<p>As has already been said, the major sources of change to Earth's orbit are interactions with other planets and passing stars. </p>

<p>We're ruling out mass loss of the Sun, so the next consideration is probably tidal interactions between the Earth and Sun. <a href="https://arxiv.org/abs/0905.3008" rel="nofollow noreferrer">This paper</a> suggests that the Earth is receding from the Sun by about 15cm/yr for this reason. This is 150 km per million years, so over a trillion years or so, would move Earth quite a bit further from the Sun. The effect would reduce as the Earth got further away from the Sun, but I think it would always be slowly receding. I haven't done the calculation, but I'm not sure it would ever reach tidal lock -- the angular momentum of the Sun is too high.</p>

<p>Independently of this, though, orbital energy and angular momentum would slowly be lost by gravitational radiation. <a href="https://en.wikipedia.org/wiki/Gravitational_wave#Compact_binaries" rel="nofollow noreferrer">This wikipedia page</a> gives a formula</p>

<p>$$t={\frac {5}{256}}\,{\frac {c^{5}}{G^{3}}}\,{\frac {r^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}.$$</p>

<p>for the time to collision ignoring tidal effects. </p>

<p>We can plug in known values for the parameters $r = 1.5 \times 10^{11} m$, $c = 3\times 10^8 m/s$, $G = 6.7 \times 10^{-11} m^3 kg^{-1} s^{-2}$, $m_1 = 6\times 10^{24} kg$, $m_2 = 2 \times 10^{30} kg$ and get  about $3.3\times 10^{30} s$ as the time to collision. I suspect that's the time until the centres of the Earth and Sun coincide, so the actual collision would be a bit earlier, but probably not enough to notice at this precision. So that's about 10 million million times the lifetime of the universe so far.</p>

<p>I can't find enough information to work out how these two effects would interact. Clearly in the first $10^{12}$ years tidal evolution would be more important, but that conserves angular momentum and slows as the Suns rotation slows, whereas gravitational radiation actually carries angular momentum away, so might dominate in the long term. On the other hand the radiation drops as a high power of the separation, so we might have the Earth moving ever more slowly away from the Sun, while radiating gravitational waves of ever weaker strength, and never reaching tidal lock or starting to move back in. </p>

CC BY-SA 4.0
2018-05-30T18:31:08.017Z
Surely tidal effects are bigger than this?
ProfRob
CC BY-SA 4.0
2018-05-31T09:57:48.463Z
Yes. Almost anything is bigger than this I admit. Since the Sun rotates considerably more than once per year, the tidal effects would transfer angular momentum from the Sun to the Earth, which would slowly be driven further from the Sun, so if we allow for tidal effects (while still ignoring other perturbations) it will never collide.
Steve Linton
CC BY-SA 4.0
2018-05-31T10:11:35.690Z
and that is the answer.
ProfRob
CC BY-SA 4.0
2018-05-31T10:14:40.727Z
I've added a discussion to the answer. I can't work out which will dominate in the very long term
Steve Linton
CC BY-SA 4.0
2018-05-31T17:33:45.000Z
@nicoleSharp the moon is receding due to tidal effects. Gravitational raisin head the opposite effect (but much, much, more slowly)
Steve Linton
CC BY-SA 4.0
2018-05-30T11:23:19.557Z
2018-05-31T10:13:54.383Z
2018-05-31T10:13:54.383Z
Steve Linton
<p>This is unpredictable.  The solar system is a chaotic system and it's long-term behavior can't be predicted.  But it's also a <em>very stable</em> chaotic system and the planetary orbits have been stable for 4 billion years and will remain stable for some millions of years into the future.  See the <a href="https://en.wikipedia.org/wiki/Stability_of_the_Solar_System" rel="nofollow noreferrer">Wikipedia article for an overview.</a></p>

<p>When we numerically integrate the orbits of the planets into the future in an ensemble (doing many integrations, each with small differences in starting values that are still within experimental error), a small fraction of the ensemble members diverge chaotically. (E.g., Mercury is eventually flung into a wild orbit which may impact Venus.)</p>

<p>The key point here is that it is intrinsically unpredictable more than some millions of years into the future -- we're definitely safe for much more than a thousand human lifetimes -- other than statistically.  But base don those statistics, it seems likely that the solar system will be disrupted due to chaotic dynamics before the Earth's orbit decays due to tidal forces or anything like that.</p>

<p>(Another presently unpredictable future is an interloping star coming close enough to disrupt planetary orbits. Stars pass through the Oort cloud on a regular basis -- one each million years or so -- but getting inside the orbit of Pluto is <em>much</em> less likely and hasn't happened yet as far as we can tell.  Nonetheless, this <em>could</em> happen any time.)</p>

