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I just found an article,

Long-Ago Supernovae Littered Earth,

which reviews evidence presented in

The locations of recent supernovae near the Sun from modelling 60Fe transport; D. Breitschwerdt et al., Nature 532, 73–76 (2016),

that there were supernova explosions in the neighbourhood of the Sun (a few tens of parsecs, if I'm reading it correctly) in the comparatively recent past (about 2 million years).

What are our chances of finding the remnants of those explosions? That is: given reasonable assumptions for the parents' velocities with respect to the solar system, and for the kick given by the supernova to the remnant, how far are these remnants likely to be from the solar system now?

In addition to that, is there any chance of finding them there using current or future astronomical surveys? I imagine the answer to the latter is "definitely not", but I might as well ask. (Note, however, that the main question is where the remnants are likely to be now, and only then how detectable they're likely to be.)

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Slim, unless the remnants are still in a close binary system. Almost all massive stars are born in binary systems and some fraction of these survive after a supernova to form neutron star or black hole binary systems. Typical kick velocities appear to be 200-500 km/s (e.g. Janka 2013 http://arxiv.org/abs/1306.0007), which translates (useful fact) to 200-500 pc/Myr, so they would be long gone from their birth location.

If they are single, or ejected from a binary they will be invisible (for a black hole) or practically invisible (for a neutron star, which would have cooled by many orders of magnitude in a million years).

So it is possible I suppose, especially when we have Gaia data, that a binarity survey of isolated high mass stars might uncover an invisible compact companion (if they were accreting we would surely already know about them; no examples are nearby), and then the space motions might trace it back to a position in Sco Cen.

NB By "remnant" I assumed you meant the compact object. There are of course large, expanding supernova remnants produced by the explosion. These fade and are mixed into the ISM on timescales of 100,000 years. There is little chance of seeing one from a 2-million year old supernova.

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IMO is 0% chance.

There is no physical evidence of SNe explosions.
They should look for $^{60}Fe$ at the sites were Iridium was found (Eltanin impact 2.5 million years ago)
There should be another solution to the $^{60}Fe$ problem if the Iridium connection proves to be good.

edit add - after 2 unreasonable down votes
The age of the events are the same.
High noble metal concentrations in a late Pliocene sediment (a 2.3-Myr-old layer in a sediment from the Antarctic Ocean contains Ir and Au) Because $^{60}Fe$ has a half-life of 2.6 Myr can be directly detected, if present, with an excess of $^{60}Ni$.

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