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Equation 20 · Part 8 · A Neutron Star Is a Redshift Standard

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ΔE/E≈veq/c=2πνR/c≈1.3×10−1(ν/500 Hz)(R/12 km)\Delta E/E \approx v_{eq}/c = 2\pi\nu R/c \approx 1.3\times10^{-1}(\nu/500\,\text{Hz})(R/12\,\text{km})
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

The mechanism behind that contradiction is simple enough to state in one line, and the paper’s authors compute it carefully because the commonly quoted version understates it by an order of magnitude. Surface rotation Doppler-broadens any spectral line by a fraction of its energy equal to the star’s equatorial velocity over the speed of light, Δ\Delta E/E ≈\approx veqv_{eq}/c = 2π\piν\nu R/c ≈\approx 1.3×\times10^{-1}(ν\nu/500\,Hz\text{Hz})(R/12\,km\text{km}) , where ν\nu is the star’s spin frequency and R its radius.

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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