Equivalent width

The equivalent width of a spectral line is a measure of the area of the line on a plot of intensity versus wavelength in relation to underlying continuum level. It is found by forming a rectangle with a height equal to that of continuum emission, and finding the width such that the area of the rectangle is equal to the area in the spectral line. It is a measure of the strength of spectral features that is primarily used in astronomy.^[1]

Definition

Formally, the equivalent width is given by the equation^[2] $W_{\lambda }=\int {F_{c}-F_{s} \over F_{c}}d\lambda =\int (1-F_{s}/F_{c})d\lambda .$

Here, $F_{c}(\lambda )$ represents the underlying continuum intensity, while $F_{s}(\lambda )$ represents the intensity of the actual spectrum (the line and continuum). Then $W_{\lambda }$ represents the width of a hypothetical line which drops to an intensity of zero and has the "same integrated flux deficit from the continuum as the true one."^[2] This equation can be applied to either emission or absorption, but when applied to emission, the value of $W_{\lambda }$ is negative, and so the absolute value is used.

In other words, if the continuum level is constant and the area under/above the emission/absorption line (compared to the continuum) is $A$ (the integral above), then $A=F_{c}\times W_{\lambda }$ (further highlighting the continuum-level dependence). Therefore, for a fixed line strength ( $A$ ), the equivalent width will be smaller for a brighter continuum.

Applications

The equivalent width is used as a quantitative measure of the strength of spectral features. The equivalent width is a convenient choice because the shapes of spectral features can vary depending upon the configuration of the system which is producing the lines. For instance, the line may experience Doppler broadening due to motions of the gas emitting the photons. The photons will be shifted away from the line center, thus rendering the height of the emission line a poor measure of its overall strength. The equivalent width, on the other hand, "measures the fraction of energy removed from the spectrum by the line," regardless of the broadening intrinsic to the line or a detector with poor resolution.^[3] Thus the equivalent width can in many conditions yield the number of absorbing or emitting atoms, by using the curve of growth.^[1]

For example, measurements of the equivalent width of the Balmer alpha transition in T Tauri stars are used in order to classify individual T Tauri stars as being classical or weak-lined.^[2] Also, the equivalent width is used in studying star formation in Lyman alpha galaxies, as the equivalent width of the Lyman alpha line is related to the star formation rate in the galaxy.^[4] The equivalent width is also used in many other situations where a quantitative comparison between line strengths is needed.

References

^ ^a ^b Carroll, Bradley; Ostlie, Dale (2007). An Introduction to Modern Astrophysics. Pearson Addison-Wesley. ISBN 978-0-321-44284-0.
^ ^a ^b ^c Stahler, Steven; Palla, Francesco (2004). The Formation of Stars. Wiley-VCH. ISBN 3-527-40559-3.
^ Spitzer, Lyman (1998). Physical Processes in the Interstellar Medium. Wiley-VCH. ISBN 0-471-02232-2.
^ Dijkstra, Mark; Westra, Eduard (2010). "Star formation indicators and line equivalent width in Lya galaxies". Monthly Notices of the Royal Astronomical Society. 401 (4): 2343–2348. arXiv:0911.1357. Bibcode:2010MNRAS.401.2343D. doi:10.1111/j.1365-2966.2009.15859.x. S2CID 67786958.

External links

Equivalent Width in the SAO Encyclopedia of Astronomy

[co-1] Carroll, Bradley; Ostlie, Dale (2007). An Introduction to Modern Astrophysics. Pearson Addison-Wesley. ISBN 978-0-321-44284-0.

[formation_of_stars-2] Stahler, Steven; Palla, Francesco (2004). The Formation of Stars. Wiley-VCH. ISBN 3-527-40559-3.

[spitzer-3] Spitzer, Lyman (1998). Physical Processes in the Interstellar Medium. Wiley-VCH. ISBN 0-471-02232-2.

[lae-4] Dijkstra, Mark; Westra, Eduard (2010). "Star formation indicators and line equivalent width in Lya galaxies". Monthly Notices of the Royal Astronomical Society. 401 (4): 2343–2348. arXiv:0911.1357. Bibcode:2010MNRAS.401.2343D. doi:10.1111/j.1365-2966.2009.15859.x. S2CID 67786958.

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