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Stellar Pulsation

Stellar Pulsation. Observations of Pulsating Stars The Physics of Stellar Pulsation Modeling Stellar Pulsation Non-radial Stellar Pulsation Helioseismology and Asteroseismology. RR Lyrae variables in the globular cluster M3 (one night ’ s observation).

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Stellar Pulsation

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  1. Stellar Pulsation • Observations of Pulsating Stars • The Physics of Stellar Pulsation • Modeling Stellar Pulsation • Non-radial Stellar Pulsation • Helioseismology and Asteroseismology

  2. RR Lyrae variables in the globular cluster M3 (one night’s observation) cfa-www.harvard.edu/~jhartman/M3_movies.html

  3. Observations of Pulsating Variables • First noticed in 1595 by David Fabricius 2nd magnitude at its brightest…would “vanish”… • O-Ceti-->Mira • Believed to be due to dark splotches on rotating star…

  4. Delta-Cephei Prototype of the classical cepheid variable star John Goodricke discovered in 1784 that the brightness of Delta-Cephei was variable with a period of about 5 days!!!! magnitude varies from 3.4 to 4.3, luminosity changes by factor of100(Dm/5) = 100(0.9/5) = 2.3

  5. Period-Luminosity Relation Stars of the “Classical Cepheid Variable” type in the Small Magellanic Cloud were observed…and found to have a strong correlation between Period and apparent magnitude… • Henrietta Swan Leavitt discovered and classified ~2400 classical cepheid variable stars • Periods 1-50 days • She plotted luminosity vs. period for a set of cepheids from the Small Magellanic Cloud and found the Period-Luminosity relation • Ability to measure Distances !!!!! Henrietta Swan Leavitt (1868-1921)

  6. Period-Luminosity Relation Notice that there is scatter…

  7. Calibration of Cepheids The nearest Cepheid is Polaris (~200 pc), too far for trigonometric parallax. d (pc) = 1/p (in arcsec) In 1913, Ejnar Hertzsprung of Denmark used least squares mean parallax to determine the average magnitude M = -2.3 for a Cepheid with P = 6.6 days. d (pc) = 4.16/slope (in arcsec/yr)(4.16 AU/yr is the Sun’s motion) www.cnrt.scsu.edu/~dms/cosmology/DistanceABCs/distance.htm

  8. Calibration of Cepheids • Relation between average V band absolute magnitude and Period

  9. Cepheid Calibration-Infrared • Improved calibration at infrared wavelengths that suffer less from extinction • Adding a color term gives further improvement

  10. How to Find the Distance to aPulsating Star • Find the star’s apparent magnitude m (just by looking) • Measure the star’s period (bright-dim-bright) • Use the Period-Luminosity relation to find the stars absolute magnitude M • Calculate the star’s distance (in parsecs) using d (pc) = 10(m-M+5)/5

  11. Pulsation Hypothesis for Brightness Variation • Shapley proposed that the observed variation in brightness and temperature caused by radial pulsations of single stars. • Rhythmically “breathing” in and out! • R varies-->causes Luminosity and temperature to vary • Sir Arthur Eddington provided theoretical framework that could explain the variations in Brightness, temperature, radius and surface velocity • Delta-Cephei: supergiant star. Radius varies by 5%-10% (~1 R). F5(hottest)-G2(coolest) • Star is brightest when its surface is expanding outward most rapidly, after it has passed through its minimum radius…phase lag Delta-Cephei

  12. The Instability Strip • Majority of pulsating stars lie in the instability strip on the H-R diagram • As stars evolve along these tracks they begin to pulsate as they enter the instability strip and cease oscillations once they leave it. DT ~ 600 – 1100 K

  13. Classes of Pulsating Stars

  14. The Physics of Stellar Pulsation • The radial oscillations of a pulsating star are the result of sound waves resonating in the stars interior • Pulsation period can be roughly estimated from how long it takes a sound wave to cross the star’s diameter • Sound speed; • Pressure: • Period: Standing sound waves in an organ pipe. Radial modes for a pulsating star

