By Rudolf Kurth

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Translated and revised from the 1986 Russian version.

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**Additional resources for Dimensional Analysis And Gr Theory In Astrophysics**

**Example text**

4), Thus X2 - f421y1( 0 ) - A22y2 (:2 -A = x2 ~ ) - ~2lf4;:~l. =o, ~ A ~ which implies the assertion. 6. 4), shows that 22 - A ~ ~ A ; : = $ ~x2 - ~ 2 1 ~ ~ l ' x l Therefore, for any real function y defined on R"-', the equation y (x2 - ~21A;:xl)= 0 is invariant under the group asserted. 7. 3): If r = n, then A = (All, A1219 and the equation Y(x - Ay) = 0 takes the form y(x1 - Ally1 - A12~2)= 09 This holds for arbitrary y1 and y2: thus, in particular, for y2 arbitary, and y1 = y:o' defined by (0) Hence Ally1 + A12y2 = 0.

Example. For a typical globular cluster, R x 30 (parsec) x 9 x 1019 (cm), V w 1 (km sec-I) = 1 x lo6 (cm sec-I). 14 is applicable, Mm V2R w 1x G w 5 (gram) x lo5 (solar masses). 16. Example. For a typical open star cluster, R =3 (parsec) V x 1 (kmsec-l) w 9 (cm) , x 10' = 1 x lo5 (cmsec-I); therefore, if Proposition 2. I4 is applicable, M w 5x lb2 (solar masses). 17. Example. Suppose a cluster of galaxies has a radius Rw2 2 lo5 (PC), 27 T H E TECHNIQUE OF DIMENSIONAL ANALYSIS a mean "internal" velocity V = 1000 (km sec- l) and contains about 1000 members.

Thus an asymptotic formula for E -r a, rather than for 5 0 is to be devised. The approach is different from the procedures applied before. It consists of three steps : Firstly, tentatively assume that ~ ( 5 = ) const 5" for some positive reala a s & +00, Secondly, the expression obtained for T will be given a concrete (though vague) physical interpretation. This justifies the [a-assumption. Thirdly, it will even suggest that or = 1 is the appropriate value for the exponent. 9. Suppose, firstly, that y(E) is asymptotically proportional to Then, T = const (vn112)- ( v ~ / G % ~as~ ~ V* ~ )co~.