5. The Baade-Wesselink Method
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1 Rolf Kudritzki SS The Baade-Wesselink Method Baade-Wesselink Method (BWM): distance determination method for stellar objects, for which the photosphere can be approximated as a spherically moving (expanding) shell R = R(t) R(t 0 )= Z t t 0 v(t)dt change of radius determined by integral of velocity 1
2 Rolf Kudritzki SS 2017 radius R, angular diameter Θ and distance d are related R d = tan 2 2 change in radius ΔR change in angular diameter ΔΘ distance d = 2 R measure ΔR from velocity and find a way to measure ΔΘ à distance 2
3 Rolf Kudritzki SS 2017 original idea: test pulsation hypothesis of Cepheids later application: calibrate absolute magnitudes of radial pulsators and supernovae 3
4 Rolf Kudritzki SS 2015 M. Groenewegen 4
5 Rolf Kudritzki SS 2017 measurement of ΔΘ = 2ΔR/d method 1: measure stellar surface flux πf λ and stellar brightness S λ S (t) = R2 (t) d 2 F (T eff (t)) changes ΔS λ à ΔΘ, F λ (t) from spectral analysis and model atmospheres or surface brightness-color relationship of G,K,M giants 5
6 Surface Brighteness Color Relationship SB V (mag) Interferometry à obs. surface brightness color relationship SBCR late type giants G, K, M σsb V (mag) σ ~ 0.03 mag DiBenedetto, 1998,2005 Groenewegen, 2004 Kervella, 2004
7 BWM distances SB[mag] = (V-K) (V-K) 0 2 SBCR σ = 0.03 mag φ[mas] = 100.2(SB V) angular diameter d[pc] = ΔR[R sun ]/Δφ[mas] distance
8 method 2: interferometry 1.55 LD Angular diam. (mas) res. (mas) Phase A B Interferometric observations of δ Cep (Mérand et al. 2005) M. Groenewegen 8 MIAPP, 17 June 2014 p.8/50
9 Rolf Kudritzki SS 2017 simple straightforward method, but the devil is in the detail. radial velocity from spectral lines radial expansion velocity v(t) 1. v rad is integral over stellar surface v(t) =pv rad (t) projection factor p, p > 1 2. hydrodynamic motion of matter caused by pulsation à very likely velocity gradients in photosphere 9
10 Interferometry! The Baade-Wesselink methods V puls or! IRSB! Angular Diameter (mas) Kervella et al., 2004, A&A, 416, 941 V rad Phase p = V puls V rad d ΔR Δθ with R(t) = rad pv dt A direct linear relation 10 Nicolas NARDETTO Naples, 2011! 2!
11 The radial velocity definition λc λg λm λm β Dor HARPS observations 11 Nicolas NARDETTO Naples, 2011! 3!
12 V The radial velocity definition A geometric effect (uniform disk) puls = 30km/s RV = 20km/s c p =1.5 0 p <= 1.5 Limb-darkening effect V puls = 30km/s RV = 21.5km/s c p = po=-0.18uv+1.50 (Nardetto et al. 2006)! 12 Nicolas NARDETTO Naples, 2011! 4!
13 HARPS observations of 10 Cepheids (P=3j à P=42j)! Model of δ Cep! 300 spectra! Thousands of spectral lines! 17 selected! Radius (Rs)! β Dor! ζ Gem! Phase! ΔRV c = a o D + b o Nicolas NARDETTO Naples, 2011! 13 8
14 The p-factor decomposition 4! 3! semi- p-factor decomposition : theoretical approach 1 Direct measure of the velocity gradient (HARPS) Amplitude of the radial velocity (km/s) 2! 1! δ Cep! Line depth p =1.39 * 0.99 * 0.96 = geometric p-factor 3 extrapolation to the photosphere 4 optical/gas layers : hydro code 2! 1! 4! 3! 14 Nicolas NARDETTO Naples, 2011! 9!
15 The Pp relation High resolution spectroscopy for Cepheids distance determination II. A period-projection factor relation N. Nardetto, D. Mourard, Ph. Mathias, A. Fokin, D. Gillet, 2007, A&A, 471, 661 Period (days) Period (days) Period (days) Geometric model HARPS observations Hydrodynamical modelling FeI 4896A! p-factor p = [ 0.06 ± 0.02]log P + [1.38 ± 0.02] Period (days) Nicolas NARDETTO Naples, 2011! 15 10!
