Earth's trajectory if the sun disappeared...?

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Earth's trajectory if the sun disappeared...?

Postby King Author » Fri Apr 10, 2015 11:23 am UTC

I know that nothing travels faster than light, and that includes the effects of gravity. I also know that the light from the surface of the sun takes about eight minutes to reach Earth.

So, if the sun suddenly disappeared, I take it the Earth would continue to orbit around the nothingness where the sun used to be for eight minutes, as if it were still there.

What then? Would the Earth continue to move in a more-or-less circular pattern around the not-sun, the path getting wider and wider, or would the planet essentially shoot off like a stone released from a sling?
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Re: Earth's trajectory if the sun disappeared...?

Postby Neil_Boekend » Fri Apr 10, 2015 12:19 pm UTC

After the 8 minute delay it would go on in an aproximately straight line because the only forces pulling it would be the other planets (Jupiter mainly) and they don't have nearly the same mass and most are further away. Sort of like a stone released from a sling.
The moon would keep spinning around the earth unless the new path of any of the planets and Earth would be intersecting lines. If for example Jupiter would get within a couple of Earth-moon distances it would most likely steal or eject our moon and probably do the same to Earth. What exactly happens depends on details.
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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Fri Apr 10, 2015 4:08 pm UTC

Neil's right, it would fly off tangentially because there's nothing keeping it in orbit.

I had wondered if maybe the time taken for spacetime to relax back to Minkowski spacetime might delay the tangential release by a little bit (on top of the expected 8 minutes) but a lengthy back-of-the-envelope calculation showed that the timescale of this relaxation is on the order of a tenth of a millisecond (so insignificant compared to the 8 minutes light-speed delay).
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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Fri Apr 10, 2015 5:52 pm UTC

If the sun disappeared, all of the planets would rocket off tangent to their former orbits. The inner planets have higher orbital velocities than the outer planets, but it seems extremely unlikely that they would be able to "catch up" to the trajectory of an outer planet in such a way as to come within a meaningfully close distance.

For example, Earth orbits at 29.8 km/s at a distance of 1 AU. Jupiter orbits at 13.1 km/s at a distance of about 5.2 AU:

jupiter and earth.png
jupiter and earth.png (4.17 KiB) Viewed 5703 times


Distances are to scale; object sizes are definitely not to scale.

Because Earth's trajectory is tangential, it will take us about 9.7 months to cross Jupiter's former orbit. By that time, Jupiter will have traveled an additional 2.23 AU along its tangential trajectory, moving it an additional 0.45 AU from the former Sun.

After 10 months, 26 days, 1 hour, 25 minutes, and 15 seconds, the Earth will have traveled 5.686 AU and Jupiter will have traveled 2.500 AU, bringing them both to the same distance of 5.773 AU from the former position of the Sun.

jupiter and earth 2.png
jupiter and earth 2.png (1.31 KiB) Viewed 5703 times


At this point, the speed of Earth relative to Jupiter will be 20.61 km/s. At that speed, it will be captured by Jupiter's gravity if it comes within 596,109 km of the gas giant. So Earth has a window of about 1.2 million kilometers; if it hits that window, Jupiter will capture it; if not, it will escape.

Let's also consider the worst-case scenario: an impact. Now, Earth is dense enough that its Roche limit with respect to Jupiter is only 63,000 km, less than the actual radius of Jupiter. So the window of impact is simply the sum of the two planet diameters, or 151,100 km.

jupiter and earth 3.png
jupiter and earth 3.png (8 KiB) Viewed 5703 times


Compared to the size of the solar system, this is NOT a very large window. From Earth's orbit, the capture window is just 0.0803 degrees wide (4.82 arcminutes) and the impact window is even more miniscule at 0.0102 degrees wide (0.612 arcminutes).

The direction of Earth's velocity vector in its orbit around the sun goes through 360 degrees every year, or 0.986 degrees every day, or 4.93 arcminutes per hour. Thus, there will only be 58 minutes, 40 seconds out of the entire year in which Earth will be aimed in the right direction to hit the capture zone, and 7 minutes, 27 seconds out of that in which Earth will be aimed in the right direction to actually hit the impact zone (Jupiter will stay in orbit for 35 minutes longer than Earth due to lightspeed delay, but this will only shift the timing of the window, not the size of the window).

Jupiter is by far the largest planet and will have a much larger capture window than any other planet; it's also the closest of the outer planets. So Earth has a much better chance of hitting Jupiter's capture window than any of the other outer planets. The chance of an inner planet hitting Earth's capture window is also very small. Hitting Jupiter's capture window seems to be our only chance for survival.

Not that it's much of a chance. The nearly-11-month journey would almost certainly reduce Earth to a completely inhospitable ice cube. I'm not entirely sure how much heat the Earth would still retain but it probably wouldn't be enough for any but the most robust life-support systems to continue operating. And even once we reached Jupiter, we'd still be in a bad spot; its thermal contraction only causes it to radiate about 51 W/m2, just 4% of the energy we are used to receiving from the sun. Of course, if we can manage to construct hydrogen fusion rockets on opposite ends of Thebe and tow it into Jupiter's upper atmosphere to turn it into a fusion candle, we'll be just fine.

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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Fri Apr 10, 2015 7:35 pm UTC

eSOANEM wrote:Neil's right, it would fly off tangentially because there's nothing keeping it in orbit.

I had wondered if maybe the time taken for spacetime to relax back to Minkowski spacetime might delay the tangential release by a little bit (on top of the expected 8 minutes) but a lengthy back-of-the-envelope calculation showed that the timescale of this relaxation is on the order of a tenth of a millisecond (so insignificant compared to the 8 minutes light-speed delay).


Welp, I realised I messed up my calculation. It's actually much shorter, about a third of a nanosecond picosecond.

Spacetime relaxation is very much definitely not anything close to significant (or indeed even really very measurable).

