity of a gravity-wave packet is the same as the phase speed, but the direction is perpendicular to the wave vector, k. This means that while the nodes of the waves of which gravity-waves packets are composed move perpendicular to themselves, the wave packets slip sidewise along the nodes. Problem: Show that the Brunt-Väiäsäla frequency for a compressible medium is dρ dρ N 2 = g − . (5.6) dp dp ad The Fundamental Modes

Problem: Consider the general class of non- compressional modes for equation (3.3) in which c and g are constant. Show that there are such modes. We call them “fundamental modes”. Show that fundamental modes have the following dispersion relation: ω2 = kg. (6.1) Plots of acoustic power as a function of ν and l for a model of the solar interior. The Spatial-Temporal Acoustic Power Spectrum

Acoustic power as a function of frequency, ν and surface wavenumber, l. Normal modes of the Sun plotted in the ν-l plane. Questions?