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

2. Hubble’s Law Cosmological Principle plus general relativity yield Friedmann’s equations. Cosmological Principle plus general relativity yield Friedmann’s equations. The most important parameter in these equations is Hubble’s constant, H 0. The most important parameter in these equations is Hubble’s constant, H 0. Empirically, we have Hubble’s law: Empirically, we have Hubble’s law: V = H 0 × D, V = H 0 × D, Where V = speed of recession and D is the distance. Where V = speed of recession and D is the distance. Measure V by Doppler shifts. Hence can measure H 0 if D is known. Measure V by Doppler shifts. Hence can measure H 0 if D is known. Find distances and recession velocities far out in the “Hubble Flow”. Find distances and recession velocities far out in the “Hubble Flow”.

3. The Extra-Galactic Distance Scale (Nearby) Measurement of distance is crucial in Astronomy. Measurement of distance is crucial in Astronomy. Most accurate distances are by parallax: due to the motion of the Earth around the Sun. Most accurate distances are by parallax: due to the motion of the Earth around the Sun. d = 1/p d = 1/p Unfortunately, this is only possible for the nearest stars (e.g α Centauri: parallax of 0.75” at a distance of 1.33pc. Unfortunately, this is only possible for the nearest stars (e.g α Centauri: parallax of 0.75” at a distance of 1.33pc. Good out to about 100pc, can only get 10% accuracy for distances out to a few pc. Good out to about 100pc, can only get 10% accuracy for distances out to a few pc. Many measured by Hipparcos (10% accuracy out to about 100pc). Many measured by Hipparcos (10% accuracy out to about 100pc). New missions SIM (10% accuracy out to about 25000pc), GAIA (10% accuracy out to about 10000pc). New missions SIM (10% accuracy out to about 25000pc), GAIA (10% accuracy out to about 10000pc). Parallax distances used to turn m into M for local stars: create local HR diagram. Parallax distances used to turn m into M for local stars: create local HR diagram. Then main sequence fitting of young open clusters eg. Pleiades and Hyades clusters. Then main sequence fitting of young open clusters eg. Pleiades and Hyades clusters.

5. The Extra-Galactic Distance Scale (Far away) Tully-Fisher relation: rate at which a galaxy spins is related to its intrinsic luminosity: faster a galaxy spins the more luminous is the galaxy. Why? Tully-Fisher relation: rate at which a galaxy spins is related to its intrinsic luminosity: faster a galaxy spins the more luminous is the galaxy. Why? For circular motion, star of mass m at radius r in a galaxy of mass M and radius R, For circular motion, star of mass m at radius r in a galaxy of mass M and radius R, mV c 2 /R = GMm/R 2, so M~V c 2 /R, mV c 2 /R = GMm/R 2, so M~V c 2 /R, The more massive a galaxy, the brighter it will be. The more massive a galaxy, the brighter it will be.

8. Cepheids and the Extra-Galactic Distance Scale Cepheids are important in that they connect the near and far distance ladder and are currently one of the most accurate primary distance scale tools. Cepheids are important in that they connect the near and far distance ladder and are currently one of the most accurate primary distance scale tools. Period-Luminosity (PL) relation. Period-Luminosity (PL) relation. Actually we have the Period-Luminosity- Color (PLC) relation. Actually we have the Period-Luminosity- Color (PLC) relation. Need Period-Mean Density theorem, Stefan-Boltzmann law and the existence of the instability strip. Need Period-Mean Density theorem, Stefan-Boltzmann law and the existence of the instability strip.

9. Cepheids and the Extra Galactic Distance Scale Period-mean density theorem plus Stefan- Boltzmann yields a relation of the form: Period-mean density theorem plus Stefan- Boltzmann yields a relation of the form: logL = a + blogT e + clogP, logL = a + blogT e + clogP, with an observational counterpart: with an observational counterpart: M v = a’ + b’(V-I) + c’logP (*) M v = a’ + b’(V-I) + c’logP (*) Currently (*) has been found in the LMC,SMC but not in our Galaxy. Why? Currently (*) has been found in the LMC,SMC but not in our Galaxy. Why? Averaging these equations over color or logT e yields, Averaging these equations over color or logT e yields, logL = c+dlogP, logL = c+dlogP, M v = c’ + d’logP. M v = c’ + d’logP.

10. Cepheids and the Extra-Galactic Distance Scale Key Project on the Hubble Space Telescope (HST) observed Cepheids in local group galaxies and used the Cepheid PL relation to get distances. Key Project on the Hubble Space Telescope (HST) observed Cepheids in local group galaxies and used the Cepheid PL relation to get distances. These distances, in turn, enabled the calibration of many secondary distance indicators like the TF relation far out in the Hubble flow: These distances, in turn, enabled the calibration of many secondary distance indicators like the TF relation far out in the Hubble flow: Estimate H 0 to 10% error. Estimate H 0 to 10% error.

