So say there was a rogue gas giant about the mass of Jupiter that, unfortunately, happened to be headed directly at Earth (or at least close enough to knock it into an orbit incompatible with life). It's coming in from way out of the plane of the solar system, so it's not going to knock anything else significantly off-kilter, but Earth is in for a spell of bad luck.

In this completely hypothetical scenario, we, your benevolent alien neighbours, had absolutely nothing to do with this, but, just as a matter of curiosity, whereabouts would you probably detect this incoming gas giant, and what kind of time range would there be between detection and impact?

In other words, at approximately what point would humanity discover a Jupiter-sized gas giant headed towards us, and how long would we have before it hit us?

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    $\begingroup$ related worldbuilding.stackexchange.com/q/72181/30492 $\endgroup$ – L.Dutch - Reinstate Monica Jan 6 at 7:57
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    $\begingroup$ This is difficult to answer because we could detect it very early but the point at which we definitely have detected it is impossible to know. But in case you have never looked at the night sky, Jupiter is often the brightest thing besides the moon and airplanes. You can see it with your own eyes very well and a simple amateur telescope will reveal moons and some structures on the planet. Perhaps looking at the sky yourself at night will help you build your world and realize just how big Jupiter is. If you want to calculate the time until impact, we need the speed btw. $\endgroup$ – Raditz_35 Jan 6 at 9:42
  • $\begingroup$ @Raditz_35 In fact, I am an amateur astronomer and own a rather large telescope, so I do have some concept of how absurdly large Jupiter is. Detection is going to be achieved at a rather long distance (although even I had underestimated how far away it could be detected, as calculated by Starfish Prime). $\endgroup$ – Gryphon Jan 7 at 0:11

Lets consider how hard it would be to spot this object with visible light astronomy. This isn't quite the right way to go about things, but it is a start. Most things are easier to see in IR than than visible light, so my final detection distance may be out be an order of magnitude (or a bit more). Note though that whilst your rogue Jupiter might be warmer than your average space rock, it won't be nearly as hot or bright as the smallest and weakest stars.

The faintest near-earth object in this JPL database is 2008 TS26, with an absolute magnitude of 33.2. We can find the absolute magnitude of Jupiter by using its geometric albedo $a$ of 0.538 and a diameter in kilometres $D$ of 142984km:

$$H = 5\log_{10}\left({1326 \over D\sqrt{a}}\right)$$

This gets us an absolute magnitude of about -9.49, which is Quite A Lot Brighter (roughly 2.543 times) that the boring rock.

2008TS26 has a semimajor axis of 1.92AU. Assuming that it is in opposition to the sun (a syzygy, an awesome word that is hard to use very often) it will have an apparent magnitude of 34.4, given that $$m = H + 5\log_{10}\left({D_{BS}D_{BO} \over D_0^2}\right) - 2.5\log_{10}\left(q(\alpha)\right)$$ where $H$ is the absolute magnitude, $D_{BS}$ is the distance from the body to the sun, $D_{BO}$ is the distance from the body to the observer, $D_0$ is the distance between Earth and the Sun and $q(\alpha)$ is something called the phase integral that I'm declaring to be 1 in this position.

With the same geometric relationship, we can rearrange the equation to find the equivalent distance of our rogue Jupiter where it would have the same apparent magnitude:

$$10^\frac{m - H + 2.5\log_{10}(q(\alpha))}{5} = D_{BS}^2 - D_{BS}$$

Leaving us with a nice quadratic to solve, giving us a $D_{BS}$ of ~24495AU, or about .387 lightyears. You haven't actually told us how fast this rogue Jupiter is travelling. Barnard's Star has the highest proper motion of nearby stars, as it is going at about 142km/s relative to us. If your rogue Jupiter had a similar speed, it would take 817 years to reach us.

(edit for Alexander: with a more pessimistic detection magnitude of 20, the detection distance becomes ~1190AU, and the transit time at 142km/s would be 40 years)

It is lucky it will take so long, because finding such cool and faint objects requires some serious hardware and dedicated telescope time. Something like WISE (Wide-field Infrared Survey Explorer), a satellite, whose main mission ran for just 10 months before the coolant ran out, or 2MASS (Two Micron All-Sky Survey) which used ground-based telescopes over a four year period would be needed, at the very least.

The most difficult bit would be spotting that the object was close and getting closer... if we were meeting it head on, it would have no proper motion and so traditional techniques for spotting nearby bodies wouldn't work and it could take multiple surveys over a long period of time to spot that it was getting brighter. If it were hitting us as right-angles to the Sun's trajectory, it would have some proper motion that would appear to reduce as time went on and it should become clear that it was on a collision course with us given a bit of extra attention.

