• Is is sometimes convenient to describe the chemical content of a star by fractional weight in different elements.

• This is usually done by:
  • X = weight of hydrogen

  • Y = weight of helium

  • Z = weight of everything else

  For the Sun, ( X, Y, Z ) = (0.70, 0.28, 0.02), though there is some uncertainty in the solar helium abundance since the sun is too cool to express He lines.

• If we assume that the distribution of the elements summarized in "Z " is universal, then we can write:
  [Fe/H] = log[ Z / X ]_* - log[ Z / X ]_Sun
  Accounting for the last term with the actual solar values, an approximate conversion between Z and [Fe/H] is (Bertelli et al. 1994, A&AS, 106, 275):
  • log(Z ) = 0.977 [Fe/H] - 1.699

  • [Fe/H] = 1.024 log(Z ) + 1.739

• If one relaxes the assumption of a universally scaled solar heavy element distribution, then the correspondence between [Fe/H] and Z depends obviously on the relative distributions of elements.
  In this case then it is more appropriate to write
  [m/H] = log[ Z / X ]_* - log[ Z / X ]_Sun

• As we shall learn, a very important modulation on the abudnance patterns derived from the relative proportions of the enrichment of the interstellar medium by supernovae of Type II versus Type Ia.
  • A prominent difference is that Type II supernova inject back into the ISM matter that is enriched with &alpha-elements, with minor amounts of iron.
    On the other hand, Type Ia SNe inject mainly Fe, with minor &alpha-element contributions.

  • Type II SNe come from massive main sequence stars that have short lives and thus happen early in the life of a stellar population, whereas Type Ia SNe come from mass exchange binaries, and take longer (> 1 Gyr) to show up and start pumping out iron.
    so that they dominate the chemical enrichment of
  • Thus, the earliest stars formed in a stellar system tend to be rich in alpha elements.
    For example, the stars in the Milky Way halo show a significant enhancement of &alpha elements compared to the Sun, with [&alpha/Fe] ~ 0.3-0.4, while more recently formed stars look more like the Sun, with [&alpha/Fe] ~ 0.0.

  So, an interesting way to account for these variances in &alpha/Fe mixtures is to use an equation that accounts for them (Salaris & Cassisi, Evolution of Stars and Stellar Populations:
  [m/H] ~ [Fe/H] + log(0.694f_&alpha + 0.306)
  where f_&alpha = 10^[&alpha/Fe].
  So, for example, for [&alpha/Fe] = 0.3, this gives:
  [m/H] ~ [Fe/H] + 0.2
  (which shows that the Fe is off as a proxy of the total metallicity content by a few 0.1 dex in this case, because of the &alpha-enhancement relative to the Sun).

• The following table and associated figure gives the abundances of the elements in the Sun, given in proportion to H. Note that it is common to give values in a commonly used system of log(A) + 12 (i.e., number of atoms per 10¹² H atoms) in order to keep most of the values positive:
  

  From The Observation and Analysis of Stellar Photospheres by David F. Gray.

  A few things to notice:
  • The general decline in abundances with atomic number is obvious.

  • Superposed on the general trend is an overall "odd-even effect", where elements that are even multiples of a He nucleus are enhanced (probably as a result of synthesis by alpha particle capture).

  • The huge drop in abundance for the light nuclei Li, Be, and B arises primarily from the instability of nuclei of mass 5, making the early creation of these elements in the universe rare, as well as the easy destruction of these elements in stars, particularly when convection drops the atmospheric nuclei to the hotter interior.

  • Elements around iron (V, Cr, Mn, Fe, Co, Ni) show enhanced abundance, forming an "iron peak". These elements have the highest binding energy, which is the energy required to remove a nucleon or the energy released when a nucleon is added.
    

    From http://rst.gsfc.nasa.gov/Sect20/A7.html.

    Indeed, elements lower than Fe release energy as fusion adds nucleons, whereas elements past iron require energy to do this.
    Thus, we produce an abundance of iron peak elements through normal nucleosynthesis.

