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Solar-like Oscillations in other Stars or The only way to test directly stellar structure theory I. Scaling Relations II. Results Asteroseismology: A relatively new field Kjeldsen & Bedding, 1995, A&A, vol. 293, 87 „Asteroseismology of solar-like stars has so far produced disappointing results. Despite repeated attempts and everincreasing sensitivity, there have been no unambiguous detection of solar-like oscillations on any star except the Sun.“ Solar-like oscillations: Any oscillation that is excited by convection in the outer part of the star analogous to the 5 minute oscillations. Only low-degree modes can be detected Unlike the sun for which we have detected mode with l up to 400, this is because the Sun is resolved. For stars for which we measure integrated light, there is a high degree of cancellation. We can only detect low degree modes, l = 0-3 The Asymptotic Limit for p-mode Oscillations For high order modes: n >> l nn,l = Dn0 (n + l /2 + e) – l (l +1 )D0 Dn0 = large spacing D0 is sensitive to the sound speed near the core e is sensitive to the surface layers The last term is related to the small frequency spacing Note: l = 0 → radial pulsations. For pure radial modes the frequency spacing is the large spacing If there is no small frequency spacing nn+1,0 = nn,2 but the gradient of the sound speed (small frequency spacing) lifts this degeneracy Definition: dn02 is the small separation as the frequency spacing of adjacent modes with l = 0 and l=2 dn13 is the small separation as the frequency spacing of adjacent modes with l = 1 and l =3 dn01 is the amount modes l = 1 are offset between the midpoint of l =0 modes on either side. If the asymptotic relation holds exactly then: D0 = dn02 6 = dn01 2 = dn01 10 The Solar Power Spectrum showing the large and small spacings: dn01 A Diagnostic Tool: Echelle Diagrams Dn l=0 l=1 l=2 nn,l = n0 + kDn + n1 n0 = a reference k = integer n1 = 0 → Dn Echelle diagrams take advantage of the fact that in the asymptotic relationship p-modes are equally-spaced in frequency. An echelle diagram basically cuts the frequency axis into chunks of Dn and stacks them on each other dn13 dn02 l=1 Radial order n, increasing Frequency l=0 l=3 l=2 0 Dn1(= Dn0/2) Modulo Frequency Dn0 The „Asteroseismic“ H-R Diagram The goal of asteroseismology is to detect enough modes to derive the large and small frequency spacings. From these you get the mass and age of the star from the „Asteroseismic“ H-R diagram Estimating the Photometric Amplitude of Oscillations (Kjeldsen & Bedding, 1995, 293, 87) For adiabatic oscillations To first order the density compression for an adiabatic sound wave dL (L ( The change in luminosity of the star is due almost entirely to temperature changes dT T ∝ ∝ bol dr = r dT T dr r v cs Estimating the Photometric Amplitude of Oscillations The adiabatic sound speed: [ cs = ln P [ ln r P r ad Ideal gas law give: P∝ rT 2 cs ∝ T ln P [ ln r [ 2 = ad 5/3 Estimating the Photometric Amplitude of Oscillations ( dL (L ~ bol vosc Assume T ≈ Teff √Teff This expresses the luminosity amplitude of stellar oscillations in terms of the velocity amplitude. Note: this is for small amplitude variations. In comparing to observations, which are made at specific wavelengths we have to take into account that the luminosity amplitude of an oscillation depends on the wavelength that it is observed. dL (L l (l bol ( ~ dL (L l (l bol l l ( ( bol ( = ( dL (L lbol Combining: ( dL (L l = vosc l Teff –1.5 = 623 nm Teff /5777K When comparing to real data a better fit is obtained using an exponent of –2 instead of –1.5. This is not unsurprising since we used an adiabatic approximation so there may be temperature corrections when comparing to real data. The revised equation becomes: ( = (l/ 550 nm) (Teff / 5777 K) l dL (L ( dL (L vosc / m s–1 bol 2 20.1 ppm vosc / m s–1 = (Teff / 5777 K) 2 1 ppm = 1 part per million (10–6) = 1.086 mmag 17.7 ppm This shows the predicted luminosity variations versus observed variations for a variety of pulsating stars. The previous expression is a good approximation even