No, just the opposite. Given any position and velocity relative to the Sun (or the Earth), an object in that position at that velocity is in an "orbit" that will be a conic section of some sort. Most of them are ellipses i.e. stable. The only way to not be in a stable orbit is to go fast enough that it becomes a parabola or hyperbola - a flyby - or slow enough that the ellipse becomes so narrow that it's just a straight line (or rather, narrow enough that it hits the thing you're orbiting).

When you throw a ball it flies in a parabola, right? But actually that parabola is just one end of a very long, narrow ellipse, with the Earth's centre as one of the foci. If you could "turn on noclip"[1] and allow the ball to fall through the Earth, it would already be in a stable orbit.

(If you want to get a better intuition for this kind of thing, play Kerbal Space Program)

[1] And if the Earth were a point mass located at its centre, which it isn't. And ignoring air resistance.

If Earth was travelling a bit slower, it would just be on different orbit. Either more eccentric or just further away from the sun. This is encapsulated in Kepler's laws [1], which describe the motion of a single planet around a single star.

If you have other planets in the system (like our solar system does), things are more complicated. Orbital stability then depends on the arrangement of other planets. You are right in pointing out that the Earth's orbit is (possibly) unstable [2] over a timescale of billions of years.

[1] http://www.physicsclassroom.com/class/circles/Lesson-4/Keple...

[2] Unstable in the sense that Earth's orbit may become eccentric enough to smash into another planet, get ejected from the solar system, or dive into the sun.

> But aren't the vast majority of possible orbits unstable?

No. Ignoring perturbations from other planets, planetary orbits are stable ellipses with the sun at one focus.

http://en.m.wikipedia.org/wiki/Kepler's_laws_of_planetary_mo...