File:Some Problems Illustrating the Forms of Nebulae (1915) (14585888600).jpg

Identifier: philtrans02048693 (find matches)
Title: Some Problems Illustrating the Forms of Nebulae
Year: 1915 (1910s)
Authors: Walker, G.
Subjects: Proceedings of the Royal Society of London Philosophical Transactions of the Royal Society
Publisher: Royal Society of London

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l form giving real positive density exists, and it may be a matterof interest to pure mathematicians to investigate this point, just as it wouldbe of value to applied mathematicians to know the most general form. Case I.—If we take (£> = (r Ja)n cos nd, ty = (r/a)n sin n6, we get P = poehY w a 2 7ryha2 ? 2n2 a? - —- sech2 n losj (r/a) n iryha2 7 a ma ,aj \r <a rel="nofollow">n -2 This is the most general case of circular symmetry. We note that when n 1 the densityis zero at the origin, rises to a finite maximum, and then diminishes andvanishes at infinity. Writing the equation in the form p/pi = 4 a r n - + a ,n -2 the curves in fig. 1 show pjpi as a function of r/a, for the four specialvalues n = 0, J, 1, and 2; a is still at our disposal while pi = 2n2jiryha2 isdetermined when n and a are fixed.
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Some Problems Illustrating the Forms of Nebulce. 413 Boyles Law involves the possibility of an infinite density, and the lawdoes not hold physically so far. In this and In other cases where asingularity occurs we may replace it by a solid nucleus arranged so as togive the proper value of V over its surface. This case of circular symmetry for n > 1 appears to have some analogy toa ring nebula. Case II—Take 4> = h~1\ogrja, i/r = b~10) 2¥ ,2 we get p = P0^ = -^-j ± (log2 (r/a) +ffi> + &»)-». The surfaces of equal density are given by P = -loo-2 Z-W+ (J^Jf*. The curve, fig. 2, shows the particular surface 02 — —log2 1 + ~, a f drawn to a scale of 10.philtrans02048693

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