<p>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 <a href="https://en.wikipedia.org/wiki/Mass_segregation_(astronomy)" rel="nofollow noreferrer">mass segregation</a> and can be viewed as an result of <a href="https://en.wikipedia.org/wiki/Thermalisation" rel="nofollow noreferrer">thermalization</a>: 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.</p>
<p>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 <a href="https://en.wikipedia.org/wiki/Dynamical_friction" rel="nofollow noreferrer">dynamical friction</a>, 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.</p>
<p>By approximating that the host dark matter halo is an <a href="https://en.wikipedia.org/wiki/Singular_isothermal_sphere_profile" rel="nofollow noreferrer">isothermal sphere</a>, <a href="https://ui.adsabs.harvard.edu/abs/2010gfe..book.....M/abstract" rel="nofollow noreferrer">Mo, van den Bosch, and White (2010)</a> estimate that the time for an object of mass <span class="math-container">$m$</span> on a circular orbit to fall to the center of a halo of mass <span class="math-container">$M$</span> is
<span class="math-container">$$
t\simeq
\frac{1.17}{\ln\Lambda}
\left(\frac{r}{R}\right)^2
\frac{M}{m}
\frac{R}{V_\mathrm{circ}},
\tag{1}\label{eq1}
$$</span>
where <span class="math-container">$r$</span> is the initial radius of the object's orbit, <span class="math-container">$R$</span> is the radius of the halo, <span class="math-container">$V_\rm{circ}$</span> is the circular velocity of the halo (uniform for an isothermal sphere), and <span class="math-container">$\ln \Lambda$</span> is the <a href="https://en.wikipedia.org/wiki/Coulomb_collision#Coulomb_logarithm" rel="nofollow noreferrer">Coulomb logarithm</a>, the logarithm of the ratio between the maximum and minimum distances at which a star could pass a dark matter particle. We can approximate
<span class="math-container">$$\Lambda\simeq \frac{R}{R_*},\tag{2}$$</span>
the ratio between the radius <span class="math-container">$R$</span> of the halo and the radius <span class="math-container">$R_*$</span> 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.</p>
<p>Now let's specialize to the Sun, which orbits within the Milky Way at a radius of about <span class="math-container">$r=8.2~\rm{kpc}$</span>. 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 <span class="math-container">$R=r=8.2~\rm{kpc}$</span> and its mass <span class="math-container">$M$</span> to be the mass below the Sun's orbit, roughly <span class="math-container">$M\simeq 10^{11}~\rm{M}_\odot$</span>. (About half of that mass is stars and not dark matter, but I'll neglect this as its an <span class="math-container">$\mathcal{O}(1)$</span> consideration.) The Milky Way's circular velocity is about <span class="math-container">$V_\rm{circ}\simeq 220~\rm{km/s}$</span>. The Sun's mass is <span class="math-container">$m=\rm{M}_\odot$</span> and its radius is <span class="math-container">$R_*=\rm{R}_\odot\simeq 2\times 10^{-11}~\rm{kpc}$</span>. Equation \eqref{eq1} becomes
<span class="math-container">$$t\simeq \frac{1.17}{\ln(4\times 10^{11})}10^{11}\frac{8.2~\rm{kpc}}{220~\rm{km/s}}\simeq 10^{17}~\rm{years}.\tag{3}$$</span></p>
<p>Therefore, the Sun is expected to sink into the center of the Milky Way in a time scale of order <span class="math-container">$10^{17}$</span> 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 &quot;<a href="https://en.wikipedia.org/wiki/Black_dwarf" rel="nofollow noreferrer">black dwarf</a>&quot; (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).</p>
<p>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. <a href="https://arxiv.org/abs/1806.09718" rel="nofollow noreferrer">If the string theorists are to be believed</a>, 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 (<span class="math-container">$10^{11}$</span> to <span class="math-container">$10^{12}$</span> years).</p>
<p>Finally, <span class="math-container">$10^{17}$</span> years is much shorter than the time scale for the Sun's inspiral into the center of the Galaxy <a href="https://en.wikipedia.org/wiki/Gravitational_wave#Binaries" rel="nofollow noreferrer">due to gravitational-wave emission</a>, which is around <span class="math-container">$(5/256)c^5 r^4/(G^3 M^2 m)\sim 10^{32}$</span> years, so I'm justified in neglecting this process.</p>

