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--- un:elliptical-galaxies [2026/02/09 07:37] – shuvo
+++ un:elliptical-galaxies [2026/02/09 07:50] (current) – shuvo
@@ Line 27: / Line 27: @@
 Expressing the luminosity in absolute magnitude, the empirical Faber-Jackson relation is:
-$$\rm log_{10}\sigma_0 = -0.1 M_B+ constant$$
+$$log_{10}\sigma_0 = -0.1 M_B+ constant$$
+==== - The Fundamental Plane ====
+As is readily apparent by inspecting observational data, there is considerable scatter in the Faber-Jackson relation. This is further reflected in the fact that the slope of the best-fit line through the data differs slightly from the idealized theoretical derivation. Apparently, the assumption that galaxies are a one-parameter family is not strictly true, which should not be surprising given the simplifications made in its development. Depending on the sample set used, the relation typically follows:
+$$L \propto \sigma_0^\alpha, \quad \text{where } 3 < \alpha < 5$$
+**Multidimensional Scaling
+**
+In an effort to find a tighter fit to the data, astronomers have introduced a second parameter into the expression: the effective radius ($r_e$). One representation of this empirical fit is:
+$$L \propto \sigma_0^{2.65} r_e^{0.65}$$
+In this framework, galaxies are visualized as residing on a two-dimensional "surface" within a three-dimensional "space" defined by the coordinates of luminosity ($L$), central velocity dispersion ($\sigma_0$), and effective radius ($r_e$).
+**Defining the Plane
+**
+Known as the Fundamental Plane, this relationship combines the contributions of a galaxy’s gravitational potential well ($\sigma_0$) with its physical scale and light output. Alternatively, the fundamental plane can be expressed in terms of the effective surface brightness ($I_e$) of a galaxy at its effective radius:
+$$r_e \propto \sigma_0^{1.24} I_e^{-0.82}$$
+The fundamental plane appears to represent the whole family of elliptical galaxies and serves as a significant constraint on theories regarding the formation and virialization of these stellar systems.