1. Variable Stars: Pulsation, Evolution and applications to cosmology Shashi M. Kanbur SUNY Oswgo, June 2007

2. Lecture II: Stellar Pulsation Most stars are intrinsically stable. Most stars are intrinsically stable. If the Sun were to contract a bit, the core gets hotter, nuclear reactions go at a faster rate creating more energy and hence more pressure halting the contraction. If the Sun were to contract a bit, the core gets hotter, nuclear reactions go at a faster rate creating more energy and hence more pressure halting the contraction. If the Sun were to expand a bit, the core gets cooler, nuclear reactions go at a slower rate creating less energy and hence less pressure halting the expansion. If the Sun were to expand a bit, the core gets cooler, nuclear reactions go at a slower rate creating less energy and hence less pressure halting the expansion. Most stars are stable against departures from hydrostatic equilbrium because of this. Most stars are stable against departures from hydrostatic equilbrium because of this. In Cepheids and RR Lyraes, upon contraction, the extra energy flowing out is “held up” in the outer layers for a short while and released when the star is expanding again. In Cepheids and RR Lyraes, upon contraction, the extra energy flowing out is “held up” in the outer layers for a short while and released when the star is expanding again. It thus amplifies the contraction or departure from hydrostatic equilbrium, at least in the outer layers. It thus amplifies the contraction or departure from hydrostatic equilbrium, at least in the outer layers. Swing analogy. Swing analogy. Held up due to opacity behaviour in outer layers. Held up due to opacity behaviour in outer layers.

3. Static Stellar Structure m(r) = mass inside a radius r. m(r) = mass inside a radius r. L(r) = luminosity at radius r. L(r) = luminosity at radius r. P(r) = total pressure at radius r (gas plus radiation) P(r) = total pressure at radius r (gas plus radiation) T(r) = temperature at radius r. T(r) = temperature at radius r. ρ(r) = density at radius r. ρ(r) = density at radius r. Spherically symmetric. Spherically symmetric. Equation of State, energy generation and opacity as a function of T,ρ. Equation of State, energy generation and opacity as a function of T,ρ.

4. Static Stellar Structure dm/dr = 4πr 2 ρ(r): mass continuity dm/dr = 4πr 2 ρ(r): mass continuity dP/dr = -Gm(r)ρ(r)/r 2 : hydrostatic equilbrium dP/dr = -Gm(r)ρ(r)/r 2 : hydrostatic equilbrium dL(r)/dr = 4πr 2 ε(r) : thermal equilbrium dL(r)/dr = 4πr 2 ε(r) : thermal equilbrium ε=ε(ρ,T): energy generation rate ε=ε(ρ,T): energy generation rate κ=κ(ρ,T): opacity κ=κ(ρ,T): opacity P = P(ρ,T): Equation of State. P = P(ρ,T): Equation of State. At r=0, L(r=0)=0, m(r=0)=0 : Boundary At r=0, L(r=0)=0, m(r=0)=0 : Boundary Ar r=R, L(r=R)=L, m(r=R)=M: Conditions Ar r=R, L(r=R)=L, m(r=R)=M: Conditions

5. Energy Transport dL(r)/dm relates to energy transport. dL(r)/dm relates to energy transport. In such stars usually by radiation or convection. In such stars usually by radiation or convection. Convection is very complicated: 3D problem – more later Convection is very complicated: 3D problem – more later Radiative Transport: diffusion approximation Radiative Transport: diffusion approximation L(r) = -(1/3κ)64πacr 4 T 4 dlnT/dm L(r) = -(1/3κ)64πacr 4 T 4 dlnT/dm

