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Chapter 7: from Theory to Observations

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Chapter 7: from Theory to Observations Chapter 7: From theory to observations Given the stellar mass and chemical composition of a ZAMS, the stellar modeling can, in principle, predict the evolution of the stellar bolometric luminosity, effective temperature Teff , and possibly even the surface chemical composition. How can these parameters be linked to observables? Outline Spectra and magnitudes The effects of interstellar extinction K-correction for high-redshift objects Spectroscopic notation of the stellar chemical composition Calibration and uncertainty Review Spectra and magnitudes One important way is the use of the Color Magnitude Diagram (CMD), which is the observational counterpart of the HRD. HRD of selected stellar evolutionary tracks CMD of the globular cluster M3 using the (dashed lines) with the same initial solar Johnson BV filters. Remember that the chemical composition. The heavy solid magnitude scale is backward wrt intensity; lines display two isochrones for the same a larger value of B − V implies a cool chemical composition and ages of 600 Myr object. (the brighter sequence) and 10 Gyr. Magnitude definition The apparent magnitude is defined as R fλSλdλ 0 mA ≡ −2:5log R 0 + mA fλ Sλdλ where fλ is the monochromatic flux (or spectrum) of a star received at 0 the top of the Earth’s atmosphere while fλ denotes the spectrum of a 0 reference star that produces a known apparent magnitude mA, which is called the ’zero point’ of the band for a given photometric system. I This system specifies a specific choice of the response function, Sλ. I Many photometric systems exist! I Measured magnitudes do depend on both the fλ and the actual photometric system used (i.e., the shape of the filter). I So in general, one should not compare magnitudes defined for different photometric systems. Various photometric systems The very popular Johnson system, for example, uses the star Vega (A0V at the distance of 7.7 pc; M = 2:14M ) to fix the zero points. It assumes that mV = 0 and that the color indices are equal to zero for the star, whereas other systems may have slightly different values. The absolute magnitude is defined as the apparent magnitude a star would have at a distance of 10 pc (if the radiation travels undisturbed from the source to the observer): " # d 2 R f S dλ M ≡ −2:5log λ λ + m0 = m − 5(log(d) − 1) A R 0 A A 10 pc fλSλdλ where d is the distance in parsec. In the cosmological context, the distance is so called luminosity distance dL. mA − MA ≡ (m − M)A is called the distance modulus. What do you think about a comparison of this Mbol value with MK ; = 3:4? The absolute bolometric magnitude is independent of any photometry system! Absolute bolometric magnitude The absolute magnitude of the Sun is then " 2 R # d f S dλ M ≡ −2:5log λ, λ + m0 : A; R 0 A 10 pc fλSλdλ By setting Sλ = 1 at all wavelengths in the above two definitions (thus the subscript change from A to bol ) and adopting the Sun as the reference star (placing it at the distance of 10 pc), we have the definition of the absolute bolometric magnitude of a star as Mbol ≡ Mbol; − 2:5log(L=L ) 33 −1 where L = 3:826 × 10 ergs s and Mbol; = 4:75. The absolute bolometric magnitude is independent of any photometry system! Absolute bolometric magnitude The absolute magnitude of the Sun is then " 2 R # d f S dλ M ≡ −2:5log λ, λ + m0 : A; R 0 A 10 pc fλSλdλ By setting Sλ = 1 at all wavelengths in the above two definitions (thus the subscript change from A to bol ) and adopting the Sun as the reference star (placing it at the distance of 10 pc), we have the definition of the absolute bolometric magnitude of a star as Mbol ≡ Mbol; − 2:5log(L=L ) 33 −1 where L = 3:826 × 10 ergs s and Mbol; = 4:75. What do you think about a comparison of this Mbol value with MK ; = 3:4? Absolute bolometric magnitude The absolute magnitude of the Sun is then " 2 R # d f S dλ M ≡ −2:5log λ, λ + m0 : A; R 0 A 10 pc fλSλdλ By setting Sλ = 1 at all wavelengths in the above two definitions (thus the subscript change from A to bol ) and adopting the Sun as the reference star (placing it at the distance of 10 pc), we have the definition of the absolute bolometric magnitude of a star as Mbol ≡ Mbol; − 2:5log(L=L ) 33 −1 where L = 3:826 × 10 ergs s and Mbol; = 4:75. What do you think about a comparison of this Mbol value with MK ; = 3:4? The absolute bolometric magnitude is independent of any photometry system! Observationally, one needs to select proper filters that are Evolution of a 1 M star with Z = 