Edit: fixed the prefix. Apparently I can't remember my SI prefixes either.
Last edited by eSOANEM on Fri Apr 10, 2015 11:11 pm UTC, edited 1 time in total.
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Re: Earth's trajectory if the sun disappeared...?

Postby King Author » Fri Apr 10, 2015 7:55 pm UTC

Neil_Boekend wrote:After the 8 minute delay it would go on in an aproximately straight line because the only forces pulling it would be the other planets (Jupiter mainly) and they don't have nearly the same mass and most are further away. Sort of like a stone released from a sling.
The moon would keep spinning around the earth unless the new path of any of the planets and Earth would be intersecting lines. If for example Jupiter would get within a couple of Earth-moon distances it would most likely steal or eject our moon and probably do the same to Earth. What exactly happens depends on details.


Oh neat, I hadn't even considered the moon. So the sun basically has no significant effect on the moon? That makes sense. Big as the sun is, the Earth is so much closer, the effect is bound to be overwhelming; compared to a blowtorch in my face, a nuclear explosion in another country won't hold a candle.

eSOANEM wrote:Neil's right, it would fly off tangentially because there's nothing keeping it in orbit.

I had wondered if maybe the time taken for spacetime to relax back to Minkowski spacetime might delay the tangential release by a little bit (on top of the expected 8 minutes) but a lengthy back-of-the-envelope calculation showed that the timescale of this relaxation is on the order of a tenth of a millisecond (so insignificant compared to the 8 minutes light-speed delay).


Whoa now what now? If there was going to be any additional effect, shouldn't it also take eight minutes, since nothing can move faster than light? Or is it that spacetime itself can shrink/expand/warp faster than light?

sevenperforce wrote:If the sun disappeared, all of the planets would rocket off tangent to their former orbits. The inner planets have higher orbital velocities than the outer planets, but it seems extremely unlikely that they would be able to "catch up" to the trajectory of an outer planet in such a way as to come within a meaningfully close distance.

For example, Earth orbits at 29.8 km/s at a distance of 1 AU. Jupiter orbits at 13.1 km/s at a distance of about 5.2 AU:

jupiter and earth.png


Distances are to scale; object sizes are definitely not to scale.

Because Earth's trajectory is tangential, it will take us about 9.7 months to cross Jupiter's former orbit. By that time, Jupiter will have traveled an additional 2.23 AU along its tangential trajectory, moving it an additional 0.45 AU from the former Sun.

After 10 months, 26 days, 1 hour, 25 minutes, and 15 seconds, the Earth will have traveled 5.686 AU and Jupiter will have traveled 2.500 AU, bringing them both to the same distance of 5.773 AU from the former position of the Sun.

jupiter and earth 2.png


At this point, the speed of Earth relative to Jupiter will be 20.61 km/s. At that speed, it will be captured by Jupiter's gravity if it comes within 596,109 km of the gas giant. So Earth has a window of about 1.2 million kilometers; if it hits that window, Jupiter will capture it; if not, it will escape.

Let's also consider the worst-case scenario: an impact. Now, Earth is dense enough that its Roche limit with respect to Jupiter is only 63,000 km, less than the actual radius of Jupiter. So the window of impact is simply the sum of the two planet diameters, or 151,100 km.

jupiter and earth 3.png


Compared to the size of the solar system, this is NOT a very large window. From Earth's orbit, the capture window is just 0.0803 degrees wide (4.82 arcminutes) and the impact window is even more miniscule at 0.0102 degrees wide (0.612 arcminutes).

The direction of Earth's velocity vector in its orbit around the sun goes through 360 degrees every year, or 0.986 degrees every day, or 4.93 arcminutes per hour. Thus, there will only be 58 minutes, 40 seconds out of the entire year in which Earth will be aimed in the right direction to hit the capture zone, and 7 minutes, 27 seconds out of that in which Earth will be aimed in the right direction to actually hit the impact zone (Jupiter will stay in orbit for 35 minutes longer than Earth due to lightspeed delay, but this will only shift the timing of the window, not the size of the window).

Jupiter is by far the largest planet and will have a much larger capture window than any other planet; it's also the closest of the outer planets. So Earth has a much better chance of hitting Jupiter's capture window than any of the other outer planets. The chance of an inner planet hitting Earth's capture window is also very small. Hitting Jupiter's capture window seems to be our only chance for survival.

Not that it's much of a chance. The nearly-11-month journey would almost certainly reduce Earth to a completely inhospitable ice cube. I'm not entirely sure how much heat the Earth would still retain but it probably wouldn't be enough for any but the most robust life-support systems to continue operating. And even once we reached Jupiter, we'd still be in a bad spot; its thermal contraction only causes it to radiate about 51 W/m2, just 4% of the energy we are used to receiving from the sun. Of course, if we can manage to construct hydrogen fusion rockets on opposite ends of Thebe and tow it into Jupiter's upper atmosphere to turn it into a fusion candle, we'll be just fine.


https://www.youtube.com/watch?v=i6l8MFdTaPE

Heh, but seriously, thanks for all that. You've raised a few questions in my admittedly-uninformed head.

1) How much do the planets vary from one another on the, erm, solar plane? I mean like, the stuff floating around Saturn deviates from the imaginary perfectly-flat disc-shaped plane we could envision surrounding it by only about two stories. That is, every ice and rock chunk is within a story of the "middle of the disc." How widely do the planets vary? Is it possible Earth could slip under or over Ju...well, maybe not Jupiter since it's so, so much larger than Earth. So yeah, nevermind >_>

2) How long did it take you to run those calcs, and how many years of studying did it take you to get to that point?

3) Is that Thebe thing a reference to something? Sounds like the setting of a scifi novel.
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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Fri Apr 10, 2015 9:09 pm UTC

King Author wrote:
eSOANEM wrote:Neil's right, it would fly off tangentially because there's nothing keeping it in orbit.

