Download Unresolved Stellar Populations

Stellar Objects: Unresolved Stellar Populations
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Unresolved Stellar Populations
Update date: December 14, 2010
Here we consider the case when individual stars are not resolved. In such a case, we may use
photometric and spectroscopic observations of integrated magnitudes, colors and spectra to
infer properties of the underlying stellar populations.
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SSP
The monochromatic integrated flux Fλ from an SSP is the sum of the fluxes from the enclosed
individual stars:
Z
Mu
Fλ (t, Z) =
fλ (M, t, Z)Φ(M )dM,
(1)
Ml
where fλ (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 (assumed to be the Salpeter IMF here). while Ml and Mu are the
lower and upper limit to the mass of the stars. The choice of Ml , as long as sufficiently small,
should not affect significantly the value of Fλ , whereas Mu is not an independent parameter
and is the function of the age.
Fig. 11.1 in the text shows the evolution of contributions from stars at different evolutionary
phases to the total bolometric luminosity of an SSP. The luminosity is contributed mostly
by stars in the MS at ages < 3 × 108 yrs, the AGB later until ∼ 2 Gyr, and then the RGB.
If one can have an observed integrated spectrum of an SSP, its comparison with Fλ (t, Z)
may allow for constraints on both t and z. But 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.
One may obtain the integrated magnitude MA in an individual photometric band A by convolving Fλ with the corresponding filter. Alternatively, one may use the bolometric correction
of individual stars, which may be provided in an SSP spectral code (e.g., Starburst99):
Z Mu
−0.4MA (M,t,Z)
MA (t, Z) = −2.5log
10
Φ(M )dM .
(2)
Ml
Of course, the relative contributions at different evolutionary phases depend sensitively on
the specific photometric bands (Fig. 11.2):
Note that the color of HB stars depend on Z.
1.1
Integrated colors
How could we use magnitudes and colors to infer the properties of SSPs?
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Table 1: Main contributors to various bands
Band
young SSP old SSP
ionizing UV MS
post-AGB and hot WD
UV
MS
blue HB
U, B
MS
MS
K
red HB
AGB/RGB
Let us first see a few general trends:
• 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 ×107 yrs and the onset of the AGB at ×108 yrs
(Fig. 11.3).
• 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. But the slope depends
strongly on metallicity; a color difference could be caused by a difference either in age
or in metallicity — the age-metallicity degeneracy (Fig. 11.4).
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 AGB and/or RGB color). But
this does not help much for a young SSP (for which the color of AGB and/or RGB also
depends on the age). There are also other potential uncertainties:
• For an SSP with mass < 106 M , the color may be subject to large statistical fluctuations caused by the small number of upper RGB and AGB stars.
• The color may 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?
1.2
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
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particular 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, the index, usually expressed in Å, is defined as
Z λ2 FI,λ
1−
Iλ =
dλ
FC,λ
λ1
(3)
where FC,λ represents the straight line connecting the mid-points of the flanking (or sideband)
pseudo-continuum fluxes, and FI,λ the observed flux over the the on-feature wavelength range
from λ1 to λ2 . For a broad molecular band, the index, measured in magnitude Imag is defined
as
Z λ2
FI,λ
1
dλ
(4)
Imag = −2.5log
λ2 − λ1
λ1 FC,λ
A widely used set of absorption-feature indices is the so-called Lick system, which defines
the wavelength intervals of the filters. Due to the small ranges of the wavelengths involved,
such indices are largely unaffected by interstellar extinction.
In general, a set of such defined indices will be used. We use the symbol wi for the indices
of an individual star, as function of the surface effective temperature Tef f , gravity g, and
metallicity Z, and Wi for the indices of an SSP, as function of the age t and metallicity Z.
It has been traditional to use semi-empirical methods to determine Wi (t, Z). Based on
observations of samples of local stars to which Tef f , g and Z are known well, fitted functions
of the indices wi (Tef f , g, Z) are obtained.
