4 Oscillations of stars: asteroseismology

Transcription

1 4 Oscillations of stars: asteroseismology The HR diagram below shows a whole variety of different classes of variable or pulsating/oscillating stars. The study of these various classes constitutes the expanding field of Asteroseismology. Figure 4.01: Classes of oscillating stars on the HR diagram

2 We can have two different physical types of oscillation. This is because we can either compress or displace parcels of gas: 1. Acoustic oscillations: These oscillations are formed by standing sound (pressure or p) waves, and result from compression of the gas. Gradients of pressure inside the star act as the restoring force for the oscillations. 2. Buoyancy oscillations: These oscillations are formed by standing buoyancy (gravity or g) waves, and result from displacement of the gas. Gravity, acting on the perturbation in density, acts as the restoring force for the oscillations. 4.1 Timescale for acoustic pulsation Boardwork 4.2 Non-radial modes We can also have non-radial modes, in which we have components of motion at right angles to the radial direction (i.e., across the surface of the star).

3 Small-amplitude oscillations of a star may then be described by spherical harmonic functions of the form: Y ( θ, φ), m l where l and m are integers, θ is co-latitude (angle from pole of star) and φ is longitude. We have: Y m l m m ( 1) c P (cosθ )exp( im ) ( θ, φ) = φ, (4.05) lm l m where P (cosθ ) are Legendre polynomials, and the l function c lm is given by: ( 2l + ) 2 1 ( l m)! c lm = 4π ( l + m)!. (4.06) The angular degree, l, measures the wavelength (or wave number) of sound waves over the surface of the star (what we shall refer to as the horizontal direction). Radial modes have l=0. If k h is the horizontal wave number and λ h the horizontal wavelength, we have: k h 2 π L l( l + 1) = = = λ R R h, (R4.07) where R is the stellar radius and

4 L = l( l + 1). (R4.08) The equivalent horizontal wavelength is therefore: 2 π 2π R λh = = L. (R4.09) k h Thus, L is very roughly the number of wavelengths along the solar circumference. Figure 4.02: Spherical harmonic oscillations of stars The azimuthal order, m, measures the number of nodes around the equator. Without rotation the frequencies would be degenerate in m, but rotation lifts this degeneracy. If

5 waves have a velocity component in the direction of the rotation, they will be advected by it (see Figure 4.03), and with this extra push we will observe a higher frequency. Similarly, waves with a component against the direction will be advected to lower frequencies. Axis of rotation Figure 4.03: Effect of rotation on the waves The angular frequencies will be displaced according to: ω nlm = ωnl0 + mω, (R4.10) where Ω is a suitable average of the angular velocity in the cavity probed by the waves.

6 4.3 Trapping of acoustic waves in stellar interiors Before we look at a quantitative description of waves in the interior, let us first consider the physics of how acoustic waves are trapped in stellar interiors (we shall concentrate on results of observations of acoustic, as opposed to buoyancy, oscillations). Waves need to be trapped in cavities to set up standing waves; with the result we see oscillations Refraction at lower turning point Let us begin by considering a wave heading back into the interior from the surface. The temperatures of the interior increase as the acoustic waves propagate inwards. Consider Figure surface increasing T b a direction of propagation Figure 4.04: the refraction of acoustic wave fronts in the solar interior

7 Since the temperature of the solar plasma increases with depth, point a on the wave front shown will travel faster than point b. This follows from the fact that the speed of sound in a medium with temperature T, pressure P, density ρ, mean molecular weight µ, and adiabatic exponent γ, (with R being the gas constant) is given by: 1 γ P 2 γ RT 2 c = = ρ µ. (R4.01) 1 As a result, the direction of propagation, which is orthogonal to the wave front, bends away from the inward radial direction Eventually, the wave is refracted back toward the surface of the Sun. The radius of deepest penetration is the lower turning point (LTP); beneath this there exists a zone of avoidance that the wave will never sample.

8 Waves launched at a steeper angle penetrate more deeply (Figure 4.05). The deeper the penetration, the longer is the horizontal wavelength λ h and the lower is the angular degree, l. Lower turning point Figure 4.05: Differential penetration of acoustic waves Reflection at upper turning point As waves head back toward the surface, they encounter a region where the density falls sharply. If the period of the wave is longer than the time it takes to traverse this region, the perturbations required to give the wave cannot be sustained over the period of the wave. The wave cannot travel outwards and we get reflection.

