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

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

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. 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. 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). 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 Spectroscopic notation of the stellar chemical composition I Theoretically, it is customary and convenient to specify the composition in terms of X, Y , and Z, as we have done so far. I 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. I The stellar metal abundances, Z, are typically traced with certain spectroscopic indicators and are usually determined differentially wrt the Sun. I 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. I 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 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].

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 ∗

Z where 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 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 ∗

Z where 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]. 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.

One can often approximately group the metals into two categories: I 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). I 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. One can often approximately group the metals into two categories: I 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). I 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. 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 If we know the angular diameter of a nearby star, θ (via interferometric observations), we can infer Teff from the bolometric ﬂux, Fbol , and the relation

F 1/4 d 1/2 T = bol eff σ R

where d/R = 2θ−1. 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 calibration and uncertainty

In addition to the chemical compositions, we need to know Teff and gravities to predict a stellar spectrum. 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 calibration and uncertainty

In addition to the chemical compositions, we need to know Teff and gravities to predict a stellar spectrum. If we know the angular diameter of a nearby star, θ (via interferometric observations), we can infer Teff from the bolometric ﬂux, Fbol , and the relation

F 1/4 d 1/2 T = bol eff σ R

where d/R = 2θ−1. Stellar model calibration and uncertainty

F 1/4 d 1/2 T = bol eff σ R

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: I Many spectral lines predicted by the models are not observed in the Sun. Or the relative strengths of many lines are not well reproduced. I 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. 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 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, Teff , be determined from the absolute bolometric magnitude of a nearby star and its diameter?