when the oscillations are nonlinear or non-adiabatic Comparison of the predictions for this simple expression to model predictions from Christensen-Daalsgard & Frandsen (1983, Sol. Physics, 82, 469) Estimating the Velocity Amplitude Stellar models suggest that the velocity amplitude scales: Vosc ~ L/M We can thus take the solar amplitudes and scale these according the values to other stars Scaling the Velocity Amplitude to other stars Amplitude Scaling Laws vosc ( dL (L l L/Lסּ (23.4 ± 1.4) cm/sec = M/Mסּ L/L( סּ4.7 ± 0.3) ppm = (l/ 550 nm) (Teff / 5777 K)2 (M/M)סּ These equations scale to values observed for the sun The Solar Power Spectrum nmax ≈ 3000 mHz Dn0 ≈ 135 mHz How do we scale these to other stars? nmax nmax +1 l =0 Dn0 l =1 Dn1 Each peak in the frequency spectrum corresponds to a harmonic mode characterized by a radial order n, and an angular degree l. For stars observed in integrated light we most likely detect only l = 0,1 nnl ≈ Dn0 (n + l/2 + e) R –1 Dn0 ≈ [2∫ dr/cs] e ~ 1.6 for the sun cs is the sound speed 0 In other words the large spacing is the inverse travel time of a sound wave passing directly through the star. It is 134.92 mHz for the Sun (i.e. about 2 hours travel time) This sound travel time is related to the global dynamical Timescale of the star: d2R dt2 GM = – R2 Msun = 2 ×1033 gm =GrR Rsun r = mean density = 7 ×1010 cm t = (G r )–½ r = 1.4 gm/cm3 The dynamical time for the sun is about 1 hour. This would be the period of radial pulsations, if they were present. Or, if you were to hit the sun, this is the fastest it could respond But recall that the adiabatic sound speed satisfies: 2 cs ∝ T R –1 Dn0 ≈ [2∫ dr/cs] 0 ≈ cs R ∝ √T R Where T is now the average internal Temperature Footnote: Equation for Hydrostatic Equilibrium 1. Hydrostatic Equilibrium dA P +dP P + dP P r + dr A dr r dm P M(r) Gravity The gravity in a thin shell should be balanced by the outward gas pressure in the cell Fp = PdA –(P + dP)dA = –dP dA dM FG = –GM(r) r2 Gravitational Force r M(r) = ∫ r(r) 4pr2 dr 0 dM = r dA dr Both forces must balance: FP + FG = 0 dP dr Pressure Force r(r)M(r) = –G r2 Some more approximations: Stellar structure equation for hydrostatic equilibrium dP GM r = – 2 dr r dP M GM = – R3 dr R2 P R GM2 = – R5 M2 P∝ R4 Ideal gas law: P∝ rKT M2 R4 ∝ M R3 M T ∝ R T Dn0 Dn0 ∝ ∝ M T∝ R √T R ½ M (R) 3 ∝ ½ r The frequency splitting is directly proportional to the square root of the mean density of the star! (M/M)סּ1/2 Dn0 ≈ (R/R)סּ3/2 134.9 mHz The power has an envelope whose maximum is at nmax ≈ 3mHz For the sun. The shape of the envelope and value of nmax is determined by excitation and damping. The acoustic cutoff, nac, defines a dynamical timescale for the atmosphere, so we expect nmax to scale as nac For frequencies above the acoustic cutoff the energy of the mode decreases exponentially with height in the atmosphere nac ∝ cs/Hp And what is the scale height of the atmosphere? Footnote #2 :Scale Height of Atmosphere Pressure dP = –gr dh F = GMm/R2 A r = Pm/kT h mg dP = – P dh kT F =grAh F/A = g r h r gravity mgh ( kT ) = P r = r exp (– h ) H P = Po exp – o Scale height H = kT/mg o ( exp – h H ) Hp is the pressure scale height where the pressure decreases exponentially: P = Po e–h/Hp Hp = kT/mg nmax ∝ nac ∝ cs/Hp cs ∝ √T M nmax ∝ g/√T ∝ 2 R √T nmax = M/Mסּ (R/R)סּ2√Teff/5777K 3.05 mHz The maximum power in the sun is seen for modes with n ≈ 21 nmax = M/Mסּ ((R/R ) √T סּ 2 eff/5777K ½ ) X 22.6 – 1.6 Summary of Scaling Relationships vosc (23.4 ± 1.4) cm/sec M/Mסּ dL L ( ( L/Lסּ = l (R/R)סּ = nmax = (4.7 ± 0.3) ppm (l/ 550 nm) (Teff / 5777 K)2 (M/M)סּ (M/M)סּ1/2 Dn0 ≈ nmax = L/Lסּ 134.9 3/2 mHz M/Mסּ 3.05 mHz (R/R)סּ2√Teff/5777K M/Mסּ ((R/R ) √T /5777K סּ 2 eff ½ ) X 22.6 – 1.6 Scaling between Dn and nmax Stello et al. 2009, MNRAS, 400, L80 Dn0 = (0.263 ± 0.009) mHz (nmax/mHz)0.772±0.005 The previous expression is useful when you do not have enough data to derive the large spacing. Fit a Gaussian to the envelope of excess power, find nmax and compute Dn0 nmax ≈ 950 mHz →Dn0 ≈ 52 mHz. This is data for Procyon, and as we shall see this is near the correct