Chapter 10: Unresolved Stellar Populations We now consider the case when individual stars are not resolved. So we need to use photometric and spectroscopic observations of integrated magnitudes, colors and spectra to infer properties of the underlying stellar populations. Simple stellar population Spectroscopy Integrated colors Absorption-feature indices CSPs Distance measurement Review
Outline Simple stellar population Spectroscopy Integrated colors Absorption-feature indices CSPs Distance measurement Review
SSP: spectroscopy The monochromatic integrated flux F λ from an SSP is the sum of the fluxes from the enclosed individual stars: F λ (t, Z ) = Mu(t) M l f λ (M, t, Z )Φ(M)dM, where f λ (M, t, Z ) is the monochromatic flux emitted by a star of mass M, metallicity Z, and age t; Φ(M)dM is the IMF; and M l and M u are the lower and upper limits to the mass of the stars. The choice of M l, as long as sufficiently small, should not affect significantly the value of F λ, whereas M u is the function of the age. Synthetic SEDs of SSPs, having Salpeter IMFs and solar abundances from Bruzual & Charlot (1993).
SSP: spectroscopy The monochromatic integrated flux F λ from an SSP is the sum of the fluxes from the enclosed individual stars: F λ (t, Z ) = Mu(t) M l f λ (M, t, Z )Φ(M)dM, where f λ (M, t, Z ) is the monochromatic flux emitted by a star of mass M, metallicity Z, and age t; Φ(M)dM is the IMF; and M l and M u are the lower and upper limits to the mass of the stars. The choice of M l, Synthetic SEDs of SSPs, having as long as sufficiently small, should Salpeter IMFs and solar abundances not affect significantly the value of F λ, from Bruzual & Charlot (1993). whereas M u is the function of the age. The comparison of an observed integrated spectrum of an SSP with F λ (t, Z ) can, in principle, constrain both t and Z.
SSP: photometry Often we just have imaging data in various photometric bands. Examples include the studies of distant galaxies and various potential SSPs in a nearby galaxy. In this case, one can obtain an integrated theoretical magnitude M A in an individual photometric band A, by convolving F λ with the corresponding filter.
SSP: photometry Often we just have imaging data in various photometric bands. Examples include the studies of distant galaxies and various potential SSPs in a nearby galaxy. In this case, one can obtain an integrated theoretical magnitude M A in an individual photometric band A, by convolving F λ with the corresponding filter. Alternatively, one may directly use the bolometric correction of individual stars, which may be provided in an SSP spectral code (e.g., Starburst99): ( Mu(t) M A (t, Z ) = 2.5log M l Relative contributions of different evolutionary phases to the total luminosity of an SSP with solar initial chemical composition and varying ages. ) 10 0.4M A(M,t,Z ) Φ(M)dM.
SSP: photometry Often we just have imaging data in various photometric bands. Examples include the studies of distant galaxies and various potential SSPs in a nearby galaxy. In this case, one can obtain an integrated theoretical magnitude M A in an individual photometric band A, by convolving F λ with the corresponding filter. Alternatively, one may directly use the bolometric correction of individual stars, which may be provided in an SSP spectral code (e.g., Starburst99): ( Mu(t) M A (t, Z ) = 2.5log M l Relative contributions of different evolutionary phases to the total luminosity of an SSP with solar initial chemical composition and varying ages. ) 10 0.4M A(M,t,Z ) Φ(M)dM The luminosity is contributed mostly by stars in the MS at ages < 3 10 8 yrs, the AGB later until 2 Gyr, and then the RGB..
Of course, the relative contributions at different evolutionary phases depend sensitively on the specific photometric bands: Relative intensity contributions of different evolutionary phases in various bands.
Of course, the relative contributions at different evolutionary phases depend sensitively on the specific photometric bands: Table: Main contributors to various bands Band young SSP old SSP UV ionizing MS post-agb & hot WD UV MS blue HB U, B MS MS K red HB AGB/RGB Relative intensity contributions of different evolutionary phases in various bands.
Of course, the relative contributions at different evolutionary phases depend sensitively on the specific photometric bands: Table: Main contributors to various bands Band young SSP old SSP UV ionizing MS post-agb & hot WD UV MS blue HB U, B MS MS K red HB AGB/RGB An SSP generally fades and becomes redder with increasing age due to the decrease of the MS extension and He-burning luminosity. Relative intensity contributions of different evolutionary phases in various bands.
