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Within a few billion years our Sun would become a red giant and destroying Mercury and Venus and perhaps even the Earth. But nevertheless our solar system still have some planets like Jupiter and Neptune orbitting around the sun, because his mass wouldn't be very different.

After another while our Sun would turn into a white dwarf. Probably his mass changed a bit, but would it be enough to lose for example Neptune?

If not, is there a prediction at what time, if ever, our 'sun' will lose (some of) his planets for the first time?

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The process whereby the Sun will become a red giant and then turn into a white dwarf is neither short nor simple. It's absolutely not the case that it will expand, blow away a chunk of its mass, contract again, and that will be it.

According to Wikipedia (emphasis mine):

For the Sun, four thermal pulses are predicted before it completely loses its outer envelope and starts to make a planetary nebula. By the end of that phase – lasting approximately 500,000 years – the Sun will only have about half of its current mass.

There's a good lengthy coverage of the last days of the Sun here. It cites a study and then concludes that

The bottom line is that the orbits are unpredictable and where the planets find themselves once the star dies is dependent on when, how much and how quickly the sun shed half its mass. This makes a major difference in how the solar system will look in the end and how easily the planets may be ejected from their orbits.

So the simple answer to your question is: as the Sun loses mass, the orbits of the outer planets will become larger. When they become wide enough, they will escape into interstellar space. If the hypothesized Planet Nine exists, it could help kick out Uranus and Neptune. Jupiter (and Mars, I guess) may or may not remain. Earth will in all likelihood be dragged into the Sun during the red giant phase as the Sun's outer layers expand to fill its orbit.

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  • $\begingroup$ I agree with a lot of what you wrote and there is some uncertainty, but I don't agree with everything. Planet 9 is thought to be less massive than Neptune and Uranus and it's very far away, so it's unlikely to have an effect on either of them. I also think that someone here, better at math an I am, could probably run a prediction on Jupiter and maybe Saturn. There is some uncertainty due to a possibility of orbital resonance, but some measure of reasonable prediction should be doable. $\endgroup$
    – userLTK
    Feb 5, 2017 at 12:54
  • $\begingroup$ @userLTK The thing with Planet 9 is not its mass but its angular momentum, as I understand - in fact one reason to hypothesize there is a Planet 9 is that it could explain the fact that the Sun's axis is tilted 6 degrees from the ecliptic. Planet 9 could also be involved in some as-yet-unexplained perturbations of Saturn measured by Cassini. $\endgroup$
    – pablodf76
    Feb 6, 2017 at 13:52
  • $\begingroup$ That's an interesting argument, and made for some fun reading, but a tilt that happens very slowly and an ejection are measurably different energy wise. An orbit on a tilt to the plane of the star isn't unstable. So, I stand by the argument that it couldn't eject Neptune or Uranus for the reasons I said: less mass than those planets and too far away. $\endgroup$
    – userLTK
    Feb 6, 2017 at 14:14

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