Spectroscopy Determination of Fundamental Parameters. I. Gravity II. Radii III. Temperature IV. Stellar Rotation

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1 Spectroscopy Determination of Fundamental Parameters I. Gravity II. Radii III. Temperature IV. Stellar Rotation

2 Measuring the Gravity in Stars As we have seen a change in surface gravity or effective temperature can change the measured abundance. How can we be sure that we have the correct pressure and temperature. Here we discuss means of measuring the gravity (pressure) and temperature. Determination of the gravity of a star is of fundamental importance to transit detection of planets. The gravity is used to infer the luminosity class and thus radius of the star which is needed to derive the radius of the planet.

3 Measuring the Gravity in Stars Direct way: Obtain Mass via spectroscopy and the Doppler effect (binary stars for example) Measure the radius by an independent means (interferometry, lunar occultations, eclipsing binaries) g = GM R 2 In principle this can only be done for very few stars must rely on spectroscopic determinations.

4 Measuring the Gravity in Stars Continuum measurements The Balmer jump is the only reliable means of using the continuum to measure the gravity D Recall that the Balmer jump is due to the change in continuous opacity across the wavelength corresponding to the ionization of the Balmer electron. (Photons no longer have the energy to ionize a hydrogen atom, the opacity decreases and you are looking deeper into the atmosphere across the jump).

5 Continuum measurements Measuring the Gravity in Stars The Balmer jump is sensitive to gravity. If D = F + /F - (balmer discontinuity), then the larger D, the better it is as a gravity indicator. The best sensitivity occurs around 7500 K. This method is not as useful for stars cooler than 6500 K because it is masked by the large number of metallic lines that appear in cooler stars.

6 Measuring the Gravity in Stars Hydrogen lines One of the more common means of inferring the gravity

7 Other strong lines Measuring the Gravity in Stars Lines like Ca II H & K, Ca I 4227 Å, Na D lines, Mg I b lines show strong pressure broadened wings in the spectra of cool stars.

8 Other strong lines: Ca I Measuring the Gravity in Stars

9 The Gravity-Temperature diagram Measuring the Gravity in Stars Weak lines can be used by comparing the two stages of ionization for the same element. However, for ionic line strength depends on P e and thus indirectly on g. Plus, you really need to know the chemical composition. Pressure (gravity) diagnostics are always temperature sensitive. Therefore one should make simultaneous solutions to the effective temperature and gravity. Use 2 lines with different response to the variables (g and T), for example lines with different excitation potentials Each line is computed for a constant elemental abundance Vary surface gravity keeping temperature constant (vice versa) to recover the observed equivalent width. The crossing point is the solution.

10 The Gravity-Temperature diagram Measuring the Gravity in Stars

11 Empirical indicators Measuring the Gravity in Stars There are two empirical indicators of surface gravity: The width of the chromospheric emission reversal of the Ca II H & K lines (Wilson- Bappu effect) The size of the Doppler broadening called macro-turbulence (next week)

12 Empirical indicators Measuring the Gravity in Stars

13 Visual binaries allow the direct determination of the mass and thus gravity, if you can measure the stellar radius.

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15 Measuring the Gravity in Stars If you have measured the surface gravity from spectral fitting, and the radius by other means, then the mass of the star is given by M = gr 2 /G If you can measure the mass of the star by other means (e.g. binary systems) as well as the radius you can get a direct measurement of the surface gravity. If you can measure the mass of the star by other means and then the gravity from spectrosctopy, then you have the stellar radius Simply put, you gravity is related to mass and radius (g, M, R). If you measure 2 you have the third and there are a several methods to get the different quantities.

16 Direct Measurements of Stellar Radii If one can measure the radius of the star one can get a direct measurement of the effective temperature and mass (gravity) of the star. If L is the observed absolute luminosity of the star, then L = 4πR 2 σt eff or, if you do not know the radius you can measure the effective temperature via spectroscopy 4 and derive the radius. This is the more common way since the direct measurement of radii can only be done for a few stars. The absolute luminosity of the star requires an accurate determination of the stellar distance. Accurate distances of the brightest stars are available from Hipparcos measurements. In the near future GAIA will have more accurate stellar distances for millions of stars.

