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Why don't the stars in a star cluster attract each other gravitationally, forming one big star? What causes a cluster to disperse the stars in it?

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"Dissipate" and "disperse" are the wrong way to approach this context. It implies the cluster undergoes a more compact state, followed by expansion - which is not the case, or not always.

First off, big blobs of gas don't form a single giant star for the same reason why the whole Arctic Ocean doesn't form a single giant iceberg - there's too much local motion for that. Instead, both gas in a proto-cluster, and water in the ocean, coalesce into smaller chunks - the individual stars, or the ice floes.

Following that, the evolution of a star cluster is very diverse. Sometimes they do undergo a sort of "compression", where there is a global motion of the stars towards the center. Other times they inflate and end up occupying a larger volume. Other clusters pulsate. Yet others are seemingly chaotic (although there's nothing truly random about the motion of each component star).

The stars in a cluster don't simply fall into each other for the same reason why the planets don't fall into the Sun: they are orbiting their common center of mass. In a cluster, each star follows a path described, in broad terms, as an orbit around the cluster's center; however, the nearest neighbors act as perturbations, and so most stars don't actually circle the center on perfect conic-section curves.

The dynamics are pretty complex and there's no one pattern to rule them all.

I suggest you download Universe Sandbox and run one of their cluster simulations.

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+1 for Universe Sandbox. – Chris White Dec 11 '12 at 9:10
Your answer seems to imply that most clusters don't actually disperse - or at least it doesn't say how they do so. In fact as most stars are born in clusters, yet are apparently isolated in the field, the opposite is true. Most clusters disperse and on a short timescale of tens of millions of years or less. – Rob Jeffries Mar 25 '15 at 16:57

I will assume you are talking about open clusters, since globular clusters hang around for many billions of years without dispersing.

Clusters form from cold clouds of molecular gas, that collapse and fragment into clusters of various sizes from $10$ to $10^5$ stars. However, we can tell from observational studies of the age distributions of clusters that they are not long-lived. Most of these clusters must disperse within about 10 million years - hence I suppose your question.

Star formation is inefficient. Typically, it is thought that only a few percent of the gas is turned into stars. That means the gravitational potential of a newly formed cluster is completely dominated by the gas. The virialisation time of this newborn cluster is short, so the stars and gas are initially in approximate equilibrium after a couple of million years. By equilibrium, we mean that the stars are in orbits around their common centre of mass. They cannot collapse further because they have kinetic energy and angular momentum, which is not easily dissipated.

What is then thought to happen is that the cluster loses the gas. Observationally we see that clusters that are older than about 5 million years are gas-free, whilst those that are younger tend to have at least some gas. The processes that cause this gas expulsion are still debated, but may include heating by the winds and ionising radiation of massive stars or the input of energy from the outflows of forming protostars, or even, after a few million years in the more massive clusters, a supernova could clear out the gas.

Once the gas is gone, then the cluster will be far out of equilibrium and will need to evolve on dynamical timescales to some sort of new equilibrium. This will result in it being less bound and may ultimately result in the disruption of the cluster, because a large fraction of stars will have velocities larger than the escape velocity of the new gas-free cluster.

Aiding these processes on longer timescales are dispersion due to two other processes. One is that as clusters evolve dynamically, the stars tend towards "equipartition" of energy. Thus lower mass stars attain higher velocity dispersions than higher mass stars. Stars in the high tails of the velocity distributions may be able to escape from the cluster's potential well. This is known as cluster evaporation. A second process is simple tidal disruption by the gravitational field of the Galaxy. The tidal radius of a cluster at the location of the Sun is roughy $1.4 (M_c/M_{\odot})^{1/3}$ parsecs. Stars that stray close to this limit are easily stripped from the cluster. For older clusters there is also the possibility of encounters with giant molecular clouds, which will also be effective in tidally stripping stars from the cluster.

As the cluster gets less and less massive, then tidal radii become smaller, thus enhancing the dispersion process.

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