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Published on November 14, 2007

Author: Nikita

Source: authorstream.com

Slide1:  Helioseismology Basic Principles of Stellar Oscillations Helioseismology Asteroseismology Slide2:  The birth of helioseismology Slide3:  A Dopplergram Add the two. No velocity is gray, postive, negative velocities appear white/dark Slide5:  I. Basic Principles of Stellar Oscillations Notation for Normal Modes of Resonance: Oscillations within a spherical object can be represented as a superposition of many normal modes , each which varies sinusoidally in time. The velocity due to pulsations can thus be expressed as: v(r,q,f,t) = ∑ ∑ ∑ n=0 l=0 m=–l ∞ ∞ l Vn(r) Ylm(q,f)e –imf Vn(r) is the radial part of the velocity displacement n,l, and m are the „quantum“ numbers of the stellar oscillations where: n is the radial quantum number and is the number of radial nodes (order) l is the angular quantum number (degree) m is the azimuthal quantum number Slide6:  The number of nodes intersecting the pole = m The number of nodes parallel to the equator = l – m The angular degree l measures the horizontal component of the wave number: kh = [l(l + 1)]½ /r at radius r Slide7:  Low degree modes High degree modes Zonal mode Sectoral mode Slide8:  Types of Oscillations To get stable oscillations you need a restoring force. In stars oscillations are classified by 3 major modes depending on the nature of the restoring force: p-modes: pressure is the restoring force (example: Cepheid variable stars) g-modes: gravity is the restoring force (example: ocean waves). As we shall see this is also related to the buoyancy force. f-modes: fundamental modes. g- or p-modes that do not have a radial node In the sun p-modes have periods of minutes, g-modes periods of hours Slide9:  Characteristic Frequencies Stellar oscillations are characterized by two frequencies, depending on whether pressure or gravity is the restoring force. Lamb Frequency: The Lamb frequency is the reciprocal time scale defined by the horizontal wavelength divided by the local sound speed: k = 2p/l Travel time t = (1/k)/c Frequency = 1/t Slide10:  Brunt-Väisälä Frequency The frequency at which a bubble of gas may oscillate vertically with gravity the restoring force: Characteristic Frequencies g ( ) 1 dlnP dr – is the ratio of specific heats = Cv/Cp g is the gravity N2 = g Where does this come from? Remember the convection criterion? Slide12:  Dr = rADr FB = –grAVDr In our case x = Dr, k = grA, m ~ rV N2 = grAV/Vr = gA = g g ( ) dlnP dr dr – 1 dlnr The Brunt-Väisälä Frequency is the just the harmonic oscillator frequency of a parcel of gas due to buoyancy Slide13:  Characteristic Frequencies The frequency of the oscillations indicate the type of the restoring force. If s is the frequency of the oscillation: s2 > Ll2, N2: For high frequency oscillations the restoring force is mainly pressure and oscillations show the characteristic of acoustic (p) modes. s2 < Ll2, N2: For low frequency oscillations the restoring force is mainly due to buoyancy and the oscillations show the characteristic of gravity waves. Ll2 < s2 < N2, Ll2 > s2 > N2 : In these regions of the star evanescent waves exist, i.e. the wave exponentially decreases with distance from the propagation region. Slide14:  Propagation Diagrams g modes cannot propagate through the convection zone. Why? Buoyancy force is a destabilizing force. Propagation diagrams can immediately tell you where the p- and g-modes propagate p-modes g-modes g-modes p-modes Decaying waves Decaying waves Slide15:  The period is determined by the travel time of acoustic waves in a cavity defined by two turning points: one just below the photosphere where the where the density decreases rapidly (reflection), and a lower turning point caused by the gradual increase of the sound speed, c, with depth (refraction). At the lower reflection point the wave is traveling horizontally and the reflection occurs at a depth d where Probing the Interior of the Sun: p-modes c = 2ps/kh Slide16:  Horizontal wavelength Decreasing density causes the wave to reflect at the surface Increasing density causes the wave to refract in the interior. Slide17:  By observing modes with a range of frequencies one can sample the sound depth with speed: Slide18:  Assymptotic Relationship for P-mode oscillations (n>> l) p-modes: nnl ≈ Dn0 (n + l/2 + e) + dn is a constant that depends on the stellar structure Dn0 = [2∫0R dr/c]–1 where c is the speed of sound (i.e. this is the return sound travel time from the surface to the core) dn = small spacing (related to gradient in sound speed) n p-modes are equally spaced in frequency Slide19:  For g-modes wave