17 Stellar Radii Speckle Photometry The diffraction limit of a 4m telescope is 0.02 arcsecs. There are a significant number of stars that have angular diameters greater than this. However, atmospheric turbulence prevents one from achieving the diffraction limit Texereau (1963), Labeyrie (1970) and others realized that the instantaneous seeing disk in the focal plane of a large telescope consists of slightly displaced diffraction-modified images of the star. These speckles indicate that the atmosphere consists of a small number of refracting elements that redirects portion of the wavefront entering the telescope aperture.

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19 Stellar Radii Speckle Photometry Averaging over a few seconds integration removes all traces of the individual speckles. One must take rapid (~millisecs) exposures. Each speckle is the convolution of the telescope instrumental profile (diffraction and other optical effects) and the true distribution of the intensity of the stellar disk. Fourier analysis recovers the disk of the star, and one simply adds the Fourier transforms of the individual speckle exposures The instrumental profile of the telescope can be obtained by observing a star known to be a unresolved. (e.g. distant giant star).

20 The speckle pattern for Vega taken with the Hale 5m telescope

21 From space one of course can reach the diffraction limit of the telescope:

22 Original image Deconvolved image Hot Spot

23 Stellar Radii Interferometry By increasing the size of your telescope, you increase the diffraction limit, but building large telescopes is difficult and requires and are technically challenging. Solution: use many small telescopes Interferometry can increase the angular resolution of your telescope and thus allow you to measure directly the angular diameter of a star

24 A Two-telescope Interferometry S s B = B cos θ A 1 x 1 B θ A 2 x 2 Delay Line 1 d 1 Beam Combiner Delay Line 2 d 2 Idealized 2-telescope interferometer

25 Keck: A Two-telescope Interferometry

26 VLTI: A Multi-Telescope Interferometer

27 Two Beam Interferometer and 2 Point sources: Two plane parallel wave fronts: φ 1 ~ e ikd 1 e -iωt φ 2 ~ e ikd 2 e -ikŝ B e -iωt φ Total = φ 1 + φ 2 ~ e -iωt (e ikd 1 + e ikd 2 e ik ŝ B ) Can show that the total power is: P = 2(1 + cos k (s B+d 1 d 2 )) s B is the path difference of the light hitting the 2 telescopes d 1 d 2 is the path difference caused by the delay lines

28 Output of 2 telescope interferometer Adjacent fringe crests projected on the sky are separated by an angle given by: Δs = λ/b

29 The Visibility Function Michelson Visibility: V = I max I min I max +I min Visibility is measured by changing the path length and recording minimum and maximum values

30 Cittert-Zernike theorem Δs ŝ o α β α, β are angles in x-y directions of the source. Δs = (α,β,0) in the coordinate system where ŝ o =(0,0,1)

31 Cittert-Zernike Theorem V(k, B) = dα dβ I(α,β) e -2πi(αu+βv) Cittert-Zernike theorem: The interferometer response is related to the Fourier transform of the brightness distribution under certain assumtions (source incoherence, small-field approximation). In other words an interferometer is a device that measures the Fourier transform of the brightness distribution.

32 Footnote: Fourier Transforms The continous form of the Fourier transform: F(s) = f(x) e ixs dx f(x) = 1/2π F(s) e ixs ds e ixs = cos(xs) + i sin (xs)

33 Footnotes: Fourier Transforms In interferometry it is useful to think of normal space (x,y) and Fourier space (u,v) where u,v are frequencies Two important features of Fourier transforms: a) The spatial coordinate x maps into a frequency coordinate 1/x (= s) Thus small changes in x map into large changes in s. A function that is narrow in x is wide in s

34 I. Background: Fourier Transforms x ν x

35 I. Background: Fourier Transforms x ν x ν

36 I. Background: Fourier Transforms sinc x ν J 1 (2πx) 2x x ν Diffraction patterns from the interference of electromagnetic waves are just Fourier transforms!