I had wondered if maybe the time taken for spacetime to relax back to Minkowski spacetime might delay the tangential release by a little bit (on top of the expected 8 minutes) but a lengthy back-of-the-envelope calculation showed that the timescale of this relaxation is on the order of a tenth of a millisecond (so insignificant compared to the 8 minutes light-speed delay).


Whoa now what now? If there was going to be any additional effect, shouldn't it also take eight minutes, since nothing can move faster than light? Or is it that spacetime itself can shrink/expand/warp faster than light?


This would be additional to the 8 minutes light travel time.

It takes 8 minutes for the earth to become aware that the sun has disappeared but then there is an additional delay as spacetime rebounds to its no-sun form (pretty much Minkowski space). This delay turns out to be completely inconsequential.

It's also worth noting that, if the sun suddenly disappeared you'd get some really funky gravitational monopole radiation which means that, whilst the planets' paths would average out to straight lines, in practice they'd do slightly wibblier things.
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Re: Earth's trajectory if the sun disappeared...?

Postby Qaanol » Fri Apr 10, 2015 10:29 pm UTC

King Author wrote:
Neil_Boekend wrote:After the 8 minute delay it would go on in an aproximately straight line because the only forces pulling it would be the other planets (Jupiter mainly) and they don't have nearly the same mass and most are further away. Sort of like a stone released from a sling.
The moon would keep spinning around the earth unless the new path of any of the planets and Earth would be intersecting lines. If for example Jupiter would get within a couple of Earth-moon distances it would most likely steal or eject our moon and probably do the same to Earth. What exactly happens depends on details.


Oh neat, I hadn't even considered the moon. So the sun basically has no significant effect on the moon? That makes sense. Big as the sun is, the Earth is so much closer, the effect is bound to be overwhelming; compared to a blowtorch in my face, a nuclear explosion in another country won't hold a candle.

Quite the contrary. The moon orbits the sun, and the earth just makes the moon wobble a bit in its orbit. In fact, the moon’s orbit around the sun is always convex: the wobble as it “orbits” earth never causes the moon to so much as curve away from the sun, let alone backtrack. The moon’s trajectory around the sun just varies between “bent slightly more, as if going around a smaller circle” and “bent slightly less, as if going around a wider circle”.

In round number, the moon is about 389 times closer to the earth than to the sun, and the sun is about 333,000 times more massive than the earth. Now 389² is about 151,000, so the sun’s pull on the moon is more than double the earth’s pull on the moon.
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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Fri Apr 10, 2015 11:15 pm UTC

King Author wrote:
Neil_Boekend wrote:The moon would keep spinning around the earth unless the new path of any of the planets and Earth would be intersecting lines. If for example Jupiter would get within a couple of Earth-moon distances it would most likely steal or eject our moon and probably do the same to Earth. What exactly happens depends on details.
I hadn't even considered the moon. So the sun basically has no significant effect on the moon?

Actually (as Qaanol sniped about), the sun has a very significant effect on the moon. In fact, from the sun's perspective, the moon doesn't seem to orbit the Earth at all; it just seems to follow approximately the same orbit as the Earth but with slight perturbations from Earth's gravity:

earthmoonloop.png
earthmoonloop.png (3.39 KiB) Viewed 5604 times


This image amplifies the distortion in the moon's orbit by a factor of about 4. In reality, the moon never loops back "away from" the sun at all; its orbit around the sun is concave at all points. Based on this, the Earth and the moon could legitimately be considered a two-planet system, but because their center of gravity lies within the Earth, the moon is generally considered a satellite of the Earth rather than a co-orbiting planet.

Of course, from Earth's perspective, the moon spins around us quite merrily. And it would continue to do so if the sun suddenly disapparated, following the same tangential trajectory:

earth moon 2.png
earth moon 2.png (2.58 KiB) Viewed 5604 times


This image actually IS to scale (though of course the Earth and moon themselves are invisible at this scale).

If we were lucky enough to hit the Jovian capture window, the moon would become problematic. In order to keep the moon as our own satellite, we would need to be orbiting at nearly 4 million km from Jupiter, more than six times outside our capture window. Within the capture window, the moon would rapidly be pulled away from us and into exclusive orbit around Jupiter. This would likely be a pretty eccentric orbit, though, and would likely meet a fiery end by smashing into one of Jupiter's other moons and showering us with a cloud of nasty debris.

sevenperforce wrote:If the sun disappeared, all of the planets would rocket off tangent to their former orbits. The inner planets have higher orbital velocities than the outer planets, but it seems extremely unlikely that they would be able to "catch up" to the trajectory of an outer planet in such a way as to come within a meaningfully close distance.

There will only be 58 minutes, 40 seconds out of the entire year in which Earth will be aimed in the right direction to hit the capture zone, and 7 minutes, 27 seconds out of that in which Earth will be aimed in the right direction to actually hit the impact zone.

Jupiter is by far the largest planet and will have a much larger capture window than any other planet; it's also the closest of the outer planets. So Earth has a much better chance of hitting Jupiter's capture window than any of the other outer planets. The chance of an inner planet hitting Earth's capture window is also very small. Hitting Jupiter's capture window seems to be our only chance for survival.

Not that it's much of a chance. The nearly-11-month journey would almost certainly reduce Earth to a completely inhospitable ice cube. I'm not entirely sure how much heat the Earth would still retain but it probably wouldn't be enough for any but the most robust life-support systems to continue operating. And even once we reached Jupiter, we'd still be in a bad spot; its thermal contraction only causes it to radiate about 51 W/m2, just 4% of the energy we are used to receiving from the sun. Of course, if we can manage to construct hydrogen fusion rockets on opposite ends of Thebe and tow it into Jupiter's upper atmosphere to turn it into a fusion candle, we'll be just fine.

Heh, but seriously, thanks for all that. You've raised a few questions in my admittedly-uninformed head.

