= Solution
The Fraunhofer approximation drops the aperture-dependent quadratic propagation phase. For illuminated extent $b$ and observation distance $R$, a sufficient paraxial condition is $b^2/(\lambda R)\ll1$, with coherent nearly plane incident illumination. Collimation followed by observation in a lens's back <focal plane> produces the same angular Fourier pattern without an astronomical propagation distance.
Let $(x,y)$ label the <aperture> field $\psi_\Sigma$ and let $(x_0,y_0)$ be transverse coordinates of the distant detector, with $R$ the axial separation. Expanding the propagation distance retains a common phase and the cross term $-(x_0x+y_0y)/R$, while dropping the <aperture>'s quadratic term. Summing the scalar Huygens contributions gives a field proportional to the <Fourier transform> at $(k x_0/R,k y_0/R)$. In one dimension, writing $q=kx_0/R$, $\widetilde\psi(q)=\int\psi(x)e^{-iqx}dx$ gives
$$
\boxed{\psi(x)=\frac1{2\pi}\int\widetilde\psi(q)e^{iqx}\,dq}.
$$
Remove the known propagation prefactor and any detector-plane quadratic phase before applying this inverse transform. Recovery needs the complex diffracted field, including phase; its intensity alone is insufficient for this inversion.
The <whole-aperture Fraunhofer distance of a coherent array> uses the full ten-dish extent, which is about $b=9D+25\ \mathrm m=7.225\ \mathrm{km}$. Thus $b^2/\lambda\simeq8.7\times10^8\ \mathrm m$, and a conventional boundary $2b^2/\lambda$ is about $1.7\times10^9\ \mathrm m$. \b[Far-field propagation must be of order a million kilometres or more], with a stronger inequality for accurate Fraunhofer behaviour. Applying the condition to only one dish or one spacing would miss the array extent.
For equal synchronized dish fields, the array sum is $\sum_{j=0}^9e^{-ij k_0D\sin\theta}$. Its intensity is
$$
\boxed{I(\theta)=I_1(\theta)
\left[\frac{\sin(10\beta)}{\sin\beta}\right]^2,\qquad
\beta=\frac{k_0D\sin\theta}{2}}.
$$
Near the centre, $\sin\theta\simeq\theta$ and the single-dish envelope $I_1(\theta)$ is nearly constant, giving the requested approximation. A uniformly illuminated circular dish instead supplies the broader envelope $[2J_1(\pi d\sin\theta/\lambda)/(\pi d\sin\theta/\lambda)]^2$, with first zero near $1.22\lambda/d=2.93\times10^{-3}\ \mathrm{rad}$. It cannot be ignored over the entire pattern.
Principal array maxima have $\sin\theta_m=m\lambda/D$, height $100I_1(\theta_m)$, with spacing approximately $7.5\times10^{-5}\ \mathrm{rad}=15.5$ arcseconds. There are narrow sidelobes and many grating orders within the dish envelope because $D\gg d$. The central maximum's first zeros obey $10\beta=\pm\pi$. By reception/transmission reciprocity, the source must therefore move through
$$
\boxed{\Delta\theta\simeq\frac{\lambda}{10D}=7.50\times10^{-6}\ \mathrm{rad}=1.55\text{ arcseconds}}
$$
from zero-order maximum to first minimum. The equatorial geometry fixes the sweep direction; no elapsed time is requested.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-2-fraunhofer-array.png]
{title=Original labelled Fraunhofer geometry, ten-element central interference lobe and grating orders under a circular-dish envelope}
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