Equivalent width measurement

We present here the results of abundance measurements of iron, calcium and nickel in four open clusters, from UVES spectra of solar type stars. A code developed by one of the authors (Francois) performs line recognization, equivalent width measurements and finally obtains the abundances by means of OSMARCS LTE model atmosphere [4]. Temperature, gravity and microturbulence velocity have to be input to the program. This is made in an automatic way for a grid of values chosen on photometric basis. Those that best reproduce excitation and ionization equilibria are selected and used, namely when no significant trend of the computed abundances is seen, neither versus the excitation potential of the line nor versus its equivalent width, and for which the abundances obtained with lines of different ionization stages of the same specie give equal results within the errors. This check is made with iron lines, we have in fact at least thirty Fe I lines in each star, and six Fell lines. 

V(w), Vc(o>) w w W wM W(SX) x, x Fourier transforms of voltages v(t), vc(t) entrance-slit width detector width equivalent width measured equivalent width equivalent width expressed as a series generalized independent variables and arguments of various functions, in places given specifically in cm-1... 

The Lyman lines of HI and the Lyman resonance bands of H2 fall in the latter category. The column density of H2 is then obtained (Carruthers, 1970) by comparing the equivalent widths measured in the stellar spectrum and in the laboratory spectrum, from which a column density NHl =1.3 (+0.9, -0.65) x... 

As with the other relic nuclides, the dominant uncertainties in estimating the primordial abundance of 7Li are not statistical, they are systematic. Lithium is observed in the atmospheres of cool stars (see Lambert (2001) in these lectures). It is the metal-poor, Pop II halo stars that are of direct relevance for the BBN abundance of 7Li. Uncertainties in the lithium equivalent width measurements, in the temperature scales for... 

The peak width (w) is the distance between each side of a peak measured at 0.6065 of the peak height. The peak width measured at this height is equivalent to two standard deviations (2o) of the Gaussian curve and, thus, has significance when... 

We are therefore developing a set of routines for an automatic or semiautomatic abundance analysis of stellar spectra based on equivalent widths (EW). The first product is DAOSPEC, a code developed by P. B. Stetson for automatic EW measurement (http //cadcwww.hia.nrc. ca/stetson/daospec/). The preliminary abundance analysis presented here is the first step of an iterative and automatic procedure under development at the Bologna Observatory. 

The equivalent widths were determined using gaussian fit and the atmospheric models were computed using OSMARCS code improved by [6,3]. When it was not possible to measure equivalent width, the abundance was directly determined by using spectrum synthesis. 

We now consider in somewhat more detail a simplified approach based on the curve of growth . For this, we ignore fine details of the observed line profile and use the equivalent width (EW) defined in Fig. 3.4, WA = f RdX or Wv = f Rdv, where R(AX) or R(Av) is the relative depression below the continuum at some part of the line. The curve of growth is a relationship between the equivalent width of a line and some measure of the effective number of absorbing atoms. Equivalent... 

Fig. 3.12. Simple (exponential) curve of growth for low-excitation Fe I lines with wavelengths between 4000 and 8700 A at the centre of the solar disk, with 6>ex = 1.00, b = lkms-1 (assuming Roo = 1), a = 0.02. Equivalent widths are from Moore, Minnaert and Houtgast (1966). gf -values are from furnace measurements by the Oxford group (Blackwell et al. 1986 and references therein).

The Na I D-lines have wavelengths and oscillator strengths A,i = 5896 A, /i = 1 /3, and X2 = 5889 A, f2 — 2/3. In a certain interstellar cloud, their equivalent widths are measured to be 230 mA and 370 mA respectively, with a maximum error of 30 mA in each case. Assuming a single cloud with a Gaussian velocity dispersion, use the exponential curve of growth to find preferred values of Na I column density and b, and approximate error limits for each of these two parameters. (Doublet ratio method.)... 

H. N. Russell analyzes solar spectrum with theoretical transition probabilities and eye estimates of line intensities. Notes predominance of hydrogen (also deduced independently by Bengt Stromgren from stellar structure considerations) and otherwise similarity to meteorites rather than Earth s crust. M. Minnaert et al. introduce quantitative measurements of equivalent width, interpreted by the curve of growth developed by M. Minnaert, D. H. Menzel and A. Unsold. 

Measurements of the total integrated absorption S = J P(x) dx may be made by using the method of equivalent widths. Such measurements are independent of instrumental broadening and are considered further in Sections ILF and II.G. 

When pressure broadening dominates, the situation is more complicated because the resulting Lorentzian profile contributes significant area far from the line center. A further complication in this case is that the Lorentzian half-width cannot be accurately calculated and must be measured in other experiments. If both Doppler and pressure broadening are present, however, and if the Lorentzian to Doppler half-width ratio is small, the correction necessitated by pressure broadening is small. In this situation an accurate value of the Lorentzian half-width may not be needed. Line strength in the case of combined Doppler and pressure broadening may be obtained from the equivalent width by the use of tables (Jansson and Korb, 1968). 