CC BY-SA 4.0
2023-05-24T11:26:25.437Z
i do not understand last paragraph. i checked out https://en.wiktionary.org/wiki/inspiral . and the first formula seems is currently broken, because its code is shown instead of render.
qdinar
CC BY-SA 4.0
2023-05-24T11:28:11.217Z
can sun fall into a more near/local black hole, and earlier?
qdinar
CC BY-SA 4.0
2023-05-24T11:28:57.830Z
i checked out https://en.wikipedia.org/wiki/Celestia and see: The time simulated by Celestia can be set to any time 2 billion years forward or backward from the present, although planetary orbits are only accurate within a few thousand years of the present day, and date arithmetic overflows at the year 5,874,774.[citation needed]
qdinar
CC BY-SA 4.0
2023-05-24T11:33:51.523Z
@qdinar I added a link for the gravitational-wave inspiral, which includes the relevant equation. I'm not sure what's wrong with equation 1 -- it fixes itself whenever I refresh the page. The probability that the Sun has a chance collision with another star, let alone a black hole, is incredibly small because the relevant objects are so small and space is so large.
Sten
CC BY-SA 4.0
2023-05-24T11:37:27.033Z
inspiral is about losing of energy due to rotation of binary objects? do you mean than that far planets (ie saturn, jupiter, etc, i think) of the sun will not fall into the sun before that time?
qdinar
CC BY-SA 4.0
2023-05-24T11:39:03.913Z
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.
Sten
CC BY-SA 4.0
2023-05-24T11:48:20.397Z
what can happen if sun flies away when m.w. is merged with andromeda? i think option 1 it will stay until it decays due to space widening, option 2 it is catched into an other galaxy. probably the option 2 is also very unlikely, like falling to other black hole. option 3 there will be another, bigger center, to which all nearby galaxies/black holes will fall, and the sun together with them. can that center be calculated now by some software?
qdinar
CC BY-SA 4.0
2023-05-24T11:52:54.937Z
i accepted this answer, but it does not mean that i fully understand it and checked its calculations.
qdinar
CC BY-SA 4.0
2023-05-24T11:59:00.887Z
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.
Sten
CC BY-SA 4.0
2023-05-24T12:03:38.157Z
@pm2ring said, in a comment to the question, that in 10^30 years it is likely that https://en.wikipedia.org/wiki/Local_Group will merge, so maybe sun will merge together?
qdinar
CC BY-SA 4.0
2023-05-24T12:06:07.880Z
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](https://en.wikipedia.org/wiki/Andromeda%E2%80%93Milky_Way_collision), no stars are expected to collide, according to that link.
Sten
CC BY-SA 4.0
2023-05-24T12:14:29.020Z
yes, i remembered now, you have already said about that in the answer. so, you say, andromeda-m.w. merger can throw the sun away from the local group, (so that it does not return). in the local group wikip. article it is said: "The group itself is a part of the larger Virgo Supercluster, which may be a part of the Laniakea Supercluster." . will not both of that 2 bigger and bigger clusters almost totally merge within themselves?
qdinar
CC BY-SA 4.0
2023-05-24T12:17:34.710Z
Due to dark energy, superclusters are never going to collapse. Everything outside the Local Group will continue to recede from us.
Sten
CC BY-SA 4.0
2023-05-24T10:25:03.427Z
2023-05-25T11:08:13.037Z
2023-05-25T11:08:13.037Z
Sten
<p>According to the Wikipedia page &quot;<a href="https://en.wikipedia.org/wiki/Timeline_of_the_far_future" rel="nofollow noreferrer">Timeline of the far future</a>&quot;, stellar remnants not ejected from their galaxies will fall into the central supermassive black holes over a time scale of around <span class="math-container">$10^{30}$</span> years. Will the Sun's remnant fall into a black hole? If so, when and into which one?</p>

CC BY-SA 4.0
2023-05-23T12:54:03.443Z
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.
Sten
CC BY-SA 4.0
2023-05-23T15:53:59.070Z
@Heopps comments are for comments. Answers are for answers. Your comment looks like an answer to me.
ProfRob
CC BY-SA 4.0
2023-05-23T16:55:13.120Z
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.
James K
CC BY-SA 4.0
2023-05-24T02:37:06.420Z
$10^{30}$ 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.
PM 2Ring
CC BY-SA 4.0
2023-05-24T11:12:07.683Z
https://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.
qdinar
<the-sun><black-hole><future>
Will the Sun fall into a black hole? If so, when and which one?
qdinar
# Will the Sun fall into a black hole? If so, when and which one?

