method

Spectral Analysis

Spectral analysis is a method for studying materials or astronomical objects by decomposing their emitted or reflected light into constituent wavelengths.
10 related articles · 6 formulas · 6 who worked here

Description

Spectral analysis is a universal key. It doesn’t require touching the object; it’s enough to capture its emission. For example, light from a distant galaxy passing through a gas cloud loses very specific colors—they are absorbed by atoms. Such an absorption spectrum resembles a barcode: each dark line corresponds to a particular element. From the shift of these lines, one can even measure how fast the object is moving toward or away from us (the Doppler effect, a change in frequency due to motion). And in the radio range, spectral analysis ‘listens’ to molecular clouds, finding complex organic molecules.

History

The origins of spectral analysis lie in 17th-century optics (Newton decomposed light with a prism). In 1802, Wollaston noticed dark lines in the solar spectrum. From 1814 to 1823, Joseph von Fraunhofer catalogued them, not knowing the cause. The decisive step was taken by Gustav Kirchhoff and Robert Bunsen in 1859, who proved that each line is produced by a specific chemical element. Thus, the chemical analysis of radiation was born. Later, Johannes Rydberg (1890) derived a formula for hydrogen lines, and quantum mechanics explained spectra based on electron transitions.

How it works

Atoms absorb or emit photons of precisely defined energies equal to the difference between the energy levels of electrons. A transition from one level to another corresponds to a spectral line at frequency ν = ΔE/h. Spectra are classified into line spectra of atoms, band spectra of molecules (rotational and vibrational transitions), and continuous spectra of hot dense bodies (thermal radiation). For analysis, dispersive instruments (prism, diffraction grating), interferometers (high resolution), and photodetectors are used. Key characteristics: line position (wavelength), intensity, width, and line shape (Doppler and Lorentz broadening). The methods are applied across the entire electromagnetic spectrum, from radio to gamma rays.

In practice

Astronomers use a star’s spectrum to learn what elements it is composed of. In medicine, tissue spectra help distinguish healthy cells from cancerous ones.

Fun fact

The spectrum is the only way to know the chemistry of distant objects: no space probe has ever visited the Sun, but we know for certain that it consists mainly of hydrogen and helium, thanks to spectral lines.

Concept graph →

Formulas

$$n_s - 1 = \frac{d\ln\mathcal{P}_\mathcal{R}}{d\ln k}$$
The scalar spectral index measures the deviation from scale invariance of primordial perturbations.
What is in the formula
n_svariablescalar spectral index — scalar spectral index
P_Rvariablepower spectrum of curvature perturbations — power spectrum
kvariablewavenumber — wavenumber reciprocal metre
d/d ln koperatorderivative with respect to natural logarithm of wavenumber — logarithmic derivative
n\equiv\frac{d\ln\mathcal{p}_\mathcal{r}}{d\lnk}=-5 · 2606.23307
n_s\approx1+\frac{5}{12}(4\frac{\dot{h}}{h^2}-\frac{\dot{\gamma}}{h\gamma})\bigg|_{t_*} · 2607.27193v1
$$\langle r^2 \rangle = \sum w_i \langle r_i^2 \rangle$$
Mean square radius expressed as weighted sum of component radii.
What is in the formula
⟨r²⟩variableTotal mean square radius — mean square radius metre squared
w_ivariableWeighting factor for component i — weighting factor
⟨r_i²⟩variableMean square radius of component i — mean square radius component metre squared
∑operatorAdds contributions from each component — summation
\langler^2_m\rangle=\frac{a_c}{a}\langler^2_m\rangle_c+\frac{2a_c}{a^2}\langler^2_{c-nn}\rangle+\frac{1}{2a}\langler^2_{nn}\rangle · 2606.28950
$$\Omega_X(k) = \frac{1}{\rho_{\rm tot}} \frac{d\rho_X}{d\ln k}$$
Defines the spectral density of a component per logarithmic frequency.
What is in the formula
Ω_XvariableSpectral density of component X — spectral density
ρ_totvariableTotal energy density of the universe — energy density joule per cubic metre
ρ_XvariableEnergy density of component X — energy density joule per cubic metre
kvariableWavenumber (logarithmic scale) — wavenumber per metre
d/d ln koperatorDerivative with respect to the natural logarithm of the wavenumber — logarithmic derivative
ω_{\rmgw}\equiv\frac{1}{ρ_{\rmtot}}\frac{{\rmd}ρ_{\rmgw}}{{\rmd}\lnk} · 2606.31571
$$R = \frac{\lambda}{\Delta\lambda}$$
Spectral resolution is wavelength divided by the smallest resolvable wavelength difference.
What is in the formula
Rvariablespectral resolution — spectral resolution
λvariablewavelength of interest — wavelength metre
Δλvariablesmallest resolvable wavelength difference — wavelength difference metre
r=\frac{λ}{δλ} · 2607.05600v2
$$\beta = \frac{\mathrm{d} \log \kappa}{\mathrm{d} \log \nu}$$
Measures how opacity changes with frequency.
What is in the formula
βvariablespectral index of opacity
κvariableopacity (absorption coefficient) — opacity square metre per kilogram
νvariablefrequency of radiation — frequency hertz
d/d log νoperatorderivative with respect to the natural logarithm of frequency — logarithmic derivative
β=\frac{{d}\log\kappa_ν}{{d}\logν} · 2607.02660v1
α=\frac{d\log(λf_λ)}{d\logλ} · 2607.04942v1
$$n_s = 1 - \frac{2}{N}$$
Spectral index deviation from scale invariance depends on e-folds.
What is in the formula
n_svariableScalar spectral index — scalar spectral index
NvariableNumber of e-folds of inflation — e folds
n_s=1-\frac{2}{n_*} · 2605.00735v1

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