Stellar Astrophysics: Pulsating Stars. Stellar Pulsation

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1 Stellar Astrophysics: Stellar Pulsation Pulsating Stars The first pulsating star observation documented was by the German pastor David Fabricius in 1596 in the constellation Cetus The star o Ceti, later called Mira, slowly faded and disappeared from sight for several months David Fabricius ( ) Mira A was later found to be a red giant AGB star of one solar mass with a small companion star Mira B Mira is a long-period variable star with a period of about 333 days during which the stars expands and contracts 1

2 Pulsating Stars The second pulsating star observation was made by John Goodricke of England in 1784 in the constellation Cepheus The star δ Cephei oscillates with a period of 5.37 days and an amplitude of less than one magnitude John Goodricke ( ) Period-Luminosity Relation Henrietta Swan Leavitt worked on the analysis of photographic plates at Harvard (Charles Pickering s lab) Comparison between plates to search for brightness variations indicating Cepheid stars Investigating pulsating stars in the Small Magellanic Cloud (SMC), part of local galaxy group 61 kpc from Earth Henrietta Swan Leavitt ( ) 2

3 Period-Luminosity Relation Leavitt found 2400 classical variable stars with periods between one and 50 days The more luminous stars have longer periods Because the distance to all stars in the SMC is about the same, the variations in apparent luminosity must be the same as the variation in the absolute luminosity Can be used for distance measurement because the measured period gives the absolute luminosity The comparison between apparent and absolute luminosity gives the distance Henrietta Swan Leavitt ( ) Period-Luminosity Relation Most precise period-luminosity relation found in the infrared waveband Less extinction due to dust H = log 10 Π W.L. Freedman, et.al., Ap. J. 679: (2008) 3

4 δ Cepheus The American astronomer Harlow Shapley ( ) explained the brightness variations by radial pulsation Eddington provided the theoretical framework for variations in Brightness Effective temperature Radius Surface velocity Doppler Shift in Pulsating Stars Supergiant stars with helium and hydrogen burning shells Doppler shifted spectral lines due to pulsation Classified as Cepheid I or II showing different spectral lines due to different chemical composition (compare to cluster populations) Population I metal-rich (classical Cepheids) Population II metal-poor 4

5 Pulsating Stars on the H-R Diagram Long-period variable stars Periods days Classical Cepheids Periods 1 50 days RR Lyrae stars Periods 1 24 hours β Cepheids Periods 3 7 hours Period-Density Relation Radial oscillations of a pulsating star result from resonating sound waves inside the star The velocity of a sound wave is related to the compressibility of the gas and its inertia (density) Using v s = B / ρ γ d V V = d P P with B = V with d P d V γ = ad C P C V Yields for the sound velocity v s = γ P / ρ 5

6 Period-Density Relation Assuming a constant density for the gas we had found for the pressure in hydrostatic equilibrium d P d R G M r ρ 4 = = π G ρ 2 r r 2 3 Integration, with P = 0 at the surface, yields 2 P ( r ) = π G ρ 2 ( R 2 r 2 ) 3 Period-Density Relation For the pulsation period we obtain d r Π R v s 0 R 2 3 d r γ π G ρ ( R 2 r 2 ) Π 3 π 2 γ G ρ The pulsation period is inversely proportional to the square-root of the star s mean density This illustrates why very tenuous supergiants have large pulsation periods For a 5 M giant of 50 R we get Π 10 days 6

7 Radial Modes of Pulsation The radial sound waves inside the star are comparable to standing waves Compare standing sounds waves inside an organ pipe to a star (a) Fundamental node (b) First overtone (c) Second overtone Radial Modes of Pulsation Motion of stellar material occurs mostly in the surface regions Some oscillations occur deep inside the star Fractional displacement of stellar material as a function of fractional radius Maximum fractional displacement for a classical Cepheid is about 0.05 to 0.1 Classical Cepheids pulsate usually in the fundamental node or first overtone 7

8 Eddington s Heat Engine The layers of stellar material do work P dv during the pulsation cycles as they expand and contract When the integral is Positive, the oscillation is driven Negative, the oscillation is damped Possible Explanations Nuclear energy generation rate Compression of the core of a star leads to temperature and density increase resulting in an increased energy production rate In reality, the pulsation amplitude is very small at the center (node) This mechanism contributes only a small fraction to the pulsation Eddington s Valve If a layer of the star becomes more opaque upon compression, energy flow to the surface will be reduced and the layer will expand If the expanding layer becomes more transparent, the trapped heat can escape and the layer fall back In reality, most regions of a star become more transparent upon compression 8

9 Explanation Eddington s valve mechanism can work in partial ionization zones S. A. Zhevakin, R. Kippenhahn, N. Baker, and J. Cox showed that in layers of partial ionization an increase in pressure leads to additional ionization and not heating The increase in density without increase in temperature leads to an increase of opacity This is called the κ-mechanism Variations of temperature and opacity for a model of a RR Lyrae star at maximum compression The He II partial ionization zone has a temperature of ~ 40,000 K Partial Ionization Zones Most stars have two partial ionization zones H partial ionization zone at 10,000 to 15,000 K Ionization of neutral hydrogen and helium (H II and He II) He II partial ionization zone at 40,000 K Second ionization of helium (He III) Calculations show that the He II partial ionization zone is primarily responsible for oscillations 9

10 Partial Ionization Zones Location of the zones determines the pulsation parameters In hot stars (7,500 K) the density at large radii is small and the mass insufficient to drive large oscillations In medium hot stars (6,500 K) more mass is available and the first overtone may be excited In cooler stars (5,500 K) the ionization zone occur deep enough to drive the fundamental node of the oscillations Helioseismology The Suns oscillations were first observed in 1962 by American astronomers R. Leighton, R. Noyes and G. Simon The Suns oscillation modes have very low amplitude very low surface velocity of 0.1 m/s low luminosity variation of 10-6 Oscillations are acoustic pressure waves Detection through Doppler-shift of spectral lines The roughly ten million modes are nonradial Non-radial oscillations can be described by spherical harmonic functions Y l m (θ, φ) 10

11 Helioseismology The main pulsation period of the Sun is about five minutes These oscillations are p-modes and are located below the photosphere in the convection zone The waves allow the study of the convection zone and it s boundary The differential rotation of the stellar matter at the surface can be probed for depth dependence with varying p-waves Computer generated p-mode oscillation l = 20, m = 16, n = 14 (Wiki) 11