6. Why do Cepheids and RR Lyraes pulsate? Pulsation is not due to variations in the rate of energy generation in the core. Pulsation is not due to variations in the rate of energy generation in the core. More to do with the variation of the rate at which this radiation can escape. More to do with the variation of the rate at which this radiation can escape. Early astronomers thought they were binary stars. Early astronomers thought they were binary stars. Harlow Shapley suggested there was an internal “breathing” mechanism. Harlow Shapley suggested there was an internal “breathing” mechanism. Radial pulsations proposed by Arthur Ritter in 1879 but overlooked until Eddington provided a mathematical formulation. Radial pulsations proposed by Arthur Ritter in 1879 but overlooked until Eddington provided a mathematical formulation. Assume pulsations are due to sound waves resonating in the stellar interior. Then, the pulsation period, Π, is Assume pulsations are due to sound waves resonating in the stellar interior. Then, the pulsation period, Π, is Π = 2R/v s, where R is the radius and v s, the sound speed is Π = 2R/v s, where R is the radius and v s, the sound speed is V s = √ γP/ρ, and γ is the ratio of specif heats for the stellar material. V s = √ γP/ρ, and γ is the ratio of specif heats for the stellar material.

7. Why do Cepheids and RR Lyraes pulsate? To work this out, use hydrostatic equilbrium, To work this out, use hydrostatic equilbrium, dP/dr = - Gm(r)ρ/r 2 = -4Gπrρ 2 /3 dP/dr = - Gm(r)ρ/r 2 = -4Gπrρ 2 /3 Integrate this between the center and surface and assume P(surface)=0, yields Integrate this between the center and surface and assume P(surface)=0, yields P(r) = 2πGρ 2 (R 2 -r 2 ) P(r) = 2πGρ 2 (R 2 -r 2 ) Substituing back into the original expression for v s and integrating between 0 and R yields approx: Substituing back into the original expression for v s and integrating between 0 and R yields approx: Π~ √ 3π/2Gγρ Π~ √ 3π/2Gγρ This is the period mean density theorem, where we take ρ to be the mean density of the star. This is the period mean density theorem, where we take ρ to be the mean density of the star. More accurate treatments of this derivation exist. More accurate treatments of this derivation exist.

8. Why do Cepheids and RR Lyraes pulsate? This is why Cepheids have longer periods than RR Lyraes. This is why Cepheids have longer periods than RR Lyraes. Cepheids are much more tenous and have a smaller mean density and hence a longer period. RR Lyraes are compact with a high mean density. Cepheids are much more tenous and have a smaller mean density and hence a longer period. RR Lyraes are compact with a high mean density. Pendulum with a short string has a shorter period than another pendulum with a longer string. Pendulum with a short string has a shorter period than another pendulum with a longer string.

9. Why do Cepheids and RR Lyraes pulsate? Suppose a layer in the outer part of the star contracts for some reason due to some momentary loss of hydrostatic equilbrium. Suppose a layer in the outer part of the star contracts for some reason due to some momentary loss of hydrostatic equilbrium. This layer heats up and becomes more opaque to radiation. This layer heats up and becomes more opaque to radiation. Radiation diffuses more slowly through the layer because of its increased opacity and heat builds up beneath it. Radiation diffuses more slowly through the layer because of its increased opacity and heat builds up beneath it. Pressure rises below the layer and eventually starts to push the layer out. Pressure rises below the layer and eventually starts to push the layer out. The layer expands, cools and becomes more transparent to radiation. The layer expands, cools and becomes more transparent to radiation. Energy can now escape from below the layer and the pressure beneath the layer drops. Energy can now escape from below the layer and the pressure beneath the layer drops. The layer falls inwards and the cycle repeats. The layer falls inwards and the cycle repeats. Need a mechanism by which opacity increases as temperature increases. Need a mechanism by which opacity increases as temperature increases. Kramer’s opacity: κ~ρT -3.5 decreases upon compression. Kramer’s opacity: κ~ρT -3.5 decreases upon compression.

10. Why do Cepheids and RR Lyraes pulsate? In hydrogen and helium partial ionization zones, temperature does not increase much upon compression since energy of compression goes into fully ionizing hydrogen and helium. In hydrogen and helium partial ionization zones, temperature does not increase much upon compression since energy of compression goes into fully ionizing hydrogen and helium. Likewise during the expansion phase, temperature does not decrease significantly because the ions release energy when they combine with electrons. Likewise during the expansion phase, temperature does not decrease significantly because the ions release energy when they combine with electrons. This is the κ mechanism. This is the κ mechanism.