1) How much do the planets vary from one another on the, erm, solar plane? I mean like, the stuff floating around Saturn deviates from the imaginary perfectly-flat disc-shaped plane we could envision surrounding it by only about two stories. That is, every ice and rock chunk is within a story of the "middle of the disc." How widely do the planets vary? Is it possible Earth could slip under or over Ju...well, maybe not Jupiter since it's so, so much larger than Earth. So yeah, nevermind >_>

Because all the planets in our solar system formed from the same rotating nebula, we all ended up with about the same orbital inclination. Due to tidal forces from the sun, Mercury's orbit is tilted at 6.34° relative to the system's invariable plane (the plane perpendicular to the total angular momentum of the system), but all the other planets are within 2.2° of it. Jupiter's orbital plane is only 1.31° offset from our own.

Unfortunately, our window is less than a tenth of a degree wide, so that 1.31° is sixteen times too much. That means that Earth's tangential trajectory vector only crosses within 600,000 km of Jupiter's orbital path for 22 days out of the year. So the precise alignment of vectors and orbital inclination will only take place for about an hour every sixteen years.

To give a better sense of scale: imagine that you are standing in the end zone of a football field and I'm sitting the stands at around the 40-yard line near the opposite end of the field. We're both blindfolded; you have a pistol and I have a paintball gun. At the same time, you and I point our guns in a random direction and fire...by sheer chance, your bullet and my paintball hit the horizontal crossbar of the opposite goalpost less than 3 inches apart. At the exact same time.

2) How long did it take you to run those calcs, and how many years of studying did it take you to get to that point?

I dunno about how long it took -- I did it over the course of a couple of hours today, I guess? And as far as studying is concerned, I have a degree in physics and spend a lot of time doing this kind of research.

3) Is that Thebe thing a reference to something? Sounds like the setting of a scifi novel.

There's a hard sci-fi Niven novel called A World Out Of Time in which the Sun is about to expand and torch the Earth, so they go to Jupiter, build a 15,000 km tower with fusion rockets on each end, and sink one end into Jupiter before igniting the "fusion candle" at both ends. The buried end holds the tower aloft against Jupiter's gravity while the exposed end serves to propel Jupiter into a closer orbit until it is able to capture Earth. Then the direction of the fusion candle is altered and the whole gas giant (with Earth in tow) is pushed back out to a safe distance.

Thebe should work well for this. It's one of the nearest moons, and it's pretty heavy; holding Thebe aloft in Jupiter would take a whopping 2.0e19 N of force. A fusion rocket would likely have an exhaust velocity on the order of 160 km/s, and because the power consumption of a rocket producing constant thrust is given by P = F*ve/2, this would result in a power output of roughly 1.6e24 W. If about 10% of this energy was radiated away as light, the Earth would receive a toasty 1357 W/m2 at a nice safe orbital distance of 306,000 km.

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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Fri Apr 10, 2015 11:24 pm UTC

eSOANEM wrote:
King Author wrote:
eSOANEM wrote:Neil's right, it would fly off tangentially because there's nothing keeping it in orbit.

I had wondered if maybe the time taken for spacetime to relax back to Minkowski spacetime might delay the tangential release by a little bit (on top of the expected 8 minutes) but a lengthy back-of-the-envelope calculation showed that the timescale of this relaxation is on the order of a tenth of a millisecond (so insignificant compared to the 8 minutes light-speed delay).


Whoa now what now? If there was going to be any additional effect, shouldn't it also take eight minutes, since nothing can move faster than light? Or is it that spacetime itself can shrink/expand/warp faster than light?


[stuff which didn't really answer your question very well]



The solar system is very much in the weak field limit of general relativity even close to the sun and certainly at earth so it's valid for us to think in terms of Newtonian-like potentials.

Before the sun disappears, I see a nice monopole 1/r potential (until I hit the surface of the sun where it becomes quadratic reaching a finite value at the origin). Immediately after the sun disappears, the potential has to respond smoothly (spacetime is a differentiable manifold so smooth functions over it such as potentials have to be smooth in time as well as in space).

The simplest choice would just be to have the potential decay exponentially to 0 everywhere but this would be a non-local effect and isn't allowed.

What you'd actually get is that the only plausible solution is for the sun to turn into an equivalent energy's worth of monopole gravitational radiation. Normally gravitational waves can only be produced by quadrupole and higher moments but the radiation from the sun has to also be spherically symmetric and the only choice for that is monopole radiation. This isn't forbidden here because we're doing such unphysical things as throwing away a bunch of mass.

Such radiation's also forbidden at large ranges in electromagnetism but should be visible in the very near field of one end of an electric dipole antenna.

Anyway, this gravitational wave travels radially outwards in a bubble at the speed of light and peels the potential away from its initial 1/r value towards its new value of 0 (ignoring the effect of the planets).

It takes 8 minutes for the earth to see the start of this wave and for the potential here to start peeling off from the initial 1/r value. It then takes on the order of a third of a picosecond for the potential to reach 0.

Once the sun has disappeared, for those 8 minutes, the earth is in fact orbiting the gravitational wave bubble but, once the 8 minutes are up, there are shells of energy density further out than the earth and so the earth feels no gravity from them (the shell theorem) and its orbit will straighten. Once the pulse of gravitational radiation has passed, the earth's path will be straight.

Of course, in practice, the gravitational wave will interact with the earth and give it some radial momentum so, unlike the Newtonian case where the earth flies out exactly tangentially, in the GR case it will have some velocity perpendicular to its initial orbit.

Edit: I also worked out why I kept getting confused with my earlier numbers where I'd been misinterpretting one of the values I introduced which was that one of the terms was actually proportional to the timescale over which the sun disappears (which has to be at least the light crossing time from the sun's centre to its surface of 2.32s). The numbers I gave are all proportional to this factor (and hence the timescale over which the sun disappears) and are correct if the sun disappears on the timescale of a second (so the fastest possible scale).
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Re: Earth's trajectory if the sun disappeared...?

