If the Sun once had an older companion star that exploded 5 billion years ago it could surely explain many things about the solar system. Such as why so many planets have hot cores that are cooling. Also why the planets formed at the same time. Ok perhaps explode is wrong. What if the companion star became a Red giant and then shed its outer layer. What was left could have ended up as a white dwarf or even a brown dwarf. If the distance between the stars was great enough it could have wandered off. If instead it was again captured by the remaining star it could now be either Neptune or Uranus.
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FWIW, there probably was an older star that went supernova in our stellar nursery, contributing matter & energy to the formation of our Solar System, but it exploded well before the Sun was formed. See https://astronomy.stackexchange.com/q/54577/16685 – PM 2Ring Oct 30 '23 at 13:52
2 Answers
No, it would not. Because the companion star would in nearly all cases leave a remnant - a white dwarf, or for heavier stars, a neutron star or black hole. We do not see such a remnant.
The reason planetary cores are hot is simple: when mass accretes the potential energy needs to go somewhere, and much of it turns into heat. In terrestrial planets there is also radioactive decay. The result is a hot mass, surrounded by a cooler universe. Hence the temperature gradient.
Planetary formation has been observed around other stars, and is reasonably quick on astronomical timescales (mere tens to hundreds of millions of years). So assuming another star does not help here.
Finally, this theory has a problem: why did a young star end up with an old companion? We know the age of the sun directly (if less firmly than when using other solar system bodies assumed to be equally old) by looking at where it is along the main sequence. Normally binaries form together, so this theory has to posit a very rare interaction making a newly formed star show up next to an old star.
In short, this is difficult to believe because of the data and understanding we have of astrophysics.

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The main part of that theory that I have a problem is the idea that dust accretes. The idea that two dust particles would accrete due to gravity alone seems a bit far fetched. The asteroid belt has been out there for a very long time and hasn't accreted. The other thing is where does all the dust come from. How many supernova would have to go off within a reasonable distance to send that much dust to a lonely star on the edge of the galaxy. – Michael Mcgarry Oct 30 '23 at 10:15
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If a star exploded and didn't leave behind a remnant how would you know. You also say planetary formation has been observed around other stars. As far as I am aware we have only recently discovered planets around other stars and they have only been noticed by the dimming of the stars light. JWST may have evidence of other planets but I am pretty sure you have never seen planetary formation. – Michael Mcgarry Oct 30 '23 at 10:41
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Finally you say that binary stars normally form together. I do not understand how that could be possible. If a gas cloud had sufficient material to form two stars at the same time the stars motion relative to each other would be zero until gravity took over and brought the stars together to form a bigger star. A more reasonable explanation would be that after the first star formed the solar wind would blow back the gas falling in to the first star until that was balanced by the gravity pulling it in. Eventually this gas would form a star but it would already be in orbit around the first star. – Michael Mcgarry Oct 30 '23 at 11:44
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5@MichaelMcgarry Dust particles don't accrete via gravitation; they are indeed too small. They do however accrete by collisions and chemistry. Once these every growing amalgams of dust get large enough they can merge gravitationally. Astronomers have observed many stellar nurseries in multiple stages of development. In some of those, they have observed what are almost certainly protoplanets in the process of forming. – David Hammen Oct 30 '23 at 12:02
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3@MichaelMcgarry - A lot of this is covered in standard astrophysics textbooks. There are big research literatures on dust dynamics, accretion, planetary formation, stellar formation etc. – Anders Sandberg Oct 30 '23 at 12:27
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3Electrical charges may well have a lot to do with the initial clumping of dust into larger chunks, but that aside, the asteroid belt isn't a good example here -- Jupiter's massive gravitational influence is what has stopped the asteroids from clumping any more than they have. The gravity of one asteroid to the next is small compared to the influence Jupiter has on them, even at that distance. They can't clump up because the pot is constantly being stirred by an external force. – Darth Pseudonym Oct 30 '23 at 15:14
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1@MichaelMcgarry There is dust scattered throughout the galaxy, mixed in with the gas. When a dense molecular cloud collapses to form a star, the dust in the cloud is part of the collapse. Most of it ends up in the star (along with the gas), but some of it ends up in the protoplanetary disk around the young star (along with some of the gas). – Peter Erwin Oct 30 '23 at 22:43
Stars and their planets form at the same time. For multiple-star systems the stars form at the same time along with any planets. There is a plethora of systems one can observe from all stages of formation.
Further, the physics of the interaction of a newly-born star with its surrounding cloud of gas and dust it was born from imply, that the proto-planets basically have to have formed by then. Subsequently these planetesimals and protoplanets only can consolidate in the late stage of planet formation by gravitational interaction (i.e. collisions, change to their orbital distance to the host star) into the planets, e.g. as we know them in our solar system today. The stellar wind of a star will quite quickly, and eventually push away all gas and dust which could faciliate any formation of planets - but also other stars.
One of the well-known regions of star-formation is the Orion nebula where you can observe star formation in all of its stages. Observing the individual planets is more challenging, if you want to detect them by something else than transits. Yet even transits are very unique in their shape of the light curve and their constant periodicity; thus there is no mistaking in the observed transits of exoplanets being transits of dark(er) bodies in front of their host stars. There are closer star systems which are quite young and which allow a direct view into the later stages of planet formation, like the beta pictoris system. There is even a video of the newly-formed planets orbiting their host star from beta pictoris, still in the not yet completely blown-away cloud of dust and debris.
All this said, a former companion star to the Sun would have had serious gravitational influence on the whole system. The current dynamics would not be explicable had a short-lived larger companion existed somewhen since the birth of the Sun and the planets. And no such remanent can be detected anywhere either.
That the Sun and the planets are of similar age is proven by isotopic analysis which hint strongly at a joint origin and chemical analysis of the oldest rocks indicate the age of the planets being about identical to the age of our Sun.

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The idea of a companion star may have come to me from reading articles such as the one you linked to about Beta Pictoris. It is described as having a debris disk which implies the result of an explosion. – Michael Mcgarry Oct 30 '23 at 13:46
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3No, a debris disk does not imply an explosion; it merely implies that bigger bodies (the prot-planets) collide and thus cause some bodies to grow, and some stuff not accreted in the collisions to remain as "debris" – planetmaker Oct 30 '23 at 14:16
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3Also a stellar "explosion" does not leave debris... it leaves hot, ionized gas - an environment in which definitely nothing forms which is close to anything solid. – planetmaker Oct 30 '23 at 14:28
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The above answers are obviously nonsense. Stellar explosions do not only leave ionized gas behind. other wise where do all the heavy elements come from. Just because when you look at a stellar explosion hundreds of light years away all you can see are clouds of gas does not mean that there are not planets sized rocks in there as well. – Michael Mcgarry Oct 31 '23 at 08:01
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If a companion Star explodes and leaves a white dwarf remnant the white dwarf would have the velocity of the original but lack the mass and so would leave the system. – Michael Mcgarry Oct 31 '23 at 12:27
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3@MichaelMcgarry "Rocks" do not form in stellar winds or in supernova explosions. Heavy elements are in the form of an ionised gas in the interstellar medium or part of sub-micron-sized dust particles . A "debris disk" is formed when pre-existing solid bodies collide and create a lot of dust, which is then what is observed as a debris disk. – ProfRob Nov 01 '23 at 14:02
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1Small correction. We know the ages of the oldest meteorites. These are similar to (but older than) the age of the oldest materials on Earth. There isn't a way to directly get the age of the Sun. – ProfRob Nov 01 '23 at 14:04