CC BY-SA 4.0
2018-05-30T06:17:41.020Z
+1, but you’re last sentence isn’t accurate. The local stellar neighbourhood is well mapped, and there’s no star coming closer than 3-4  light years of us in the foreseeable future.
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-05-30T06:23:23.260Z
@Chappo That was what I thought too, unless a star gets catapulted towards us from one of our group of local galaxies.
Peter Johnmeyer
CC BY-SA 4.0
2018-05-30T06:32:08.157Z
Yes, that’s possible, but it would have to have a huge peculiar velocity to unexpectedly appear in our local neighbourhood. Even if it was heading straight for us at 0.1 c, it would take 200 years to cover 20 ly. There would be plenty of warning!
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-05-30T12:26:16.280Z
@Chappo - What you wrote in your first comment quite incorrect. [Gliese 710](https://en.wikipedia.org/wiki/Gliese_710) is predicted to come within less than a quarter of a light year of the Sun in 1.3 million years. [Scholz's Star](https://en.wikipedia.org/wiki/Scholz%27s_Star) is thought to have come within 0.8 light years of the Sun about 70000 years ago.
David Hammen
CC BY-SA 4.0
2018-05-30T12:28:47.917Z
The local neighborhood is *not* well-mapped towards the dim red end of the main sequence where stars are most numerous. We haven't even completely excluded the possibility that the Sun has a distance brown dwarf companion! (Though that's pretty unlikely.) And in a million years stars with the low peculiar velocity of 10 kps will travel thirty light years and a high velocity star can travel a light year in just a 500-1000 years.
Mark Olson
CC BY-SA 4.0
2018-05-31T06:31:04.963Z
@DavidHammen Yes, my error in using “foreseeable future” when in my own mind I was thinking the next 1000 years or so.
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-05-31T06:59:25.710Z
I’d had in mind [this chart](https://en.wikipedia.org/wiki/Ross_248?wprov=sfti1) in which the nearest approach of any star in the next 80,000 years is Ross 248. In 33,000 years from now it will pass within 3.024 ly of the Sun before moving away again.
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-05-31T07:14:00.170Z
@MarkOlson the issue is whether a star could pass within Pluto’s orbit **“at any time”** (your statement). I think this gives an inaccurate impression of the likelihood there’s a very dim undiscovered star within 0.5 parsecs of us.
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-05-31T10:17:28.407Z
That the Sun has a distant brown dwarf companion has been excluded.
ProfRob
CC BY-SA 4.0
2018-05-30T02:32:28.957Z
2018-05-30T02:32:28.957Z
Mark Olson
<p>Hypothetically speaking (once again), if the Sun stays the same size forever (i.e. it somehow maintains a stable hydrogen supply and the helium product just disappears), how long would it take before the decay of the Earth's orbit sends it plummeting into the Sun and disintegrating?</p>

CC BY-SA 4.0
2018-05-30T06:09:26.083Z
Peter, it’s preferable that you wait a day or two before awarding the “tick” for your preferred answer. Mark’s answer is a good one, but a few hours isn’t enough time for the Astronomy SE community to comment on any errors or provide alternative or more detailed answers. :-)
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-05-30T06:22:17.913Z
@Chappo Aight, ty for the heads-up.
Peter Johnmeyer
CC BY-SA 4.0
2018-05-30T16:28:24.123Z
You could also write why you think the Earth should spiral into the sun. If you have a certain mechanism in mind, that would help answer it.
AtmosphericPrisonEscape
<orbit><solar-system><orbital-migration>
Hypothetically, if the sun stays the same size forever, when will the Earth fall into it?
Peter Johnmeyer
# Hypothetically, if the sun stays the same size forever, when will the Earth fall into it?

(Asked by: username_0 on 2018-05-30)

Hypothetically speaking (once again), if the Sun stays the same size forever (i.e. it somehow maintains a stable hydrogen supply and the helium product just disappears), how long would it take before the decay of the Earth's orbit sends it plummeting into the Sun and disintegrating?

**Tags:** orbit, solar-system, orbital-migration
> **Comments:**
> **username_1**: Peter, it’s preferable that you wait a day or two before awarding the “tick” for your preferred answer. Mark’s answer is a good one, but a few hours isn’t enough time for the Astronomy SE community to comment on any errors or provide alternative or more detailed answers. :-)  
> **username_0**: @Chappo Aight, ty for the heads-up.  
> **username_2**: You could also write why you think the Earth should spiral into the sun. If you have a certain mechanism in mind, that would help answer it.  

---

## 2 answers

### Answer by username_3 on 2018-05-30. Score: 5 | Accepted answer

As has already been said, the major sources of change to Earth's orbit are interactions with other planets and passing stars.

We're ruling out mass loss of the Sun, so the next consideration is probably tidal interactions between the Earth and Sun. [This paper](https://arxiv.org/abs/0905.3008) suggests that the Earth is receding from the Sun by about 15cm/yr for this reason. This is 150 km per million years, so over a trillion years or so, would move Earth quite a bit further from the Sun. The effect would reduce as the Earth got further away from the Sun, but I think it would always be slowly receding. I haven't done the calculation, but I'm not sure it would ever reach tidal lock -- the angular momentum of the Sun is too high.

Independently of this, though, orbital energy and angular momentum would slowly be lost by gravitational radiation. [This wikipedia page](https://en.wikipedia.org/wiki/Gravitational_wave#Compact_binaries) gives a formula

$$t={\frac {5}{256}}\,{\frac {c^{5}}{G^{3}}}\,{\frac {r^{4}}{(m\_{1}m\_{2})(m\_{1}+m\_{2})}}.$$

for the time to collision ignoring tidal effects.