0:001 sensitive to the desirable and Y = 0:246 from the ZAMS until the tip of the RGB, displayed in the HRD and measurements, particular for Teff . various CMDs. Bolometric correction The bolometric correction to a given photometric band A is defined as BCA ≡ Mbol − MA: If the stellar bolometric luminosity is known (from a model), one can predict the corresponding MA. In practice, tables of bolometric corrections and color indices are available, for a grid of gravities and Teff that cover all the major phases of stellar evolution, and for a number of chemical compositions. Interpolations among the grid points provide the sought BCA for the model. Bolometric correction The bolometric correction to a given photometric band A is defined as BCA ≡ Mbol − MA: If the stellar bolometric luminosity is known (from a model), one can predict the corresponding MA. In practice, tables of bolometric corrections and color indices are available, for a grid of gravities and Teff that cover all the major phases of stellar evolution, and for a number of chemical compositions. Interpolations among the grid points provide the sought BCA for the model. Observationally, one needs to select proper filters that are Evolution of a 1 M star with Z = 0:001 sensitive to the desirable and Y = 0:246 from the ZAMS until the tip of the RGB, displayed in the HRD and measurements, particular for Teff . various CMDs. Outline Spectra and magnitudes The effects of interstellar extinction K-correction for high-redshift objects Spectroscopic notation of the stellar chemical composition Calibration and uncertainty Review With this extinction-corrected magnitude, mA;0, together with the distance module, we can then calculate the absolute magnitude, A (= MA). The effect of extinction on a color index (A − B) is (A − B) = (A − B)0 + E(A − B) where E(A − B) = AA − AB is called the color excess or reddening. The effects of interstellar extinction If the extinction (Aλ) is important, the observed flux is then −τλ −0:4Aλ fλ = fλ,0e ≡ fλ,010 Defining AA as R −0:4Aλ −0:4A 10 fλ,0Sλdλ 10 A ≡ R ; fλ,0Sλdλ assuming certain Aλ and fλ,0 (as functions of λ), we have mA = mA;0 + AA: The effect of extinction on a color index (A − B) is (A − B) = (A − B)0 + E(A − B) where E(A − B) = AA − AB is called the color excess or reddening. The effects of interstellar extinction If the extinction (Aλ) is important, the observed flux is then −τλ −0:4Aλ fλ = fλ,0e ≡ fλ,010 Defining AA as R −0:4Aλ −0:4A 10 fλ,0Sλdλ 10 A ≡ R ; fλ,0Sλdλ assuming certain Aλ and fλ,0 (as functions of λ), we have mA = mA;0 + AA: With this extinction-corrected magnitude, mA;0, together with the distance module, we can then calculate the absolute magnitude, A (= MA). The effects of interstellar extinction If the extinction (Aλ) is important, the observed flux is then −τλ −0:4Aλ fλ = fλ,0e ≡ fλ,010 Defining AA as R −0:4Aλ −0:4A 10 fλ,0Sλdλ 10 A ≡ R ; fλ,0Sλdλ assuming certain Aλ and fλ,0 (as functions of λ), we have mA = mA;0 + AA: With this extinction-corrected magnitude, mA;0, together with the distance module, we can then calculate the absolute magnitude, A (= MA). The effect of extinction on a color index (A − B) is (A − B) = (A − B)0 + E(A − B) where E(A − B) = AA − AB is called the color excess or reddening. I The extinction law, Aλ=AV , is usually determined empirically, where AV is the extinction in the Johnson V band. Notice the 2175 A˚ bump. I The ratio RV ≡ AV =E(B − V ) is nearly a constant and equal to 3.1 for stars in the Galaxy. I More accurately, AV can be determined in a color-color plot, e.g., U − B vs. B − V . I For large reddening (e.g., toward the Galactic center), one needs to account for the non-constancy of Aλ and hence the substantial spectral change in a given filter range. Outline Spectra and magnitudes The effects of interstellar extinction K-correction for high-redshift objects Spectroscopic notation of the stellar chemical composition Calibration and uncertainty Review I One claim for the origin of the term ”K correction” is the correction as a Konstante (German for ”constant”) used by Carl Wirtz. I To determine the appropriate K-correction, one obviously needs to know the shape of the object intrinsic spectrum, or its evolution with time (e.g., related to the evolution of the stellar content, which may be modeled theoretically for a galaxy). K-correction for high-redshift objects I When an observing object (typically a galaxy) has a non-negligible redshift, one also needs to make the so-called K-correction, converting a measurement to an equivalent measurement in the rest frame of the object.
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