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Star formation (SF), the process by which stars are born, is thought to be understood in outline, but questions remain, perhaps more than for models of subsequent stellar evolution. Clouds sufficiently dense will undergo gravitational collapse to achieve the density to trigger fusion, but among the remaining questions is how and when this cloud density arises.
Molecular clouds (cold gas, gas that has cooled sufficiently to form molecules) apparently sometimes develop patches dense enough to collapse into stars, the areas of such activity known as stellar nurseries or star-forming regions. Some such dense regions may result from collisions of clouds, or even whole galaxies. Movement of cold gas, which would (sometimes) produce such high density can be due to nearby supernovae and/or radiation pressure from nearby early stars (radiation driven implosion or RDI). For any of this to happen, there must exist available cold gas within the galaxy and galaxy-accretion of cold gas from the intergalactic medium increases the probability of star formation and can trigger it as well.
As stars form, any nearby dust will be heated. This spreads the heat energy, giving it a larger surface and lower temperature, with a black-body spectrum concentrated at longer wavelength electromagnetic radiation such as radio. As a consequence, the presence of recent star formation can result in short wavelengths from the early stars produced, but what is sometimes observable is longer wavelengths from heated dust surrounding the star formation. As a general rule regarding galaxies, the more infrared it emits (a sign of heated dust), the higher the star formation rate. Star formation in distant galaxies is of interest as the amount of star formation appears to have grown and diminished over the history of the universe (star formation history) and mechanisms that might do that are of interest. H-alpha and the HI line are used to measure SF in distant galaxies. Lyman alpha can also indicate distant SF and Lyman-alpha emitters are presumed to have a very high SF rate.
The specifics of star formation of early stars are not as well understood as for lower-mass stars because radiation pressure would seem to limit the process. Theories include merging lower-mass stars, or that the radiation is anisotropic, i.e., less in some direction(s), the lesser directions constituting a "hole in the wall" through which gas accretion can continue.
As reflected in the initial mass function's greater-than-2 exponent, the vast majority of star formation, by mass, is in later, lower mass stars, e.g., much more mass amongst M-type stars than O-type stars. This means that one useful sign of star formation in distant galaxies, the colors of early stars, is imperfect because if some star formation regions don't trigger whatever it is that creates high mass stars, they could be missed.
The term quenching is used to indicate the cessation of star formation, e.g., in a galaxy, which would be said to become quiescent. For the long term, a star forming region is assumed to cease due to gas heating from hot stars and supernovae (star formation feedback), followed by cooling and settling and perhaps triggers from nearby events. Thus a long-term star formation rate must be time-averaged over periods of high and low star formation. Such an oscillation is termed episodic star formation. Dwarf galaxies show evidence of this, e.g., through their stellar demographics and the mechanism by which this happens is of interest.
The term star and planet formation (SPF) covers star formation and planet formation as well, which is believed to happen during the first few million years of a star's life.