= Longitudinal coherence estimate
The <coherence length from a finite spectral band> is most safely estimated by converting its endpoints to <frequency>, because the quoted band is broad:
$$
\Delta\nu=c\left(\frac1{275\,\mathrm{nm}}-\frac1{875\,\mathrm{nm}}\right)
\simeq7.48\times10^{14}\,\mathrm{Hz}.
$$
The reciprocal-bandwidth <optical coherence time> and corresponding longitudinal <optical coherence length> are
$$
\tau_c\sim\frac1{\Delta\nu}\simeq1.34\times10^{-15}\,\mathrm s,\qquad
\boxed{\ell_\parallel\sim c\tau_c=
\frac{(275)(875)}{875-275}\,\mathrm{nm}
\simeq401\,\mathrm{nm}=0.40\,\mu\mathrm m.}
$$
The given <temperature> is consistent with the spectral peak: <Wien's displacement law> gives $\lambda_{\rm peak}=2.898\times10^{-3}\,\mathrm{m\,K}/5778\,\mathrm K\simeq502\,\mathrm{nm}$. Using the common narrow-band estimate $\lambda_{\rm peak}^2/\Delta\lambda$, with $\Delta\lambda=600\,\mathrm{nm}$, gives about $0.42\,\mu\mathrm m$. Its closeness here does not justify the narrow-band approximation for arbitrary broad spectra.
The $0.40\,\mu\mathrm m$ result is a bandwidth-scale estimate. A flat spectral density per unit <frequency> would make it the first-zero <optical coherence length>; a truncated <Planck law> spectrum is not flat and would have a different correlation function. Neither the stated endpoints nor the word “coherence” specifies an exact visibility threshold. Propagation to Earth sets the angular size and transverse scale, but does not by itself narrow the solar <optical spectrum> or lengthen this longitudinal scale.
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