Today in Astronomy 241: radial stellar pulsation
Today’s reading: Carroll and Ostlie pp. 541-556, on Review of types of pulsating stars The “distance ladder” pulsating stars: Cepheids, RR Lyrae stars, and W Virginis stars The ε, κ, and γ mechanisms and the physics of pulsation Pattern of one particular p- “One-zone” model of mode solar oscillation, linear radial pulsation amplitude greatly exaggerated. (GONG/NSO)
22 March 2005 Astronomy 241, Spring 2005 1 Actually, all stars are thought to pulsate.
Even the Sun, for which the strongest modes comprise the famous “5-minute oscillation.” And that’s a good thing. Stellar pulsations have two crucial uses in astrophysics: Probes of stellar structure. The frequencies of oscillation modes are very sensitive to the internal structure and state of stars. Standard candles. For some stars the oscillation frequency is related to the star’s luminosity. If one knows the relationship, one can use a measurement of the frequency and average apparent flux to calculate the star’s distance.
22 March 2005 Astronomy 241, Spring 2005 2 Notable stellar pulsations
Type Period Population Radial or nonradial Long-period 100-700 days I, II R variables (e.g. Mira) Classical Cepheids 1-50 days I R W Virginis stars 2-45 days II R (type II Cepheids) RR Lyrae stars 1.5-24 hours II R δ Scuti stars 1-3 hours I R, NR β Cephei stars 3-7 hours I R, NR ZZ Ceti stars 10-1000 I NR seconds
22 March 2005 Astronomy 241, Spring 2005 3 HR diagram of pulsating stars: the instability strip
Cepheids
W Vir stars
RR Lyr stars
δ Scu stars
ZZ Cet stars
A. Gautschy and H. Saio 1995
22 March 2005 Astronomy 241, Spring 2005 4 Physics of stellar pulsation
Rough estimate of pulsation period: sound crossing time
γ P v = Adiabatic sound speed S ρ 2 PG=−πρ22(Rr2) (at constant density) 3 R dr 3π Π=2 ≈ ∫ vG2γρ 0 S ==10 days for MM5 ::, R=50R
22 March 2005 Astronomy 241, Spring 2005 5 Physics of stellar pulsation (continued)
Paradigms for radial-mode stellar pulsation: standing waves in wind instruments and organ pipes thermodynamic heat engines Driving: heat enters gas near maximum compression Damping: heat dissipates (leaves the gas) Types of pulsation: •ε-mechanism: in core of star, driven by temperature and energy-generation rate rise toward center. •κ-mechanism: opacity increases with increasing density; thus soaks up radiative energy near maximum compression. (Usually this mechanism dominates.) •γ-mechanism: same as κ, plus conduction. 22 March 2005 Astronomy 241, Spring 2005 6 Physics of stellar pulsation (continued)
One-zone model with linearization of equations of motion: specialize to small oscillations, write P = P0+δP, ignore terms in (δP)2 or higher. dR2 GMm mR=− +4π 2P , linearized, becomes dt22R dG2 2 Mm mR()δδ=+R84πRPδR+πR2δP 2300 0 dt R0 Assume adiabatic compression and expansion (i.e. polytrope equation of state): γ γ ⎛⎞4π 3 PV = P⎜⎟R =constant ⎝⎠3
22 March 2005 Astronomy 241, Spring 2005 7 Physics of stellar pulsation (continued)
Resulting equation of motion for δR:
dG2 M ()δγRR=−(3 −4) δ 23 dt R0 Sinusoidal solution: angular frequency is given by GM ωγ2 =−(3 4) 3 R0 and the period is 2π Π= 4 πρG (3γ− 4) 3 0
22 March 2005 Astronomy 241, Spring 2005 8 Today’s in-class problems
Problems 14.7 and 14.8. Answers and/or secrets to problems done last time: 13.11 (a) With ε as the energy released per nuclear decay, dN LN==εε−λe−λt =−ελN dt 0 dL = ελ2N dt dd log e dL But ()log Le==()log ln L , so dt dt L dt d ()log Le=−λλlog =−0.434 . dt
22 March 2005 Astronomy 241, Spring 2005 9 Today’s in-class problems
13.11 (b) From the definition of absolute bolometric magnitude:
MMbol =−Sun 2.5log (LL: )
MMbol − Sun ⇒=log LL−log : 2.5 d 1 dM ⇒=()log L −bol dt 2.5 dt dM Thus bol ==()2.5 (0.434)λλ1.086 . dt
22 March 2005 Astronomy 241, Spring 2005 10