Postby Neil_Boekend » Sat Apr 11, 2015 2:36 pm UTC

I didn't know that the moon orbited the sun instead of the earth. However when the sun is gone, what else is it going to do? It seems unlikely that it would crash or escape
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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Sat Apr 11, 2015 2:53 pm UTC

Neil_Boekend wrote:I didn't know that the moon orbited the sun instead of the earth. However when the sun is gone, what else is it going to do? It seems unlikely that it would crash or escape

Nah, it will follow the same trajectory as Earth, with the same perturbations. So it will weave back and forth along the same path, alternately crossing in front of us and behind us, so from Earth's perspective it will still appear to be orbiting us just the same.

On the topic of dropping Thebe onto the surface of Jupiter and holding it aloft with fusion rockets to provide us with energy...

The mass flow will be around 1.25e14 kg/sec. Of course, only around 5% of that will be actual fusion reactant; the rest will be reaction mass in the form of inert gas. So we should expect our Second Sun to consume around 6,300 billion tonnes of Jupiter's atmosphere every second. That's a lot of mass...but not in comparison to Jupiter. Jupiter's atmosphere is about 75% hydrogen by mass, or 1.4e27 kg; our Second Sun could burn for 722,000 years before consuming just 10% of the planet's available hydrogen. By that time, waste heat would have long since warmed Jupiter enough to serve as a heat source all on its own.

We could probably place our fusion candle around the equator, so that it swung by Earth every 11 hours. Of course, Earth would also be spinning, so the days would be a little mixed up. But other than that, the possibilities are pretty nice. We could manually control the latitude of the fusion candle to simulate seasonal variations and so forth.

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Re: Earth's trajectory if the sun disappeared...?

Postby gmalivuk » Sat Apr 11, 2015 4:34 pm UTC

Qaanol wrote:Quite the contrary. The moon orbits the sun, and the earth just makes the moon wobble a bit in its orbit. In fact, the moon’s orbit around the sun is always convex: the wobble as it “orbits” earth never causes the moon to so much as curve away from the sun, let alone backtrack. The moon’s trajectory around the sun just varies between “bent slightly more, as if going around a smaller circle” and “bent slightly less, as if going around a wider circle”.

In round number, the moon is about 389 times closer to the earth than to the sun, and the sun is about 333,000 times more massive than the earth. Now 389² is about 151,000, so the sun’s pull on the moon is more than double the earth’s pull on the moon.
And yet, the moon is well within Earth's sphere of influence, meaning the path of the Moon in a reference frame centered on Earth can be fairly well approximated only considering Earth's gravity.
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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Sat Apr 11, 2015 5:12 pm UTC

gmalivuk wrote:
Qaanol wrote:Quite the contrary. The moon orbits the sun, and the earth just makes the moon wobble a bit in its orbit. In fact, the moon’s orbit around the sun is always convex: the wobble as it “orbits” earth never causes the moon to so much as curve away from the sun, let alone backtrack. The moon’s trajectory around the sun just varies between “bent slightly more, as if going around a smaller circle” and “bent slightly less, as if going around a wider circle”.

In round number, the moon is about 389 times closer to the earth than to the sun, and the sun is about 333,000 times more massive than the earth. Now 389² is about 151,000, so the sun’s pull on the moon is more than double the earth’s pull on the moon.
And yet, the moon is well within Earth's sphere of influence, meaning the path of the Moon in a reference frame centered on Earth can be fairly well approximated only considering Earth's gravity.

These considerations are further complicated by the fact that we have a big-ass moon compared to all the other planets.

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Re: Earth's trajectory if the sun disappeared...?

Postby scarecrovv » Sun Apr 12, 2015 9:35 am UTC

sevenperforce wrote:At this point, the speed of Earth relative to Jupiter will be 20.61 km/s. At that speed, it will be captured by Jupiter's gravity if it comes within 596,109 km of the gas giant. So Earth has a window of about 1.2 million kilometers; if it hits that window, Jupiter will capture it; if not, it will escape.

Why would Earth be captured? As it falls into Jupiter's gravity well it will pick up speed which it will subsequently use to escape right? In order to be captured the Earth's orbital energy around Jupiter needs to go somewhere. Is there an energy sink that I'm not seeing?

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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Mon Apr 13, 2015 5:44 am UTC

scarecrovv wrote:
sevenperforce wrote:At this point, the speed of Earth relative to Jupiter will be 20.61 km/s. At that speed, it will be captured by Jupiter's gravity if it comes within 596,109 km of the gas giant. So Earth has a window of about 1.2 million kilometers; if it hits that window, Jupiter will capture it; if not, it will escape.

Why would Earth be captured? As it falls into Jupiter's gravity well it will pick up speed which it will subsequently use to escape right? In order to be captured the Earth's orbital energy around Jupiter needs to go somewhere. Is there an energy sink that I'm not seeing?

If the Sun disappears, Jupiter's Hill Sphere will admittedly expand quite dramatically. But the escape velocity of 1 Jupiter mass at a distance of 596,109 km is 20.61 km/s. Inside, it is smaller; outside, it is larger.

You say, "in order to be captured the Earth's orbital energy around Jupiter needs to go somewhere" but that's not quite true. It is the earth's orbital energy (technically, the momentum or inertia) which would keep it in orbit around Jupiter. Jupiter exerts tremendous force on passing mass; it will pull into orbit any object moving at less than the escape velocity for the distance at which it is passing.

If Earth were to pass by Jupiter at less than ~600k km, it would indeed be accelerated, but it would be accelerated in a curve, so the magnitude of its tangential velocity would remain constant.

Another way of thinking about it is that even if Earth were to gain speed falling toward Jupiter, it would lose speed as it moved away from Jupiter (since it would be pulled "back") and thus eventually fall back toward Jupiter and into orbit.