It's coming in from way out of the plane of the solar system, so it's not going to knock anything else significantly off-kilter,

You can't just fling a Jupiter-mass through the inner system and assume that everything but Earth is going to be just A-OK. It is going to have a non-trivial effect on the orbits of all the inner worlds and the asteroid belt. How disruptive this would be I couldn't say, so you'll have to run it through a gravity simulator and see for yourself.

With regards to Rob's answer and brown dwarf stars, they're not quite comparable to Jupiter-type large gas giants. A binary brown dwarf system has been discovered a mere 6.5 lightyears away, in the form of Luhman 16. The stars might not be hot hydrogen-fusing things like most stars are, but they still have quite high surface temperatures... over 1000K. Jupiter, by comparison, is a mere 165K, and at least some of that will be contributed by solar heating (though admittedly not very much). The Stefan-Boltzmann law shows that radiated power per unit area from a black body is proportional to the fourth power of the object's temperature, which means that 1000K brown dwarf is ~1350 times more powerful an emitter as a 165K gas giant of the same size (and warmer brown dwarfs will be slightly larger, too, and so emit a higher total power). Brown dwarfs may not be much larger than Jupiter, but they can be much hotter and therefore much easier to spot with IR telescopes.

I suspect my .387ly estimate is a little pessimistic, and whilst I wouldn't be too surprised to be out by an order of magnitude, being out by two would be surprising for such a small and relatively cold object.

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    $\begingroup$ This answer seems to be exploring the absolute best-case scenario - humans will be studying particular region of space long to notice that this object is moving. In real life Oumuamua was discovered only when its apparent magnitude has peaked at about 20 (while according to your "best case" calculations we can detect objects at 34.4). $\endgroup$ – Alexander Jan 6 at 17:48
  • $\begingroup$ @Alexander the answer is exploring the fact that it could be detectable from Quite A Long Way Away, and even given very high speeds there would be centuries in which to launch or perform additional survey missions and review their results, and future surveys are likely to be better than the ones we've managed to date. The answer doesn't consider the possibility of detection techniques improving beyond the present day. $\endgroup$ – Starfish Prime Jan 6 at 17:50
  • $\begingroup$ @Alexander I also used visible light astronomy as an example, which isn't as effective as infrared at spotting these things, so there's an entirely reasonable chance that my detection range may be too short by an order of magnitude, giving many more centuries in which to detect the body. $\endgroup$ – Starfish Prime Jan 6 at 17:51

I expect we could see it from along way away. If we look at brown dwarfs, which have higher masses than Jupiter but cruically will have about the same temperature, then looking in the infra red we have seen brown dwarfs out to a few tens of light years (restricting ourselfs to the t dwarfs which are the closest match to a Jupiter) https://en.m.wikipedia.org/wiki/List_of_brown_dwarfs

The bigger issue will be if we have a telescope pointing at the right place at the right time. We have only just discovered a brown dwarf practically next door (https://en.m.wikipedia.org/wiki/WISE_J0521%2B1025) at 5 light years away simply because space is big and we have limited telescope time.

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  • $\begingroup$ That's 5 parsecs (about 16 light-years). Still very close, but not that close. $\endgroup$ – AlexP Jan 6 at 10:47
  • $\begingroup$ The golden point in your answer is that space is big and unless you are spending really large amounts of money looking for specific kinds of objects, you'll never notice most of them. (Detect them, yes. Notice them as being special and interesting? Not so much. There's just too many objects in the sky to have a decent chance of accidentally stumbling on things.) $\endgroup$ – Mark Olson Jan 6 at 14:57
  • $\begingroup$ There's also a rather large area in the sky that we don't typically look into at all, except with some very narrow-focused observatories that are not designed to observe things like planets at all. If the rogue planet is a sungrazer, with an orbit that stayed behind the sun, then SOHO's corona cameras will be our first and only warning, and the planet would only be 1AU from us, traveling over 615 km/s (escape velocity), reaching Earth in about 3-4 days. $\endgroup$ – Ghedipunk Jan 6 at 16:23
  • $\begingroup$ @Ghedipunk given the timescale of the wide field surveys, and the time it will take for a rogue planet to smite us, it'd spend plenty of time well away from the sun in the sky. Also, 615km/s? that's faster than galactic escape velocity. You're off by an order of magnitude for solar escape velocity. $\endgroup$ – Starfish Prime Jan 6 at 17:03
  • $\begingroup$ @StarfishPrime, Wikipedia puts the solar escape velocity at 617.5 km/s... Google gives 615 km/s. On average, the sun is denser than a galaxy, so having a higher escape velocity is expected. $\endgroup$ – Ghedipunk Jan 6 at 17:06

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