What does an adaptive primary do?

An adaptive primary is a concentrating mirror that redirects sunlight to a fixed focus—no matter where the sun appears in the sky. A mirror that accomplishes this task must adapt its curvature as the sun rises higher or sets lower in the sky. To a first approximation, the curvature of the mirror is toric, that is, the mirror is always approximately shaped like a small patch on the surface of a torus. The adaptation required is slight, the changes in curvature will be scarcely visible to a viewer looking sideways at the mirror.

In the animation above—rendered in POV-Ray using Mega POV—the zenith distance of the sun varies from 15°  to 75°, while the adaptive primary acquires a toric curvature by the thin-shell bending of an initially spherical mirror. The POV-Ray scene description file for the animation is here.

In addition to changing its curvature, an adaptive primary needs to follow the sun in two angular dimensions: turning to face the sun's azimuth while also tilting its normal to one half the sun's zenith distance. (At the most an adaptive primary only needs to be tilted 45° away from horizontal.)

Toric curvature of an adaptive primary

Bending a thin-shell, initially spherical, primary mirror to accommodate the changing angle of solar incidence, θ,  induces a toric curvature in the mirror.

The primary mirror of a two-mirror heliostat must function over a wide range of the solar incidence angle θ: from θ ≈ 0° at summer noon, to θ ≈ 45° at sunrise and sunset. A unchanging mirror profile cannot handle this wide range of incidence angle. It is necessary for the primary mirror to change shape only slightly. Luckily, simple thin-shell bending gives the correct range of profiles.

In the thin-shell bending of a surface, its Gaussian curvature—the product of the curvatures in the two principal planes (the planes that reveal respectively the maximum and minimum curvatures)—is constant. In a spherical mirror these to curvatures are equal, so as a thin-shell spherical mirror bends, one principal curvature increases while the other decreases in the inverse proportion.

Consider a spherical mirror reflecting sunlight. The plane that contains both the sun and the surface normal at the center of the mirror is called the meridional plane. The plane that likewise contains the surface normal at the center of the mirror, but is disposed perpendicular to the meridional plane is called the sagittal plane. When the primary mirror is adapted correctly to the changing angle of solar incidence, the meridional and sagittal planes are also the principal planes of curvature. In particular, as the angle of solar incidence, θ, increases, the radius of curvature in the meridional plane must increase as 1/cosθ, while the radius of curvature in the sagittal plane will, correspondingly, decrease as cosθ.

In the open-source ray-tracing program POV-Ray, a torus is specified by its major radius and its minor radius (as identified in the illustration above.) Thus, in this system, a sphere of radius 1 is also a torus with major radius 0, and minor radius 1.

For correct adaptation to a changing angle of solar incidence θ, we need the meridional radius of curvature (which is equal to the major toric radius plus the minor toric radius) to be proportional to 1/cosθ; and therefore, under the constraint of thin-shell bending, the sagittal radius of curvature (which is equal to the minor toric radius) will be proportional to cosθ.