Of course, the relative contributions at different evolutionary phases depend sensitively on the specific photometric bands: Table: Main contributors to various bands Band young SSP old SSP UV ionizing MS post-agb & hot WD UV MS blue HB U, B MS MS K red HB AGB/RGB Relative intensity contributions of different evolutionary phases in various bands. An SSP generally fades and becomes redder with increasing age due to the decrease of the MS extension and He-burning luminosity. These general trends are briefly interrupted, especially when the K band is involved, by the appearance of a large number of red He-burning stars at several 10 7 yrs and the onset of the AGB at 10 8 yrs.
SSP: colors Time evolution of selected integrated colors of SSPs with various metallicities. Optical color evolution tends to be complex at young age (< 1 Gyr; different ages may be inferred for the same color!) and becomes monotonic later. A color difference could be caused by a difference either in age or in metallicity the age-metallicity degeneracy; the slope of AGB and/or RGB depends strongly on metallicity. This degeneracy may be minimized by including near-ir bands in constructing the color indices (e.g., J-K is sensitive to the metallicity-dependent color of HB, AGB, and/or RGB, whereas B-K to the ratio of MS to AGB/RGB stars).
SSP: colors Time evolution of selected integrated colors of SSPs with various metallicities. Optical color evolution tends to be complex at young age (< 1 Gyr; different ages may be inferred for the same color!) and becomes monotonic later. A color difference could be caused by a difference either in age or in metallicity the age-metallicity degeneracy; the slope of AGB and/or RGB depends strongly on metallicity. This degeneracy may be minimized by including near-ir bands in constructing the color indices (e.g., J-K is sensitive to the metallicity-dependent color of HB, AGB, and/or RGB, whereas B-K to the ratio of MS to AGB/RGB stars). But this still does not help much for a young SSP (for which the color of AGB and/or RGB also depends on the age). A color-color plot can be helpful.
Various simulated realizations (squares) of the integrated colors for two SSPs of 10 6 M, [Fe/H]=-1.27 and ages of 0.6 and 10 Gyr, respectively. The spread of the realizations is primarily due to the Poisson uncertainty of the number of HB, AGB, and/or RGB stars. The dashed lines mark the ages (0.6, 1, 3, 6, 10, and 14 Gyr), while the solid lines are for [Fe/H](-1.79, -1.27, -0.66, and 0.06).
There are also other potential uncertainties: Various simulated realizations (squares) of the integrated colors for two SSPs of 10 6 M, [Fe/H]=-1.27 and ages of 0.6 and 10 Gyr, respectively. The spread of the realizations is primarily due to the Poisson uncertainty of the number of HB, AGB, and/or RGB stars. The dashed lines mark the ages (0.6, 1, 3, 6, 10, and 14 Gyr), while the solid lines are for [Fe/H](-1.79, -1.27, -0.66, and 0.06). For an SSP with mass < 10 6 M (e.g., a relatively low-mass globular cluster), the color may be subject to large statistical fluctuations caused by the small number of upper RGB and AGB stars. The color may be affected by the possible presence of blue HB stars not accounted for in the theoretical calibration (e.g., due to the binary evolutionary effect, depending on dynamic processes in a globular cluster or in the central region of a bulge).
There are also other potential uncertainties: Various simulated realizations (squares) of the integrated colors for two SSPs of 10 6 M, [Fe/H]=-1.27 and ages of 0.6 and 10 Gyr, respectively. The spread of the realizations is primarily due to the Poisson uncertainty of the number of HB, AGB, and/or RGB stars. The dashed lines mark the ages (0.6, 1, 3, 6, 10, and 14 Gyr), while the solid lines are for [Fe/H](-1.79, -1.27, -0.66, and 0.06). For an SSP with mass < 10 6 M (e.g., a relatively low-mass globular cluster), the color may be subject to large statistical fluctuations caused by the small number of upper RGB and AGB stars. The color may be affected by the possible presence of blue HB stars not accounted for in the theoretical calibration (e.g., due to the binary evolutionary effect, depending on dynamic processes in a globular cluster or in the central region of a bulge). What might be the better way to break the age-metallicity degeneracy?
Absorption-feature indices The idea is to measure individual absorption features that are sensitive to the age or metallicity of the underlying stellar population. These measurements are based on images with a set of narrow-band filters of a few Å wide each, on and off a feature.