11. Assumptions and Methods I Cepheid PL relation in V and I bands is well characterized by Cepheids in the Large Magellanic Cloud (LMC): Cepheid PL relation in V and I bands is well characterized by Cepheids in the Large Magellanic Cloud (LMC): M v = -2.760[±0.03](logP – 1) – 4.218[±0.02], σ V = ±0.16. M v = -2.760[±0.03](logP – 1) – 4.218[±0.02], σ V = ±0.16. M I = -2.962[±0.02](logP – 1) – 4.904[±0.01], σ V =±0.11. M I = -2.962[±0.02](logP – 1) – 4.904[±0.01], σ V =±0.11. Assume this is true universally, except perhaps for a metallicity dependence of the zero point. Assume this is true universally, except perhaps for a metallicity dependence of the zero point. The LMC is the “calibrator galaxy” The LMC is the “calibrator galaxy” Observe Cepheids in the target galaxy. Observe Cepheids in the target galaxy.

12. Distance Measurement with LMC Cepheid PL Relation Distance modulus Calibrated (LMC) PL relation Data in target galaxy Tanvir (1997)

13. Assumptions and Methods II Metallicity correction: change distance modulus by a certain amount determined empirically. Metallicity correction: change distance modulus by a certain amount determined empirically. Observe in V and I bands to correct for reddening: reddening of light due to interaction with interstellar dust. Observe in V and I bands to correct for reddening: reddening of light due to interaction with interstellar dust. Use V and I bands simulatneously to correct for reddening Use V and I bands simulatneously to correct for reddening μ 0 = μ V – 2.45(μ V – μ I ) μ 0 = μ V – 2.45(μ V – μ I ) Assume all Cepheids in the target galaxy are at the same distance. Assume all Cepheids in the target galaxy are at the same distance. Assume μ 0 = 18.5 magnitudes and there is no depth effect in the LMC. Assume μ 0 = 18.5 magnitudes and there is no depth effect in the LMC.

17. Future I KP estimate of H 0 ~ 72±8km/s/Mpc. KP estimate of H 0 ~ 72±8km/s/Mpc. CMB estimate of H 0 ~ 71 ± 3km/s/Mpc. CMB estimate of H 0 ~ 71 ± 3km/s/Mpc. But estimating many other cosmological parameters, such as Ω, can only be estimated through a convolution with H 0. But estimating many other cosmological parameters, such as Ω, can only be estimated through a convolution with H 0. An independent estimate of H 0 accurate to a few percent can help this situation. An independent estimate of H 0 accurate to a few percent can help this situation. Need a more accurate Cepheid distance scale. Need a more accurate Cepheid distance scale.

18. Future II Different calibrator galaxy? NGC 4258 has an accurate geometric distance due to water maser measurements: μ 0 ~ 29.28 ±0.15 mag. Different calibrator galaxy? NGC 4258 has an accurate geometric distance due to water maser measurements: μ 0 ~ 29.28 ±0.15 mag. Known more accurately than the distance to the LMC. Known more accurately than the distance to the LMC. Recent progress on the metallicity dependence of the zero point of the PL relation. Recent progress on the metallicity dependence of the zero point of the PL relation. But what about change of slope with period? But what about change of slope with period?

19. Future III Strong statistical, observational and theoretical evidence to indicate that in the LMC, the following model is more consistent with the data: Strong statistical, observational and theoretical evidence to indicate that in the LMC, the following model is more consistent with the data: M v = a + blogP, logP < 1, M v = a + blogP, logP < 1, M v = a’ + b’logP, logP > 1. M v = a’ + b’logP, logP > 1. In B,V,I,J,H, marginally linear in K. In B,V,I,J,H, marginally linear in K. Effects estimates of H 0 to 1-2%. Effects estimates of H 0 to 1-2%.

21. Future IV Why, in terms of Physics? Why, in terms of Physics? What, in terms of what effect on the distance scale? What, in terms of what effect on the distance scale? How in terms of how widespread this is? How in terms of how widespread this is? Combination of statistical/theoretical/numerical/obse rvational work. Combination of statistical/theoretical/numerical/obse rvational work.

22. RR Lyraes Population II (Z=0.001 to Z=0.0001) Population II (Z=0.001 to Z=0.0001) Periods of the order of hours, M ~ 0.55Msun to 0.65Msun. Periods of the order of hours, M ~ 0.55Msun to 0.65Msun. Hundreds of solar luminosities. Hundreds of solar luminosities. T eff ~6000K to 7000K T eff ~6000K to 7000K RRab, RRc, RRd (fundamental, first overtone and double mode oscillators RRab, RRc, RRd (fundamental, first overtone and double mode oscillators Mostly found in globular clusters Mostly found in globular clusters All stars in a cluster at the same distance and same age but chemically homogeneous. All stars in a cluster at the same distance and same age but chemically homogeneous. But stars evolve at different rates. But stars evolve at different rates. Globular clusters in halo: distribution of ages in a globular cluster can tell us how halo collapsed. Globular clusters in halo: distribution of ages in a globular cluster can tell us how halo collapsed.

