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 T eff, 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 (dashed lines) with the same initial solar chemical composition. The heavy solid lines display two isochrones for the same chemical composition and ages of 600 Myr (the brighter sequence) and 10 Gyr. CMD of the globular cluster M3 using the Johnson BV filters. Remember that the magnitude scale is backward wrt intensity; a larger value of B V implies a cool object.

Magnitude definition The apparent magnitude is defined as ( ) fλ S λ dλ m A 2.5log + m f 0 A 0 λ S λ dλ where f λ is the monochromatic flux (or spectrum) of a star received at the top of the Earth s atmosphere while fλ 0 denotes the spectrum of a reference star that produces a known apparent magnitude ma 0, which is called the zero point of the band for a given photometric system. This system specifies a specific choice of the response function, S λ. Many photometric systems exist! Measured magnitudes do depend on both the f λ and the actual photometric system used (i.e., the shape of the filter). 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 m V = 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): M A 2.5log [ ( d 10 pc ) 2 ] fλ S λ dλ + m f 0 A 0 λ S λ dλ = m A 5(log(d) 1) where d is the distance in parsec. In the cosmological context, the distance is so called luminosity distance d L. m A M A (m M) A is called the distance modulus.

Absolute bolometric magnitude The absolute magnitude of the Sun is then M A, 2.5log [ ( d 10 pc ) 2 ] fλ, S λ dλ + m f 0 A 0 λ S λ dλ. By setting S λ = 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 M bol M bol, 2.5log(L/L ) where L = 3.826 10 33 ergs s 1 and M bol, = 4.75.

Bolometric correction The bolometric correction to a given photometric band A is defined as BC A M bol M A. If the stellar bolometric luminosity is known (from a model), one can predict the corresponding M A. In practice, tables of bolometric corrections and color indices are available, for a grid of gravities and T eff that cover all the major phases of stellar evolution, and for a number of chemical compositions. Interpolations among the grid points provide the sought BC A for the model. Observationally, one needs to select proper filters that are sensitive to the desirable measurements, particular for T eff. Evolution of a 1 M star with Z = 0.001 and Y = 0.246 from the ZAMS until the tip of the RGB, displayed in the HRD and various CMDs.

The effects of interstellar extinction If the extinction (A λ ) is important, the observed flux is then Defining A A as f λ = f λ,0 e τ λ f λ,0 10 0.4A λ 10 0.4A A 10 0.4A λ f λ,0 S λ dλ, fλ,0 S λ dλ assuming certain A λ and f λ,0 (as functions of λ), we have m A = m A,0 + A A. With this extinction-corrected magnitude, m A,0, together with the distance module, we can then calculate the absolute magnitude, A (= M A ).

The extinction law, A λ /A V, is usually determined empirically, where A V is the extinction in the Johnson V band. Notice the 2175 Å bump. The ratio R V A V /E(B V ) is nearly a constant and equal to 3.1 for stars in the Galaxy. More accurately, A V can be determined in a color-color plot, e.g., U B vs. B V. For large reddening (e.g., toward the Galactic center), one needs to account for the non-constancy of A λ and hence the substantial spectral change in a given filter range.

K-correction for high-redshift objects 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. One claim for the origin of the term K correction is the correction as a Konstante (German for constant ) used by Carl Wirtz. 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).

Spectroscopic notation of the stellar chemical composition Theoretically, it is customary and convenient to specify the composition in terms of X, Y, and Z, as we have done so far. But not all element abundances can be directly measured, even spectroscopically, for a star. Low mass stars, for example, have typically too cold atmosphere to show any helium spectral lines. The stellar metal abundances, Z, are typically traced with certain spectroscopic indicators and are usually determined differentially wrt the Sun. One needs to be careful about the specific version of the solar compositions; e.g., Z=0.0194 (Anders & Grevesse 1989) while Z=0.0122 (Asplund et al. 2005). Oxygen abundance, in particular, differs by up to about one third among various versions commonly used versions. An element number abundance is defined as [Fe/H] log[n(fe)/n(h)] log[n(fe)/n(h)] Here Fe is chosen partly because its lines are prominent and easy to measure.