value Stellar Oscillations (or not) of Procyon Procyon is a bright (mv = 0.36) star in the winter sky that is slightly evolved off the main sequence Spectral Type: F5 IV Vosc = 0.8 m/s Teff = 6450 K dL/L ≈ 2 x 10–5 Mass = 1.42 Mסּ Dn0 ≈ 54 mHz Radius = 2.07 Rסּ nmax ≈ 1 mHz L = 7.03 Lסּ P ≈ 17 min nmax ≈ 11 First possible detections of p-mode oscillations with radial velocity measurements were made with a fiber fed spectropgraph: FOE More convincing evidence… Martic et al. 1999 found convincing evidence for p-mode oscillations in Procyon using ELODIE and the simultaneous Th-Ar for radial velocity measurements Excess power is in the same frequency range as found by Brown et al. Most probable large frequency spacing ≈ 55 mHz Even more convincing evidence… 2-site campaign using ELODIE and AFOE but… Microvariability and Oscillations of STars1 MOST is a 15cm telescope (Canada‘s First Space Telescope) designed to study stellar oscillations. It can continuosly observe stars for up to 60 days. In 2004 MOST observed Procyon for 32 days. 1 PI: Jaymie Matthews, also known as Matthew‘s Own Space Telescope MOST found no evidence of solar-like oscillations in the photometry of Procyon casting doubt on the radial velocity results. Expected power of oscillations at 1 mHz Top panels: simulated power spectra of oscillations for Procyon and with 3 time scales for the mode lifetimes. In the lower panel nose has been added to the simulated data to reach the noise level of MOST. Conclusion #1: MOST could not have detected the pulsations even if they were present Bedding et al. A&A, 432, L43, 2005 The „power density“ of the MOST observations is significantly higher than for the EW measurements of Kjeldsen et al. 1999, and for the Sun. Conclusion #2, the noise level of MOST is too high. A multi-site campaign from 9 observatories was conducted on Procyon in 2008 Above: the radial velocity measurements from the various sites (black: TLS). Left: preliminary power spectrum Stellar Oscillations h Boo Spectral Type: G0 IV Vosc = mv´= 2.68 dL/L ≈ 2.5 x 10–5 (25 ppm) Teff = 6050 K Dn0 ≈ 36 mHz Mass = 1.6 Mסּ nmax ≈ 0.61 mHz Radius = 2.8 Rסּ P ≈ 27 min L = 9.5 Lסּ nmax ≈ 8 1.4 m/s Looking for oscillations through temperature changes L ∝ T4 (assuming constant radius). dL/L ∝ 4 dT/T. The strength of the Balmer lines of hydrogen are sensitive to temperature and the variations are expected to be 6 ppm. Advantage: you do not need a high resolution spectrograph to measure the strength of the hydrogen lines! Power Spectrum of Temperature Variations of h Boo Echelle diagram based on equivalent width (temperature) variations of Hydrogen lines Echelle diagram combining temperature and radial velocity measurements Stellar Oscillations b Hya Spectral Type: G2 IV Teff = 5774 K Mass = 1.1 Mסּ Radius = 1.87 Rסּ L = 3.53 Lסּ Vosc ≈ 0.8 m/s dL/L ≈ 15 ppm Dn0 ≈ 55 mHz nmax ≈ 1 mHz P ≈ 17 min nmax ≈ 11 Radial velocity measurements from a multi-site campaign (HARPS + UCLES). Black are HARPS, red are UCLES Vosc ≈ 0.8 cm/s Stellar Oscillations a Cen A Spectral Type: G2 IV Teff = 5810 K Mass = 1.1 Mסּ Radius = 1.22 Rסּ L = 1.5 Lסּ Vosc = 0.3 m/s dL/L ≈ 6.3 ppm Dn0 ≈ 105 mHz nmax ≈ 2.2 mHz P ≈ 7.5 min nmax ≈ 11 Radial velocity measurements taken with an iodine cell: Power spectrum l=2 l=0 l=1 Bouchy & Carrier 2003 Dn0 ≈ 105.5 mHz Power spectrum of data taken with a different instrument shows power at the right frequency range Bazot et al. 2007, A&A, 470, 295 Radial Velocity Measurements of a Cen A with HARPS Power Spectrum Bazot et al. 2007, A&A, 470, 295 Large separation as a function of frequency for l=0,1,2 modes Small separation as a for l=0 And a questionable claim of rotational splitting: nnlm ≈ nm,l ± mW Bazot et al. 2007, A&A, 470, 295 For l=2 This gives W ≈ 1 mHz → P = 11.5 days, but estimated rotational period is 28.8 days Stellar Oscillations a Cen B Spectral Type: K1 V Vosc = 0.13 m/s mv = 1.33 dL/L ≈ 3 ppm Teff = 5260 K Dn0 ≈ 160 mHz Mass = 0.90 Mסּ nmax ≈ 3.9 mHz Radius = 0.86 Rסּ P ≈ 4.3 min L = 0.5 Lסּ nmax ≈ 20 The Radial Velocity and Power Spectrum of a Cen B Eggenberger et al. 2004, A&A, 417, 235 