OFAD Modeling

In an astrograph, the spherical sky is presumed to be projected onto a flat plate perpendicular to the beam of the astrograph. In this ideal case, the distance r₀ from the optical axis to a point on a plate is given by  r₀=f tan(A), where f is the focal length of the astrograph and A the angular distance from the optical axis to the same point.

Correctors such as the PFADC deviate from this model via radial pincushion distortion, called optical field angle distortion (OFAD), which varies as a function of the distance from the optical axis. We will represent distortion using the model in Chiu's 1976 paper

r = f tan(A)[1+d₃ tan²(A) +d₅ tan⁴(A)]

Here, r is the measured distance from the center of the field to the image in the detector plane. Distortion is modeled via the third and fifth order dimensionless distortion coefficients d₃ and d_5.

The inverted model

r₀ = r b₃ r³ +b₅ r⁵

is usually preferred for analyzing plates, along with the image scale S, usually expressed in arcsec/mm. The two models are equivalent for our purpose and easily interchangeable via the simple relations

f = 206265/S,   d₃ = -b₃f² and d_{5 =} (3b_3² -b₅) f⁴