The focal egg

Focal egg for an array of telescopic heliostats with constant beaming angle β, outer radius ro, and inner radius ri = 0.2 * ro.

The egg of the Emperor Penguin is shaped something like a focal egg.

Definition: In solar engineering, a focal egg is an imagined directionally-selective blackbody radiator whose radiative emission would replicate the solar flux distribution at the center of a field of heliostats. 

Definition: In solar engineering, a minimal absorber, is the smallest absorber that can absorb all of the solar flux at the center of a field of heliostats.

It may seem that we have great freedom in where we point the beam of each telescopic heliostat. This is not true if we must use the same optics over the whole array, because the optics will have been optimized for a standard beaming angle (i.e., a particular angular elevation of the central ray of the beam.) Since considerable costs may be incurred in using different optics in different parts of the field, a better strategy is to use the same optics at their designed-for beam elevation over as much of the field as possible, perhaps aiming heliostats in the outer portions of the field slightly low, and heliostats in the inner portions of the field slightly high.

Consider an infinite field of telescopic heliostats, all of them having the same beam spread, σ, and beaming angle, β, where β ≈ σ. (Note that β is the elevation of the central ray, not the lowest ray which was used in an earlier post.) The focal zone in this case is an inverted cone with its apex in the center of the field at the same height as the secondaries. Since β ≈ σ, this focal cone has the property base ≈ altitude, making the angle at the apex of the cone about 2 * arctan(1/2) = 53°.

The smallest absorber (gray) that can intercept all of the radiation from a field of telescopic heliostats is a segment of a cone (apex angle slightly more than 53°) terminated by spherical caps. The focal egg is a directional blackbody radiator that mimics the array's flux distribution near the focus.

A practical heliostat field begins at some inner radius r_i and terminates at some outer radius r_o. To minimize the number of parameters in these calculations, we will arbitrarily assume r_i = 0.2 * r_o.

Since the beaming angle, β, and the beam spread, σ, are the same over the whole field, and also σ ≈ β, any light coming from the secondaries is confined to a passband of elevation angles between 0.5 β and 1.5 β. This passband of elevation angles is a fixed characteristic of the light coming from the array until it encounters the beam-down optics.

Suppose we take a ride up the axis of the focal zone. To keep the calculations simple, β is in radians, and we use the approximation tanβ ≈ β.

At first we see no sunlight, because looking as far up as sunlight from the secondaries could possibly be coming from (0.5 β below the horizon,) we can't see any secondaries.

The first sunlight from secondaries will be visible when we reach a height above the secondary mirrors, h, of

h = r_i * 0.5 β = 0.1 βr_o

We will see full flux—the same flux as from an infinite heliostat array—when secondaries fill our passband, our elevation-limited field of view, at a height of

h = r_i * 1.5 β = 0.3 βr_o

The infinite array model will continue to represent what we see until we begin to see desert beyond the farthest ring of secondaries at

h = 0.5 βr_o

The last sunlight from secondaries will be seen when the farthest secondaries are just too far below us to be seen at the steepest angle of elevation:

h = 1.5 βr_o.

We will try to model the distribution of flux near the focus by the contrivance of a directional blackbody radiator—its absorption and emission limited to the elevation passband mentioned above—whose back radiation would cancel the flux from the heliostats. With a finite heliostat array, the cancellation cannot be perfect because there will be locations, especially near the top of the focal zone, where the passband will not be filled: we will see desert where an infinite array would have heliostats. Viewed from  these locations, the directional blackbody, or focal egg, must have a reduced radius to compensate for the partially filled passband of the light from the heliostats.  The focal egg will therefore be somewhat smaller than the minimal absorber, the smallest body that could intercept all of the light coming from the array.

In the height range where the infinite array model can be applied, 0.3 βr_o < h < 0.5 βr_o, the focal zone is an inverted cone. Below the conical range, h < 0.3 βr_o, the flux falls to zero at 0.1 βr_o. Above the conical range, h > 0.5 βr_o, the flux falls to zero at 1.5 βr_o. As illustrated in the upper drawing, the focal egg can be approximated as a short segment of a cone terminated with ellipsoidal caps. The purpose of low-aiming the outer portion of the heliostat field, and high-aiming  the inner portion, would be to make the ellipsoidal caps more oblate.

If the optics of our system are perfect, the focal egg, with its directionally-limited thermal radiation, will recreate the heliostat array's flux distribution when the egg's temperature is that of the surface of the sun. With the oculus also at the sun's temperature, the beam-down optics will need to sited on the locus of satisfying the thermodynamic constraint that the two radiant fluxes are equal. From anywhere on that locus, the oculus will subtend the same solid angle as the portion of the focal egg seen within the passband. Assuming the size of the heliostat array is a given, the only parameters available to vary are the oculus radius and its z coordinate. Hopefully, some combination of those two parameters gives a locus approximately coincident with one of the family of parabolas satisfying the geometric constraint.