Fundamental plane (elliptical galaxies)

Correlations

The following correlations have been empirically shown for elliptical galaxies:

• Larger galaxies have fainter effective surface brightnesses (Gudehus, 1973). Mathematically speaking: ${\displaystyle R_e\propto ⟨I{⟩}_e^{-0.83\pm 0.08}}$ (Djorgovski & Davis 1987), where ${\displaystyle R_e}$ is the effective radius, and ${\displaystyle ⟨I{⟩}_e}$ is the mean surface brightness interior to ${\displaystyle R_e}$.
• As ${\displaystyle L_e=\pi ⟨I{⟩}_e{R}_e^2}$ measuring observable quantities such as surface brightness and velocity dispersion, we can substitute the previous correlation and see that ${\displaystyle L_e\propto ⟨I{⟩}_e⟨I{⟩}_e^{-1.66}}$ and therefore: ${\displaystyle ⟨I{⟩}_e\sim L^{-3/2}}$ meaning that more luminous ellipticals have lower surface brightnesses.
• More luminous elliptical galaxies have larger central velocity dispersions. This is called the Faber–Jackson relation (Faber & Jackson 1976). Analytically this is: ${\displaystyle L_e\sim {\sigma }_o^4}$. This is analogous to the Tully–Fisher relation for spirals.
• If central velocity dispersion is correlated to luminosity, and luminosity is correlated with effective radius, then it follows that the central velocity dispersion is positively correlated to the effective radius.

Usefulness

The usefulness of this three dimensional space ${\displaystyle \left(\log R_e,⟨I{⟩}_e,\log {\sigma }_o\right)}$ is studied by plotting ${\displaystyle \log R_e}$ against ${\displaystyle \log {\sigma }_o+0.26{\mu }_B}$, where ${\displaystyle {\mu }_B}$ is the mean surface brightness ${\displaystyle ⟨I{⟩}_e}$ expressed in magnitudes. The equation of the regression line through this plot is:

${\displaystyle \log R_e=1.4\log {\sigma }_o+0.36{\mu }_B+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.}$

or

${\displaystyle R_e\propto {\sigma }_o^{1.4}⟨I{⟩}_e^{-0.9}}$.

Thus by measuring observable quantities such as surface brightness and velocity dispersion (both independent of the observer's distance to the source) one can estimate the effective radius (measured in kpc) of the galaxy. As one now knows the linear size of the effective radius and can measure the angular size, it is easy to determine the distance of the galaxy from the observer through the small-angle approximation.

Variations

An early use of the fundamental plane is the ${\displaystyle D_n-{\sigma }_o}$ correlation, given by:

${\displaystyle \frac{D_n}{\text{kpc}}=2.05{\left(\frac{{\sigma }_o}{100\text{km}/\text{s}}\right)}^{1.33}}$

determined by Dressler et al. (1987). Here ${\displaystyle D_n}$ is the diameter within which the mean surface brightness is ${\displaystyle 20.75{\mu }_B}$. This relationship has a scatter of 15% between galaxies, as it represents a slightly oblique projection of the Fundamental Plane.

Fundamental Plane correlations provide insights into the formative and evolutionary processes of elliptical galaxies. Whereas the tilt of the Fundamental Plane relative to the naive expectations from the Virial Theorem is reasonably well understood, the outstanding puzzle is its small thickness.

Interpretation

The observed empirical correlations reveal information on the formation of elliptical galaxies. In particular, consider the following assumptions

• From the virial theorem the velocity dispersion ${\displaystyle \sigma }$, characteristic radius ${\displaystyle R}$, and mass ${\displaystyle M}$ satisfy ${\displaystyle {\sigma }^2\sim GM/R}$ so that ${\displaystyle M\sim {\sigma }^{2}R}$.
• The relation between luminosity ${\displaystyle L}$ and mean surface brightness (flux) ${\displaystyle I}$ is ${\displaystyle L\propto I{R}^2}$.
• Assume homology which implies a constant mass to light ratio ${\displaystyle M/L}$.

These relations imply that ${\displaystyle M\propto L\propto I{R}^2\propto {\sigma }^{2}R}$, therefore ${\displaystyle {\sigma }^2\propto IR}$ and so ${\displaystyle R\propto {\sigma }^2{I}^{-1}}$.

However, there are observed deviations from homology, i.e. ${\displaystyle M/L\propto L^{\alpha }}$ with ${\displaystyle \alpha =0.2}$ in the optical band. This implies that ${\displaystyle M\propto L^{1+\alpha }\propto I^{1+\alpha }{R}^{2+2\alpha }\propto {\sigma }^{2}R}$ so ${\displaystyle R\propto {\sigma }^{2/(1+2\alpha )}{I}^{-(1+\alpha )/(1+2\alpha )}}$ so that ${\displaystyle R\propto {\sigma }^{1.42}{I}^{-0.86}}$. This is consistent with the observed relation.

Two limiting cases for the assembly of galaxies are as follows.

• If elliptical galaxies form by mergers of smaller galaxies with no dissipation, then the specific kinetic energy is conserved ${\displaystyle {\sigma }^2=}$constant. Using the above-mentioned assumptions implies that ${\displaystyle R\propto I^{-1}}$.
• If elliptical galaxies form by dissipational collapse then ${\displaystyle \sigma \propto (GM/R{)}^{1/2}}$ increases as ${\displaystyle R}$ decreases for constant ${\displaystyle M}$ to satisfy the virial theorem and ${\displaystyle M\propto L\propto I{R}^2}$ implies that ${\displaystyle R\propto I^{-0.5}}$.

The observed relation ${\displaystyle R_e\propto ⟨I{⟩}_e^{-0.83\pm 0.08}}$ lies between these limits.