If the element distribution is assumed to follow the solar mixture, the conversion from Z to [Fe/H] is then given by ( ) ( ) ( ) Z Z Z [Fe/H] = log log = log + 1.61 X X X where Z X is the metal to hydrogen mass ratio. The dominant X can typically be considered approximately a constant. The above equation can then be simplified into ( ) Z [Fe/H] = log Z Typical errors of the spectroscopic determinations of [Fe/H] are of the order of at least 0.10 dex. If the assumption of a universal scaled solar metal mixture needs to be relaxed, the above equations can still be used, but the left side now refers to the ratio of the total number abundance of metal to hydrogen, [M/H].

One can often approximately group the metals into two categories: The α-elements and Fe elements. α-elements (mainly O, Ne, Mg, Si, S, Ca, and Ti) are mostly the products of core-collapsed SNe (including Type II and Ib,c). The Fe elements are mostly from Type Ia SNe. Type Ia SNe start to explore and contribute to the chemical composition of the ISM much later ( 1 Gyr) than core-collapsed SNe do. As a result, old metal-poor stars are typically α-enhanced in the halo and bulge of our Galaxy ([Fe/H]< 0.6; [α/fe] 0.3-0.4). For these α-enhanced mixtures, the approximate relationship can be generally given by [M/H] [Fe/H] + log(0.694 10 [α/fe] + 0.306). On the other hand, younger stellar generations (like our Sun) are characterized by a metal mixture with a small α/fe ratio wrt the oldest stars.

Stellar model calibration and uncertainty In addition to the chemical compositions, we need to know T eff and gravities to predict a stellar spectrum.

Stellar model calibration and uncertainty In addition to the chemical compositions, we need to know T eff and gravities to predict a stellar spectrum. If we know the angular diameter of a nearby star, θ (via interferometric observations), we can infer T eff from the bolometric flux, F bol, and the relation where d/r = 2θ 1. T eff = ( Fbol σ ) 1/4 ( ) 1/2 d R But, this expensive approach can be applied only to some of nearby stars, which cover a narrow range of the composition, but is certainly useful to test the stellar models. To determine the gravity, we need to measure the mass and radius. How could these parameters be done?

Stellar model uncertainty With the above parameters measured, one can test/calibrate the theoretical stellar evolution models against observed spectra. There are two main shortcomings of current models: Many spectral lines predicted by the models are not observed in the Sun. Or the relative strengths of many lines are not well reproduced. Existing convective model atmosphere is still treated with the MLT. More sophistical models are needed that cover all the relevant evolutionary, mass, and chemical ranges. The resultant uncertainties are typically about several % in magnitude. Another approach to judge the adequacy of the models is to compare the predictions of different models.

Review Key concepts: apparent and absolute magnitudes, photometric system, zero points, distance modulus, absolute bolometric magnitude, bolometric correction, color excess, extinction law, K-correction. 1. Please draw a CMD or HRD diagram and compare star concentrations for a flux or volume limited sample. How to construct an honest sample of stars for the study of their evolution? 2. What is the reference star used for defining an absolute bolometric magnitude? Is it the same as for a magnitude in the Johnson photometric system? 3. How can the apparent K-band magnitude of a star in the Galaxy be corrected if we know the color excess E(B V ) along the line of sight? 4. What may be the procedure to estimate the apparent magnitude of a star of a certain type in the Galactic center from a stellar model? 5. Interesting questions to muse: What are the pros and cons of the CMD and spectroscopic approaches? What about other possible approaches (e.g., SED and color-color diagrams)? 6. How may the effective surface temperature of a star, T eff, be determined from the absolute bolometric magnitude of a nearby star and its diameter?