en>Tabletop: Spell enlosing => enclosing (4)

2013-09-12T04:12:12Z

Spell enlosing => enclosing (4)

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	~~Friends contact her Felicidad~~ and ~~her husband doesn't like~~ it at ~~all~~. ~~Alabama has always been~~ his ~~house~~. ~~Bookkeeping~~ is ~~what she does~~. ~~Bottle tops gathering~~ is the ~~only pastime his spouse doesn~~'~~t approve of~~.<br><br>~~Here is my web site;~~ [http://~~www~~.~~Muh~~-~~Tang~~-~~clan-onyxia~~.~~de/index~~.~~php?mod~~=~~users~~&~~action~~=~~view~~&id=~~7622 extended auto warranty~~]		[[Image:Faber Jackson.png\|thumb\|450px\|right\|[[Velocity dispersion]] (y-axis) plotted against [[absolute magnitude]] (x-axis) for a sample of [[elliptical galaxies]], with the Faber–Jackson relation shown in blue.]]
			The '''Faber–Jackson relation''' is an early [[empirical]] [[power-law]] relation between the [[luminosity]] <math>L</math> and the central [[star\|stellar]] [[velocity dispersion]] <math>\sigma</math> of [[elliptical galaxy\|elliptical galaxies]], first noted by the astronomers [[Sandra M. Faber]] and [[Robert Jackson (scientist)\|Robert Earl Jackson]] in 1976. The original relation can be expressed mathematically as:

			:<math>
			L \propto \sigma^ \gamma
			</math>

			where the index <math>\gamma</math> is observed to be approximately equal to 4, but depends on the range of galaxy luminosities that is fitted. The Faber–Jackson relation is now understood as a projection of the [[fundamental plane (elliptical galaxies)\|fundamental plane]] of elliptical galaxies. One of its main uses is as a tool for determining distances to external galaxies.

			== Theory ==

			The [[Gravitational binding energy\|gravitational potential]] of a mass distribution of radius <math>R</math> and mass <math>M</math> is given by the expression:

			:<math>
			U=-\alpha \frac{GM^2}{R}
			</math>

			Where α is a constant depending e.g. on the density profile of the system and G is the [[gravitational constant]]. For a constant density, α = <math>\frac{3}{5}</math>

			The kinetic energy is (recall <math>\sigma</math> is the 1 dimensional velocity dispersion. Therefore 3<math>\sigma</math><sup>2</sup> = V<sup>2</sup>):

			:<math>
			K = \frac{1}{2}MV^2
			</math>

			:<math>
			K = \frac{3}{2}M \sigma^2
			</math>

			From the [[Virial Theorem]] (<math>2 K + U = 0</math> ) it follows

			:<math>
			\sigma^2 =\frac{1}{5}\frac{GM}{R}
			</math>.

			If we assume that the mass to light ratio, <math>M/L</math>, is constant, e.g. <math>M \propto L</math> we can use this and the above expression to obtain a relation between <math>R</math> and <math>\sigma^2</math>:

			:<math>
			R \propto\frac{LG}{\sigma^2}
			</math>.

			Let us introduce the surface brightness, <math>B = L/(4 \pi R^2)</math> and assume this is a constant
			(which from a fundamental theoretical point of view, is a totally unjustified assumption) to get

			:<math>
			L=4\pi R^2 B
			</math>.

			Using this and combining it with the relation between <math>R</math> and <math>L</math>, this results in

			:<math>
			L \propto 4\pi\left(\frac{LG}{\sigma^2}\right)^2B
			</math>

			and by rewriting the above expression, we finally obtain the relation between luminosity and velocity dispersion:

			:<math>
			L \propto\frac{\sigma^4}{4\pi G^2 B}
			</math>,

			that is

			:<math>
			L \propto \sigma^4
			</math>.

			When account is taken of the fact that massive galaxies originate from homologous merging, and the fainter ones from dissipation, the assumption of constant surface brightness can no longer be supported. Empirically, surface brightness exhibits a peak at about <math>M_V=-23</math>. The revised relation then becomes

			:<math>
			L \propto \sigma^{3.1}
			</math>

			for the less massive galaxies, and

			:<math>
			L \propto \sigma^{15.0}
			</math>

			for the more massive ones. With these revised formulae, the fundamental plane splits into two planes inclined by about 11 degrees to each other.

			Even first-ranked cluster galaxies do not have constant surface brightness. A claim supporting constant surface brightness was presented by astronomer [[Allan R. Sandage]] in 1972 based on three logical arguments and his own empirical data. In 1975, Donald Gudehus showed that each of the logical arguments was incorrect and that first-ranked cluster galaxies exhibited a standard deviation of about half a magnitude.

			==Estimating distances to galaxies==
			Like the Tully-Fisher relation, the Faber–Jackson relation provides a means of estimating the distance to a galaxy, which is otherwise hard to obtain, by relating it to more easily observable properties of the galaxy. In the case of elliptical galaxies, if one can measure the central stellar velocity dispersion, which can be done relatively easily by using [[spectroscopy]] to measure the [[Doppler shift]] of light emitted by the stars, then one can obtain an estimate of the true luminosity of the galaxy via the Faber–Jackson relation. This can be compared to the [[apparent magnitude]] of the galaxy, which provides an estimate of the [[distance modulus]] and, hence, the distance to the galaxy.

			By combining a galaxy's central velocity dispersion with measurements of its central surface brightness and radius parameter, it is possible to improve the estimate of the galaxy's distance even more. This standard yardstick, or "reduced galaxian radius-parameter", <math>r_g</math>, devised by Gudehus in 1991, can yield distances, free of systematic bias, accurate to about 31%.

			==See also==
			* [[Fundamental plane (elliptical galaxies)]]
			* [[Tully–Fisher relation]]
			* [[M-sigma relation]]

			==External links==
			* [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1976ApJ...204..668F&db_key=AST&data_type=HTML&format=&high=4481ca366126254 The original paper by Faber & Jackson]
			*[http://adsabs.harvard.edu/abs/1991ApJ...382....1G Gudehus's revision of the Faber–Jackson relation]

			{{DEFAULTSORT:Faber-Jackson relation}}
			[[Category:Elliptical galaxies\|-]]
			[[Category:Extragalactic astronomy]]

en>ClueBot NG: Reverting possible vandalism by 63.247.165.234 to version by 87.99.27.160. False positive? Report it. Thanks, ClueBot NG. (972104) (Bot)

2012-03-23T12:54:10Z

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Friends contact her Felicidad and her husband doesn't like it at all. Alabama has always been his house. Bookkeeping is what she does. Bottle tops gathering is the only pastime his spouse doesn't approve of.<br><br>Here is my web site; [http://www.Muh-Tang-clan-onyxia.de/index.php?mod=users&action=view&id=7622 extended auto warranty]

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en>Tabletop: Spell enlosing => enclosing (4)

en>ClueBot NG: Reverting possible vandalism by 63.247.165.234 to version by 87.99.27.160. False positive? Report it. Thanks, ClueBot NG. (972104) (Bot)