= Finite-response stellar intensity correlations
{title2=$g^{(2)}_{\rm obs}(B,0)-1=p\rho_1\rho_2\beta|\gamma(B)|^2$}
An exact teaching model makes the peak suppression explicit. Assume the narrowband field correlation separates as $g^{(1)}_{12}(B,u)=\gamma(B)g_t^{(1)}(u)$ after delay compensation. Average each channel's <light intensity> over the same rectangular gate of duration $T$. The two time integrals have triangular relative-delay weight
$$
W_T(u)=\begin{cases}(T-|u|)/T^2,&|u|\leq T,\\0,&|u|>T,\end{cases}
\qquad \int_{-\infty}^{\infty}W_T(u)\,du=1.
$$
Integrating the <Siegert relation> against this weight gives
$$
\beta=\int_{-\infty}^{\infty}W_T(u)|g_t^{(1)}(u)|^2\,du,
\qquad 0\leq\beta\leq1.
$$
The factor $p$ is one for a selected <polarization of waves> and $1/2$ for two independent equally populated polarizations. The signal fraction $\rho_j=s_j/(s_j+b_j)$ uses mean stellar signal $s_j$ and independent additive background $b_j$ in each detector. The <photon correlation loss and background model> then supplies $\rho_1\rho_2$. This assumes no correlated <sky background> or electronic coupling.
For the original model $|g_t^{(1)}(u)|^2=e^{-|u|/\tau_0}$, elementary <integration> gives
$$
\beta=\frac{2\tau_0}{T}-2\left(\frac{\tau_0}{T}\right)^2(1-e^{-T/\tau_0}).
$$
The area-based <optical coherence time> of this intensity envelope is $\tau_c=\int|g_t^{(1)}(u)|^2du=2\tau_0$. For $T\gg\tau_0$, $\beta\approx\tau_c/T$; for $T\ll\tau_0$, $\beta\to1$. With the chosen $\tau_0=1\,\mathrm{ps}$, $T=50\,\mathrm{ps}$, $p=1$, and $\rho_1=\rho_2=0.8$, $\beta\approx0.0392$ and $g^{(2)}_{\rm obs}(0,0)=1.025088$. At $B=10\,\mathrm m$ the stipulated disk gives approximately $1.013617$. A nearly unit measured $g^{(2)}$ can therefore conceal a substantial $V^2$.
This rectangular-gate calculation specifies a counting model. An analog instrument's amplifier <bandwidth> and impulse response instead weight the fluctuations it actually passes; the Sirius amplifiers were bandpass devices. Both require measuring the response and calibrating the zero-baseline excess. When $p$, $\rho_j$, and the temporal response are unchanged, the ratio of excesses recovers $V^2$:
$$
\frac{g^{(2)}_{\rm obs}(B,0)-1}{g^{(2)}_{\rm obs}(0,0)-1}=|\gamma(B)|^2.
$$
A sufficiently unresolved calibration source can supply the denominator; dividing the full $g^{(2)}$ values would give the wrong result. Narrower optical <bandwidth> lengthens <optical coherence time>, whereas wider usable electronic <bandwidth> can resolve faster fluctuations; these are distinct bandwidths. Background changes, unequal spectral responses, drift, or averaging over changing <interferometric baselines> can invalidate a common calibration. Resolving a disk means establishing a reproducible decline of calibrated correlation, supported by those controls.
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