To begin with, we will start off by applying Newton's law of motion and Newton's law of universal gravitation together to find that

But we're not finished. This doesn't tell us much about the shape of a planet's orbit, although all the pieces are there. This is the general solution, and it could be an orbit of any of the possible shapes (though we can't be sure what they are yet) or any of the possible orientations. We're interested in knowing the shape, of course, so we want to restrict the possible orientations.

To do that, we'll take a special case. It makes sense to have perihelion -- that is, closest approach to the Sun -- at time t = 0. We'll restrict the orientation so that, when perihelion occurs, the planet lies along the zero radian line from the Sun (or, in Cartesian terminology, along the positive x-axis) -- that is, theta = 0. At this point, r, the position vector of the planet, will have only a component in the positive x-axis. We'll also assume that the planet orbits the Sun counterclockwise, through increasing measures of angles. If this is the case, then the velocity v at the instant of perihelion should be orthogonal to the position vector r, and it should have only a component in the positive y-axis.

According to our expression for v, we have a scalar times the vector quantity theta + C.

Now we are ready to finish up the problem. We can dot both sides of the equation with theta and get:

... Well, almost. Kepler's first law actually states that planets follow the paths of ellipses. An ellipse is only one type of conic section. So why is an ellipse allowed while the others are not?

Because the others are allowed -- we just don't call them planets. When Kepler said planet, he meant a body which returns to our skies over and over again. This means that the curve representing the orbit must be closed -- that is, it must continue to retrace itself over and over. The only two conic sections which are closed are the circle and the ellipse (the circle is just a special case of the ellipse anyway). Thus, any body in the skies we see over and over must be orbiting the Sun in an ellipse.

The other two conic sections -- the parabola and hyperbola -- are open curves and correspond to a position where the body has sufficient velocity to escape from the Sun's gravity well. The body would approach the Sun from an infinite distance, round the Sun rapidly, and then recede away into the infinite abyss, never to be seen again.

So we have proved an extension of Kepler's first law: A body influenced by the Sun's gravity follows a path defined by a conic section with the Sun at one focus.