Absorption-feature indices The idea is to measure individual absorption features that are sensitive to the age or metallicity of the underlying stellar population. These measurements are based on images with a set of narrow-band filters of a few Å wide each, on and off a feature. This technique is particularly useful to map out a large number of SSPs in a single set of images.
Absorption-feature indices The idea is to measure individual absorption features that are sensitive to the age or metallicity of the underlying stellar population. These measurements are based on images with a set of narrow-band filters of a few Å wide each, on and off a feature. This technique is particularly useful to map out a large number of SSPs in a single set of images. Such an individual absorption feature may represent a single atomic line or a molecular band. For a narrow feature labeled as i, the index (or EW), usually expressed in Å, is defined as W i = λ2 λ 1 ( 1 F ) I,λ dλ (1) F C,λ For a broad molecular band, the index, measured in magnitude I mag, is defined as [ ( ) ] 1 λ2 F I,λ I mag,i = 2.5log dλ λ 2 λ 1 F C,λ λ 1
A widely used set of absorption-feature indices is the so-called Lick system, which defines the wavelength intervals of the filters. Such indices are largely unaffected by interstellar extinction because of the small wavelength ranges. Many of the indices, such as Fe5270, are sensitive to the SSP metal content in cool stars, like ones in RGB and AGB, and suffer comparably minor changes due to age. Others, such as Hβ, Hγ, and Hδ indices, are mostly sensitive to the temperature of TO stars, hence the age, and largely unaffected by the chemical abundances. With proper choices of the indices, one can break the age-metallicity degeneracy. Sketch of the definition of four Balmer line indices and a representative stellar spectrum. The central passbands are shown as boxes (filled in for the case of the narrower F definition); pseudo-continuum sidebands are displayed as horizontal strokes at the average flux level. The pseudo-continuum used for the index measurement is drawn as a dashed line between the flanking sidebands.
Semi-empirical absorption indices We use the symbol w i for the indices of an individual star, as function of the surface effective temperature T eff, gravity g, and metallicity Z. W i for an SSP is a function of the age t and metallicity Z. Semi-empirical methods are traditionally used to determine W i (t, Z ). Based on observations of samples of local stars to which T eff, g and Z are known well, fitted functions of the indices w i (T eff, g, Z ) are obtained.
Semi-empirical absorption indices We use the symbol w i for the indices of an individual star, as function of the surface effective temperature T eff, gravity g, and metallicity Z. W i for an SSP is a function of the age t and metallicity Z. Semi-empirical methods are traditionally used to determine W i (t, Z ). Based on observations of samples of local stars to which T eff, g and Z are known well, fitted functions of the indices w i (T eff, g, Z ) are obtained.
Semi-empirical absorption indices We use the symbol w i for the indices of an individual star, as function of the surface effective temperature T eff, gravity g, and metallicity Z. W i for an SSP is a function of the age t and metallicity Z. Semi-empirical methods are traditionally used to determine W i (t, Z ). Based on observations of samples of local stars to which T eff, g and Z are known well, fitted functions of the indices w i (T eff, g, Z ) are obtained. The caveats of using the absorption indices include the uncertainties due to non-solar patterns of the chemical abundances and to the potential presence of hot HB stars. Theoretical calibration of the Hβ-Fe5270 diagram for the stellar age and metallicity, compared with observed data for a sample of Galactic and extragalactic globular clusters. The typical observational error bars are also shown.
w i may be approximated as ( ) fi,i w i = 1 (λ 2 λ 1 ) fc,i where f I,i and f C,i are the average on-feature and continuum fluxes for an individual star. From the measurement of w i, one can then estimate ( fi,i = 1 w ) i fc,i, (2) λ 2 λ 1
w i may be approximated as ( ) fi,i w i = 1 (λ 2 λ 1 ) fc,i where f I,i and f C,i are the average on-feature and continuum fluxes for an individual star. From the measurement of w i, one can then estimate ( fi,i = 1 w ) i fc,i, (2) λ 2 λ 1 For an SSP with an assumed IMF, one can generate stars with various masses with certain t and Z.
w i may be approximated as ( ) fi,i w i = 1 (λ 2 λ 1 ) fc,i where f I,i and f C,i are the average on-feature and continuum fluxes for an individual star. From the measurement of w i, one can then estimate ( fi,i = 1 w ) i fc,i, (2) λ 2 λ 1 For each star with a mass (j), the stellar evolution model gives T eff and g, and hence f C,i theoretically and f I,i (j) from Eq. 2. For an SSP with an assumed IMF, one can generate stars with various masses with certain t and Z.
