Question: This is part of a series of questions.

*Context*: a friend of mine is writing a novel about a rogue planet around the mass of Mars passing by the solar system before continuing its journey in interstellar space (it must not be captured by the Sun). Given sufficient heads-ups, Earth sends a research mission to land on it, study it for as long as possible, and return.

We would like Earthians to have as much time as possible on the rogue planet.

**How slow and close could a Mars-like planet pass near the Solar System to be plausible and not cause catastrophic Earth orbit perturbation?**

I am guessing that there is no theoretical lower speed limit, as it can barely have the escape velocity of its birth system, then be slowed down by more stars behind it that in front of it. (right?) Or maybe a theoretical limit is the solar system escape velocity, which is around 700km/s, as even an almost stale object relative to the solar system will fall at that speed?

Answer: 
"Passing through the solar system" I guess means in this case "passing through the inner solar system". If it only passes the Kuiper belt or the Oort cloud, there would be no serious perturbations (no "showering the inner solar system" or the like").

When passing through the inner solar system there will be no perturbation either, unless it passes close to a planet, which is not likely. The trajectory of the object will in most cases be highly inclined to the plane of the solar system which diminishes the probability of a close encounter.

The escape velocity depends on the distance from the Sun. The value of 700 km/s refers to escape velocity from the surface of the Sun. At Earth's orbit it is something around 42 km/s.

An interstellar object will typically enter the solar system with a delta-v of 20-30 km/s, but the value can be much larger - 200 km/s and more - or smaller. As it approaches the Sun it will accelerate due to the Sun's gravity and pass by in a hyperbolic trajectory. How fast it is at any point is the (vectorial) sum of its initial velocity and its acceleration by the Sun. Its maximum velocity will therefore depend on how close it gets to the Sun - the closer it gets the faster it will become. Unless its initial delta-v was close to zero it will almost always have escape velocity. The capture of interstellar objects by the Sun is possible but certainly very rare.

**How close could a Mars-like planet pass near the Solar System to be plausible and not cause catastrophic Earth orbit perturbation?**

As close as you like.
Considering that the solar system is a multi-body system, we are sure it is a chaotic system in the long run. For the solar system, its [Lyapunov time](https://en.wikipedia.org/wiki/Lyapunov_time) is 5 million years.

This means that the effect of any perturbation on the system cannot really be reliably predicted past 5 million years. Plus what are you going to do with the Oort cloud bodies disturbed by the passage of the planet and that will likely end up showering the inner solar system?

For sure any body which can shoot through the solar system will need to have at least the sun escape velocity.

The additional problem is that landing on something moving that fast is going to be tricky: first you need to reach that velocity to be able to land, and then slow down to get back on Earth. Considering that the Apollo mission had to struggle being in the range of 11 km/s, you can see the challenge you are up to.