propagation is generally only possible in regions of the Sun below the convection zone. A particular g-mode is trapped in regions where its frequency s is less than the local buoyancy frequency N. The upper and lower reflection points of any given cavity correspond to where N has approached s. G-modes thus sample the Brunt-Väisälä frequency, N, as a function of depth The g-modes all share the reflection point near the base of the convection zone and their amplitudes decay throughout that zone (evanescent). Since the decay rate increases with l only low degree modes are likely to be detected in the atmosphere. Probing the Interior of the Sun: g-modes Slide20:  Assymptotic Relationship for G-mode oscillations (n>> l) g-modes: n l=0 l=1 l=2 P0 g-modes are equally spaced in Period Pn,l ≈ n + ½ l + g [l(l+ 1)]½ P0 P0 = 2p2 ∫ [ 0 rc N dr r ] –1 Slide21:  Excitation of Modes Normally, when a star undergoes oscillations dissipative forces would cause the oscillations to quickly damp out. You thus need a driving force or excitation mechanism to sustain the oscillations. Two possible mechanisms: e Mechanism: The energy generation depends sensitively on the temparature. If a star contracts the temperature rises and the energy generation increases.This mechanism is only important in the core, and is not an important mechanism in the Sun. Slide22:  Excitation of Modes k Mechanism: If in a region of the star the opacity changes, then the star can block energy (photons) which can be subsequently released in a later phase of the pulsation. Helium and and Hydrogen ionization zones of the star are normally where this works. Consider the Helium ionization zone in the interior of a star. During a contraction phase of the pulsations the density increases causing He II to recombine. Neutral helium has a higher opacity and blocks photons and thus stores energy. When the star expands the density decreases and neutral helium is ionized by the emerging radiation. The opacity then decreases. This mechanism is reponsible for the 5 minute oscillations in the Sun. Slide23:  II. Helioseismology The solar 5 min oscillations were first thought to be just convection motion. Later it was established that these were acoustic modes trapped below the photosphere. The sun is expected to have millions of these modes. The amplitude of detected modes can be as small as 0.2 m/s Slide24:  Currently there are several thousands of modes detected with l up to 400. These are largely the result of global networks which remove the 1-day alias. These p-mode amplitude have a Gaussian distribution centered on a frequency of 3 mHz Slide25:  To find all possible pulsaton modes you need continuous coverage. There are three ways to do this. Ground-based networks: Telescopes that are well spaced in longitude. South pole in Summer Spaced-based instruments Slide26:  GONG: Global Oscillation Network Group Big Bear Solar Observatory: California, USA Learmonth Solar Observatory: Western Australia Udaipur Solar Observatory: India Observatory del Teide: Canary Islands Cerro Tololo Interamerican Observatory: Chile Mauno Loa: Hawaii, USA BiSON: Birmingham Solar Oscillation Network Carnarvon, Western Australia Izaqa, Tenerife Las Campanas, Chile Mount Wilson, California Narrabri, New South Wales, Australia Sutherland, South Africa Slide27:  L1 is where gravity of Earth and Sun balance. Satellites can have stable orbits with minimum energy use Slide28:  The p-mode Fourier spectrum from GOLF, using a 690-day time series of calibrated velocity signal, which exhibits an excellent signal to noise ratio. Slide29:  The low-frequency range of the p modes from above spectrum, showing low-n order modes. Slide31:  Rise to low frequency due to stochastic noise of convection Slide35:  The Small Frequency Spacing Slide36:  The Small Frequency Spacing Normally modes of different n and l that differ by say –1 in n and +2 in l are degenerate in frequency. In reality since different l modes penetrate to different depths this degeneracy is lifted. nn,l = Dn0 (n + l/2 + d) – A,h,e depend on the structure of the star The small separation is sensitive to sound speed gradients Slide37:  Echelle Diagrams nn,l = n0 + kDn + n1 n0 = a reference k = integer n1 = 0 → Dn nn,l = n0 + kDn n1 0 Dn l=n l=n+1 An echelle diagram basically cuts the frequency axis into chunks of Dn and stacks them on each other Slide38:  Results from Helioseismology There are two ways of deriving the internal structure of the sun Direct Modelling Computationally easy Results depend on model Inversion Techniques Model independent Computationally difficult Slide39:  Sound Speed: P-modes give information about the sound speed as a function of