9 This implies a critical period or equivalently a critical frequency. This critical frequency is called the acoustic cutoff frequency, ω ac : = c 2H ω, (R4.11) ac / where: RT H = µ g. (R2.14) If ω > ω ac a wave can escape the surface layers. If, on the other hand, ω < ω ac the wave cannot propagate any further, and will undergo reflection. This latter class of waves are the ones that can set up standing waves and oscillations. 4.4 Wave equation for waves in stellar interiors Boardwork

10 4.5 Musical Instrument Analogy Let us begin with the analogy of a simple 1-D pipe, which runs from z = 0 to L. We look at two pipes: a pipe open at both ends, and another closed at one end and open at the other. Figure 4.06: Waves in 1-D pipes In the fully open pipe, the boundary conditions are such that: λ The fundamental (first harmonic) has L = 2

11 The first overtone (2 nd harmonic) has The second overtone (3 rd harmonic) has And so on An expression for all the overtones is: (R4.18) where n is an integer. We can re-write this as: L k z 2π n + 1 = λ, λ 2 where k z is the wavenumber. So, we have: k L = k dz = ( n +1) π. (R4.19) z + 1 L = n λ, 2 L z This is a classic interference condition: φ = k dz = ( n +α) π, L z (4.20) where α is a constant. The value of this constant is fixed by the boundary conditions. It is 1 for a fully open pipe, and 0.5 for a semi-closed pipe. L = λ L = 3λ 2 Can we apply this analogy to the case of an oscillating star? The answer is yes. Here, we consider the interference

12 condition in the radial direction (recall Equations 4.12 and 4.13), and hence the radial wavenumber: R φ = k dr = ( n +α) π. r t t r (R4.21) Here, the limits on the integral are the lower and upper turning points for the waves. The value for α is about 1.5 in stars, as we now go on to show for the Sun Duvall Law Do oscillations of stars obey the simple interference condition above? Tom Duvall was the first person to show that they do (using data on the oscillations of the Sun). We start here with: kl = ( n + α ) π. We assume that locally the waves behave as plane waves. This means that: ω = ck. We may therefore re-write the interference condition as: ωl = n + c ( α ) π.

13 The travel time across the cavity (pipe) is just: τ = So, we have: Now: L c. + α τ = n π. ω When a wave reaches the lower boundary of the cavity, it must by definition be moving horizontally The horizontal component of the speed is just ω/k, and this must equal sound speed, c, at this depth. So, any two modes having the same value of ω/k must penetrate to the same depth, and be bounded by the same cavity If the cavities are identical, then the travel time τ will be the same. The implication of the above is that: τ = f ω k.

14 This means that: n + α π = ω f ω k. A plot of n + α π vs. ω / k should collapse onto a ω single curve, if the oscillations of a star obey the simple interference condition. Duvall found this to be the case, provided α~1.5 (Figure 4.07 shows results with real data on the Sun). The fact that the constant α differs from the values for simple pipes is not surprising. α = 1.5 n + α π ω ω / k Figure 4.07: Demonstration of the Duvall Law

15 4.5.2 Frequency Spectrum Let us consider the fully open pipe. Its first three frequencies (fundamental F; harmonics H; overtones O) are given by: ν ν ν =ν F H1 = =ν O 1 H 2 = =ν O 2 H 3 = c. 2L 2c. 2L 3c. 2L These frequencies follow the relation: c ν (4.22) 2L n. n = The frequency spacing between successive overtones is: c ν = 2L. (4.23)

16 Figure 4.08: Frequency spectra of 1-D pipes Might we expect to see a similar pattern in stars, i.e., a spectrum of overtones showing a regular spacing in frequency? The answer again is yes, although the spacing is not exactly uniform (but not far off). To get a description that includes non-radial modes, we go straight to the solution of oscillations in a sphere (albeit one which has a homogeneous structure). Oscillations of a uniform, non-rotating sphere obey: c ν nl (2n + l) 4R. (4.24)

17 This assumes clamped boundary conditions, where the displacement is zero at the centre and surface. This solution is an asymptotic approximation. The exact solutions are Bessel functions, and the spacing between successive overtones is not exactly uniform. However, the higher is n, the more accurate the above approximation becomes. We can re-write Equation 4.24 in terms of the frequency spacing between successive overtones: c ν = 2R, (R4.25) to give: l ν nl ν n + 2. (4.26) If instead we solve the oscillation equations for the case of free boundary conditions (edges not clamped), we get: c 1 ν nl 2n + l + 4R 2, (4.27) and so: l 1 ν nl ν n (4.28)

18 We may write the solution in the more general form: l ν nl ν n + + ε 2, (4.28) where the value of ε depends upon the boundary conditions. Stars can be shown to follow the same asymptotic relation. There is also an extra term because stars are not uniform, so we have: l ν nl ν n + + ε Dl( l + 1) 2, (4.29) where D reflects that properties vary with depth. In fact, D is very sensitive to conditions in the core of the star. For stars, ε takes values between 1 and 2. The value of D is approximately 2 µhz or less. There is one final addition to the equation, because stars rotate: ν ν n + l + ε Dl( l 2 + 1) + m ν nlm rot. (4.30) Here, ν rot is the frequency splitting due to the rotation.