w i may be approximated as ( ) fi,i w i = 1 (λ 2 λ 1 ) fc,i where f I,i and f C,i are the average on-feature and continuum fluxes for an individual star. From the measurement of w i, one can then estimate ( fi,i = 1 w ) i fc,i, (2) λ 2 λ 1 For each star with a mass (j), the stellar evolution model gives T eff and g, and hence f C,i theoretically and f I,i (j) from Eq. 2. We may then get approximate model indices W i (t, Z ) as ( W i (t, Z ) = 1 Σ ) j f I,i (j) (λ 2 λ 1 ) Σ j fc,i (j) where the sum is over all the stars in the SSP. For an SSP with an assumed IMF, one can generate stars with various masses with certain t and Z.
w i may be approximated as ( ) fi,i w i = 1 (λ 2 λ 1 ) fc,i where f I,i and f C,i are the average on-feature and continuum fluxes for an individual star. From the measurement of w i, one can then estimate ( fi,i = 1 w ) i fc,i, (2) λ 2 λ 1 For an SSP with an assumed IMF, one can generate stars with various masses with certain t and Z. For each star with a mass (j), the stellar evolution model gives T eff and g, and hence f C,i theoretically and f I,i (j) from Eq. 2. We may then get approximate model indices W i (t, Z ) as ( W i (t, Z ) = 1 Σ ) j f I,i (j) (λ 2 λ 1 ) Σ j fc,i (j) where the sum is over all the stars in the SSP. The resultant indices W i (t, Z ) can then be compared with the observed indices to constrain t and Z.
Outline Simple stellar population Spectroscopy Integrated colors Absorption-feature indices CSPs Distance measurement Review
Composite stellar population One can fit an observed high-resolution spectrum of a CSP (O λ ) with a linear synthesis spectrum a sum of single stellar populations with different stellar ages and metallicities, using a program like Starlight: M λ = M λ0 (Σ i x i b i,λ 10 0.4(A λ A λ0 ) ) G(v, σ) where M λ0 is the normalization, b i,λ is the spectrum of the ith SSP, and G(v, σ) denotes the convolution with the line-of-sight stellar motions which may be modeled with a Gaussian centered at a velocity v and with a dispersion σ.
Composite stellar population One can fit an observed high-resolution spectrum of a CSP (O λ ) with a linear synthesis spectrum a sum of single stellar populations with different stellar ages and metallicities, using a program like Starlight: M λ = M λ0 (Σ i x i b i,λ 10 0.4(A λ A λ0 ) ) G(v, σ) where M λ0 is the normalization, b i,λ is the spectrum of the ith SSP, and G(v, σ) denotes the convolution with the line-of-sight stellar motions which may be modeled with a Gaussian centered at a velocity v and with a dispersion σ. The weight to the synthesis contribution of the ith SSP, x i, is determined from minimizing the χ 2 defined as χ 2 = Σ λ [(O λ M λ )w λ ] 2, where w 1 λ is the error in O λ.
Composite stellar population One can fit an observed high-resolution spectrum of a CSP (O λ ) with a linear synthesis spectrum a sum of single stellar populations with different stellar ages and metallicities, using a program like Starlight: M λ = M λ0 (Σ i x i b i,λ 10 0.4(A λ A λ0 ) ) G(v, σ) where M λ0 is the normalization, b i,λ is the spectrum of the ith SSP, and G(v, σ) denotes the convolution with the line-of-sight stellar motions which may be modeled with a Gaussian centered at a velocity v and with a dispersion σ. The weight to the synthesis contribution of the ith SSP, x i, is determined from minimizing the χ 2 defined as χ 2 = Σ λ [(O λ M λ )w λ ] 2, where w 1 λ is the error in O λ. With the best fitting synthesized spectrum, one can obtain the flux-weighted age and metallicity as logt = Σ i x i logt i ; logz = Σ i x i logz i.