  15. The Period – Mean Density Relation Period –luminosity relationship density period incr incr

  16. Eddington’s Thermodynamic Heat Engine • Mechanism that powers these standing waves are powered by the layers of gas that expand and contract • It the net positive work done in a cycle is positive the oscillations will be driven…but how • A layer becomes more opaque during compression. Dam up energy flowing toward surface and push layers outward • As layer expands becomes transparent would fall back down to repeat cycle…

  17. But this does not work for most stellar material! Why? The opacity is more sensitive to the temperature than to the density, so the opacity usually decreases with compression (heat leaks out). But in a partial ionization zone, the energy of compression ionizes the stellar material rather than raising its temperature! In a partial ionization zone, the opacity usually increases with compression! Partial ionization zones are the direct cause of stellar pulsation.

  18. Opacity Effects • Partial ionization zones have increased opacity under compression • Layer will trap energy and be lifted

  19. hydrogen ionization zone (H H+ and He He+) T = (1 – 1.5) x 104 K • helium II ionization zone (He+ He++) T = 4 x 104 K C C If the star is too hot, the ionization zones will be too near the surface to drive the oscillations. This accounts for the “blue edge” of the instability strip. The “red edge” is probably due to the onset of convection. f u n d a m e n t a l 1 s t o v e r t o n e n o p u l s a t I o n

  20. www.univie.ac.at/tops/dsn/texts/nonradialpuls.html

  21. Modeling Stellar Pulsation Consider the adiabatic, radial pulsation of a gas-filled shell. Linearize the equation of motion by setting to get Brad Carroll Weber State University physics.weber.edu/carroll/GoodVibrations/Good_Vibrations2.ppt

  22. For Adiabatic motion and Also Set Plug into

  23. This results in  If g < 4/3, s is imaginary  dynamical instability If g > 4/3, the oscillation period is

  24. For g =5/3 and r=1.41 g cm-3 for the Sun, Compare this with the time for sound to cross a star’s diameter: Estimate! and P

  25. Nonradial Oscillations Pulsational corrections df to equilibrium model scalar quantities f0 go as (the real part of) l = 0 radial m > 0 retrograde m < 0 prograde m = 0 standing http://gong.nso.edu/gallery/images/harmonics

  26. Smith, The Astrophysical Journal, 240, 149, 1980 to Earth In a rotating star, frequencies are rotationally split (~ Zeeman). Si III l = 2, m = 0, -1, -2

  27. Two Types of Nonradial Modes www.astro.uwo.ca/~jlandstr/planets/webfigs/earth/slide1.html

  28. p modes a surface gravity wave

  29. g modes

  30. Seismology and Helioseismology 5-minute p15 mode with l = 20 and m = 16 www.geophysik.uni-muenchen.de /research/seismology Courtesy NOAO

  31. GONG (Global Oscillation Network Group) a six-station network of extremely sensitive and stable velocity imagers located around the Earth to obtain nearly continuous observations of the Sun's "five-minute" oscillations

  32. SOHO (Solar and Heliospheric Observatory) • Michelson Doppler Interferometer • (MDI) • measures vertical motion of • photosphere at one million points • can measure vertical velocity • as small as 1 mm/s

  33. click 5 hours of MDI Medium-l data 96/09/01 Measurements of Frequencies of Solar Oscillations from the MDI medium-l Program by E.J. Rhodes, Jr., A.G. Kosovichev, P.H. Scherrer, J. Schou & J. Reiter sohowww.nascom.nasa.gov/publications/CDROM1/papers/rhodes/

  34. 5-minute p modes have a very low amplitude, ~ 10 cm/s • dL/L ~ 10-6 • incoherent superposition of 10 million modes p2 p1 f sohowww.nascom.nasa.gov /publications/CDROM1/papers /rhodes/