11. Why do Cepheids and RR Lyraes pulsate? Increased temperature gradient between partial ionization zones and surrounding layers causes more heat to flow into them increasing ionization. Increased temperature gradient between partial ionization zones and surrounding layers causes more heat to flow into them increasing ionization. This is the γ mechanism and reinforces the κ mechanism. This is the γ mechanism and reinforces the κ mechanism.

12. Partial Ionization Zones Important ones are Important ones are Hydrogen ionization zone, Hydrogen ionization zone, H H + + e -, H H + + e -, At a temperature of around T ~ 10000K. At a temperature of around T ~ 10000K. Helium ionization zones, Helium ionization zones, He He + + e -, close to the H ionization zone and He He + + e -, close to the H ionization zone and He + He ++ + e - at a temperature of around 4×10 4 K. He + He ++ + e - at a temperature of around 4×10 4 K.

13. Partial Ionization Zones The pulsation properties of a star are determined primarily by where the partial ionization zones are located with respect to the mass distribution of the star. The pulsation properties of a star are determined primarily by where the partial ionization zones are located with respect to the mass distribution of the star. This is determined by the temperature of the star and hence the location of the star on the HR diagram. This is determined by the temperature of the star and hence the location of the star on the HR diagram. Too close to the surface and there is not enough mass to drive the pulsations effectively. Too close to the surface and there is not enough mass to drive the pulsations effectively. Too deep in the star and convection becomes efficient thus transporting energy out of the partial ionization zone on compression. Too deep in the star and convection becomes efficient thus transporting energy out of the partial ionization zone on compression. This is why there is an instability strip. This is why there is an instability strip.

15. The Instability Strip This strip corresponds to the range of temperatures for which the partial ionization zones are located in the right place to sustain stellar pulsations. This strip corresponds to the range of temperatures for which the partial ionization zones are located in the right place to sustain stellar pulsations. The blue and red edges of the instability strip. The blue and red edges of the instability strip. Red edge defined by a theory of time dependent turbulent convection (deepest layer in the star at which partial ionization zones can sustain pulsation. Red edge defined by a theory of time dependent turbulent convection (deepest layer in the star at which partial ionization zones can sustain pulsation. Blue edge: defined by the highest layer at which partial ionization zones can sustain pulsation. Blue edge: defined by the highest layer at which partial ionization zones can sustain pulsation. But not all stars in instability strip pulsate. But not all stars in instability strip pulsate.

17. Pulsation Periods Pulsation is a transient phenomenon. Pulsation is a transient phenomenon. As a star evolves of the main sequence, it crosses the instability strip and starts to pulsate. As a star evolves of the main sequence, it crosses the instability strip and starts to pulsate. As observational methods improve, smaller and smaller amplitude pulsators will be discovered. As observational methods improve, smaller and smaller amplitude pulsators will be discovered. Long Period Variables: 100-700 days Long Period Variables: 100-700 days Classical Cepheids: 1-100 days Classical Cepheids: 1-100 days W Virginis: 2-45 days W Virginis: 2-45 days RR Lyraes: 1.5-24 hours RR Lyraes: 1.5-24 hours δ Scutis: 1-3 hours δ Scutis: 1-3 hours β Cepheids: 3-7 hours β Cepheids: 3-7 hours ZZ Cetis: 100-1000 seconds ZZ Cetis: 100-1000 seconds

24. Luminosity and the partial Ionization zones HeII partial ionization zone is the main driving zone, HeII partial ionization zone is the main driving zone, Bu H ionization zone important as well for driving and Bu H ionization zone important as well for driving and Produces an observable phase lag between maximum brightness and minimum radius. Produces an observable phase lag between maximum brightness and minimum radius. Max brightness at minimum radius is to be expected on a purely adiabatic approach. Max brightness at minimum radius is to be expected on a purely adiabatic approach. Max. brightness when mass between H ionization zone and star’s surface is a minimum. Max. brightness when mass between H ionization zone and star’s surface is a minimum. This occurs slightly after maximum compression or minimum radius. This occurs slightly after maximum compression or minimum radius.

26. Non-Radial Oscillations Can also model small departures from hydrostatic equilbrium as Can also model small departures from hydrostatic equilbrium as