16 And when using the cross-correlation method? Rolf Kudritzki SS 2015 High resolution spectroscopy for Cepheids distance determination of the cross-correlation on the p-factor and the c-velocity N. Nardetto, W. Gieren, P. Kervella, P. Fouqué, J. Storm, G. Pietrzynski, D. Mourard, D. Queloz, 2009, A&A, 502, 951 V. Impact 5% first moment of the spectral line (RVc) cross-correlation p = [0.08 ± 0.05]log P + [1.31± 0.06] Conclusions: 1 the static approach lead to an overestimation of the distances of 10% 2 consistent with HST parallaxes measurements Mérand et al ; Groenewegen slope PL (Milky Way) = slope PL (LMC) P. Fouqué et al., 2007, A&A, 476, Nicolas NARDETTO Naples, 2011! 11!
17 Rolf Kudritzki SS 2015 An Example V, K, RV phased light curves for AQ Pup Groenewegen17 MIAPP, 17 June 2014 p.19/50
18 θ vs. R gives distance. AQ Pup Groenewegen 18 MIAPP, 17 June 2014 p.20/50
19 G13: Martin Groenewegen, 2013 Distance to the LMC should be independent of Period Distance to the LMC should be independent of (mean) (V K) colour Slope of the M VorK P relation should be the same as the m VorK P one LMC distance (corrected for "plane") versus Period for p =1.33 Groenewegen 19 MIAPP, 17 June 2014 p.22/50
20 p(p ) dependence log P = p d LMC = 45.5 kpc Groenewegen 20 MIAPP, 17 June 2014 p.23/50
21 For the LMC sample we have expanded the sample to 36 stars by obtaining radial velocity curves for 22 LMC Cepheids based on observations with the HARPS spectrograph on the ESO 3.6m telescope. (Storm et al. 2011) 21 Storm, 2014, MIAPP WS
22 LMC photometry Persson et al. (2004) OGLE-III Soszynski et al (2008) 22 Storm, 2014, MIAPP WS
23 Radial velocity curves for galactic Cepheids using the 1.2m STELLA-I robotic telescope on Tenerife with the SES fiber-fed echelle spectrograph. (Storm et al. 2011) 23 Storm, 2014, MIAPP WS
24 The new constraints on the p-factor, p = αlog(p) + β N09 HB86 Best estimate: p= log(p) 24 Storm, 2014, MIAPP WS
25 Comparison of distance moduli from HST parallaxes (Benedict et al. 2007) and the IRSB technique as a function of log(p). The dispersion is 0.14mag. 25 Storm, 2014, MIAPP WS
26 Distances to the LMC Cepheids using the revised p-factor relation The distance to the LMC is measured to be (m-m)o=18.45(+-0.04) 26 Storm, 2014, MIAPP WS
27 results BWM Nardetto et al., 2009 p = logP (m M) LMC = ± 0.02mag Groenewegen, 2014 p = logP (m M) LMC = ± 0.02mag Storm et al., 2014 p = logP (m M) LMC = ± 0.04mag 10% uncertainty 27
28 A Systematic Error for Richard Anderson, 2014 BWM error source: irregularities in v rad Baade-Wesselink Distances 25 p RV (t)dt d = 2 Δ R(t) Δ θ(t) Anderson, 2014 MIAPP WS Δ d/d = Δ R 1 Δ R 2 7 % 1 Δ R 1 Independent of p-factor 28
29 Modulation! 26 N = 125 Δ d /d 1 15 % Anderson, 2014 MIAPP WS RIA 2014, A&A letters, accepted 29
30 27 Cycle-to-cycle Modulation in l Car N = 324 Δ d /d 1 6 % Anderson, 2014 MIAPP WS 30
31 RS Pup: an Extreme Case 28 N = 439 Δ d /d 1 7 % Anderson, 2014 MIAPP WS 31
32 ΔΘ Modulation? MOST SZ Tau 30 Yes in space-based photometry Interferometry? (1-2% precision required) ΔR and ΔΘ must be observed contemporaneously (similar puls. cycle) mag l Car Evans et al. (2014) phase Kervella et al. (2004) Anderson, 2014 MIAPP WS 32
33 Rolf Kudritzki SS
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