19 4.6 Origins of acoustic oscillations There are two principal mechanisms by which acoustic oscillations are excited Kappa Mechanism Here, acoustic waves are generated by favourable changes in opacity. The mechanism relies on the fact that as a layer in the interior is compressed, and temperatures within it increase, the opacity goes up. This closes a valve and suppresses the outward flow of radiation, and pressures build. When the layer eventually expands, it will do so beyond the radius it would have reached without closure of the valve. When the layer has expanded, temperatures drop and so does the opacity. The valve opens : Radiation streams out, and when the layer contracts it will fall below the radius it would have without the valve opening.

20 We get these conditions in ionization zones in the outer parts of stars. The zones must lie within the right range of physical parameters to drive the oscillations: If the zones lie too deep, conditions will tend to be adiabatic, and efficient energy exchange is not possible. If the zones are too shallow, the density will be too low and there will not be enough inertia to drive the oscillations. Figure 4.01: Classes of oscillating stars on the HR diagram

21 Oscillations can be excited to very high amplitudes, e.g., leading to a several-per-cent change in radius and the luminosity of the star (a good analogy is pushing a child on a swing ). The oscillations most strongly excited in Kappa-driven stars are low overtone modes, e.g., the fundamental mode and low overtones. Non-radial modes are not usually observed. Figure 4.01 shows different classes of oscillation. Oscillating stars driven by the Kappa mechanism for ionization zones of He and H lie in the classic instability strip (Cepheids and RR Lyrae stars). Another example is Beta Cephei stars, which are heavy main sequence stars where ionization zones associated with iron group elements drive the oscillations The only major class of oscillating stars in Figure 4.01 not driven by the Kappa mechanism are the solar-like oscillators

22 4.6.1 Solar-like mechanism The Sun was the first star in which solar-like oscillations were observed (hence the name!). Figure 4.09: Solar-like excitation in convective envelopes Here, acoustic waves are excited by turbulence in the outermost layers of subsurface convection zones. Stars must therefore be cool enough to have efficient convection in their outer layers (see Section 2.2.3). A good analogy is a bell in a sandstorm, where lots of small impacts drive the bell to resonance, while many grain

23 impacts also damp the resonance. This limits the oscillations to very weak amplitudes (typically a few partsper-million in luminosity, or a few tens of cm/s in Doppler velocity). A very rich spectrum of oscillations is excited by this stochastic mechanism: we see many overtones, which tend to be high-order overtones. 4.7 Observing the oscillations The oscillations are usually observed in one of two ways Oscillations in intensity We may observe the change in intensity that results from compressions of the gas. Assuming a black body, we have: L R 2 T 4. The amplitude of the oscillations in luminosity (bolometric) is therefore given by: δl δr δt = L R T.

24 If we assume that changes in L are dominated by changes in T, we have that: δl δt L T. Stochastically excited oscillations are also damped intrinsically by the convection, and hence tend to show smaller amplitudes than their Kappa-driven cousins. A main-sequence star, like the Sun, shows oscillations with δl / L ~ a few parts in Cepheids can show oscillations with amplitude δl / L ~ 0.1! Oscillations in Doppler velocity We may also observe the Doppler shifts of atoms in the photospheres of the stars (that are displaced periodically by the oscillations). Can we relate the amplitudes expected in Doppler velocity to those expected in intensity? This is useful for comparing observations of different types, and for understanding the physics of the oscillations and by implication the physics in

25 the near-surface layers. We assume any such relation should hold for both classes of oscillations, since the intrinsic nature of the oscillations is the same (i.e., acoustic in nature). We start from: L R 2 T 4. The amplitude of the oscillations in luminosity (bolometric) is therefore given by: δl L δr δt = R T. If we assume that changes in L are dominated by changes in T, we have that: δl δt L T. Next, we assume that the oscillations are adiabatic, i.e., there is no exchange of energy between different internal layers during the pulsation. Adiabatic conditions imply that:

26 γ P ρ. We therefore have that: γ ρt ρ, 1 T ρ γ. So: δt T δρ δρ [ γ 1] ρ ρ, (4.31) This assumes that the adiabatic constant does not alter. Now, to first order, the density compression for an adiabatic sound wave will satisfy: δρ ρ v c, (4.32) where v is the velocity and c is the speed of sound. Then we remember that: c = So: γ P ρ c 2 T. 1 2 = γ R µ T 1 2, (R4.01) We can then put all this together, to give:

27 δ L T v v δ L T c T. (D4.33) So the bolometric luminosity amplitude and Doppler velocity amplitude are expected to be related to one another via the square root of temperature. Since we observe the oscillations in the surface layers, the appropriate temperature to use to scale from one to another is the effective temperature, T eff. 4.8 Some fundamentals: solar-like acoustic oscillators v max Figure 4.11: Acoustic oscillation spectrum of the Sun. The dotted line shows the envelope of power, and is there to guide the eye

28 Boardwork Figure 4.12: Part of spectrum, showing large (red) and small (blue) frequency spacings.

29 Figure 4.13: Cavities of two modes of similar frequency Figure 4.14: The JCD diagram: δ 0,2 versus ν

30 4.9 Some Case Studies Boardwork