What is required for a satellite to remain positioned 400 km above the north pole, thus allowing it to appear essentially stationary relative to the surface of the earth? Could this best be achieved by placing the satellite in a solar orbit with the same period as the earth, only 400 km above the pole?
Could this best be achieved by placing the satellite in a solar orbit with the same period as the earth, only 400 km above the pole?
No. At least, not without it continuously applying thrust against Earth.
At a distance of only 400km above the surface, Earth's gravity (acceleration toward Earth) will still be nearly that of the value at Earth's surface (9.8 m/s/s).
If you could "magically" place an object geostationary to Earth, 400km straight up from the North Pole and release it there, it won't stay there. Yes, technically, it will be in orbit around the Sun, just like Earth is. But, it won't be in orbit around the Earth, so will respond to Earth's gravity field with a downward acceleration of pretty much 1g.
There is no way for an object to orbit the Earth such that it would appear to be stationary over a pole. There are orbits which allow an object to "linger" at high latitudes, but they are actually in highly inclined, highly elliptical orbits (see Molniya orbit); they can be geosynchronous - appearing in the same region of the sky most of the time, and repeatedly describing the same path in the sky (from the point of view of a "fixed" ground-based observer), but they are by no means geostationary.
About Geostationary Orbits
Gravity follows the inverse-square law; the greater the distance, the smaller the force. At low altitudes such as where the ISS orbits, an object must travel at about 7.6km/s to remain in orbit (completing one orbit about every 90 minutes). The Moon, circling the Earth once a month at a distance of about 239,000 miles, travels in its orbit at about 1km/s. At about 22,000 miles, an object would have to travel about 3km/s, completing one orbit every 24 hours. At just the right speed and distance, the orbit will be synchronous with Earth's rotation. If the orbit is also circular and at zero inclination to the equator, an object in that orbit will appear to be at a fixed point in the sky, always directly over some specific point on the Earth's surface. For more on this, see this question: How do communication satellites remain positioned above a particular region? and this Wikipedia article.
4,000,000km but not 400km. Even if you ignore Earth there's no solar orbit that stays 400km above the North pole. It's going to swing back and forth over the course of a year, from 400km above the North Pole to 400km above the South Pole. In practice Earth's gravity would rapidly pull it down.
A million km up it's possible to have a pole-hovering satellite but it's not actually in orbit, but using a solar sail to hover. That can only be done when you're far enough out that the minuscule force of a solar sail is enough to counteract the gravity of the body you are trying to hover over. This actually has a practical use--you could put a TV satellite over the pole to serve customers too far north to be served by the equatorial satellites. The patent has expired, you're free to build one.
Edit: The patent: https://www.google.com/patents/US5183225