Eric Joseph Bubar Stellar Atmosphere/Interiors Portfolio CHAPTER : CURVES OF GROWTH Short/Simple Definitions: Curve of Growth: Plot of equivalent widths versus number of absorbing atoms that created that line. Equivalent Width: Width of a rectangle of height equal to a continuum flux and total area equal to that under a spectral line. More on equivalent Widths: FIGURE 1: EQUIVALENT WIDTH Fc = Continuum Intensity (Flux) Level Fλ = Intensity (Flux) in a spectral line profile Source: http://cosmos.swin.edu.au/entries/equivalentwidth/equivalentwidth.html?e=1 Consider a spectral line with an area A. Draw a rectangle with a height equal to the continuum level of the flux, Fc. The equivalent width is the width necessary to give the rectangle an area equal to the spectral line. A = Spectral line profile area = Fc x b To get a working form for the spectral line equivalent width, we can normalize it to the continuum. The spectral line depth, normalized to the continuum is:
Line Depth = Fc Fλ Fc The equivalent width, or area A under the line, is found by adding all of the flux from the line depth across a spectral line (equal to the integrated flux of the spectral line). EW= F C F λ F C A note on line profile shapes. There are two basic shapes of line profiles that we will be considering in our curve of growth formulation. They are: 1) The doppler profile is also known as thermal broadening. The doppler profile is gaussian shaped and dominates formation of line cores. 2) The lorentzian profile also known as pressure broadening. This lorentzian profile dominates the wings of lines. These two line profiles can be combined into a single shape known as a Voigt profile (see figure below). These so called voigt profiles can be used to accurately model many spectral features, although the Gaussian shape typically gives the desired accuracy in a spectral line analysis. This is because the lines of interest in spectral analysis are typically weak or saturated lines which have very little contribution from the wing (or lorentzian) portion of a line. FIGURE 2: LINE PROFILES Source:http://www.wavemetrics.com/products/igorpro/dataanalysis/peakanalysis/multipeak fittingpix/dittypes.png
Curves of Growth: A full scale mathematical treatment could be performed on this Voigt function to derive the full curve of growth (for a full treatment see Mihalas, Stellar Atmospheres). The treatment here will derive only the first portion of the curve of growth and will involve a simple absorption model where a slab of cool absorbing material (F ν ) rests in front of a slab of continuum flux material (Fc). F ν =F C e The above equation basically indicates that the absorbed flux is the continuum flux times some exponential decay which depends on the optical depth, where optical depth is approximated to be: Where, L = ν L ρ dx= Nα dx L =A N N αdx E A= N E., is the number of atoms of some element E divided by the number of atoms of hydrogen. This shows us that: A. If we take a small optical depth (aka: a weak line), <<1 F ν =F C e =F C 1 = F C F ν F C A EW A We have now shown that the number of absorbing atoms is proportional to the equivalent width, by using the line profile depth equation. This is the first segment of the curve of growth. By similar means, the other two segments of the curve can be derived (Mihalas, Stellar Atmospheres) giving the full curve of growth in 3 segments:
For Weak Lines: For Moderate Strength Lines: For Strong Lines: 1) EW A 2) EW ln A 3) EW A FIGURE 3: THEORETICAL CURVE OF GROWTH 3) Strong Line 2) Moderate/Saturation 1) Weak Line Source: Gray, Stellar Photospheres The preceding is a sample, theoretical curve of growth. For a star, a given element should have a single abundance. However, various features of lines make some stronger than others, allowing determination of a curve of growth. Consider a weak line as before. If we establish our number abundances using the Boltzman equation: N j g n N = u T 1 θχ n Where Nj is the number atoms in the jth level, N is the total number of absorbing atoms, g n is the statistical weight of atoms in the nth energy level, u(t) is the partition function, θ is the standard 54/Teff and χ n is the excitation potential. We can rewrite our abundance N as: where A is defined as above. N=A N j N E g n u T 1 θχ n
We can take this weak line approximation to represent the equivalent width of our line as: w= F C F λ F C = B κ υ l υ dυ= B Nαdυ κ υ ρ where B is a constant, κ is the total absorption coefficient (in units of area per absorber), ρ is the mass density, and l is the line absorption coefficient (which is equal to the number of absorbing atoms, N, multiplied by the atomic absorption coefficient, α which has units of area per absorber). The result, when we perform the integral over the absorption coefficient and when we include f values (where f values account for quantum mechanical effects in the population of levels) is: w=c πe 2 λ 2 f N mc c κ ν w=c e mc π 2 λ 2 c where the C is now equal to the B/ ρ. f κ ν A N j N E g n u T 1 θχ n Now we divide both sides by lamda to normalize Doppler phenomena and take the logarithm of both sides to find: log w =log λ D log A+log λ + log g n f θχ n log κ ν where D= Cπe 2 mc 2 N j N E u T This makes it evident that the equivalent width, w, depends on the abundance (A: the number of atoms of element E compared to number of atoms of hydrogen), the wavelength of the line, the gf values (a quantum mechanical quantity that accounts for the populations of different states for a given atom), excitation potentials, temperature (which is contained in the θ term) and the total absorption coefficient. The program that we utilize (MOOG) for abundance analysis performs this type of curve of growth analysis (ABFIND routine). The program accepts the wavelengths, element name, and log(gf) values to calculate abundances in this curve of growth. To determine final abundances also requires use of the SAHA and Boltzmann equations, to convert a particular ionization to a total abundance.
A Note on Microturbulence: Notice the flat (saturation) part of the curve of growth. Small scale variations in line profiles (of order of the size of an atom) can result in spreading absorption over a wider spectral band. These wider absorptions delay the onset of the saturation portion of the curve of growth. This results in a shift in the y axis of the curve of growth. The following figure depicts this shift. The y axis of this figure is the log of the equivalent width while the x axis gives the log of the number of absorbing atoms per area element. Notice how a higher value for microturbulence (b=4, 12 and 4 km/s respectively), causes a shift upwards in the curves (i.e. it delays the onset of the flat saturated portion of the curve of growth. Adapted from: http://www.physics.sfsu.edu/~lea/courses/grad/cog.pdf