= Blackbody peak and detector response
Consider a hypothetical <star> whose emitted <specific intensity> is exactly that of a uniform $5000\,\mathrm K$ <blackbody>. Assume transparent propagation with negligible <Doppler shift> and <interstellar extinction>, so geometric dilution multiplies every spectral interval by the same factor. The stipulated <temperature> gives
$$
\lambda_{\max}=0.579554391\,\mu\mathrm m,\qquad
\nu_{\max}=293.946288\,\mathrm{THz},\qquad
c/\nu_{\max}=1.01988857\,\mu\mathrm m.
$$
Using the converted <frequency> peak as though it maximized $B_\lambda$ would return $b/(c/\nu_{\max})\simeq2841.26\,\mathrm K$, a substantial error caused solely by mixing spectral conventions.
Now stipulate a <photodetector> with constant <quantum efficiency> $\eta$ over the relevant <wavelengths>, collecting equal narrow <wavelength> bins. For fixed accepted <projected area> times <solid angle> $G$, with dilution included in $G$, its expected <photon> count rate density is
$$
\frac{d\dot N}{d\lambda}=\eta G\frac{\lambda B_\lambda}{hc}.
$$
This follows by dividing the received <power> by the <photon> <energy> $hc/\lambda$. It peaks where $x=4(1-e^{-x})$, at $x=3.9206903949\ldots$, giving $0.73394057\,\mu\mathrm m$ for the same model <star>. Constant <quantum efficiency> is not constant energy <spectral responsivity>: the latter includes the <wavelength>-dependent <photon> <energy> conversion. For example, $G=10^{-12}\,\mathrm{m^2\,sr}$, $\eta=0.5$ and a $1\,\mathrm{nm}$ bin centered at $0.600\,\mu\mathrm m$ give approximately $1.93\times10^{10}$ detected <photons> per second in the narrow-bin approximation. These are calculated model values, not observations of a named <star>.
A more general linear detector channel has output
$$
S_i-S_{i,0}=G\int_0^\infty R_i(\lambda)B_\lambda(T)\,d\lambda,
$$
where $S_{i,0}$ is the background reading and $R_i$ includes spectral selection, optical transmission and detector <spectral responsivity>. A raw channel maximum need not locate either <Planck function> maximum. Recovering a <temperature> requires an appropriate calibrated response and an emission model. For a real <stellar atmosphere>, <opacity> and <radiative transfer> can reshape the spectrum; <interstellar extinction> can reshape it again. A single peak does not establish a unique physical <temperature> without those assumptions.
\Image[/wien-coordinate-detector-comparison.png]
{title=One model <star>, three peak conventions, and a spectral measurement chain}
{description=All curves use the same $5000\,\mathrm K$ <Planck function> and are normalized separately. The $B_\nu(c/\lambda)$ curve retains density per <frequency> while displaying converted <wavelength>. The photon curve assumes constant <quantum efficiency> in equal narrow <wavelength> bins. The lower diagram shows a generic measurement sequence, not a reconstruction of a particular historical apparatus.}
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