29. Globular Cluster HR diagrams A: Main Sequence (H burning in the core) A: Main Sequence (H burning in the core) B: Red Giant Branch (H shell burning) B: Red Giant Branch (H shell burning) C: Helium Flash occurs here (Onset of core He burning) C: Helium Flash occurs here (Onset of core He burning) D: Horizontal Branch Steady State core He burning D: Horizontal Branch Steady State core He burning E: Hertzsprung gap: Separates Blue HB and Red HB E: Hertzsprung gap: Separates Blue HB and Red HB F: White Dwarfs: Outer envelope lost, Carbon core remains. F: White Dwarfs: Outer envelope lost, Carbon core remains.

32. Globular Cluster Ages Absolute ages: fit theoretical stellar evolutionary isochrones to CMD diagrams. Absolute ages: fit theoretical stellar evolutionary isochrones to CMD diagrams. Stellar Isochrone: locus of points on a HR diagram which have the same age for all masses. Stellar Isochrone: locus of points on a HR diagram which have the same age for all masses. CMD: color-magnitude diagram: HR diagram: T eff vs. L. CMD: color-magnitude diagram: HR diagram: T eff vs. L. Need transformation between color and T eff, magnitude and L (bolometric correction). Need transformation between color and T eff, magnitude and L (bolometric correction). Luminosity of the main sequence turnoff (MSTO): M V (TO) is a good stellar clock to determine globular cluster ages. Luminosity of the main sequence turnoff (MSTO): M V (TO) is a good stellar clock to determine globular cluster ages. But: uncertainty of 0.02mag. In color means an error of 0.1 mag. In derived distance modulus and hence an error of 1.5 Gyr in age. But: uncertainty of 0.02mag. In color means an error of 0.1 mag. In derived distance modulus and hence an error of 1.5 Gyr in age. M V (TO) is the bluest point on the main sequence for a gc: same color over a large range in magnitude. M V (TO) is the bluest point on the main sequence for a gc: same color over a large range in magnitude. M V (BTO) is brighter than the main sequence but 0.05 mag. redder – easier to measure, twice as accurate. M V (BTO) is brighter than the main sequence but 0.05 mag. redder – easier to measure, twice as accurate.

33. Globular Cluster Ages Can measure relative cluster ages and leave the zero point for a separate problem. Can measure relative cluster ages and leave the zero point for a separate problem. Brightness difference between HB and MSTO Brightness difference between HB and MSTO Brightness of HB set by the core mass of stars evolving up the GB. Brightness of HB set by the core mass of stars evolving up the GB. But as the cluster ages, its MSTO decreases, but core mass of stars going up the GB tends to be the same. But as the cluster ages, its MSTO decreases, but core mass of stars going up the GB tends to be the same. As a cluster ages, its HB stars tend to be bluer at constant brightness, but the MSTO decreases with increasing age. Thus As a cluster ages, its HB stars tend to be bluer at constant brightness, but the MSTO decreases with increasing age. Thus ΔM V (HB-MSTO) increases with cluster age. ΔM V (HB-MSTO) increases with cluster age. But metallicity also plays a role: But metallicity also plays a role: M V (HB) = a+b[M/H] M V (HB) = a+b[M/H] M V (HB) ~ M V (RR Lyraes in GC) M V (HB) ~ M V (RR Lyraes in GC) Also degeneracy between age and distance: Can simulate an older of younger GC by moving it closer or further away. Also degeneracy between age and distance: Can simulate an older of younger GC by moving it closer or further away. Thus need to know the distance to GC’s Thus need to know the distance to GC’s Thus need to know M V (RR Lyraes) Thus need to know M V (RR Lyraes)

34. RR Lyraes and GC ages Getting M V (RR) amounts to obtaining a population II distance scale. Getting M V (RR) amounts to obtaining a population II distance scale. But how? Don’t obey a PL like Cepheids excpet perhaps in the K? But how? Don’t obey a PL like Cepheids excpet perhaps in the K? Stellar pulsation modesl: obsrevation with theory. Stellar pulsation modesl: obsrevation with theory. Light curve structure against M V relation. Light curve structure against M V relation. Metallicity complication. Metallicity complication.

35. Oosterhoff Dichotomy Two types of GC’s: OoI, OoII, based on mean period their fundamental mode RR Lyrae stars. Two types of GC’s: OoI, OoII, based on mean period their fundamental mode RR Lyrae stars. Why? And what implications for age/distance estimation? Hot topic right now. Why? And what implications for age/distance estimation? Hot topic right now. Period-Color/Amplitude Color relations as a function of phase. Period-Color/Amplitude Color relations as a function of phase.