Astronomy · Reading astronomical spectra

Line broadening and overlapping features

Separate physical and instrumental contributions to spectral-line shapes, and use a controlled Gaussian example to show why a measured width need not have one unique origin.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Explain what a line profile and a declared width measure.
Distinguish broadening, blending, instrumental resolution and sampling.
Use Gaussian variance addition only within its stated model and identify what independent calibration can resolve.

The shape contains more than a centre wavelength

A line profile describes the signal across a spectral feature. Alongside its position and integrated area, astronomers can measure its shape and width. A width must name its definition; different definitions do not give the same number even for one profile.

Two useful definitions are a Gaussian standard deviation σ and the full width at half maximum, abbreviated FWHM. For a Gaussian excess or deficit, FWHM ≈ 2.355σ. For absorption, “half maximum” here refers to half the depth below the local continuum, not half the continuum flux.

Physical processes can broaden a feature. A distribution of line-of-sight particle velocities gives a distribution of Doppler shifts; thermal motions are one example. Pressure-related interactions, finite state lifetimes, rotation and unresolved motions can contribute other shapes or widths.

These mechanisms are not interchangeable. Some model profiles are Gaussian, some have extended wings, and a rotationally broadened profile need not be Gaussian. Inferring temperature from one width requires separating other contributions under an appropriate physical model.

A measured feature includes the spectrometer

An instrument spreads an ideal narrow input over a finite spectral response. A measured profile therefore reflects both the incoming spectrum and the instrument’s line-spread behaviour. Calibration observations help determine that response; a narrow plotted feature is not automatically the untouched source profile.

A convolution is a mathematical way to express such spreading: each small part of the input contributes a shifted copy of a response kernel. In a simple linear model, a normalised kernel redistributes the signal without arbitrarily adding its integrated area. This model must be checked against the actual measurement system.

Spectral sampling specifies where data values are recorded. Finer sample spacing can describe a broadened feature with more points, but does not by itself produce greater resolving power. Likewise, interpolation cannot separate two features the original data do not distinguish.

A blend occurs when contributions from different transitions or components overlap. Its apparent centre and width can change as the component strengths change. There is no universal rule that two features become resolved merely because their nominal centres differ by a nonzero amount.

Two different width allocations can produce the same profile

Use an explicitly synthetic model: a continuum-normalised absorption profile centred at vacuum wavelength 500 nm, with a Gaussian deficit and equivalent width W = 0.03 nm. Let its intrinsic standard deviation be σi. The normalised flux is f(λ) = 1 − [W/(√(2π)σi)] exp[−(λ − 500 nm)²/(2σi²)].

Here f is dimensionless, λ and σi use the same wavelength unit, and the prefactor makes the integrated deficit equal W. The chosen widths keep f nonnegative. This is a mathematical flux profile, not a physical saturation law for an absorbing atmosphere.

Convolve it with a normalised Gaussian instrument kernel of standard deviation σinst. In this restricted model the observed Gaussian variance is σobs² = σi² + σinst². The continuum stays one, the line centre stays 500 nm and the complete-line equivalent width stays 0.03 nm.

Model A has σi = 0.04 nm and σinst = 0.03 nm. Model B reverses those two values. Both give σobs = √(0.0016 + 0.0009) nm = 0.05 nm. Their observed FWHM is about 0.1177 nm and central deficit about 0.2394. With the same centre and area, their full observed profiles coincide.

The instrument width is deliberately unknown between these candidate models. If an independent calibration establishes σinst, that particular ambiguity is reduced. For example, σinst = 0.03 nm with σobs = 0.05 nm gives σi = √(0.05² − 0.03²) nm = 0.04 nm, within the model and its uncertainty.

Original Gaussian convolution example with shared centre 500 nm and equivalent width 0.03 nm. All listed widths are standard deviations, not FWHM.
Candidate	Intrinsic σi	Instrument σinst	Observed σobs
A	0.04 nm	0.03 nm	0.05 nm
B	0.03 nm	0.04 nm	0.05 nm

Two candidates, one broadened profile

Scroll sideways to see the full diagram. Its explanation is below.

