Accelerating towards the center of your orbit If you are in a perfectly circular orbit and accelerate perpendicular to your velocity (directly towards the center of the orbit), what would happen to your orbit? Would it just shift in the direction of acceleration?
Is there a different result if the orbit is not perfectly round?
The basis for this question comes from me playing Kerbal Space Program.  I never really considered orbital mechanics before.
 A: This answer assumes the mass of the orbiting body is much smaller than the object being orbited. This is quite valid for artificial satellites around Earth.
If the original orbit is circular with radius $R$, then doing what you describe would cause the new orbit to be elliptical (non-circular) with semi-major axis $a=R$.
Let's say the force is applied over a relatively short time period to make things easier to visualize. The force is perpendicular to the motion, so no work is done on the object+planet system. This means the total energy $E=-GMm/(2R) < 0$ of the system doesn't change.
Now, since the applied force changes the velocity of the orbiting body, but can't change the total energy of the system, then we're left concluding that the orbit is now non-circular with the same energy $E=-GMm/(2R)$. Elliptical orbits are the only other allowed orbital shapes with negative energy, so the orbit is elliptical.
The more general expression for the energy in elliptical orbits is $E=-GMm/(2a)$, where $a$ is the semi-major axis length. This implies an elliptical orbit with semi-major axis $a=R$ since the energy didn't change.
When I say elliptical, I mean a bound orbit with eccentricity between 0 and 1. That is, non-circular and non-parabolic.
A: If you are moving in a circle, your acceleration is already perpendicular to your velocity.
Here is a diagram for position and velocity when you move in a circle.

You're at the blue dot. The red arrow is your position vector; it points from the center of the circle to you. The green arrow is your velocity vector; it points in the direction you're moving.
Your velocity is the rate of change of your position. Acceleration is defined to be the rate of change of your velocity. So the relationship between position and velocity is exactly the same as the relationship between velocity and acceleration.
To get from position to velocity, you rotate by 90 degrees. Therefore, to get acceleration, you rotate the velocity vector by 90 degrees. Rotating the green arrow another 90 degrees gives and acceleration pointing back in towards the center of the circle.
If your position vector has length $r$ and velocity vector length $v$, you multiplied the length by $v/r$ to go from one to the other. Since going from velocity to acceleration is the same as going from position to velocity, we multiply $v$ by $v/r$ again to get the acceleration $a = v^2/r$.
To summarize, when you're moving in a circle, you have acceleration $a = v^2/r$ pointing in towards the center of the circle, perpendicular to your velocity.
If you made the acceleration bigger than $v^2/r$, you would start to spiral inwards towards the center. If you made the acceleration smaller, you'd spiral out. As long as the acceleration is perpendicular to the velocity vector, only the direction of motion changes, not the speed.
In other situations, it is hard to answer your question because it depends on what the acceleration will be as a function of time. Generally in physics, we give some rule for the acceleration, so that the acceleration depends on where you are. Perhaps the most famous rule is that the acceleration points towards the center of the circle with magnitude proportional to $1/r^2$. In that case, you find conic section orbits - circles, ellipses, parabolas, and hyperbolas that obey Kepler's Laws.
Another possible rule is again that the acceleration points towards the center with magnitude proportional to $r$. Then you find circles and ellipses as the solutions, although they go at different rates than before. In this case, all orbits have the same period. This is called the harmonic oscillator.
If you invent other rules, you can get different behaviors. These two rules are somewhat special in that they produce simple, closed orbits.
A: You end up generating an ellipse.
