Determination of equivalent width (W)

The equivalent width W is a measure of the „quantity“ of light, which is cut out from the continuum of a star within an absorption line by absorption processes. It is based on the local relative flux of the continuum. Geometrically it is identical with the area of the considered line under the normalized continuum.

Mathematically, the W is defined by the following equation: With

• lambda_1 and lambda_2 = lower and upper wavelength limit of the observed line
• F_c = relative flux of the continuum
• F_lambda = relative flux of the line

According to this equation, no normalization of the spectrum is needed to determine the W to, but it has the curve of the continuum under the line to be known (the quasicontinuum F_c (lambda)). According to the definition a W of a absorption line is a positive number, the area under an emission line is counted negatively. In reality, we do not have continuously measured the wavelengths lambda functions of F and Fc, but with discrete values of the individual pixels that correspond to a wave length interval. Thus the integration is replaced by a summation over the pixels (dispersion elements h_lambda), which form the line within the line limits.

h_lambda = dispersion in Angstrom/pix
M= number of pixels within the line interval
F_j / F_c = flux normalized by the continuum for pixel j

Below you see the calibrated, but not normalized spectrum of lam Cep, an O-star. In addition to the Natriumdublett (5889 and 5896 angstrom) the strongly rotationally broaden HeI5876 is found. The MIDAS command ‚integrate / line‘ calculated the W of the line. With 2 mouse clicks the integration area has been set and the continuum between them modeled by a straight line (This is the dashed line at the foot of the absorption line at about 50,000 ADU). The printout of the executed command documented the specifications (start and end points of the integrated line = 5858.55 to 5879.09 Angstrom) and the calculated W = 0.9785 Angstrom.

Midas 012> INTEGR/LINE lamCep20080827_CIV
X_start (pix/world) X_end (pix/world) Pixel sep.
Line+Cont. Continuum Line Line/Cont Equiv. w.
—————————————————————-
5858.55 724.787 5879.09 881.230 0.131279
978934. 0.102791E+07 -48974.2 -0.476446E-01 0.978505

In this example, the quasi-continuum in the line could be set straight.

In most cases due to the curvature of the spectra many quasi-continuum points are chosen. They generate then a polynomial representation of the continuum. The reference continuum is then a curve. This definition of the quasi-continuum from many grid points can be both in MIDAS (OPA, SMS), as well VSpec done by mouse clicks. Therefore individual assessments are incorporated into the course of the resulting continuum model. The result is no longer measured result, but an interpreted result.

The advantage of the concept of equivalent width W is its independence from the resolution R of the spectrograph. Therefore „line intensities“ of different observers with different spectrographs (and resolution R) are directly comparable, for the amateur scene, an important aspect.

Below the complex H alpha line of bet Lyr is in emission. The exposures was integrated with OPA in MIDAS. The lower straight line is the adopted continuum. The calculated W = -14.28 angstrom is negative because it is an emission line. The W is geometrically the surface between the quasicontinuum and the normalized spectrum, the red bordered area. It was also determined the V / R ratio. It is the intensity ratio of the maximum of the short-wavelength peak (V = violet) and the maximum of the long-wavelength peak (R = red), a number that often change over time and therefore is of interest, especially in case of Be stars with their rotating equatorial gas discs.

Below a spectrum of P Cyg, the brightest LBV in the northern sky (LBV = Luminous Blue Variable). The star has many lines in emission, resulting in his extended hot envelope under the influence of intense ultraviolet light from the very hot stellar surface. The broad H alpha line shows the typical P Cyg profile, an issue which is accompanied by a sharp short-wave shifted absorption. The latter is generated in the strong, by the light pressure to the outside driven fast stellar wind. The two peaks at 6580 angstroms are produced by carbon CII. The line with P Cyg profile at 6678 Angstrom is one of HeI.

The emission intensity of the H alpha line is unusually large, W = -68.8 angstroms. In the maximum of the line the object shines around 20 times brighter as the photosphere of the star! Here is also shown that the W include all components of the line. It includes both wings of the „foot“ of the line, which are produced by Thompson scattering of photons emitted in the gas envelope by the electrons of the wind plasma. These photons are missed in the core of the emission line. They need to be included in the W, to collect the total line intensity of emitted H alpha photons.

Here it is clear that the evaluation of a spectrum and its lines needs a basic knowledge about the physics of the object. Otherwise naive misinterpretations are inevitable.