We can plug in known values for the parameters $r = 1.5 \times 10^{11} m$, $c = 3\times 10^8 m/s$, $G = 6.7 \times 10^{-11} m^3 kg^{-1} s^{-2}$, $m\_1 = 6\times 10^{24} kg$, $m\_2 = 2 \times 10^{30} kg$ and get about $3.3\times 10^{30} s$ as the time to collision. I suspect that's the time until the centres of the Earth and Sun coincide, so the actual collision would be a bit earlier, but probably not enough to notice at this precision. So that's about 10 million million times the lifetime of the universe so far.

I can't find enough information to work out how these two effects would interact. Clearly in the first $10^{12}$ years tidal evolution would be more important, but that conserves angular momentum and slows as the Suns rotation slows, whereas gravitational radiation actually carries angular momentum away, so might dominate in the long term. On the other hand the radiation drops as a high power of the separation, so we might have the Earth moving ever more slowly away from the Sun, while radiating gravitational waves of ever weaker strength, and never reaching tidal lock or starting to move back in.

> **Comments:**
> **username_4**: Surely tidal effects are bigger than this?  
> **username_3**: Yes. Almost anything is bigger than this I admit. Since the Sun rotates considerably more than once per year, the tidal effects would transfer angular momentum from the Sun to the Earth, which would slowly be driven further from the Sun, so if we allow for tidal effects (while still ignoring other perturbations) it will never collide.  
> **username_4**: and that is the answer.  
> **username_3**: I've added a discussion to the answer. I can't work out which will dominate in the very long term  
> **username_3**: @nicoleSharp the moon is receding due to tidal effects. Gravitational raisin head the opposite effect (but much, much, more slowly)  

---

### Answer by username_5 on 2018-05-30. Score: 3

This is unpredictable. The solar system is a chaotic system and it's long-term behavior can't be predicted. But it's also a *very stable* chaotic system and the planetary orbits have been stable for 4 billion years and will remain stable for some millions of years into the future. See the [Wikipedia article for an overview.](https://en.wikipedia.org/wiki/Stability_of_the_Solar_System)

When we numerically integrate the orbits of the planets into the future in an ensemble (doing many integrations, each with small differences in starting values that are still within experimental error), a small fraction of the ensemble members diverge chaotically. (E.g., Mercury is eventually flung into a wild orbit which may impact Venus.)

The key point here is that it is intrinsically unpredictable more than some millions of years into the future -- we're definitely safe for much more than a thousand human lifetimes -- other than statistically. But base don those statistics, it seems likely that the solar system will be disrupted due to chaotic dynamics before the Earth's orbit decays due to tidal forces or anything like that.

(Another presently unpredictable future is an interloping star coming close enough to disrupt planetary orbits. Stars pass through the Oort cloud on a regular basis -- one each million years or so -- but getting inside the orbit of Pluto is *much* less likely and hasn't happened yet as far as we can tell. Nonetheless, this *could* happen any time.)

> **Comments:**
> **username_1**: +1, but you’re last sentence isn’t accurate. The local stellar neighbourhood is well mapped, and there’s no star coming closer than 3-4  light years of us in the foreseeable future.  
> **username_0**: @Chappo That was what I thought too, unless a star gets catapulted towards us from one of our group of local galaxies.  
> **username_1**: Yes, that’s possible, but it would have to have a huge peculiar velocity to unexpectedly appear in our local neighbourhood. Even if it was heading straight for us at 0.1 c, it would take 200 years to cover 20 ly. There would be plenty of warning!  
> **username_6**: @Chappo - What you wrote in your first comment quite incorrect. [Gliese 710](https://en.wikipedia.org/wiki/Gliese_710) is predicted to come within less than a quarter of a light year of the Sun in 1.3 million years. [Scholz's Star](https://en.wikipedia.org/wiki/Scholz%27s_Star) is thought to have come within 0.8 light years of the Sun about 70000 years ago.  
> **username_5**: The local neighborhood is *not* well-mapped towards the dim red end of the main sequence where stars are most numerous. We haven't even completely excluded the possibility that the Sun has a distance brown dwarf companion! (Though that's pretty unlikely.) And in a million years stars with the low peculiar velocity of 10 kps will travel thirty light years and a high velocity star can travel a light year in just a 500-1000 years.  
> **username_1**: @DavidHammen Yes, my error in using “foreseeable future” when in my own mind I was thinking the next 1000 years or so.  
> **username_1**: I’d had in mind [this chart](https://en.wikipedia.org/wiki/Ross_248?wprov=sfti1) in which the nearest approach of any star in the next 80,000 years is Ross 248. In 33,000 years from now it will pass within 3.024 ly of the Sun before moving away again.  
> **username_1**: @MarkOlson the issue is whether a star could pass within Pluto’s orbit **“at any time”** (your statement). I think this gives an inaccurate impression of the likelihood there’s a very dim undiscovered star within 0.5 parsecs of us.  
> **username_4**: That the Sun has a distant brown dwarf companion has been excluded.  

---

