How to tell if an orbit is closed or opened?

Let's assume we have an equation of orbit which is

$$\frac{1}{r} = u_o + u_1 \cos \frac{3 \theta}{2}$$

Do I have to know the value of eccentricity of the orbit to understand the shape of the orbit? I mean if the orbit is circular or ellipse, the orbit would be closed. But my problem is that how it is different to a bound orbit.

I have found a theory that says $\Delta \phi = 2 \pi \frac{m}{n}$ can be used to find a orbit is closed or not. It says the $2 pi$ has to be a rational function. Could you explain it more?

Or it is better to use any other equation?

• – AccidentalFourierTransform Nov 10 '16 at 20:32
• Thanks, I have edited my question according to your comment. :) Could you please tell me what is $u_o$ and $u_1$ in the equation? – Numerical Person Nov 10 '16 at 20:39
• The easiest way to find out the shape is to plot the orbit. – sammy gerbil Nov 10 '16 at 21:05
• – sammy gerbil Nov 10 '16 at 23:05
• But how to find that the orbit is closed or not from that equation? – Numerical Person Nov 11 '16 at 4:57

$r=\frac{1}{\cos \left(\frac{3 \theta }{2}\right)+2}$ we get a graph like But if $u_0 \leq u_1$, then the denominator will go to zero at some point, so the radius will go to infinity, and so the path won't loop back on itself.