If two stars of any type were to form near each other, how closely can they form before something prevents them from being two distinct stars?
Astrophysics – Smallest Possible Distance Between Two Stars Explained
astrophysicsbinary-starsdistancestars
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There are numerous distance indicators used for within the galaxy. The most common way is by using intrinsic magnitude. By knowing how bright an object would be if we were close, we can determine how far away it is by how dim it is. There are many types of stars where we have a rough idea of how bright they should be due to characteristics of the star:
Cephied Variables: The original type of variable star that was used by Hubble to determine the distance to the Andromeda Galaxy.
RR Lyrae Variable: Like the Cephied variable, but usually dimmer.
Type 1a Supernova: These guys, unlike the first two, are cataclismic variables. Essentially a binary white dwarf slowly accretes matter from its binary till it reaches the Chandrashankar Limit, after which point it explodes in a very characteristic way (since the mass at the time of explosion is roughly constant).
Main Sequence Stars: Generally less accurate than the first 3, there are some types of main sequence stars which are used to find distances in a similar way.
There are a few other ways we can measure distances:
Perpendicular Movement: For example there is a "light echo" from SN 1987A which is essentially light from the supernova interacting with dust around the old star. Since this echo should be expanding at the speed of light, we can tell how far away the nova is by the angular velocity of the light.
Relative Velocity in a Moving Cluster: (see dmckee's answer)
Tulley-Fisher relation: A relationship between the luminosity of the galaxy and it's apparent width. Can be used as a decent distance calculator.
Faber-Jackson Relation: Similar to Tulley-Fisher, relates luminosity with radial velocity dispersion rate.
EDIT: Some more information about redshifts.
The whole relationship between redshift and distance was in fact established by Hubble by relating distance to Cephied variables (I believe) with redshift. Later on it was made more precise using supernova, which are brighter and can be seen from much father away (I think recent supernova can be occasionally seen around Z=2, while Cephieds are all Z<1). Within a galaxy, redshift cannot be used directs since the "peculiar velocity," the velocity within the galaxy, completely overshadows the effects of universe expansion on which Hubble's Law is based. Redshift within the galaxy is useful for certain other techniques.
EDIT: corrected a few minor errors.
It would be possible, but very unlikely, since the orbits wouldn't be stable.
Try to take a look at this visualization of the gravitational potential of a binary star system (from the Wikipedia Roche Lobe entry):
If the planet orbits just one of the stars, its orbit will be inside one of the lobes of the thick-lined figure eight at the bottom part, analogous to a ball rolling around inside one of the "bowls" on the 3D-figure. Such an orbit will be stable, just like the Earth's around the sun (bar perturbations from other planets, but let's leave them out for now), and there will be many different orbital energies for which this is true.
The same goes for an orbit around both stars: the planet will have many different energy levels at which it would simply experience the two stars' gravity combined as the gravity of one single body (and in which case the figure wouldn't apply, since it would be practically unaffected by the two stars orbiting each other).
In order to orbit in a figure eight, you have to imagine that the ball has to roll across the ridge between the two indentations in the 3D part of the figure. It is clear that this is possible, but also intuitively clear that this would only be possible for a narrow range of orbital energies (a little less and it would go into one of the holes, a little more and it would simply just orbit them both), and that it would not be a stable orbit. The ball would have to roll in an orbit where it exactly passes the central saddle point at the ridge (L1) in order to stay stable, the tiniest little imperfection will get it perturbed even further away from its ideal trajectory.
Your 5-body system could possibly be timed in such a way that it would work, but it would suffer the same fundamental flaw, and as far as I can see, it would also introduce even more sources of instability into the system.
This is, by the way, the gravitational potential in the rotated coordinate system, and you can see from the symmetry of the system that the coreolis preference you mention is not present. A simple symmetry argument should convince you of the same, though: Assume the system is rotating clockwise. This should allegedly give you a preference for one of the stars. But if you now let the system continue, while you rotate yourself 180 degrees up/down, it will now be rotating counter clockwise, which should give a coreolis preference for the other star, which of course cannot be the case, since there is no preferred up/down direction in a system like this.
Best Answer
You can look at databases of binary stars to tell you what the range of orbital periods/separations of stars currently are. Unfortunately that isn't going to answer your question because many short-period binary systems have evolved to be that way - e.g. the short-period cataclysmic variable stars or the "contact" binaries known as W UMa systems, where the stars are actually touching each other and have a common envelope. Binaries can also be made "harder" by interactions with third bodies, especially during their early lives if they are born in dense star clusters, or the orbits may be shrunk or the stars may even merge due to gas-induced orbital decay during the early embedded phase of the formation of a star cluster (Korntreff et al. 2012) . Hence there are many older short period binary systems with periods shorter than one day, where the components are barely separated, or are not separated at all.
What you need to do is look for a catalogue of binary objects in the youngest star forming regions or star clusters, where you might reasonably assume there has been little time for interaction (to some extent it depends whether you include early life in a star forming environment as part of the birth process or not).
It is actually quite hard to do surveys of this kind. You need repeated, high resolution spectroscopic measurements of quite faint objects in order to find the doppler shifts due to binary motion. But some work is out there. Meibom et al. (2006) search for spectroscopic binaries in the young(ish) clusters M34 and M35 (250 and 150 Ma respectively). They find 6 binaries in these clusters which have separations smaller than 0.12 au and periods below 13 days. The shortest period binary has a $0.9M_{\odot}$ primary with a period of 2.25 days. Nevertheless, the authors concede that these binaries may have been hardened by close third-body encounters.
Morales-Calderon et al. (2012) pursued a different approach; performing photometric monitoring in the infrared to search for eclipsing binaries in the very young Orion Nebula cluster (age about 2 Ma). They found six binaries with periods between 3.9 and 20.5 days.
There are then quite a few other individual results and studies. I can't immediately locate any good compilation, but this is the closest binary that a brief search uncovered: Bakis et al. (2011) analysed the binary IM Mon, probably a 10 Ma old system in Orion. They find an orbital period of only 1.19 days and primary and secondary masses of 5.5 and 3.3$M_{\odot}$. Their radii are about 30% of the separation between the stellar centres (estimated to be $a=9.77R_{\odot} = 0.045$ au). The comparative youth and mass of this binary suggest it was probably "born this way".