• The following plots compare the abundances in the Sun to those in the Earth.
  

  From http://www.fas.org/irp/imint/docs/rst/Sect20/A7.html.

  The Earth abundance plot is actually in terms of mass fraction, and shows that oxygen and iron are the most abundant elements by mass for Earth (including the atmosphere).

• We will return to the topic of nucleosynthesis and chemical evolution later.

• For example, the following plot shows the variation in the MS, SGB, and lower RGB for stellar populations of the same age but different metallicities:
  

  Isochrones for a set of stellar populations all of age 16 Gyr, but with different [Fe/H] = -2.03, -1.78, -1.48, -1.26. From Bolte (1993, in Galaxy Evolution: The Milky Way Perspective, PASP. Conf. Ser. 49, ed. S. Majewski).

• A nice, lucid discussion of the UBV case is given in Mihalas & Binney, Section 3.6.
• Another version of this is given in the original paper on UV-excess by Wildey, Burbidge, Sandage & Burbidge (1962, ApJ, 135, 94).

Line blocking as a function of stellar type in top three panels. Bottom two panels shows change in the LBC for two stars of same spectral type but different metallicity. From Mihalas & Binney (1980).

• If we start with stars of no metallicity and therefore no metal lines, and then gradually increase increase abundances to solar type, the star should become redder in B-V, but even redder in U-B, simply on the basis of the line blanketing effect.

• But this is not the entire story. We also have several competing effects.
  One of these compensating effects that comes into play to interpret the relative color changes is backwarming:
  • All other things being equal, the same amount of energy still has to leave the star.
  • Can only do so between lines in spectrum.
  • Thus, the continuum emission levels become elevated, and make the star (in continuum) appear as if it is a star of hotter BB temperature.

• Backwarming effect mitigates and competes with line blanketing effect.
  • U band: (line blanketing) >> (backwarming)
  • B band: (line blanketing) >~ (backwarming)
  • V band: (line blanketing) <~ (backwarming)

  Thus, in the (B-V,U-B) two-color diagram, the line blanketing vectors are nearly vertical -- showing that (U-B) is more strongly changed than (B-V) for given change in metals.


Blanketing vectors for stars of different temperatures from line-free colors (shown by crosses) to solar metallicity locus (shown by the curve). From Wildey et al. (1962).

• We say that metal-poor stars show an "Ultraviolet Excess" or ``UVX'' (comparatively more UV flux) than metal-rich stars - which have a lot more UV absorption lines
• Quantifying "Ultraviolet Excess":
• Traditionally adopt the locus of stars in the Hyades open star cluster to represent ``solar metallicity" .
• Note, however, actual Hyades [Fe/H] ~ +0.25 (i.e., a bit more metal rich than the Sun...)!

• The UVX, written as:
  
  is the difference of a star's (U-B) color from that of a Hyades star of the same (B-V) color.
  
  Modified from Wildey et al. (1962) and Mihalas & Binney (1980).

• Wildey et al. (1962; Table 4) tabulate blanketing vectors (U-B), (B-V), and corresponding (U-B) for each B-V.



• Connection between (U-B) and [Fe/H]...
  • ... unfortunately, the connection of the size of (U-B) to the size of [Fe/H] is a function of B-V!!
  

  Isometallicity lines from Sandage (1969). Note that the ``guillotine'' represents the limit of UVX for stars with no metals.

  • To bring some uniformity to the translation of (U-B) to [Fe/H], Sandage (1969, ApJ, 158, 1115 ) invented a normalization of (U-B) for different B-V.
    The normalization is such that we correct all (U-B) to the (U-B) of a star with the same metallicity but (B-V)=0.6.
    This table from Sandage (1969) gives the corrections for interpolation to the ``standard" (U-B)_0.6.
  See also Box 5.4 in Binney & Merrifield for a description of this correction.

