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Let's take a look at the trajectory of variety probe missions.

^{New Horizons and Ultima Thule will be 4.1 billion miles away when it visits the Kuiper Belt object. This chart shows the path of New Horizons compared to other probes that have left the solar system. Credit: NASA/Johns Hopkins University Applied Physics Laboratory [1]}

I wonder, why all these probes tend to explore outer system were launched to go outwards of the eliptical plane instead of go upwards or downwards? What I'm talking about here is go upwards or downwards that is considerably close to 90 degree. I was told the eliptical plane of any solar system tends to stay relatively uniform, with only Kuiper Belt Objects (KBOs) showing bizarre inclinations. The closest thing we'd find "below" the Earth would be an Oort Cloud object or outer star system, is it accurate to said so? What would we find if we go straight upwards and downwards?

Point of interest

¹ Nola Taylor Redd, Space.com Contributor, January 2 2019, NASA's New Horizons Just Made the Most Distant Flyby in Space History. So, What's Next?

Starting out from Earth, you have the free 30 km/s from Earth's movement around the Sun, which is in the plane of the ecliptic. To get far out of the plane you either have to boost a similar amount "up" or "down" (which is beyond the capability of current rockets) or go via one of the gas giants, and use its gravity to change course. So at least to start with you don't really have a choice.

One example of using a gas giant to change plane was the Ulysses probe, which, although it only barely got into the Outer Solar System, did orbit at almost 90 degrees to the ecliptic (80.2 degrees, in fact) in order to get a look at the North and South poles of the sun and the radiation and magnetic fields coming from them.

It is important to realize that space probes aren't really useful for finding objects in deep space. Space is so empty that a probe sent in a random "exploratory" direction would have a negligible chance of detecting an object orbiting the sun. The best way to find objects outside the ecliptic is to look for them using really large Earth-based or orbital telescopes. And there doesn't appear to be much out there.

All of the probes on their way out of the solar system were meant to explore current or former planets, which are all close to the plane of the ecliptic. That really is the reason the probes are close to that plane.

It would be of interest to sample the far reaches of the Sun's magnetosphere in different directions, but it hasn't been judged worth the expense.

Finally, it is easier to send probes out within the plane of the ecliptic because we can use the motions of the planets to give a speed boost. The Earth's motion of about 30 km/s is a nice start, but it's useless if you are heading directly towards the ecliptic north pole. And you can't use the outer planets as gravitational slingshots if you are going perpendicularly away from them.

We've had 5 flyby missions to the outer solar system so far. All of them had primary missions at one or more planets. That set the main constraints for their trajectories. Anything after the last planetary encounter was secondary.

For Voyager 2, for instance, the Neptune flyby was aimed at a close encounter with Triton, which reduced the possible exit trajectories:

You also see Voyager 2 now has a significant angle relative to the ecliptic.

The others:

• Voyager 1 had moons to visit at Saturn


• for the Pioneers, I don't think an interstellar mission was considered at all


• New Horizons had to fly by Pluto, so it was limited to Pluto's plane (Pluto is not large enough to change NH's course significantly)

What would we find if we go straight upwards and downwards?

There are almost no targets of interest outside the ecliptic. No planets or other bodies we can visit until you get to the Oort cloud, which is 1000 AU out (10x further than the Voyagers are now after 40+ years of traveling).
The only reason to choose a trajectory perpendicular to the ecliptic, is to have another measurement point for the solar wind. But spending an entire mission to 100 AU on just that would be expensive for very limited return.

A study for an interstellar mission is underway, but that too is planned to include a KBO visit so it'll be constrained to the ecliptic.

Staying in the same plane as the planets of the solar system allows us to use them for propulsion by means of a slingshot maneuver. Actually this Wikipedia page even has gifs showing the Voyagers as an example, so I guess it doesn't make much sense for me to go into detail here: https://en.wikipedia.org/wiki/Gravity_assist

But in essence there is no reason to not use such maneuvers for additional acceleration and as mentioned before going perpendicular to the plane would require propulsion systems we don't have (yet). Even if we had such technology, there is still no reason to not use that free energy anyway, since these probes have no specific destination.

Mostly because launching upwards would be useless to us, as we have extremely limited deep space travel capabilities.

Besides that; all rockets are launched along our equator anyways, the pull and movement of the Earth allows for an easier launch, and if we launch along our familiar plain we are able to utilize gravity assists from other celestial objects.

Gravity assists are a big reason, they give a huge boost to the rocket being launched and they give a flyby opportunity (great for research).