Composite stellar population One can fit an observed high-resolution spectrum of a CSP (O λ ) with a linear synthesis spectrum a sum of single stellar populations with different stellar ages and metallicities, using a program like Starlight: M λ = M λ0 (Σ i x i b i,λ 10 0.4(A λ A λ0 ) ) G(v, σ) where M λ0 is the normalization, b i,λ is the spectrum of the ith SSP, and G(v, σ) denotes the convolution with the line-of-sight stellar motions which may be modeled with a Gaussian centered at a velocity v and with a dispersion σ. The weight to the synthesis contribution of the ith SSP, x i, is determined from minimizing the χ 2 defined as χ 2 = Σ λ [(O λ M λ )w λ ] 2, where w 1 λ is the error in O λ. With the best fitting synthesized spectrum, one can obtain the flux-weighted age and metallicity as logt = Σ i x i logt i ; logz = Σ i x i logz i. The estimated age is likely biased towards young or metal-poorer sub-population, while the metallicity estimate towards older or more metal-rich population.
Outline Simple stellar population Spectroscopy Integrated colors Absorption-feature indices CSPs Distance measurement Review
Distance measurement What are the ways to measure the distance to an unresolved stellar population (e.g., a nearby elliptical galaxy)?
Distance measurement What are the ways to measure the distance to an unresolved stellar population (e.g., a nearby elliptical galaxy)? One effective way is the so-called Surface Brightness Fluctuation (SBF) technique, based on photometric observations. With present capabilities, this technique is applicable to distances in the range of 10-100 Mpc.
Distance measurement What are the ways to measure the distance to an unresolved stellar population (e.g., a nearby elliptical galaxy)? One effective way is the so-called Surface Brightness Fluctuation (SBF) technique, based on photometric observations. With present capabilities, this technique is applicable to distances in the range of 10-100 Mpc. Consider a target population with stars in a range of intrinsic luminosities L i (or fluxes f i = L i /4πD 2 ) and with an average stellar number density n i per unit area across the observed face at the distance D. For an image of this population with an angular resolution δφ, each resolution element will contain an average number of unresolved stars with L i : Ni = n i (δφd) 2. The average total flux F received from the stars in the element is n F = Σ i Ni f i = Σ i L i δφ 2 i 4π with a variance σf 2 = Σ i N i fi 2 n = Σ i L 2 i δφ2 i. (4πD) 2
Therefore, σ2 F F = L 4πD 2, with L Σ i n i L 2 i Σ i n i L i. From theory, one can determine L (the so-called SBF luminosity) for an SSP or CSP, provided its age and metallicity or the SFH are known, and an IMF is adopted. Once L is determined, the observed σ2 F F distance D. immediately provides the The method clearly works best when observing the external parts of galaxies, where the number of stars is low and Poisson fluctuation are larger. The SBF in the K band is very weakly dependent of age, with variations of the order of 0.2 mag for a change of age form 1 to 15 Gyr, and so is good for determining the distance to a old CSP (a spheroid).
Therefore, σ2 F F = L 4πD 2, with L Σ i n i L 2 i Σ i n i L i. From theory, one can determine L (the so-called SBF luminosity) for an SSP or CSP, provided its age and metallicity or the SFH are known, and an IMF is adopted. Once L is determined, the observed σ2 F F distance D. immediately provides the The method clearly works best when observing the external parts of galaxies, where the number of stars is low and Poisson fluctuation are larger. The SBF in the K band is very weakly dependent of age, with variations of the order of 0.2 mag for a change of age form 1 to 15 Gyr, and so is good for determining the distance to a old CSP (a spheroid). The above method uses only the first and second momentum of the intensity distribution of the stellar light. In principle, one can directly model the distribution, which would be less biased by the presence of any outliers.
Outline Simple stellar population Spectroscopy Integrated colors Absorption-feature indices CSPs Distance measurement Review
Review Key concepts: age-metallicity degeneracy, absorption-feature indices 1. For an unresolved SSP, how is the stellar type that dominates the K-band luminosity changes with age? How different is when the optical luminosity is considered? 2. What are the stellar evolutionary stages that cause the sudden increases of the K-band brightness of a simple stellar population at its age of several 10 7 yrs and of several 10 8 yrs? 3. Why may the inclusion of a near-ir color (e.g. J-K) help to break the age-metallicity degeneracy for a relatively old SSP? 4. What are the advantages in using absorption-feature indices than broadband colors in breaking the age-metallicity degeneracy? 5. What kinds of absorption features tend to be good age or metallicity indicators? 6. Why is it difficult to determine the SFH of a unresolved CSP?