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I'm aware that the Milky Way has a dark matter 'halo' around it, presumably a spherically symmetric distribution.
But I'm completely ignorant regarding the theories explaining dark matter... Is there any reason to not expect a star-sized object to also be made of dark matter?
I know they'll be extremely difficult to detect, but I'm wondering if it's even physically possible to exist.
This depends on what you think dark matter is composed of.
One possibility is you could think dark matter is composed of MACHOS, in which case they are viewed as being composed of ordinary baryonic matter. There is some thought that these objects could actually be very cold neutron stars or black holes, in which case they are already dead stars well past the point of fusion. However, they could also be simply just smaller, colder objects such as asteroids, in which case it is entirely possible that over time, if their current seemingly stable orbits were sufficiently disturbed, they could collapse with other matter as part of normal star formation. This is similarly the case with RAMBOS. I think it is not controversial to think that a portion of dark matter is in fact composed by some MACHO or RAMBO like objects, since we have ample analogies of massive objects (like asteroids and planetoids) existing in the bounds of our own solar heliopause even if such things are extremely rarefied and difficult to detect.
More plausible and probably the leading candidate, is the idea that dark matter is composed of WIMPS. This term is sometimes misunderstood to include massive neutrinos, but even massive neutrinos are not massive enough to be good dark matter candidates, and are not WIMPS. WIMPS, while sharing similar interaction properties as neutrinos, are much more massive, and are most commonly associated with particles predicted by theories with supersymmetry. Since these objects only interact via the weak force, the characteristic scale of the interactions are significantly larger than that of the strong force, which is the force typically associated with star formation.
Hubble has actually managed to discern large dark matter structures, and NASA has been able to publish "photos" (which are actually false color images of detected distributions) showing dark matter. So while these objects are able to create structures on large scales, such as dark matter filaments, these are not structures we have full theories of. If the hypothesis is correct that these are composed of WIMPs, then these will not collapse to form stars in a regular sense (barring a general change in the definition of a star). However, it has been hypothesized that given enough time, complex structures could emerge, but at this point it is more speculation until we have proof of what dark matter is composed of.
See Is dark matter really present around the sun? and the links in the comments to the question.
It's hard to make a star from dark matter because there isn't any way for the infalling particles to lose their kinetic energy. Ordinary matter can shed energy as EM radiation, but dark matter doesn't feel the EM force.
If dark matter was self interacting, (and there is some evidence that it is self interacting) then it might form star-like clumps. Some groups even think that dark matter might be sort of copy of all the standard model particles.
If those star sized and massed clumps of dark matter were to pass through a nebula, it would (most likely) trigger star formation. And there seems to be evidence that dark matter does trigger starbursts.
https://www.sciencedaily.com/releases/2016/03/160309140048.htm