• Armed with normalized (U-B)_0.6, we may now estimate the [Fe/H] for any star.
  Several schemes have been given:
  • Mihalas & Binney (1980, eqn. 3-42) give the simple relation ``for modest values of (U-B)_0.6":
    [Fe/H] = [Fe/H]_Hyades - 5(U-B)_0.6
    

  • Laird et al. (1988, AJ, 95, 1843) gives (see Binney & Merrifield Box 5.4):
    (U-B)_0.6 ~ -0.0776 + (0.01191-0.05353[Fe/H])^1/2

• For example: figures from Eggen & Sandage (1962, ApJ, 136, 735) showing the color-absolute magnitude diagram for stars sorted by their ultraviolet excess showing a correlation of (U-B) with subluminosity compared to, say, the Hyades.
  

  Figures from Eggen & Sandage (1962).

• Note that the above figures are very hard to make (properly) because requires having trigonometric parallaxes for these subdwarfs.
  • Because subdwarfs are associated with Population II there are very few of them near the Sun, where reliable parallaxes can be obtained!
  • This is a continuing problem in astronomy, and the situation is only marginally better after Hipparcos, which contributed only a few more parallaxes for subdwarfs.
  

  Figure from Reid (1999, ARAA, p.203) showing the current state of the art after Hipparcos obtained parallaxes for a few more subdwarfs. A concerted effort by the US Naval Observatory to do CCD parallaxes of subdwarfs has also contributed to this endeavor.

  • Figures like those above are critical components to the distance scale problem, because they are used to get distances to globular clusters from main sequence fitting.
  • There is a serious statistical bias (called the Lutz-Kelker bias) inherent to creating a figure like the above out of the few subdwarfs we have managed to be able to get parallaxes for (a problem we will return to in the distance scale section of the course).

  
  
• What is the origin of the subdwarf effect?
  • Recall that in the V band, backwarming dominates line blanketing as we add metallicity to a star.
  • Thus, as we make a star more metal-poor (i.e. as we deblanket it), by the backwarming effect one would think that not only does the star become bluer in (B-V) [and (U-B)], but it also should get fainter in V:
  

  Color-magnitude vector for subdwarfness by Wildey et al. (1962).

  So, from the combination of stellar atmospheric blanketing/backwarming effects one would think that the main sequence locus, for increasingly metal poor populations moves left (blueward) and slightly down (fainter) in the CMD, something like this:
  
  • But there is also a competing stellar interiors effect that as one decreases the metallicity of a star, the opacity is reduced. This makes the overall energy output at the surface greater and the luminosity of the star increases.
    

  • Thus, ironically, "subdwarfs" are actually more luminous than their higher metallicity counterparts of same mass, and so "subdwarfs" is therefore a misnomer!
    • The truth of the matter is that while the subdwarf "sequence" lies below the high metallicity main sequence, subdwarf and solar metallicity stars at the same color have different mass, and subdwarfs of the same mass are bluer and brighter than their counterparts.

  • Similar atmosphere/interior effects manifest themselves along other stellar loci in the CMD, for example, the reason why red giant branches vary for clusters of different abundance.
  

  Isochrones for a set of stellar populations all of age 16 Gyr, but with different [Fe/H] = -2.03, -1.78, -1.48, -1.26. From Bolte (1993, in Galaxy Evolution: The Milky Way Perspective, PASP. Conf. Ser. 49, ed. S. Majewski).

• Connection between (U-B)_0.6 and M_V
  • One can use the Wildey et al. (or other) calibrations of blanketing effects and make the direct connection between ultraviolet excess and subdwarfness.
  • One calibration of the relation by Laird et al. (1988, AJ, 95, 1843) gives (see Binney & Merrifield, P. 277):
    M_V = 0.862 [-0.6888 (U-B)_0.6 + 53.14 (U-B)_0.6² - 97.004 (U-B)_0.6³]
  • Approximate relation in terms of [Fe/H], by Reid & Majewski (1993, ApJ, 409, 635):
    M_V ~ -0.87 [Fe/H]