(Asked by: username_0 on 2023-05-23)

According to the Wikipedia page "[Timeline of the far future](https://en.wikipedia.org/wiki/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?

**Tags:** the-sun, black-hole, future
> **Comments:**
> **username_1**: 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.  
> **username_2**: @Heopps comments are for comments. Answers are for answers. Your comment looks like an answer to me.  
> **username_3**: 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.  
> **username_4**: $10^{30}$ 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.  
> **username_0**: https://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.  

---

## 1 answers

### Answer by username_1 on 2023-05-24. Score: 6 | Accepted answer

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](https://en.wikipedia.org/wiki/Mass_segregation_(astronomy)) and can be viewed as an result of [thermalization](https://en.wikipedia.org/wiki/Thermalisation): 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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Singular_isothermal_sphere_profile), [Mo, van den Bosch, and White (2010)](https://ui.adsabs.harvard.edu/abs/2010gfe..book.....M/abstract) 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](https://en.wikipedia.org/wiki/Coulomb_collision#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](https://en.wikipedia.org/wiki/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](https://arxiv.org/abs/1806.09718), 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](https://en.wikipedia.org/wiki/Gravitational_wave#Binaries), 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.

> **Comments:**
> **username_0**: i do not understand last paragraph. i checked out https://en.wiktionary.org/wiki/inspiral . and the first formula seems is currently broken, because its code is shown instead of render.  
> **username_0**: can sun fall into a more near/local black hole, and earlier?  
> **username_0**: i checked out https://en.wikipedia.org/wiki/Celestia and see: The time simulated by Celestia can be set to any time 2 billion years forward or backward from the present, although planetary orbits are only accurate within a few thousand years of the present day, and date arithmetic overflows at the year 5,874,774.[citation needed]  
> **username_1**: @qdinar I added a link for the gravitational-wave inspiral, which includes the relevant equation. I'm not sure what's wrong with equation 1 -- it fixes itself whenever I refresh the page. The probability that the Sun has a chance collision with another star, let alone a black hole, is incredibly small because the relevant objects are so small and space is so large.  
> **username_0**: inspiral is about losing of energy due to rotation of binary objects? do you mean than that far planets (ie saturn, jupiter, etc, i think) of the sun will not fall into the sun before that time?  
> **username_1**: 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.  
> **username_0**: what can happen if sun flies away when m.w. is merged with andromeda? i think option 1 it will stay until it decays due to space widening, option 2 it is catched into an other galaxy. probably the option 2 is also very unlikely, like falling to other black hole. option 3 there will be another, bigger center, to which all nearby galaxies/black holes will fall, and the sun together with them. can that center be calculated now by some software?  
> **username_0**: i accepted this answer, but it does not mean that i fully understand it and checked its calculations.  
> **username_1**: 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.  
> **username_0**: @pm2ring said, in a comment to the question, that in 10^30 years it is likely that https://en.wikipedia.org/wiki/Local_Group will merge, so maybe sun will merge together?  
> **username_1**: 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](https://en.wikipedia.org/wiki/Andromeda%E2%80%93Milky_Way_collision), no stars are expected to collide, according to that link.  
> **username_0**: yes, i remembered now, you have already said about that in the answer. so, you say, andromeda-m.w. merger can throw the sun away from the local group, (so that it does not return). in the local group wikip. article it is said: "The group itself is a part of the larger Virgo Supercluster, which may be a part of the Laniakea Supercluster." . will not both of that 2 bigger and bigger clusters almost totally merge within themselves?  
> **username_1**: Due to dark energy, superclusters are never going to collapse. Everything outside the Local Group will continue to recede from us.  

---

