How to show that planetary orbits are conic sections

Question

It's all very fun to use differential equations to show that the path of a particle follows a parabola under constant acceleration, a circle when the angular momentum is just enough to allow for zero radial acceleration and an ellipse when it is in a general orbit. (still haven't done the hyperbola).

But a question that has always bugged me is why do these objects all follow conic sections under gravity? Intuitively, it seems like the inverse square law in polar coordinates is responsible. But an object accelerating under constant force still takes on a parabola. Is this related to the fact that angular momentum is constant under gravity due to zero torque? (Kepler's second law)

Answer 1

Here's my take on the question:

First I assume that by "orbit" you really mean the image of the orbit, because the parametrized orbit (as a function of time) itself contains more information describing your physical system, for instance its time derivative(velocity).

An important feature for planetary orbits is conservation of mechanical energy, so that its position also determines its kinetic energy, so in this case position is the minimal information you need to reconstruct your physical system.

Now conic sections also have convenient alternative descriptions involving "distances", for instance an ellipse is the set of points that has the same total distance to the two foci, and parabolas are the set of points that are equidistant from both the directrix and the focus. This provides us with the necessary tools to parametrize both potential energy and kenetic energy at the same time.

For example consider a celestial body orbiting a planet. Given their relative positions and the velocity of the celestial body, think of the planet as the first focus parametrizing potential energy, and a second focus parametrizing kinetic energy. This second focus will lie on the circle around the celestial body with radius its kinetic energy, and its position on this circle will be determined by the direction of velocity, so that the velocity is tangential to the resulting eclipse.

The same exercise can of course also be done for objects under a constant force.

Answer 2

The planetary orbits are conics section in the approximation of a central attractive gravitational force from a fixed Sun.

The fact that in this case the orbits are conic sections depends not only by the fact that the force depends only on $r$ as $F=kr^n$ (central force ) but also from the fact that $ n=-2$ for the gravity.

The nature of the conic depends from the total specific energy $E_T$: we have an ellipse if the $E_T<0$ ( the system is closed), a parabola if $E_T=0$ (an open critical system) or an hyperbola if $E_T>0$ (open system).

There are plenty of resources in Internet about this topic (as suggested by Mark McClure or as here).

The case of a falling body in a constant force field is different from these situations because here we does not have a central force, and the fact that the solution is a parabola has the simple motivation that the equations of motion are second ordere differential equations and a function that has a constant second derivative is a second degree polynimial.