Stellar Oscillations m Ara Spectral Type: G3 IV-V Teff = 5813 K Mass = 1.1 Mסּ Radius = 1.3 Rסּ L = 1.91 Lסּ [Fe/H] = +0.3 Vosc = 0.4 m/s dL/L ≈ 8 ppm Dn0 ≈ 95 mHz nmax ≈ 2 mHz P ≈ 8.4 min nmax ≈ 17 2x metals as the sun A 14 Mearth planet around m Ara was discovered as part of an asteroseismic run. RV variation with long term variation due to planet Full data set Short time segment showing oscillations l= 1 l= 2 l= 0 l= 3 Power spectrum Dn0 = 90 mHz The Planet-Metallicity Connection These are stars with metallicity [Fe/H] ~ +0.3 – +0.5 Valenti & Fischer There is believed to be a connection between metallicity and planet formation. Stars with higher metalicity tend to have a higher frequency of planets. The Planet-Metallicity Connection Two scenarios have been proposed to explain the high metallicity of planet hosting Scenario 1: The high metal content is primordial and reflects the abundance in the star and thus the proto-planetary disk. A higher metal content implies that the planets are easier to form (core accretion theory) → the high metal abundance forms more planets Scenario 2: The high metal content is only on the surface layers of the star and result from the accretion of planetary bodies onto the star → the planets cause the high metalic abundance The two different scenarios should produce different asteroseismic (acoustic) signals. Over-metallic star Accretion model A slightly better fit is provided with the accretion model, however the main difference is the cross-over of the l =0,2 modes at 2.5 mHz. Unfortunately, this is beyond the highest frequency of the observations. One day asteroseismology may provide the answer to the planet-metallicity effect. Stellar Oscillations n Ind Spectral Type: G0III Teff = 5300 K Mass = 0.9 Mסּ Radius = 3 R L = 5.5 Lסּ Vosc = 1.4 m/s dL/L ≈ 34 ppm Dn0 ≈ 25 mHz nmax ≈ 0.3 mHz P ≈ 55 min nmax ≈ 6 Stellar Oscillations n Ind Dno = 24.25 mHz Best fits to the Teff, Radius, Mass, and Age of n Ind from model fitting to the observed frequencies What about Main Sequence stars? Stellar Oscillations t Cet Spectral Type: G8 V Teff = ~5400 K Mass = 0.9 Mסּ Radius = 0.79 Rסּ (interferometry) L = 0.49 Lסּ Vosc = 0.13 m/s dL/L ≈ 3 ppm Dn0 ≈ 178 mHz nmax ≈ 4.4 mHz P ≈ 3.8 min nmax ≈ 31 Note: t Cet is often used as a radial velocity standard star by planet search programs HARPS data Instrumental effects! Always have a control star! Power spectra after correcting instrumental effects Echelle Diagram for t Cet Interferometry gives a radius of 0.79 Rסּ. From the large spacing one gets a mass of 0.783 ± 0.012 M סּgood to 1.6% Comparison of observed nmax to those predicted by the scaling relationships: The „Asteroseismic“ H-R Diagram t Cet a Cen B m Ara Procyon bHya a Cen A hBoo The models of course need refinement but one can say that most of these stars are evolved. Asteroseismic Targets Most asteroseismic targets have been evolved stars because these produce the highest amplitudes. Also showing solar-like (p-mode) oscillations, but will be discussed later: • rapidly oscillating Ap stars • K giant stars • solar-like oscillations from Space Missions: expected amplitudes are 5-20 ppm. Such precision can only be obtained from space. Space also offers the possibility for continuous coverage Stellar Oscillations Network Group SONG plans a series of 1-m telescopes equipped with spectrographs + iodine absorption cells and spread across the globe. Summary 1. Scaling relationships work remarkably well for predicting the amplitude and frequencies of solar-like oscillations in other stars over a wide range of amplitudes, periods, spectral types, etc. 2. About a dozen solar-type stars have been studied with stellar oscillations using ground based observations. These have exclusively used the radial velocity method. Photometric amplitudes are expected to be ≈ 10–5 → you need to go into space 3. From the ground multi-site campaigns are the most effective means of studying stellar oscillations 4. Asteroseismology it the best means of deriving the mass, radius, effective temperature, helium and heavy element fraction and internal structure of the star.