37 V(s) = 2J 1 (πas)/πas Stellar Angular Diameters: J 1 is the first order Bessel function Note: as it should be, this is also the diffraction pattern of a circular aperture. For measuring stellar diameters you only need 2 telescopes. In fact a common practice is to take a single telescope and use a mask with two holes in it

38 Note: Each baseline measures only one point on the visibility function. Often, you cannot sample all baselines.

39 Historical Note: Michelson was the first to measure the diameter of Betelgeuse. In 1920, he and Francis Pease mounted a 6 m interferometer on the front of the 2.5 metre telescope at Mount Wilson Observatory. measured the angular diameter of α Orionis at = 47 mas. The current value is 55 mas. Large uncertainty due to limb darkening.

40 Stellar Radii Lunar Occultations As the moon occults a star it can be used as a knife edge to create a diffraction pattern. The occultation pattern is recorded as a function of time, and the time coordinate is converted to angular diameter using the angular velocity of the moon relative to the earth: References: Williams, J.D. ApJ, 1939, 89, 467 Nather, R.E. & Evans, D.S. 1970, AJ, 75, 575 Schmidtke et al. 1986, AJ, 91, 961.

41 Lunar occulations are one of the best ways to measure stellar diameters. Unfortunatley, the stars have to lie in the path of the moon!

42 Stellar Radii Bolometric Method 4πd 2 F ν = 4πR 2 F ν F ν is the observed flux of the star, F ν is the flux at the stellar surface, R is the radius of the star, d is the distance to the star R d = F ν dν ( σteff 4 ( ½ Log R = log d 0.2(m v BC) 2log T eff + constant If we ratio everything to solar units (m = (K סּ = סּ T, סּ Log R = log d + 0.2BC 2log T eff 0.2m v

43 Stellar Radii Eclipsing Binaries from CoRoT Eclipsing Binaries can also yield stellar radii. These are relatively few, but they can be used to calibrate main sequence radii

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45 Measuring Stellar Masses Uncertain log g and R (depends on how good gravity and radius is) Spectral type More certain Dynamical mass (binary) Asteroseismology

46 Dynamical Masses from Spectroscopic binaries Measure the Doppler shift of the individual spectral components, apply Keplers law and viola! You have the ratio of the stellar masses!

47 Stellar Oscillations: Asteroseismology Stellar oscillations probe the interior of the star and give you the mean density

48 P-mode Oscillations in the Sun P-mode oscillations (pressure is the restoring force) are equally spaced in frequency with an interval given by: Δν M1/2 R 3/2 µhz

49 Stellar Oscillations in β Gem Above shows the RV measurements of β Gem. The solid line represents a 17 sine component fit. The false alarm probability of these modes is < 1% and most have FAP < The rms scatter about the final fit is 1.9 m s 1

50 The Oscillation Spectrum of Pollux The p-mode oscillation spectrum of β Gem based on the 17 frequencies found via Fourier analysis. The vertical dashed lines represent a grid of evenly-spaced frequencies on an interval of 7.12 µhz

51 The Mass of Pollux Frequency spacing: Δν M1/2 R 3/2 µhz The radius of β Gem is well determi n ed through Δνinterferometric 0 = 7.12 µhz measurements: (l = 0) M = R 1.89 = 8.8 M סּ R סּ log g = 2.82 From spectroscopy: log g = 2.70

52 With enough oscillation frequencies you can derive everything with asteroseismology! 1/2 (M/M ) סּ Δν (R/R 3/2 µhz ) סּ סּ ν max = M/M סּ 3.05 mhz (R/R2) סּ T eff /5777K

53 Stellar Temperatures Indicators of stellar temperatures: 1. Slope of Paschen continuum: log(f 4000 /F 7000 ) log T eff 2.3 If you can measure the continuum slope between 4000 and 7000 Å to 2.3% then you have the temperature to 1%

54 2. Color indices Stellar Temperatures

55 3. The Balmer Jump Stellar Temperatures

56 4. Metallic Lines Stellar Temperatures: Line Ratios Line depth ratios can be used to measure accurately the temperature of a star. One choses two lines of different temperature sensitivities. These should be very close in wavelength so as to minimize errors due to continuum placement. The effects of rotational broadening and macroturbulence should effect both lines equally.

