Design of large steerable antennas

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Abstract
The design of large, steerable, single reflectors is investigated in general, in order to find the basic prin- ciples involved and the most economical solutions. (Let X be shortest wavelength, D the antenna diameter, W the antenna weight; p the material density, S the maximum stress, E the modulus of elasticity, C the thermal expansion coefficient; Ps the survival surface pressure, Po the observational wind pressure, A the tem- perature differences; and dimensionless constants.) There are three natural limits for tiltable antennas: first, the thermal deflection limit, x= CAD= 2.4 cm (D/100 m); second, the gravitational deflection limit, X = (p/E)D2 =5.3cm (D/100 m)2 third, the stress limit, D = S/p =620 m. Let each antenna be a point in a D, x plane. The part of the plane permitted by the three limits then can be divided into four regions accord- ing to what defines the antenna weight. First, the gravitational deflection region (W governed by p/E); second, the wind deflection region (governed by p0/E); third, the survival region (governed by p,/S) ; fourth, the minimum structure region (stable self-support). Formulas are derived for the regional boundaries, and for W (D,X) within each region. Some aspects of economy are considered. There is a most economical X for any D. Radomes can give ad- vantage within the wind deflection region only, which means for D 8 cm. An economical antenna with D =150 m and X =20cm should cost about four million dollars. There are four means of passing the gravitational limit. First, avoiding deflections by not moving in eleva- tion angle (fixed-elevation transit telescope). Second, fighting the deflections with motors (Sugar Grove). Third, canceling the deflections with levers and counterweights (large optical telescopes). Fourth, designing a structure which deforms unhindered, but which deforms one paraboloid of revolution into another one (homologous deformation). It is proved that exact homology solutions do exist, and an explicit solution is given for two dimensions.

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