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{zCos[e] + x Cos [f ] Sin [ e ] + y Sin [e] Sin[f] xCos[e] Cos [f ] - z Sin [e] + y Cos [e] Sin[f] y Cos[f] - x Sin[f]}

Cos[f] Sin[e] Cos[e] Cos[f] - Sin[f] Sin[e] Sin[f] Cos[e] Sin[f] Cos[f] Cos[e] - Sin[e] 0

The results (output) are the product of Eq. (2.1) and the transposed matrix of matrix A, respectively.

The equation of the paraboloid is a) in spherical coordinates p = 2 f / (1 + cos y) = f sec2 (y / 2) (2.4)

b) in cartesian coordinates r2 = x2 + y2 = 4 f(f + z)

where the paraboloidal coordinates (u, v, x) are connected to the cartesian ones by and c is the azimuthal angle about the z-axis.

In this coordinate system the paraboloid's surface is described by u = constant (or v = constant). At the vertex V we have v = 0, hence z = y u2 = f and thus u = V"2 f. Also dp = vdv for u = constant. The infinitesimal surface element is dS = VU2 + v2 uvdv d^ = 2Vf p dp dc = 2cos(-2)p dp dc (2.7)

where we have used Eq. (2.4). On eliminating p we obtain for the surface element and the surface of the paraboloid with aperture half-angle Y0 follows as the integral of Eq. (2.8)

f2 A sec3 ( Y--M - 1] = 8t f2 [{1 + (-df M2}T- 1]. (2.9)

Here we have used the alternative expression tan(y/ 2) = r / 2 f, (r2 = x2 + y2),

which is easy to derive from Eq. (2.4), considering that sin y = r / p. From this also follows tan ( -2- M = 4f. (2.10b)

The angle Y0is called the "aperture (half-)angle" of the paraboloid, i.e. the angle between the axis and the "edge-ray" from FP. The "depth" D of the paraboloid is given by w = f ( ff M2 = f (tan ^ )2- (2-11)

The following relations also hold r/ f sin y=-------5-, (2.12)

The formulas for the secondary reflectors in spherical coordinates are hyperboloid (Cassegrain) rs = (c2 - a2 )/(a + c cos y) (2.14)

ellipsoid (Gregorian) rs = (c2 - a2 )/(c - a cos y) (2.15)

where c and a are the usual parameters describing the conic sections (see above).

The Cassegrain system is characterised by the magnification factor m, connected to the eccentricity e = c / a of the hyperboloidal secondary by the relations m = (e + 1) / (e - 1) (2.16a)

The "equivalent paraboloid" of the Cassegrain antenna is given by tan f__r_

tan 2 = 2 m f from which follows tan F- = -Tf , (2.17)

2 4mf' v y where F0 is the opening half-angle of the secondary reflector seen from the Casse-grain focus. Hence we have m = tan ( Y- )/tan ( % ) (2.18)

where c is the "focal length" and ds the diameter of the hyperboloid. The distance between primary and secondary focus of the Cassegrain system is fc = 2 c and the distance from primary focus to secondary vertex is l = c - a (see Fig. 2.1). The following relations hold fc = 2c l = c - a = c (e - 1)/e a = c / e = c (m - 1)/(m + 1)

In the special case where the secondary focus coincides with the vertex of the primary reflector (i.e. f = f), we have the situation that Eq. (2.17) can be written as tan HFo / 2) = ds / 4 f, leading to the simple expression m = dp / ds.

Table 2.1. Geometry of the ALMA antenna

Parameter

Symbol

Magnitude

Prime reflector diam.

dp

12 m

Primary focal length

fp

4.8 m

Primary focal ratio

fp / dp

0 -1

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