57 Stellar Temperatures: Line Ratios

58 Stellar Temperatures: Line Ratios To use line depth ratios one must calibrate these against effective temperature (B V): Line depth ratios can yield effective temperatures accurate to about 10 K. One can measure temperature variations in one star to about ΔT 1 K

59 Note: This method only works for stars of spectral type G K

60 The Rotation Profile v = Ω R Doppler shift = yω x xω y But there is no x-component of Ω due to choice of coordinate system: v z = xωsin i Sign convention: +v z receding (redshift) v z approaching (blueshift) Largest velocity occurs at limbs where v L = RΩ sin i = v eq sin i. R is the radius of the star and v eq is the equatorial velocity

61 The Rotation Profile The flux of the star is still given by: F ν = I ν cos θ dω But now I ν has to be Doppler shifted by the radial velocity on the star: dω = da/r 2, da is the increment of surface area on the star, the increment on the apparent disk is dxdy = dacosθ F I ν = ν dxdy R 2 The proper way is to calculate I ν at each location on the stellar disk using a local profile generated by a model atmosphere which takes into account limb darkening. You the apply the appropriate Doppler shift and integrate over all points. I will present an analytical solution following Gray.

62 The apparent disk of the star can be thought of as a series of strips parallel to the projection axis each having a Doppler shift of xωsini: V = V rot x V = +V rot V = 0

63 The Rotation Profile Define the intrinsic profile at any point on the disk as H(Δλ) = H(v) = I ν /I c = ratio of intensity of any point in the spectrum to the continuum intensity. The flux profile from a rotating star is: F ν F c = H(v) I ccos θ dω I ccos θ dω H(v) depends on the disk position through the Doppler shifts: F ν = H(v v z) I c cos θ dω F ν = H(v v z ) I c dxdy R 2

64 The Rotation Profile Integration limits: x: R to +R y: y 1 to +y 1 y 1 = (R 2 x 2 ) ½ = R [ 1 ( v z v eq 2 ½ [ ( F ν = H(v v z) R +R y 1 +y 1 I c dy R dv z v L

65 The Rotation Profile Define G(Δλ): G(Δλ) = G(v z ) = 1 v L y 1 y 1 I c dy/r I c cos θ dω v z v L = 0 v z > v L Normalized profile: F ν F c = H(v v z ) + G(v z ) dv z = H(v) *G(v z ) * is the convolution Remember that v can also be replaced with Δλ

66 Footnote: Convolution Convolution f(u)φ(x u)du = f * φ f(x): φ(x):

67 φ(x-u) a 1 a 2 a 3 g(x) a 3 a 2 Convolution is a smoothing function a 1 Note: f*g F G In Fourier space the convolution is a muliplication of the individual Fourier transforms

68 The Rotation Profile Important result: The profile of a rotationally broadened spectral line is merely the convolution of the flux profile from a non-rotating star convolved with the rotation profile, so long as the shape of the line profile does not change across the stellar disk. The more rotation, the broader the rotation function G. For large rotation rates, it dominates the line shape. This means that one does not have to do a disk integration, one merely does a convolution of a simple profile. Including limbdarkening: I c /I c 0 = (1 ε) + ε cosθ I c 0 is the specific intensity at disk center and ε 0.6 for the sun. θ is the limb angle. The denominator of G(v z ) then becomes I c cos θ dω = π I c 0 (1 ε/3)

69 The Rotation Profile And the numerator Use cos θ = [R 2 (x 2 + y 2 )] ½ /R and the fact that (A 2 y 2)½ dy = ½[y(A 2 y 2 ) ½ + A 2 sin 1 (y/a)] this becomes

70 The Rotation Profile G(λ) If ε = 0, the second term is zero and the function is an ellipse. If ε =1 the first term is zero and the rotation function is a parabola