Rogue Planet

Junji Ito's Hellstar Remina actually is about this, a small rogue planet that shows up and goes really, really fast Like by the time it's discovered vs when it shows up is in a span of weeks or months. People....don't handle impending annihilation very well, as you can imagine. It's a really good story, it's both Lovecraftian and "maybe the real monsters was humans all along." There's also the show Salvation where not-Elon Musk tries to do stuff to save humanity from an approaching comet (meteor?). The various world governments know about this and keep the information from the public as long as possible to not incite mass panic. Plus stuff like Deep Impact, Armageddon etc.

So when we find exoplanets, the darkness/temperature of it really doesn't matter. We look for the dimming of a star as the planet passes in front of it, and this should happen at a regular period if it's a planet orbiting said star. I would imagine that a Very Big object (because even a Pluto sized planet is still a Very Big thing compared to comets/asteroids/oort stuff) would "cover"/block out a larger and larger part of the night sky as it approaches us. We would see this at multiple angles as we go around the sun, so someone, somewhere, at some point is going to do the math and figure out what's going on. How long this would take depends on how fast it's going and when we first notice it: unless it pops out of a wormhole we're going to have decades of time. And by "we" I mean scientists/governments, as they would absolutely hide this from people. Maybe some backyard astronomers might think this is true and there would be some level of disinformation campaigns.

It's also possible that this could be known for centuries, too, and make an alt-history story about how the world deals with this. It always amazes me that once America went to the moon we sorta just...gave up really doing stuff in space. It's been 50 years and we haven't put humans anywhere besides low earth orbit. We went from the first airplane to jets within 50 years because we saw the utility of this, especially in weaponry. But if we knew that we had a limited amount of time to engineer our way to live on another planet (or to blow up the rogue planet) then things would be a lot more The Jetsons-y than it is right now. One of the most interesting parts of science that I don't see in a lot of spec fic is the acquisition of materials such as rare earths. There are certain minerals that have only one or two mines/sources in the entire world, and if one of those countries decides to not share that with anyone else, well...I mean look at Bolivia. Imagine if the world at large actually knew about Vibranium, Wakanda would either be controlling the entire planet or it would have been burnt to the ground by colonizers. Areas that mine/produce/refine essential materials/components for space tech would either constantly be at war or the most powerful country in the world.

It's hard to guess how the world, culturally, would be different. Maybe doomsday cults are really popular. Maybe Christianity is absolutely the sole religion because of Revelations, or maybe Norse mythology because of Ragnarok. Maybe no one belives in god but they worship science. Maybe the governments drug the populations so they remain oblivious to the situation. Do whatever the heck you want for your story, since the options are so wide that anything is really possible.

Now if you want to know what would actually happen to the Earth, including how things like Jupiter might change things, then the Universe Sandbox sim game would be a good research tool. Should be a good approximation for your story! No one is expecting you to do the actual math for this.

 

Pluto, roughly speaking, is about 40 'Astronomical Units' (AU) from the Sun, and takes around 220 years to complete one orbit. Hence, five years and change to travel one AU. Pluto, at that distance, is also faint, requiring a good quality telescope to spot.

Pluto's orbit is also 'tilted' with respect to the other planets 'plane of the ecliptic.' No reason to believe a genuine rogue planet would be within the 'plane of the ecliptic' at all.

However, over the past couple decades, there have been multiple efforts, backed by solid math, to find 'planets' past the orbit of Pluto. Thus far, these rather exhaustive surveys have yielded no true planets, but rather a collection of large asteroids termed 'Centaurs,' the largest of which have diameters on the order of several hundred miles (or maybe a thousand or so - been ages since I looked into this).

My evaluation: given that there are almost always multiple large scale sky surveys ongoing at any given time, and given that several of these surveys are oriented towards finding planets (well, typically asteroids), odds are good that any intruding Pluto sized rogue planet would be spotted well before reaching Pluto's orbit. SOP would be to cross check with older surveys, locate it there, and compute an orbit. How soon said rogue would take to reach Earth depends on its exact trajectory and velocity; barring something really weird, probably at least a decade plus, likely a couple times that.

 

This was the back story to Arthur C Clarke's Rendezvous with Rama. If people on Earth had or could quickly build a network of telescopes with current technology they'd discover the planet at or just after it had come inside the orbit of Uranus - any further out than that and it would to some degree be a matter of luck if they picked it up that soon. After that it would depend on how fast the rogue planet was moving and also what effect the gravity of the Sun had on it's approach.