Physical Meaning of Cone used in Conic Section for Orbital Mechanics Does the polar angle (complement of $\theta$ below) of a cone which intersects a plane to yield a conic section have a physical meaning in orbital mechanics?  

Note that the angle of incident planes, which form parabolas, ellipses, etc., is not the polar angle of the cone.
Here's an example.  Consider the initial case of an elliptical orbit illustrated below:

Now, consider the mass of the central body greatly increasing.  To conserve momentum, the elliptical orbit would shrink in area.  In fact, the polar angle of the cone would decrease to reflect this proportional reduction in area:

[Note the smaller orbit and more narrow cone, despite keeping the same angle of the slicing plane as before.]
Is, therefore, the polar angle all and only a function of the mass of the orbiting bodies?
 A: The polar angle of the plane (the angle of the plane with respect to the symmetry axis) should relate to the eccentricity of the orbit. For $90^o$, the section is a circle and the eccentricity is zero. For an angle between the $90^o$ and the angle of the cone, you will get an ellipse with an eccentricity $0<\epsilon<1$. In both of these cases, the total mechanical energy of the system is negative.
If the angle is equal to the angle of the cone, you will have a parabola, and the total mechanical energy is zero.
At an angle less than the angle of the cone, you have a hyperbola, and the total mechanical energy is positive.
The kinetic energy and potential energy and angular momentum will all affect the related angles.
A: Ellipses, parabolas and hyperbolas are both:


*

*Defined as the conic sections you speak of: work out the intersection between the cone $\vec{R}.(\cos\phi\,\hat{X} + \sin\phi\,\hat{Z}) = \sin\theta\,|\vec{R}|$ and the plane $z=const$ where $\theta$ is your polar angle and $\phi$ the angle between the cone's axis of symmetry and the slicing plane and you'll find that the equation for $x$ and $y$ is a general quadric form;

*The path solving $\ddot{\vec{R}} = -\frac{G\,M}{|R|^3} \vec{R}$, i.e. the path of a point mass with a central attractive force proportional to the inverse square of the distance a.k.a. the Newtonian two-body celestial body orbit model: solve this when the initial velocity is contained in the $X\wedge Y$ plane and you'll see that its the same general quadric form.
So, most certainly, the two are precisely the same notion, just different ways of stating it.
