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According to the Wikipedia page "Timeline of the far future", stellar remnants not ejected from their galaxies will fall into the central supermassive black holes over a time scale of around $10^{30}$ years. Will the Sun's remnant fall into a black hole? If so, when and into which one?

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    $\begingroup$ This is answerable under the assumption of no new physics. The Sun's remnant would gradually inspiral due to gravitational wave emission and (probably) dynamical friction with the dark matter. The time scale is ridiculously long, as the link indicates. $\endgroup$
    – Sten
    Commented May 23, 2023 at 12:54
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    $\begingroup$ @Heopps comments are for comments. Answers are for answers. Your comment looks like an answer to me. $\endgroup$
    – ProfRob
    Commented May 23, 2023 at 15:53
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    $\begingroup$ What is unclear about the quote? "Stellar remnants [implicitly including the sun's] not ejected from their galaxies will fall into the central supermassive black holes over a time scale of around." So either the sun's remnant will be ejected, or it will fall into the central black hole. $\endgroup$
    – James K
    Commented May 23, 2023 at 16:55
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    $\begingroup$ $10^{30}$ years is a long time. It's highly likely that within a few hundred billion years all the galaxies in the Local Group will have merged, and their SMBHs will have merged too. $\endgroup$
    – PM 2Ring
    Commented May 24, 2023 at 2:37
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    $\begingroup$ youtu.be/uD4izuDMUQA "A Journey to the End of Time". i just want to share a video i know exists and maybe interesting to readers of this page. i considered linking to this from the question, but linked to wikipedia only. $\endgroup$
    – qdinar
    Commented May 24, 2023 at 11:12

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Particles in galaxies can exchange energy by two-body gravitational interactions, like gravitational slingshots. Such an interaction results in one particle gaining energy, so that it rises to a higher orbit or even potentially is ejected from the system, and the other particle losing energy and sinking toward the center. However, in systems of particles with unequal masses, the heavier particles are more likely to sink while the lighter particles are more likely to rise. This phenomenon is known as mass segregation and can be viewed as an result of thermalization: kinetic energy (and hence temperature) tends to equalize between the different particle species, but heavier particles need much less velocity to have the same kinetic energy as lighter particles, and so the net energy exchange is from heavy to light particles.

Stars are thought to be much heavier than individual dark matter particles. Thus, stars will tend to sink toward the centers of galaxies. The time scale for a star to sink can be estimated by appealing to classic results on dynamical friction, which is another way to interpret the cause of mass segregation. A star passing through a sea of dark matter particles tends to pull those particles toward it, but since the star is moving, this results in the dark matter particles collecting in its wake. The gravity of those particles then pulls the star back, slowing it.

By approximating that the host dark matter halo is an isothermal sphere, Mo, van den Bosch, and White (2010) estimate that the time for an object of mass $m$ on a circular orbit to fall to the center of a halo of mass $M$ is $$ t\simeq \frac{1.17}{\ln\Lambda} \left(\frac{r}{R}\right)^2 \frac{M}{m} \frac{R}{V_\mathrm{circ}}, \tag{1}\label{eq1} $$ where $r$ is the initial radius of the object's orbit, $R$ is the radius of the halo, $V_\rm{circ}$ is the circular velocity of the halo (uniform for an isothermal sphere), and $\ln \Lambda$ is the Coulomb logarithm, the logarithm of the ratio between the maximum and minimum distances at which a star could pass a dark matter particle. We can approximate $$\Lambda\simeq \frac{R}{R_*},\tag{2}$$ the ratio between the radius $R$ of the halo and the radius $R_*$ of the object, which we take to be a star (so not an idealized point mass). This is justified because dark matter particles that pass inside a star would be deflected to a lesser degree than was assumed in the derivation of equation \eqref{eq1}, with zero deflection if they pass through the star's center.

Now let's specialize to the Sun, which orbits within the Milky Way at a radius of about $r=8.2~\rm{kpc}$. Since the mass above the Sun's orbit cannot affect its dynamics, let's take the radius of the relevant portion of the Galaxy to be $R=r=8.2~\rm{kpc}$ and its mass $M$ to be the mass below the Sun's orbit, roughly $M\simeq 10^{11}~\rm{M}_\odot$. (About half of that mass is stars and not dark matter, but I'll neglect this as its an $\mathcal{O}(1)$ consideration.) The Milky Way's circular velocity is about $V_\rm{circ}\simeq 220~\rm{km/s}$. The Sun's mass is $m=\rm{M}_\odot$ and its radius is $R_*=\rm{R}_\odot\simeq 2\times 10^{-11}~\rm{kpc}$. Equation \eqref{eq1} becomes $$t\simeq \frac{1.17}{\ln(4\times 10^{11})}10^{11}\frac{8.2~\rm{kpc}}{220~\rm{km/s}}\simeq 10^{17}~\rm{years}.\tag{3}$$

Therefore, the Sun is expected to sink into the center of the Milky Way in a time scale of order $10^{17}$ years. This is much longer than the Sun's lifetime as a fusion-powered star, and it's also longer than time for the remnant white dwarf to cool into a "black dwarf" (assuming no new physics). It's also much longer than the time it will take for the Andromeda galaxy (and the dwarf galaxies of the Local Group) to merge with the Milky Way, so the Sun would accrete onto the resulting combined supermassive black hole (although there remains the remote possibility that the Sun could be ejected during the Milky Way-Andromeda merger).

This time scale is also ten million times longer than the present age of the Universe. A huge uncertainty about such far-future predictions is that it is easily possible for new physics to present themselves over such a long time span. In fact we live in a very interesting time, cosmologically, because dark energy is just starting to become dominant. If the string theorists are to be believed, the onset of dark energy domination could suggest that the physics of the Universe are likely to change significantly within perhaps only 10 to 100 times the present age of the Universe ($10^{11}$ to $10^{12}$ years).

Finally, $10^{17}$ years is much shorter than the time scale for the Sun's inspiral into the center of the Galaxy due to gravitational-wave emission, which is around $(5/256)c^5 r^4/(G^3 M^2 m)\sim 10^{32}$ years, so I'm justified in neglecting this process.

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    $\begingroup$ i do not understand last paragraph. i checked out en.wiktionary.org/wiki/inspiral . and the first formula seems is currently broken, because its code is shown instead of render. $\endgroup$
    – qdinar
    Commented May 24, 2023 at 11:26
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    $\begingroup$ As the Sun orbits the Milky Way, it emits gravitational waves, which carry away some of its orbital energy. This is another process that causes the Sun to sink to the center of the Galaxy. The point of the last sentence, however, is that we can neglect this process because it's much slower than dynamical friction due to dark matter. $\endgroup$
    – Sten
    Commented May 24, 2023 at 11:39
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    $\begingroup$ If the Sun is ejected from the Local Group, it is highly unlikely to ever encounter another galaxy (let alone a star or black hole). Dark energy accelerates cosmic expansion too much for unbound objects to cross each other in the far future. $\endgroup$
    – Sten
    Commented May 24, 2023 at 11:59
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    $\begingroup$ The Local Group galaxies are expected to merge over a much shorter time scale than that, of order 5-10 billion years. But in the whole Milky Way-Andromeda collision, no stars are expected to collide, according to that link. $\endgroup$
    – Sten
    Commented May 24, 2023 at 12:06
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    $\begingroup$ Due to dark energy, superclusters are never going to collapse. Everything outside the Local Group will continue to recede from us. $\endgroup$
    – Sten
    Commented May 24, 2023 at 12:17

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