User:Tiny green/Bessel

The image, $$I$$, is produced by convolving a theoretical image, $$D$$, of the belt of Orion with a point distribution function, $$F$$, corresponding to a lens with most of its central area blocked out.
 * $$I = D * F$$

The point distribution function is
 * $$F(\theta \; | \; \delta, \beta) = \frac{1}{\left(1-\delta^2\right)^2} \left[ \frac{2 J_1(\frac{\pi\theta}{\beta})}{\frac{\pi\theta}{\beta}} - \delta^2\frac{2 J_1(\delta \frac{\pi\theta}{\beta})}{\delta \frac{\pi\theta}{\beta}} \right]^2$$

where $$\theta$$ is the angular distance to the center of the light source, $$\delta$$ is the proportion of the area of the lens that is blocked out, $$\beta=\lambda/D$$ is the ratio of the wavelength of the light to the diameter of the lens, and $$J_1$$ is the Bessel function of first order given by
 * $$J_1(x) = \frac{1}{\pi} \int_{0}^{\pi} \cos (\tau - x \sin \tau) d\tau$$

The parameters used are
 * $$\delta=0.999$$
 * $$\beta=\pi/10$$ * pi/180?