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I read that elliptical galaxies stabilize due to the chaotic motion of stars. As I understand that means that the stellar motions are more nearly random in direction. So they perform independent movements along all major axes a,b,c (in case of a triaxial system). In contrary to spiral systems where they stabilize through orderly rotation. Are these statements correct? If yes, how can I understand the process of stabilization due to chaotic motion in ellipticals. What I think I understood is that the stars are considered as a collision-free gas that is relaxed through violent relaxation (redistribution of kinetic energy so that a thermal equilibrium is created).

If the orbits of the stars are random in ellipticals, how is it that I can see rotation, through red- and blueshifts of the respected upper and lower parts of the observed galaxies in the post I made here: How to extract galaxy spectra for different radii in Python for spectra taken by long slit spectrograph?

Thank you for your clarifications..

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    $\begingroup$ “Random” motions are not automatically “chaotic” motions. (And chaotic motions can be confined to very limited regions of space.) $\endgroup$ Jan 5, 2023 at 11:07

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I wouldn't say that elliptical galaxies "stabilize due to the chaotic motion of their stars", but ellipticals are the result of galaxies merging, and depending on the orientation of those galaxies' angular momentum, the end result — after it has stabilized, or virialized — will usually be mostly random motions of the stars.

If the net angular momentum is not zero (and the galaxy hasn't been able to rid itself of excess angular momentum), then in addition to the random motion there will be some ordered motion. This will usually (but not always) correlate with the flattering (described by the E0–E7 labels).

Thus, it is correct that ellipticals and spirals are predominantly supported by random motions along all axes and ordered rotation, respectively. But the distinction isn't necessarily as clear-cut as you may think: Ellipticals may have an overall rotation (especially dwarf ellipticals; e.g. Pedraz et al. 2002), and spirals also have random motion of their stars in their plane.

You can measure the bulk rotation velocity $V_\mathrm{rot}$ across a galaxy and compare it to the velocity dispersion $\sigma_V$. For a given galaxy you can then determine the ratio $V_\mathrm{rot} / \sigma_V$: If it's larger than 1, then the galaxy is rotation-dominated, whereas if it's smaller than 1 it's dispersion-dominated.

For a galaxy like the Milky Way, $V_\mathrm{rot} \sim 200\,\mathrm{km}\,\mathrm{s}^{-1}$ while $\sigma_V \sim 20\,\mathrm{km}\,\mathrm{s}^{-1}$, so MW is clearly rotation-dominated with a ratio of $V_\mathrm{rot} / \sigma_V \sim 10$.

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I think there are two parts to your question. First, what is meant here by the term stabilize, and second does chaotic motion stabilize elliptical galaxies (and if so how)?

Meaning of 'stabilize': As long as they are not strongly interacting, galaxies are (to very good approximation) in virial equilibrium, which implies (virial theorem for self-gravitating systems) $$ \langle v^2\rangle = GM/R, $$ where $\langle v^2\rangle$ is the overall mean-square velocity, $M$ the total mass, and $R$ a typical radius. Thus, the stars (and other objects including dark-matter particles) must move by just the right amount. If virial equilibrium is not satisfied, the galaxy either collapses (if $\langle v^2\rangle < GM/R$) or expands/explodes (if $\langle v^2\rangle > GM/R$), i.e. is unstable.

The overall mean-square velocity $\langle v^2\rangle$ is the density-weighted mean of the local mean-square velocity $\overline{v^2}$: $$ M \langle v^2\rangle = \int\mathrm{d}^3x\,\rho\,\overline{v^2}. $$ Moreover, we can always express the (local) mean-square velocity as $$ \overline{v^2} = \overline{\vec{v}}^2 + \sigma^2, $$ the sum of the square of the mean velocity $\overline{\vec{v}}$ (accounting for ordered motions) and the velocity dispersion $$ \sigma^2 = \overline{\left(\vec{v}-\overline{\vec{v}}\right)^2} $$ (accounting for un-ordered or random motions).

Stability by random motions? There are various ways in which stars can move to achieve virial equilibrium. One possibility, realized in disc galaxies, is to move on co-planar near-circular orbits (with the same sense of rotation). In this case $\sigma\ll|\overline{\vec{v}}|\approx v_{\mathrm{rot}}$, and hence $\overline{v^2}\approx v_{\mathrm{rot}}^2$: the galaxy is stabilized by rotation.

Another option is to move on non-circular (elliptic or rosette-shaped) orbits with random orbital planes and senses of rotation. In this case $\overline{\vec{v}}=0$ such that $\langle\sigma^2\rangle=GM/R$, i.e. the galaxies is stabilized by random motions. For elliptical galaxies $|\overline{\vec{v}}|\ll\sigma$, and hence $\langle v^2\rangle\approx\langle\sigma^2\rangle$: they are largely stabilized by random motions (even if some rotate).

Stability by chaotic motions? The term random motion refers to unordered motion, i.e. the deviation of individual velocities $\vec{v}$ from the (local) average velocity $\overline{\vec{v}}$. However, the term chaotic motion means something completely different, namely motion on chaotic orbits. The vast majority of orbits occupied in galaxies are regular and not chaotic (if the majority of orbits were chaotic, they could not support an aspherical shape, whilst spherical galaxies contain no chaotic orbits). Hence, the statement that elliptical galaxies are supported by chaotic motion is WRONG.

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