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Sometimes people talk about the "first passage" of an object falling in toward or beginning an orbit around a more massive object. I'm specifically thinking about this phrase in the context of satellite galaxies falling in toward a more massive "central" galaxy (like LMC/SMC toward the Milky Way, or the many smaller satellite galaxies around M87, the central galaxy of the Virgo cluster).

What exactly does "first passage" mean, in the language of orbital mechanics? Is it like the time or argument of periapsis (point of closest approach in an orbit)? Don't you need to know how the orbit will change with time to define the argument/time of periapsis (since periapsis can change with time) -- so how is "first passage" different? I'm being reminded of some kind of impact parameter maybe.

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The usual language of orbital mechanics doesn't always translate nicely to the orbits of galaxies, but 'first passage' or 'first pericentre' or 'first infall' (or several other terms, used more or less interchangeably) is well defined. The figure below, taken from Bertschinger (1985), shows a typical trajectory for a galaxy which becomes a satellite of a more massive system. The model discussed in that paper is simple and analytic, e.g. the orbit shown is exactly radial, the Universe is taken to be EdS, but these details don't matter to the definition. Scaled orbital trajectory in a self-similar infall model. At $t=0$ the satellite is at $r=0$, i.e. at the same location as its future host system. However, this is the moment of the Big Bang, so everything is at the same location as everything else. One can't really talk about the satellite being gravitationally bound to its host. Between the Big Bang and the point I've labelled 'turnaround' the two systems are flying apart, more or less in the Hubble flow. For suitable initial conditions, though, the gravitational attraction of the 'host' system will eventually slow the 'satellite' enough for it to decouple from the expansion and briefly come to rest at 'turnaround'. The turnaround radius is well-defined in cosmological theory; you should be able to find a discussion in any introductory text which covers the FLRW metric. After this the satellite falls toward its soon to be host, drawn in by gravity. Eventually it will have it's first pericentric passage, the first time at which its radial velocity relative to the host system switches sign from negative to positive. In the illustrated case of a perfectly radial orbit this happens at $r=0$, but would be further out for an orbit with some tangential velocity component.

The first pericentre is a significant event in the 'life' of a galaxy: it's likely the first time the system will experience strong tidal and/or ram-pressure forces, and after about this time it will be very difficult for the system to accrete additional gas.

Being careful to answer your specific question, the 'first pericentre' is defined as I described above. The first passage is a bit more vague, and could refer to any or all of the time between turnaround and first apocentre (which is the first peak in the plot after first pericentre). Typically I wouldn't talk about first passage until the satellite has entered ~the virial radius of the host.

Note that the $t_{\rm ta}$ and $r'_{\rm ta}$ in the graph labels are not constants, so one has to be careful of the interpretation. I picked this particular graphic just because it is very simple and let me pick out the specific points I wanted to make above.

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