You're probably thinking of a gravity assist, which is typically used by spacecraft to speed up or slow down. But this speed-up or slow-down is determined relative to the Sun, and it is accomplished by bleeding off a small fraction of the planet's momentum. The loop around a planet takes a certain amount of time, and so as long as you are always above escape velocity, you can be "pulled along" by the planet's motion around the Sun (not by its inherent gravity well) during your loop and thus gain a bit of speed. You can also use the loop to leverage the Oberth effect if you want to make an engine burn during your flight. In the case of the Earth, though, this doesn't really apply; its mass is non-negligible in comparison to Jupiter's mass and so you don't really have the same situation at all. Not to mention the absence of the Sun in this particular case.

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Re: Earth's trajectory if the sun disappeared...?

Postby gmalivuk » Mon Apr 13, 2015 12:04 pm UTC

scarecrovv is right, though. WIthout other forces, Earth either escapes or orbits Jupiter based only on its distance and speed relative to Jupiter. DIrection doesn't matter for escape, except inasmuch as crashing into something makes it harder to escape.

There is no way for Earth to be magically captured into a close circular orbit around Jupiter unless some other mechanism exists to shed most of the relative velocity.

Edit: and what do you mean it will only accelerate if it gets within 600km? Earth would accelerate toward Jupiter along the entire trajectory, because gravity has infinite range.
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Re: Earth's trajectory if the sun disappeared...?

Postby cjameshuff » Mon Apr 13, 2015 12:39 pm UTC

sevenperforce wrote:If the Sun disappears, Jupiter's Hill Sphere will admittedly expand quite dramatically. But the escape velocity of 1 Jupiter mass at a distance of 596,109 km is 20.61 km/s. Inside, it is smaller; outside, it is larger.


That's backwards, escape velocity is lower at greater distances.


sevenperforce wrote:You say, "in order to be captured the Earth's orbital energy around Jupiter needs to go somewhere" but that's not quite true. It is the earth's orbital energy (technically, the momentum or inertia) which would keep it in orbit around Jupiter. Jupiter exerts tremendous force on passing mass; it will pull into orbit any object moving at less than the escape velocity for the distance at which it is passing.


And any object moving at greater than escape velocity will always remain at greater escape velocity unless there's an interaction that transfers momentum to a third body, aerobraking, tidal losses, etc.

If the sun were to disappear while Earth and Jupiter were at their closest, the initial distance would be about 4 AU and the relative velocity about 17 km/s. Escape velocity would be 650 m/s...Earth couldn't come close to being captured. If you change the timing to make Earth pass close to Jupiter, you just increase the initial distance and velocity, making the trajectory even further from a closed orbit.

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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Mon Apr 13, 2015 3:58 pm UTC

gmalivuk wrote:scarecrovv is right, though. WIthout other forces, Earth either escapes or orbits Jupiter based only on its distance and speed relative to Jupiter. DIrection doesn't matter for escape, except inasmuch as crashing into something makes it harder to escape.

There is no way for Earth to be magically captured into a close circular orbit around Jupiter unless some other mechanism exists to shed most of the relative velocity.

Well, I never suggested a "close circular orbit"; I was assuming it would be highly elliptical. But yeah, you're right -- I forgot to account for the fact that Earth would already be in a hyperbolic orbit with respect to Jupiter.

I wonder if Earth could bleed off some momentum by slingshotting Jupiter's existing moons to kingdom come? Let's see...

Ganymede: 1.4819e23 kg, 10.88 km/s
Callisto: 1.0759e23 kg, 8.204 km/s
Io: 8.9319e22 kg, 17.334 km/s
Europa: 4.8e22 kg, 13.74 km/s

Taken together, that's 3.93e23 kg of moon with an average velocity of 11.96 km/s. Maximum momentum transfer would require that each moon make a 180-degree hyperbolic orbit around Earth, contributing twice its Jovian orbital momentum to Earth. If this could be managed, Earth would (best-case scenario) lose 1.57 km/s, which won't help it at all.

Guess we'd have to hope for a gravity assist from Saturn or something.

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Re: Earth's trajectory if the sun disappeared...?

Postby oxoiron » Mon Apr 13, 2015 7:57 pm UTC

Answers like this are why I keep coming back to these forums. God bless you, sevenperforce, for taking the time to draw those diagrams and do those maths. I love that you weren't satisfied with the simple "It would fly away in a straight line" answer.
Spoiler:
sevenperforce wrote:If the sun disappeared, all of the planets would rocket off tangent to their former orbits. The inner planets have higher orbital velocities than the outer planets, but it seems extremely unlikely that they would be able to "catch up" to the trajectory of an outer planet in such a way as to come within a meaningfully close distance.

For example, Earth orbits at 29.8 km/s at a distance of 1 AU. Jupiter orbits at 13.1 km/s at a distance of about 5.2 AU:

jupiter and earth.png


Distances are to scale; object sizes are definitely not to scale.

Because Earth's trajectory is tangential, it will take us about 9.7 months to cross Jupiter's former orbit. By that time, Jupiter will have traveled an additional 2.23 AU along its tangential trajectory, moving it an additional 0.45 AU from the former Sun.

After 10 months, 26 days, 1 hour, 25 minutes, and 15 seconds, the Earth will have traveled 5.686 AU and Jupiter will have traveled 2.500 AU, bringing them both to the same distance of 5.773 AU from the former position of the Sun.

jupiter and earth 2.png


At this point, the speed of Earth relative to Jupiter will be 20.61 km/s. At that speed, it will be captured by Jupiter's gravity if it comes within 596,109 km of the gas giant. So Earth has a window of about 1.2 million kilometers; if it hits that window, Jupiter will capture it; if not, it will escape.

Let's also consider the worst-case scenario: an impact. Now, Earth is dense enough that its Roche limit with respect to Jupiter is only 63,000 km, less than the actual radius of Jupiter. So the window of impact is simply the sum of the two planet diameters, or 151,100 km.

jupiter and earth 3.png


Compared to the size of the solar system, this is NOT a very large window. From Earth's orbit, the capture window is just 0.0803 degrees wide (4.82 arcminutes) and the impact window is even more miniscule at 0.0102 degrees wide (0.612 arcminutes).