Original analytic synthetic Gaussian convolution example, not observed data. Each curve is dimensionless continuum-normalised flux f at vacuum wavelength, with continuum 1, common centre 500 nm and complete-line equivalent width, or integrated deficit area, W = 0.030 nm. All three plots have exactly the same axis limits and drawing scale: wavelength 499.8 to 500.2 nm and flux 0 to 1.05, with zero and continuum 1 explicitly shown. Candidate A has an intrinsic Gaussian absorption deficit with standard deviation 0.040 nm and an instrument-kernel standard deviation of 0.030 nm. Candidate B has intrinsic standard deviation 0.030 nm and instrument standard deviation 0.040 nm. Their intrinsic central depths are 0.2992067103 and 0.3989422804 respectively. The dashed blue curve identifies A and the solid orange curve B. Kernel boxes give instrument widths separately; no kernel is plotted on the flux axis. Each instrument kernel is a Gaussian density in inverse nanometres, normalised to integral 1. Analytic convolution preserves the constant continuum and complete-line deficit area in this particular linear model. Both candidates give observed variance 0.0025 square nanometres and observed standard deviation 0.050 nm. One shared green output curve therefore represents both full observed model profiles, with centre 500 nm, depth 0.2393653682, minimum flux 0.7606346318 and FWHM 0.1177410023 nm. FWHM means the full width at half the deficit depth below continuum 1. Instrument width is deliberately not independently known between the two candidate models; an independent response calibration can distinguish them. Gaussian variance addition is conditional on this declared Gaussian convolution and is not a universal rule for physical line shapes, arbitrary blends or nonlinear absorption. No actual instrument response, temperature, rotation, abundance or composition is inferred.

An equation is useful when its assumptions remain visible

The variance-sum equation applies to the declared Gaussian convolution. Do not apply it unchanged to arbitrary blended lines, Lorentzian profiles, rotational kernels or a nonlinear saturated transfer problem. A good fit under one model does not establish that every physical contribution is Gaussian.

An inferred intrinsic width also inherits measurement and response uncertainty. If the observed and instrument widths are nearly equal, their squared difference can be small and sensitive to those uncertainties. A naive subtraction can even give a negative result, signalling that the assumed exact numbers cannot support that inference.

Real spectral analyses can fit template spectra after applying an instrument response and compare alternative parameters. Gaia DR 3’s processing documentation illustrates the roles of broadening, noise and template mismatch. It does not justify treating one broadening parameter as a unique measure of temperature or rotation in every source.

The next lesson uses this example to ask what a spectrum actually identifies. A calibrated instrument, additional transitions and better modelling can strengthen the conclusion; the purpose of the ambiguity is to specify which extra information matters.

Worked examples

Infer a width after an independent response calibration

Within the Gaussian model, σobs = 0.05 nm and independently calibrated σinst = 0.03 nm. What intrinsic width follows?

Square the values: σobs² = 0.0025 nm² and σinst² = 0.0009 nm².
Subtract variances: σi² = 0.0016 nm².
Take the positive square root: σi = 0.04 nm.

An intrinsic standard deviation of 0.04 nm under the stated Gaussian assumptions. This is not obtained by subtracting the widths themselves.

Why the two outputs really coincide

Why do candidates A and B give an identical measured profile rather than merely the same width?

Their convolved Gaussian widths are both 0.05 nm.
Their centres are both 500 nm and integrated deficits both 0.03 nm.
For this specified Gaussian family, centre, width and area fix the profile, including its depth.

All defining output quantities agree, so their synthetic observed curves coincide. Their unknown input allocations differ.

Try the reasoning

Is σi = 0.05 − 0.03 = 0.02 nm the correct Gaussian deconvolution?
Show the answer and reasoning
No.
Variances add in this model. Subtract squares and then take the square root to obtain 0.04 nm.
Does replacing each recorded sample with ten interpolated values increase the instrument’s resolving power tenfold?
Show the answer and reasoning
No.
Interpolation changes the displayed sampling, not the original spectral information.
For a Gaussian absorption feature, does FWHM here use half its deficit depth or half the continuum itself?
Show the answer and reasoning
Half the feature’s deficit depth.
The local continuum is the reference; the line width concerns the profile relative to it.
Does one broad measured line uniquely establish a high temperature?
Show the answer and reasoning
No.
Instrument response, other motions, blends and other broadening processes can contribute. A physical model and additional constraints are needed.

What to carry forward

A line profile combines physical structure and measurement response; width requires a definition.
Gaussian variance addition is a useful conditional model, not a universal rule for every spectral feature.
Independent response calibration and additional features can distinguish otherwise ambiguous explanations.

References

Teaching resources and research supporting this lesson’s explanations and examples.