Ion abundances can be obtained either from the equivalent widths of the absorption lines, or from their residual intensity at line center, r. The two parameters are basically equivalent in that they are directly related to each other, although the use of is preferable because it is directly proportional to abundance in a model-independent way when the lines are unsaturated, e.g., finite spectral resolution increases the observed residual line intensity, but not the equivalent width. In the supernova the lines are all easily resolved, and the residual intensity at the bottom of the lines is more readily measured than W, hence we use ri to obtain the abundances. 

In the optical region a detailed measurement of the line profiles is usually difficult, and for absorption lines it has therefore become usual to measure an absorption line by its equivalent width Wv defined in frequency units (Wv =... 

Absorption lines are defined relative to the continuum. In the case of a resolved line, one may describe the line in terms of its depth relative to the local continuum across the line, say R(AA) = F[(A )/FC where AA is the wavelength measured from line centre, Fc is the flux in the continuum, and F) is the flux at a wavelength within the line. The total absorption by the line obtained by integrating R(AA) over the line profile is known as the equivalent width W. When a line profile is not resolved yet unaffected by blending from neighbouring lines, the equivalent width is independent of the resolution even though the line profile is set by the instrumental profile and not by the intrinsic stellar profile. 

The equivalent width like the line depth is, of course, determined by the ratio of line to continuum fluxes. This ratio is a key to understanding how the equivalent of a line of an atom or ion of element E provides the abundance E/H without the measurement of a line of hydrogen. Consider a normal atmosphere in which temperature increases with increasing depth. Suppose the line-absorbing atoms (or ions) and continuum-absorbing particles are similarly distributed with depth. If the number density of line absorbers is... 

Development of the W — fN H relation, the CoG, beyond the weak-line limit depends on the profile of the absorption coefficient <)>(AA). An extreme form for the profile is effective at illustrating this point. Suppose (AA) = a for AA = AXD and 0 for AA > AAj> Normalization of provides the relation connecting the constant a, the width AXd, and the /-value - the derivation is left as an exercise for the student With increasing fN H, I(AX)/Iq falls within the line to its minimum value of zero. At which point, the equivalent width has saturated at W = 2AAd- Note that, unlike W in the weak-line limit, the CoG beyond the weak-line portion depends on the shape of the line absorption coefficient - here, the width AAd. This dependence means that conversion of a measured W to JNlH for realistic absorption coefficient profiles demands observational or theoretical knowledge of the absorption coefficient s profile. This requirement plus the reduced sensitivity of W to /NlH make this part of the CoG less attractive for abundance determinations. This stretch of the CoG is variously referred to as the flat, Doppler or saturated part. 

Sneden, Gratton, Crocker 1991, McWilliam et al. 1995). Additional advantages are the convenient rest wavelengths of the Zn II and Cr II transitions, which at z = 2 — 3 (where DLAs are most numerous in current samples) are redshifted into a easily observed portion of the optical spectrum, and the inherently weak nature of these lines which ensures that they are nearly always on the linear part of the curve of growth, where column densities can be derived with confidence from the measured equivalent widths (e.g. Bechtold 2002). 

Figure 2.22. Peak width measurement yields as the readout a A/ value that, depending on the selected detector response level, would be more (Level 2) or less sensitive (Level 1). The peak width decreases with decreasing analyte concentration, yet it can be supplemented in this low range by peak height measurement (cf. the vertical arrows). Note that at high analyte concentrations (bold lines), peak height does not yield a meaningful response, while the peak width does. The response curves are spectrophotometric readouts, recorded of the colorimetric determination of iron(II) by orthophenantroline in a two-line FIA system. The injected solutions contained 2, 4, 6, 8, 10, 20, 40, and 80 ppm iron, while the orthophenantroline concentration was equivalent to 12 ppm iron.

It might appear that peak width merely serves as range extension. Yet, since peak width is related to a time span, which in turn is related through linear flow velocity to volumetric rate, peak width measurements allow flow titrations to be performed in a novel way. Therefore, similarly to classical batch titrations, FIA titrations encompass a domain of determinations, which cannot be performed in any other way, because they are based on consumption of an equivalent amount of reagent and, therefore, titrations yield different information than a direct measurement (pH measurement versus titration of a mixture of a weak and strong acid). 

Rule 8. All FIA titrations are based on peak-width measurementy but not all peak-width measurements are FIA titrations. The difference is that FIA titrations are based on location of a pair of equivalence points by using indicator or self-indicating chemical reactions, while peak-width measurements rely on a time of flight of a dispersed zone measured at a selected level of detector response. 

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