  35. Theory (curves) vs. Data (circles) Libbrecht, Space Science Reviews, 47, 275, 1988

  36. Some Results for the Sun • base of convection zone at 0.714 Rsun, where T = 2.18 x 106 K • mass fraction of helium at surface is Y = 0.2437 • helioseismologically measured sound speed and calculated sound speed for standard solar model agree to within 0.1% www.sns.ias.edu/~jnb/Papers/Preprints/solarmodels.html

  37. Rotational Frequency Splittingin Solar p-Mode Power Spectra l = 20 Liebbrecht, The Astrophysical Journal, 336, 1092, 1989

  38. The Sun’s Internal Rotation • angular velocity profile • in the solar interior inferred • from helioseismology (b) angular velocity plotted as a function of radius for several latitudes Brandenberg, arXiv:astro-ph/0703711, 2007

  39. Delta Scuti Stars q2 Tauri • A to early F stars • Periods 30 min to 8 hrs • radial, nonradial p (sometimes g) Poretti et al, The Astrophysical Journal, 557,1021, 2002

  40. DAV White Dwarfs • hydrogen atmospheres • mass ~ 0.6 Msun • r ~ 106 g cm-3 • Te = 10,000 K – 12,000 K • periods = 100 – 1000 s nonradial g-modestrapped in hydrogen surfacelayer • hydrogen partial ionizationzone drives the DAV oscillations dr/r < surface center > log(1-r/R) Winget et al, The Astrophysical Journal Letters, 245, L33, 1981

  41. G191-16 G185-32 G191-16 G185-32 very complex! McGraw et al, The Astrophysical Journal, 250,349,1981

  42. Don Winget predicted that the helium partial ionization zone could drive oscillations in DB (helium atmosphere) white dwarfs with Te ~ 19,000 K Te ~ 26,000 K rotationally split frequencies Winget et al, The Astrophysical Journal Letters, 262, L11, 1982.

  43. White Dwarf Seismology • Verify theories of white dwarf structure • Determine white dwarf rotation rates • Calibrate cooling rates: Pg 1/T white dwarf cosmochronology! white dwarfs are fossil stars theoretical cooling rates + observed # white dwarfs of different luminosities history of star formation!

  44. Detailed Asteroseismology of Other Stars The COROT (COnvection, ROtation, and planetary Transits) satellite was launched on December 27, 2006. Equipped with a 27-cm diameter telescope and a 4-CCD camera sensitive to tiny variations of the light intensity from stars. smsc.cnes.fr/COROT/

  45. Pulsating Stars are Heat Engines The Otto cycle. 1. In the exhaust stroke, the piston expels the burned air-gas mixture left over from the preceding cycle. 2. In the intake stroke, the piston sucks in fresh air-gas mixture. 3. In the compression stroke, the piston compresses the mixture, and heats it. 4. At the beginning of the power stroke, the spark plug fires, causing the air-gas mixture to burn explosively and heat up much more. The heated mixture expands, and does a large amount of positive mechanical work on the piston. www.lightandmatter.com/html_books/0sn/ch05/ch05.html

  46. In 1918, Arthur Stanley Eddington proposed that pulsating stars are heat engines, transforming thermal energy into mechanical energy. He proposed two mechanisms: • Energy MechanismEddington suggested that when the star is compressed, more energy is generated by sources in the stellar core. Ineffective. The core pulsation amplitude is very small. • Valve Mechanism“Suppose that the cylinder of the engine leaks heat and that the leakage is made good by a steady supply of heat. The ordinary method of setting the engine going is to vary the supply of heat, increasing it during compression and diminishing it during expansion. That is the first alternative we considered. But it would come to the same thing if we varied the leak, stopping the leak during compression and increasing it during expansion. To apply this method we must make the star more heat-tight when compressed than when expanded; in other words, the opacity must increase upon compression.”

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