depth. The sound speeds in the mid-region of the radiative zone were found to be off by 1%. This suggested that the opacity below the convection zone was underestimated. This has since been confirmed by new opacities. Slide41:  Deviations of the sound speed from the solar model red is positive variations (hotter) and blue is negative variations (cooler regions). From SOHI MDI data. Slide42:  Possibly due to increased turbulence Deviations of the observed sound speed from the model. The differences are mostly less than 0.2% Slide43:  Mixing length theory Simple convection (mixing length theory) does not adequately model observed frequencies Slide44:  Z=0.018 Z=0.02 Chemical Composition: Low abundance model solution to the neutrino problem can be rejected One explanation to the solar neutrino problem is that the metal abundance (Z) in the interior is lower. But lower Z moves the models away from the observed frequencies Slide45:  Depth of the convection Zone Early helioseismic results showed that the depth of the convection zone was 50% greater than current models The mixing length parameter a = 2-3 (l= aHp, Hp is the scale height) Negative values In convection zone W = 1 – g = 1 – 5/3 = –0.67 Slide46:  Rotation: With no rotation all m modes from a given l are degenerate. Rotation lifts this degeneracy and the m. For an l=1, m = –1,0,+1. Thus rotational splitting will be a triplet. Analogy: Zeeman splitting of energy levels of atom. l = 1 stellar oscillator with modes split into triplets by rotation. Slide49:  The internal rotation of the Sun Rotation period in days The sun shows differential rotation throughout the convection zone, and almost solid body rotation in the radiative zone Slide51:  G-modes propagate only in the radiative core and are evanescent in the convection zone. The amplitude of these waves exponentially decay while passing through the convection zone. Consequently, their amplitudes at the observable surface is expected to be small. G-modes are important in that they can probe the interior of the sun all the way down to the core (r = 0). P-modes can only get to about r=0.2 Rסּ There have been many claims for detecting solar g-modes, but none have been verified. Theoretical work suggest that the amplitudes of these modes at the surface should be 0.01 – 5 mm/s. It is easy to see why these have not been detected. The search for these, however, continues. Slide52:  When observing the sun we have the advantage of having lots of photons and a resolved disk. When searching for 5-min like observations in solar type stars we are looking in integrated light. Because of cancellation effects only the lowest degree modes can be detected. Asteroseismology Low degree: Fewer numbers of +/– regions, cancellation is less High degree: larger numbers of +/– regions, cancellation is more Slide53:  Scaling to solar values Kjeldsen & Bedding, Astronomy and Astrophysics, 293, 87, 1995 M, R in solar units Asteroseismology Slide54:  3.05 mHz Slide55:  Peak power at P = 7 min so either the Mass is smaller than the sun, or the radius is larger , or both Slide56:  Pollux (from M. Zechmeister) 3 hrs R = 8.8 Rסּ (from interferometry) M ≈ 1.4 Mסּ (from scaling relations) L ≈ 33 Lסּ (from brightness and distance) V ≈ 5 m/s (from scaling relations) dL/L ≈ 0.0001 (predicted) dL/L ≈ 0.0001 (MOST) Slide57:  Przybylski‘s Star (HD 101065) Velocities taken with HARPS HD 101065 is an pulsating Ap star (12 min) with a large inclined dipole field. Slide58:  Przybylski‘s Star (HD 101065) Slide59:  Effective temperature = 6538 K Luminosity = 5.88 Lסּ Mass = 1.5 Mסּ Magnetic field = 8600 Gauss From scaling law R = 1.78 Rסּ Slide60:  CoRoT A space telescope (27 cm) designed to obtain precise light curves of stars for asteroseismology and extrasolar planet searches (transits) for 150 days continuously in one field. Launched 27 Dec 2006 French mission with partners from Germany, Austria, Belgium, Spain, and ESA German CoIs (Tautenburg, DLR, Köln) First results just released Slide61:  The CCDs of CoRoT Die COROT-Mission: Focal Plane:  Die COROT-Mission: Focal Plane Defocusd images on asteroseismology CCDs Slide63:  From the CoRoT press release Slide64:  Science from press release: What can we say? Equally spaced in frequency, these are p-modes Large spacing Dn0 = 2 ×42.5 = 85 mHz Dn0 Assuming ≈ 1 Mסּ and using expression for spacing, R = 1.36 Rסּ. This is an evolved star. Consistent with 10.4 min period. Maximum power at nmax = 1.65 mHz = 0.00165 c/s Order of maximum power nmax ≈ 18 Slide65:  And lets not forget CoRoT transits CoRoT Transit light curve Ground Based transit light curve (different star)

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