Better than you. I’ve derived all this information with two hypothetical assumptions which you have me. First is a orbital period of 100,000 years, and second is that it comes close enough to earth to have a gravitational effect. You’re absolutely correct that Kepler’s third law gives the average distance of a planet from the sun. But want to know a secret? The average distance on an ellipse from one of its foci is exactly equal to the semi-major axis.
https://www.farmingdale.edu/faculty/sheldon-gordon/RecentArticles/average-distance-in-ellipse.doc
Now, going back to the law;
https://wikimedia.org/api/rest_v1/media/math/render/svg/91c1aecc8260cb84e57b79debfa52f1fe6712f56
Where alpha is the semi-major axis and T is orbital period.
Solving for a T of 100,000 years gives us a semi-major axis of 2152 AU. Since we need this planet to get close to earth, that puts the the perihelion at 1 AU from the sun and makes the aphelion then 4303 AU from the sun, which is 6.8% of a light I originally said 10% so please forgive me for exaggerating a touch since I was doing the math in my head. You can quibble about where to put the perihelion, doesn’t have to be exactly 1 AU but as you can see it makes a very minimal amount of difference when it comes to the perihelion.
Knowing these parameters (semi-major axis), we can calculate speed at any point along its path (based on distance from the sun), specifically at perihelion it’s going 42.1 km/sec.
You know why that’s important? Because it’s within a hairs breadth of escape velocity. You see, the longer you go in orbital period, and the more eccentric an object’s orbit, the closer you get to escape velocity. And by close, I mean minute factions of a percent away in this hypothetical.
Which is why this is so unlikely. The orbital velocity of an object in your hypothesis has to be so precise. Any less, and it’s orbital period is much, much lower. Any more and it leaves the solar system entirely.
And apogee only refers to the high point of an orbit around Earth. The proper term for something around the sun is aphelion. A generic term would be apoapsis.
Now, want to tell me how I don’t understand what I’m talking about again?