The direction of Earth's velocity vector in its orbit around the sun goes through 360 degrees every year, or 0.986 degrees every day, or 4.93 arcminutes per hour. Thus, there will only be 58 minutes, 40 seconds out of the entire year in which Earth will be aimed in the right direction to hit the capture zone, and 7 minutes, 27 seconds out of that in which Earth will be aimed in the right direction to actually hit the impact zone (Jupiter will stay in orbit for 35 minutes longer than Earth due to lightspeed delay, but this will only shift the timing of the window, not the size of the window).

Jupiter is by far the largest planet and will have a much larger capture window than any other planet; it's also the closest of the outer planets. So Earth has a much better chance of hitting Jupiter's capture window than any of the other outer planets. The chance of an inner planet hitting Earth's capture window is also very small. Hitting Jupiter's capture window seems to be our only chance for survival.

Not that it's much of a chance. The nearly-11-month journey would almost certainly reduce Earth to a completely inhospitable ice cube. I'm not entirely sure how much heat the Earth would still retain but it probably wouldn't be enough for any but the most robust life-support systems to continue operating. And even once we reached Jupiter, we'd still be in a bad spot; its thermal contraction only causes it to radiate about 51 W/m2, just 4% of the energy we are used to receiving from the sun. Of course, if we can manage to construct hydrogen fusion rockets on opposite ends of Thebe and tow it into Jupiter's upper atmosphere to turn it into a fusion candle, we'll be just fine.
By the way, do not take as being facetious any of what I just wrote. I genuinely like the fact that on occasion forumites give way more detail than necessary when answering questions. Lord knows I've done it myself more than once. I find your passion for details refreshing.
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Re: Earth's trajectory if the sun disappeared...?

Postby Copper Bezel » Tue Apr 14, 2015 1:22 am UTC

Yup. The What If? blog questions could technically be answered in a single sentence in most cases, too. That's very much not the point, which is why this stuff is awesome. = )
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Re: Earth's trajectory if the sun disappeared...?

Postby sevenperforce » Tue Apr 14, 2015 5:05 am UTC

Well, I messed up the whole "hyperbolic trajectory" thing pretty badly. But taking that into account, what else can be determined?

With an initial distance of 4.742 AU and an initial relative velocity of 20.51 km/s, it's trivial to solve for the parameters of the hyperbolic trajectory. Impact will happen within a window of about 1.23 degrees, which means that there's a full 33 hours and 14 minutes out of each year when Earth's velocity vector puts it on a collision course with Jupiter.

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Re: Earth's trajectory if the sun disappeared...?

Postby Dopefish » Tue Apr 14, 2015 5:17 pm UTC

I was thinking about something like this recently due to someone insisting that the sun doesn't affect us in terms of gravity since we're so much closer to the earth.

If (via magic) we as humans stopped being influenced by the suns gravity, but the earth continued to be influenced by the suns gravity as normal, what would happen to us? Would those on one hemisphere find themselves getting pancaked against the earth and those on the other shooting off, or would it be not quite so extreme so we'd still be able to ride the earth without it getting yanked away by the sun?

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Re: Earth's trajectory if the sun disappeared...?

Postby Nicias » Tue Apr 14, 2015 5:26 pm UTC

If I asked Alpha correctly:

www.wolframalpha.com/input/?i=(mass+of+ ... s+of+earth)*(radius+of+earth%2Fdistance+to+sun)^2

The acceleration due to the sun is 5.996×10^-4 g. So, you wouldn't even notice.

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Re: Earth's trajectory if the sun disappeared...?

Postby Xanthir » Wed Apr 15, 2015 5:17 am UTC

I mean, it's really not hard to figure out. Do you feel significantly different at noon vs midnight, in a "something external is pulling/pushing on me" way? If not, then the sun doesn't have much of an effect; if it did, you'd feel it pull up at noon and "push" down at midnight.
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Re: Earth's trajectory if the sun disappeared...?

Postby Dopefish » Wed Apr 15, 2015 5:43 am UTC

That's less exciting than I'd hoped. The difference in gravity due to the sun due to being on one side or the other of the earth (as in day/night cycle) would intuitively be small, but the difference between that average amount of gravity (which turns out to be tiny, but I didn't know that apriori) and none at all isn't immediately obvious to my intuition.

The calculations are relatively easy to be fair and I was just lazy, but eh. I was sortof thinking that humans were orbiting the sun in the same way the moon is orbiting the sun, where in both cases it's intuitive that the earth's gravity is the more relevant thing, despite it being the case there's more force on the moon via sun than on the moon via earth, so maybe there were some surprises with humans wanting to go tangent to the earths orbit while the earth continued to orbit and things happening there, but it seems not to be the case.

(The person arguing no influence at all was arguing truly 0 influence though, and not just a negligible influence, so there was still definitely some lacking foundation for the debate.)

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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Wed Apr 15, 2015 11:42 am UTC

Xanthir wrote:I mean, it's really not hard to figure out. Do you feel significantly different at noon vs midnight, in a "something external is pulling/pushing on me" way? If not, then the sun doesn't have much of an effect; if it did, you'd feel it pull up at noon and "push" down at midnight.


This isn't really a good argument. The difference in gravity between noon and midnight is the tidal effect of the sun which decreases far more rapidly than the gravitational force from the sun.
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Re: Earth's trajectory if the sun disappeared...?

Postby PeteP » Wed Apr 15, 2015 11:50 am UTC

Dopefish wrote:That's less exciting than I'd hoped. The difference in gravity due to the sun due to being on one side or the other of the earth (as in day/night cycle) would intuitively be small, but the difference between that average amount of gravity (which turns out to be tiny, but I didn't know that apriori) and none at all isn't immediately obvious to my intuition.

