12. Physical Parameters from Stellar Spectra. Fundamental effective temperature calibrations Surface gravity indicators Chemical abundances

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1 12. Physical Parameters from Stellar Spectra Fundamental effective temperature calibrations Surface gravity indicators Chemical abundances 1

2 Fundamental Properties of Stars Temperature (T) Radius (R) Chemical Composition Mass (M) Surface Gravity (g) Luminosity (L) Density (ρ) g = GM/R 2 L R 2 T 4 ρ M/R 3 What can we measure with spectroscopy? Age 2

3 Fundamental Properties of Stars Temperature (T) Radius (R) Chemical Composition Mass (M) Surface Gravity (g) Luminosity (L) Density (ρ) g = GM/R 2 L R 2 T 4 ρ M/R 3 What can we measure with spectroscopy? Age(?) 3

4 1. Temperature Indicators The condition of radiative equilibrium requires that the flux at any given depth remains constant: In plane-parallel geometry r R = const and in analogy to the black body radiation, from the Stefan-Boltzmann law we define the effective temperature: 4

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6 Optical Long-Baseline Interferometry Teff can only be measured by knowing a stars angular size. Problem: Sun at 5 pc is ~7 x smaller than the diffraction limit of Keck. Enter the magic of interferometry: constructive interference b destructive interference 6

7 Optical Long-Baseline Interferometry Visibility = Contrast of the observed interference ( Fringe ) pattern V = (Imax+Imin)/ (Imax-Imin) For a point source at infinity, V=1 (perfect interference) destructive interference constructive interference 7

8 Optical Long-Baseline Interferometry R=λ/2b! destructive interference constructive interference 8

9 Optical Long-Baseline Interferometry Extended Source R=λ/2b! destructive interference Visibility < 1 constructive interference 9

10 Optical Long-Baseline Interferometry Michelson measured the angular size of Betelgeuse to be ~0.05 arcseconds; combined with it s parallax, the radius was determined to be 800 x 10 6 km (roughly 3x the perihelion distance of Mars). First stellar diameter measurement! 10

11 Center for High-Angular Resolution Astronomy S W E Mt. Wilson Observatory, CA11

12 Center for High-Angular Resolution Astronomy S 300 meters! 300 meters! W 300 meters! E Mt. Wilson Observatory, CA 12

13 1-m telescope 100-inch Vacuum tubes grad student 13

14 Interferometry of a Solar-like Star Visibility^2 Model 0.7 mas = ~2x10 6 times smaller than the apparent size of the Sun Baseline/Wavelength White et al

15 Interferometry Spectrophotometry θ = ± mas Fbol = ± 5.5 pw m -2 Teff = /- 59 K White et al

16 Empirical H-R Diagram Challenge: this can really only be done for V<8 mag stars von Braun & Boyajian (2018) 16

17 Infrared Flux Method Idea: Compare bolometric flux to monochromatic flux measured at infrared wavelengths, solve for Teff. Φ is calculated from model atmospheres, but insensitive to Teff in the IR. In practice, take ratio between bolometric and monorchromatic flux: Blackwell et al

18 Infrared Flux Method Calibrated against interferometric angular diameters Weak dependence on surface gravity and metallicity Depends on corrections for interstellar reddening! Casagrande et al

19 Color-Temperature Relations More general: measure photometric colors and compare against independently measured temperatures. Empirical calibration, depends on reddening! Casagrande et al

20 Color-Temperature Relations Casagrande et al

21 Spectroscopic Temperatures Gray 2005 Many options, with general problem that line profiles are degenerate with other parameters. Example: Hydrogen lines (also Balmer jump!) 21

22 Spectroscopic Temperatures Gray 2005 Alternative: line-depth ratios of the same elements, which removes degeneracy with chemical abundances. Gradually looses sensitivity for < 4000 K. 22

23 2. Surface Gravity Indicators Gray 2005 Collisional broadening increases with higher surface gravity. Hydrogen lines sensitive for > 7500K. For FGK stars, strong line such as Ca II H & K, Na I D, Mg I b are used. 23

24 2. Surface Gravity Indicators Gray 2005 Problem: degeneracies with temperature. Can be partially solved by analyzing lines with different EW sensitivities, e.g. by varying excitation potential 24

25 2. Surface Gravity Indicators Huber et al. (2013) Torres et al. (2012) Better solution: use externally constrained gravity (e.g. using asteroseismology, eclipsing binaries) to remove degeneracy. Multiple solutions how to implement this exist! 25

26 3. Chemical Abundances Several "schools" of spectroscopic analysis techniques: 1) Line by line analysis (old school) Measure equivalent widths for individual lines, perform curve of growth analysis, iterate with Teff & log(g). Require excitation and ionization equilibrium for individual lines of the same species. 2) Spectral Synthesis (new school) Fit fully sythesized, multidimensional grid (Teff, log(g), [Fe/H], vsin(i), [α/fe],...) directly to observed spectra. 3) Data driven modeling (hip school) Forget model atmospheres, fit functional forms trained with "labels" from independent methods 26

27 Line by Line Analysis "Classical" spectroscopy. Identify and measure individual line parameters across stellar spectrum. Examples: MOOG ( VWA ( 27

28 Line by Line Analysis Advantage: Disadvantage: requirement of excitation and ionization equilibrium removes degeneracies of atmospheric parameters detailed, often manual analysis, hard to apply to large number (1000+) spectra 28

29 Spectral Synthesis Current state-of-the-art. Very high dimensional grids available, including methods to account for wavelength covariances Examples: SME ( ispec ( Starfish ( 29

30 Spectral Synthesis Synthesis Synthesis Line-by-Line Constrained - Derived Spectral synthesis methods are more prone to stellar parameter covariances. Always best to have external calibrations! 30

31 Data Driven Modeling (aka The Cannon) flux at each pixel set of stellar labels model coefficient vector Using training set, for which ln is known, optimize Θ: For any given observation, optimize labels: Ness et al. (2015) 31

32 Data Driven Modeling [Fe/H] log(g) Teff Advantage: Disadvantage: applicable to very large datasets only as good as your training sample! Ness et al. (2015) 32

33 Data Driven Modeling Age Map based on spectroscopy! Ness et al. (2016), Ho et al. (2017) 33

34 Stellar Spectroscopy in the ~2020's Maunakea Spectroscopic Explorer SDSS-V 34

35 Wrap Up 1. Introduction: The power of modern quantitative spectroscopy 2. Basic assumptions for classic stellar atmospheres 3. Transport of Energy: Radiation 4. Transport of Energy: Convection 5. Atomic radiation processes 6. Stellar Spectral Classification, Excitation and ionization 7. LTE versus Non-LTE 8. Spectral Line Formation 9. Stellar Winds 10. Line Formation in Expanding Atmospheres 11. X-Ray, IR, Radio Excess and Stellar Wind Diagnostics 12. Physical Parameters from Stellar Spectra 35

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