Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.arcetri.astro.it/~nesti/pdfs/band_3and4_alma_elliptical_mirrors.pdf Дата изменения: Fri Sep 27 10:09:26 2002 Дата индексирования: Sat Dec 22 06:23:11 2007 Кодировка: Поисковые слова: внешние планеты

Band 3 and 4 ALMA elliptical mirrors
By Renzo Nesti, July 2002 Introduction This document is intended to give the detailed configuration of the elliptical mirrors used in the warm optics of band n. 3 and n. 4 of the ALMA receiver. The design have been already frozen and reported in [1]. An upgrade has been successively done and reported in the document `Band4optics2001jan10.pdf' by Sekimoto. This document is intended to give more detailed data to simplify mechanical study. Mirrors for band 3 and 4 The mirrors used in these two bands are two couples: one ellipsoidal shape and one plane. We assume that the surface to be realized is obtained as the intersection of the upper part of ellipsoid or the plane with an appropriate cylinder which is orthogonal to the vacuum vessel top plate (more details in the following table). Band number Mirror Ellipsoid Plane Ellipsoid Plane Generating cylinder diameter [mm] 163 115 109 100

3 4

Ellipsoid The ellipsoidal mirror is a portion of ellipsoid obtained by the revolution of an ellipse around its major axis. The minor axis of the ellipse is orthogonal to the vacuum vessel top plate. In this paragraph we point out the detailed parameters of this ellipse for both band 3 and 4. The reference sketch for the ellipse is drawn in fig. 1.

Figure 1. Generating ellipse for the ellipsoidal mirrors of ALMA bands 3 and 4 The x = -c straight line must coincide with the horn axis in the drawing of the top of the dewar The known parameters reported in [1] and in the document `Band4case1+2+4.pdf' are given in the following table. Band n. 3 4 ri [mm] 682.55 471.47 re [mm] 190.75 147.43 [deg] 50 65

From the above data it is possible to obtain the ellipse. The focal length of the mirrors is given by ri re ri + re We can then calculate the inter-focal distance 2c, the ellipse axis a, b and the eccentricity e by using f=


2c = ri2 + re2 - 2ri re cos( ) 2a = ri + re b = a2 - c2 c e= a The full set of parameters for the two ellipses is summarized in the following table. Band n. 3 4 ri [mm] 682.55 471.45 re [mm] 190.75 147.43 [deg] 50 65 d [mm] 163 109 f [mm] 149.09 112.31 c [mm] 289.35 215.20 a [mm] 436.65 309.44 b [mm] 327.02 222.35 e 0.66 0.70

The equation of the ellipse can be given both in cartesian and in polar coordinates: x a
2 2 2 2

+ =

y b

2 2

=1 1- e
2 2

cartesian polar


a

1 - [e cos( )]

By using the above equations the coordinates of the point S and Q of fig. 1 can be easily found. Method I With the assumption that xQ + xS = -2c (i.e. the end point abscissas of the mirror are symmetric with the horn axis) the values given in the following table come out. Band n. 3 4 Method II With the assumption that the edges of the elliptical mirror are almost-uniformly tapered the following system of equation results. xQ - x S = d xQ + c y
Q

xQ [mm] -207.85 -160.70

xS [mm] yQ [mm] yS [mm] -370.85 287.60 172.64 -269.70 190.01 109.01

Q [mm] 354.84 248.86

Q [deg] 125.86 130.22

S [mm] 409.06 290.90

S [deg] 155.04 157.99

=-

c+x ys

S

yQ xQ = -a 1 - b

2

y x S = -a 1 - S b The solution is obtained numerically and results are given in the following table. Band n. 3 4 xQ [mm] -190.68 -148.43 xS [mm] yQ [mm] yS [mm] -353.68 294.19 191.79 -257.43 195.10 123.38 Q [mm] 350.58 245.14 Q [deg] 122.95 127.26 S [mm] 402.33 285.47 S [deg] 151.53 154.39

2

References [1] J. W. Lamb et al. `ALMA receiver optics design,' ALMA memo n. 362, 11 Apr. 2001