The definition for wi = Iλ (Eq. 3), for example, in a single star case may approximated as
f¯I,i
wi = 1 − ¯
(λ2 − λ1 )
(5)
fC,i
where f¯I,i and f¯C,i are the average on-feature and continuum fluxes for an individual star.
From this equation, we obtain
wi
¯
f¯C,i
(6)
fI,i = 1 −
λ2 − λ1
To obtain the model indices Wi (t, Z) of an SSP, we may approximate Eq. 3 as
Σj f¯I,i (j)
Wi (t, Z) = 1 −
(λ2 − λ1 )
Σj f¯C,i (j)
(7)
where the sum is over all the stars in the SSP. For a star of certain mass, as implied by j,
we can first infer f¯C,i as well as Tef f and g from the stellar evolutionary model for any t and
Z and then f¯I,i from Eq. 6. The resultant Wi (t, Z) can then be compared with the observed
indices to constrain t and Z.
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With proper choices of the indices, one can break the age-metallicity degeneracy. Many
of the indices such Fe5270 are sensitive to the SSP metal content in cool stars, like 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.
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.
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CSPs
For a population of unknown SFH, one may get some sort of mean age and mean metallicity[1].
The method is based on fitting an observed high-resolution spectrum of a CSP (Oλ ) with a
linear synthesis spectrum of a set of single stellar populations with different stellar ages and
metallicities, using a program like Starlight[2]. The synthesis spectrum is expressed as:
Mλ = Mλ0 (Σi xi bi,λ rλ ) ⊗ G(v, σ)
(8)
where Mλ0 is the normalization of the spectrum at a chosen wavelength λ0 , bi,λ is the spectrum of the ith SSP, and rλ ≡ 10−0.4(Aλ −Aλ0 ) represents the reddening term. G(v, σ) is the
line-of-sight stellar motions which may be modeled with a Gaussian centered at a velocity v
and with a dispersion σ. ⊗ denotes the convolution operator. The weight to the systhesis
contribution of the ith SSP, xi , is determined from minimizing the χ2 defined as:
χ2 = Σλ [(Oλ − Mλ )wλ ]2 ,
(9)
where wλ−1 is the error in Oλ . After finding the best fitting synthesized spectrum, one may
obtain the flux-weighted age and metallicity as:
hlogt∗ i = Σi xi logti
(10)
hlogZ∗ i = Σi xi logZi
(11)
and
where ti and Zi denote the age and metallicity of the ith SSP. The estimated age is likely
biased towards young or metal-poorer sub-population, because it tends to dominate the light,
while the metallicity estimate towards older or more metal-rich population, which tends to
show strong metal lines.
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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)
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technique, based on photometric observations. With present capabilities, this technique is
applicable to distances in the range of 10-100 Mpc.
We consider a target population with stars in a range of intrinsic luminosities Li (or fluxes
fi = Li /4πD2 ) and with an average stellar number density ni 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 Li :
N̄i = ni (δφD)2 . The average total flux F̄ received from the stars from the element will be
F̄ = Σi N̄i fi = Σi
with a variance
σF2
=
Σi N̄i fi2
ni Li δφ2
4π
ni L2i δφ2
= Σi
(4πD)2
(12)
(13)
Therefore, we have
L̄
σF2
,
=
4πD2
F̄
(14)
Σi ni L2i
.
Σi ni Li
(15)
with
L̄ ≡
From theory, one can determine L̄ (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 L̄ is determined,
σ2
the observed F̄F immediately provides the distance D.
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).
References
[1] Bica, E. 1988, A&A, 195, 76
[2] Cid Fernandes, R., Sodré, L., Schmitt, H. R., & Leão, J. R. S. 2001, MNRAS, 325, 60
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Review
Key concepts: age-metallicity degeneracy, absorption-feature indices
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 ∼ 107 yrs and of several ∼ 108
yrs?
Stellar Objects: Unresolved Stellar Populations
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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?
What are the advantages in using absorption-feature indices than broadband colors in breaking the age-metallicity degeneracy?
What kinds of absorption features tend to be good age or metallicity indicators?
Why is it difficult to determine the SFH of a unresolved CSP?