Spectral Index Calculator

Compute a radio source's spectral index α from two flux density measurements, classify it as steep, flat, or inverted, and extrapolate the predicted flux at any other frequency — with optional uncertainty propagation.

Spectral index calculator

Uses the standard radio-astronomy convention S_ν ∝ ν^α — a negative α means flux density drops with increasing frequency (steep spectrum), positive means it rises (inverted). Some literature defines α with the opposite sign; always check before comparing quoted values.

Measurement 1

Frequency ν₁

Flux density S₁

Measurement 2

Frequency ν₂

Flux density S₂

α = -0.800001

Steep spectrum

Log-log plot — a power law is a straight line here, and its slope is α. Solid = between your two measurements; dashed = extrapolated beyond them.

Predict flux density at a third frequency

Frequency ν₃

Predicted S₃

728.36 Jy

Radio sources don’t shine at one frequency — they have a whole spectrum, and how their brightness changes across it carries real physics. A single number, the spectral index α, captures the shape of that spectrum, and whether it’s steep, flat, or rising tells you something about the underlying emission mechanism before you’ve done anything more sophisticated than compare two flux measurements.

The definition

A source’s flux density S_ν as a function of frequency ν is modeled as a power law:

S_\nu \propto \nu^{\alpha}

Given two measurements, S1 at ν1 and S2 at ν2, this rearranges to

\alpha = \frac{\ln(S_2 / S_1)}{\ln(\nu_2 / \nu_1)}

That same α then predicts the flux density at any third frequency ν3 via S3 = S1·(ν3/ν1)^α — useful for planning an observation at a frequency you haven’t measured yet, or checking whether a source’s known spectral shape is consistent with a new detection.

A note on sign convention. This calculator uses S_ν ∝ ν^α, standard in radio survey catalogs (NVSS, VLSS, and most others). Some literature instead defines α through S_ν ∝ ν^-α — the opposite sign — so a quoted “α = 0.8” can mean a rising or a falling spectrum depending on the source. Always check convention before comparing values across papers.

What the shape of the spectrum tells you

α ≈ 0 (flat spectrum): common in compact, optically-thick AGN cores — a superposition of many self-absorbed synchrotron components peaking at different frequencies can conspire to look nearly flat overall.
α ≈ -0.5 to -1.2 (steep spectrum): the classic signature of optically-thin synchrotron emission from extended structures like radio lobes and jets — the regime most extragalactic radio sources fall into.
α ≲ -1.2 (ultra-steep spectrum): used deliberately as a search criterion for high-redshift radio galaxies, since spectral steepening correlates statistically with distance.
α > 0 (inverted spectrum): the source is optically thick at the observed frequencies — young, compact, or self-absorbed sources whose emission hasn’t yet freed itself from its own opacity.

These bands are a useful rule of thumb, not a hard boundary — real spectra can curve, and the same source can sit in different categories at different frequency ranges.

Uncertainty and extrapolation

Frequencies are normally known far more precisely than flux densities, so this calculator treats ν1, ν2, and ν3 as exact and propagates only the flux density uncertainties you supply. The resulting uncertainty on α scales inversely with how far apart ν1 and ν2 are in log-frequency — measurements close together in frequency give a much noisier α than ones spread widely apart, for the same flux precision. Extrapolating to ν3 compounds α’s uncertainty further, growing with how far ν3 sits from your anchor point — a genuine feature of projecting a power law outward, not an artifact of the calculation.