Incoming Rogue Planet! When do we see it?

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.

posted almost 5 years ago

Starfish Prime‭

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