The distance isn't much different but if the Sun is on the other side of the earth it pulls you down if it's on your side of the earth it pulls you up. (Of course it isn't usually directly above or below you, but you see my point.)

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Re: Earth's trajectory if the sun disappeared...?

Postby speising » Wed Apr 15, 2015 11:55 am UTC

eSOANEM wrote:
Xanthir wrote:I mean, it's really not hard to figure out. Do you feel significantly different at noon vs midnight, in a "something external is pulling/pushing on me" way? If not, then the sun doesn't have much of an effect; if it did, you'd feel it pull up at noon and "push" down at midnight.


This isn't really a good argument. The difference in gravity between noon and midnight is the tidal effect of the sun which decreases far more rapidly than the gravitational force from the sun.


ah, i'd think the difference is actually twice the suns gravity.

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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Wed Apr 15, 2015 2:05 pm UTC

Ah, wait, I forgot that people's orientation is different at noon and midnight. You're right; it would be twice the sun's gravity.
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Re: Earth's trajectory if the sun disappeared...?

Postby gmalivuk » Wed Apr 15, 2015 2:09 pm UTC

But if we and the Earth are both affected by the Sun's gravity, why would there be that difference?
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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Wed Apr 15, 2015 2:17 pm UTC

we're not in freefall though so we do (in principle) feel an actual force.
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Re: Earth's trajectory if the sun disappeared...?

Postby Neil_Boekend » Wed Apr 15, 2015 2:26 pm UTC

eSOANEM wrote:we're not in freefall though so we do (in principle) feel an actual force.

What do you mean? Do you mean that the earth is not in free fall with respect to the sun? What else is an orbit?
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Re: Earth's trajectory if the sun disappeared...?

Postby gmalivuk » Wed Apr 15, 2015 2:37 pm UTC

If we and the Earth were both affected by a flat 1g gravitational field, and nothing were magically holding the entire planet against that field, we wouldn't feel any difference between the top and bottom of the planet. The only way we could feel a difference would be if we already "turned off" the effect on one or the other (of us vs the planet).
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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Wed Apr 15, 2015 3:03 pm UTC

Yep, you're right gmal.

I'm confusing myself. We may not be in freefall about the earth but are around the sun (up to tidal effects).
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Re: Earth's trajectory if the sun disappeared...?

Postby jaap » Wed Apr 15, 2015 3:29 pm UTC

I once did that same calculation for the moon (i.e. take twice the gravitational force it exerts to get the difference between when it's overhead and when it is opposite). This was in order to show that it would have virtually no effect on the blood circulation of a human being. It indeed wouldn't (the result was on the order of the weight of one extra grain of rice to a 1kg bag).

I hadn't realised until the discussion above that this was incorrect. The moon accelerates the earth too as they orbit around their common centre of gravity.
It doesn't make any difference to the conclusion, as the tidal effect is necessarily much smaller still, but it's always good to learn when you're wrong.

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Re: Earth's trajectory if the sun disappeared...?

Postby addams » Thu Apr 16, 2015 1:34 am UTC

eSOANEM wrote:
King Author wrote:
eSOANEM wrote:Neil's right, it would fly off tangentially because there's nothing keeping it in orbit.

I had wondered if maybe the time taken for spacetime to relax back to Minkowski spacetime might delay the tangential release by a little bit (on top of the expected 8 minutes) but a lengthy back-of-the-envelope calculation showed that the timescale of this relaxation is on the order of a tenth of a millisecond (so insignificant compared to the 8 minutes light-speed delay).


Whoa now what now? If there was going to be any additional effect, shouldn't it also take eight minutes, since nothing can move faster than light? Or is it that spacetime itself can shrink/expand/warp faster than light?


This would be additional to the 8 minutes light travel time.

It takes 8 minutes for the earth to become aware that the sun has disappeared but then there is an additional delay as spacetime rebounds to its no-sun form (pretty much Minkowski space). This delay turns out to be completely inconsequential.

It's also worth noting that, if the sun suddenly disappeared you'd get some really funky gravitational monopole radiation which means that, whilst the planets' paths would average out to straight lines, in practice they'd do slightly wobblier things.

I'd expect a wobble, too.
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Re: Earth's trajectory if the sun disappeared...?

Postby Goemon » Sun Apr 19, 2015 2:27 am UTC

eSOANEM wrote: It then takes on the order of a third of a picosecond for the potential to reach 0.


How did you calculate that?
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Re: Earth's trajectory if the sun disappeared...?

Postby eSOANEM » Sun Apr 19, 2015 9:34 am UTC

I started in the Minkowski metric with a 1° perturbation due to a small mass which, at time t=0 "disappears" with μ́=-μ δ (initially, I tried it with δ as a δ function which led to confusion, but approximating μ̇ as linear, δ's just 1/τ where τ is the characteristic decay time of the perturbing mass). As the Earth is verrrry much in the weak field regime, this approximation should be reasonably accurate.

Then (staying in the local inertial cartesian co-ordinates you'd normally use in MInkowski space) calculate the connection coefficients. Then, you can use the EFEs to calculate Γ/(d/dt Γ) which gives you a timescale for the decay of space around Earth back to Minkowski spacetime. The timescale for the metric to decay won't be exactly this of course, but they should be related by a constant prefactor of order unity which, when we're talking timescales isn't really relevant.
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Re: Earth's trajectory if the sun disappeared...?

Postby addams » Mon Apr 20, 2015 1:17 am UTC

jaap wrote:I once did that same calculation for the moon (i.e. take twice the gravitational force it exerts to get the difference between when it's overhead and when it is opposite). This was in order to show that it would have virtually no effect on the blood circulation of a human being. It indeed wouldn't (the result was on the order of the weight of one extra grain of rice to a 1kg bag).

I hadn't realised until the discussion above that this was incorrect. The moon accelerates the earth too as they orbit around their common centre of gravity.
It doesn't make any difference to the conclusion, as the tidal effect is necessarily much smaller still, but it's always good to learn when you're wrong.

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