Could an ejected "extra ice-giant" still be lurking in distant solar orbit? BACKGROUND

Hot Jupiters are thought to have migrated inwards, implying that another giant planet has been ejected in order to conserve the orbital momentum of those planetary systems. The number of interstellar "vagabond planets" has been estimated to be as many as the number of stars. And simulations show that if the Solar System started out with an extra gas giant next to Jupiter which was ejected, then one can end up with a very good fit with today's Solar System. 

QUESTIONS

Where would such a hypothetical ejected ice giant be today? Should it be closer or further away than hot Jupiter-ejected planets because it is much smaller? Could it stay in an Oort cloud type orbit during several billion years (maybe ~15 orbits around the Milky Way) without being pulled off by passing stars? At what distance could we today observe a Uranus sized planet? Should it stay in the ecliptical plane or could its orbit have become greatly inclined when it was ejected?
Since I'm a hobbyist, I usually wait to see if someone a bit smarter wants to answer first, but I can give a couple thoughts on this.  


  Hot Jupiters are thought to have migrated inwards, implying that
  another giant planet has been ejected in order to conserve the orbital
  momentum of those planetary systems.


In the article you posted (I'll pull the quote)


  The conservation of orbital momentum dictates, as planetesimals move
  from the outer disk inward, that Saturn, Uranus and Neptune must move
  outward. This process is known as the planetesimal driven migration


The conservation of orbital momentum he's talking about happens when objects cross each others orbits and interact, one gains orbital energy, the other loses energy.   That's not the only way planetary orbits can change.   Over time, there's tidal effects, orbital resonance and a sun can lose mass leading to gradual drifting or perhaps a sun can gain mass if the solar system passes through a dense dust/gas cloud.

But mostly what we're talking about here is a form of gravity assist as 2 planets get close enough that they kind of swing past each other, one moves in the other out.

More on that here:   http://science.howstuffworks.com/nomad-planet2.htm

Now, were gas giants ejected from our solar system, according to this question, somebody said no:

Is there any evidence that the Gas Giant planets in our solar system are experiencing orbital migration?

But, this article, and the article you posted that suggests otherwise, that it probobly did happen based on models, but when the solar system was young, like about 4 billion years ago, this model suggests the Solar system could have lost 1 or 2 ice giants.

http://www.space.com/13584-extra-giant-planet-solar-system.html

and we could still lose Mercury (though we probobly won't).   http://www.universetoday.com/14032/could-jupiter-wreck-the-solar-system/


  Where would such a hypothetical ejected ice giant be today? Should it
  be closer or further away than hot Jupiter-ejected planets because it
  is much smaller?


Tough call.  Using basic gravity assist mathematics, the closer to the sun the faster an object could be ejected from the solar system so a hot Jupiter probobly eject inner planets at a faster velocity on average than more distant Jupiter, but the bigger factor is the size of the star system, not the size of the planet.   The more massive the star, the faster the orbital velocities, the faster the Ejection is likely to be, even after accounting for escape velocity.   

it's also important that the object being ejected is smaller than the object doing the ejecting, for example, Jupiter could pull Mercury away from the sun but Mercury couldn't pull Jupiter into the sun.   More massive wins the tug of war, but how much smaller the ejected object needs to be isn't that relevant.  Jupiter could throw a Neptune sized planet out of a solar system at almost the same speed as it could throw a Mercury sized planet out of a solar system.

Now, as to where it would be, planets aren't likely to escape a solar system at too fast a speed.   Jupiter orbits the sun at 47,000 KM/Hour and an optimal gravity assist could eject something way from Jupiter at 94,000 KM/Hour, (plus any velocity of the object's own, which would depend largely on eccentricity, but since it would cross Jupiter's path, probably wouldn't be hugely different than Jupiter's), and if you subtract escape velocity from the sun at that distance (47,000 * square root of 2), or 66.4 KM/Hour, your left with a maximum exit the solar system velocity of, roughly 28,000 KM/Hour, plus a bit - and that's maximum.  Average escape velocity would likely be less, but lets say 20,000 KM/Hour, which works out to be 1 light year every 50,000 years.  considering it left the solar system an estimated 4 billion years ago, that's 80,000 light years, which from where we are, is pretty much the far end of the Milky way and who can say how it's orbital velocity might have changed over that time as it passed by other stars, but basically, it could be anywhere in the Milky way, and, if you consider, our sun might have come from the Sagittarius dwarf galaxy, not the Milky way, it could be practically anywhere, even ejected from the Milky way.  4 billion years is a long time, even for a slow ejection.

or, it could be orbiting another sun - article on that here:  http://www.universetoday.com/94656/rogue-planets-can-find-homes-around-other-stars/


  Could it stay in an Oort cloud type orbit during several billion years
  (maybe ~15 orbits around the Milky Way) without being pulled off by passing stars? 


Entirely possible though billions of years is a long time.   Passing stars usually don't pass very close, but I'm sure it's happened at least a few times in the past 4 billion years.

close enough to disrupt the inner planets - that's different and more rare.   Close enough to disrupt the Oort cloud, probobly happens every so often - but I wouldn't want to guess how often.


  At what distance could we today observe a Uranus sized planet? Should
  it stay in the ecliptical plane or could its orbit have become greatly
  inclined when it was ejected?


it depends if we know where to look.    If we saw it pass in-front of a star and we could plot it's path, then a number of light-years.   Looking into the blackness of space, much less distance, and it would depend on the planet's reflectivity too.   I wouldn't even know what to guess, but Eris, for example is 1/20th the diameter of Uranus and it's one of the larger Kuiper belt objects.   With relative ease, they could likely spot a Uranus sized object at well over 20 times the distance of Eris.   Now, if the question is, what's the furthest a Uranus sized object could be and not have been seen yet - that's hard for me to say but an object that size and at that distance would likely have it's own Trojans and I find it hard to believe it would have gone unnoticed.

as to "would it stay in an elliptical plane" - not necessarily.   A gravitational assist around a planet can alter the elliptical plane depending on the angle of approach.

hope that's not too wordy.  I can try to tidy up later.