How can you determine the nature of an object from its flight path angle, velocity, and radius alone?

Question

Assuming that a defense system is capable of determining an object's flight path angle ($\gamma$), velocity (v), and radius (r), how could one differentiate between an ICBM, a satellite, or a solar probe?

One could begin with the energy equation:

$$\epsilon = \frac {v^2}{2} -\frac{\mu}{r}$$

where $\mu$ is the gravitational parameter $G(M+m)$, and the sign of $\epsilon$ would help to differentiate the objects, since if:

$$\epsilon<0 = closed$$ $$\epsilon \geq 0 = open$$

Thus a negative value would indicate a closed orbit, meaning that the object is either an ICBM or satellite, while a positive value would indicate an open orbit, meaning that the object is a solar probe.

However, in cases where $\epsilon$ is negative, how could one use the flight path angle to differentiate between the two options?

Answer 1

Given the radial distance $r$, velocity $v$, flight path angle $\gamma$ in radians, gravitational parameter $\mu$, and specific orbital energy $\epsilon$,

the specific relative angular momentum is

$$h= \|{\overrightarrow{r} \times \overrightarrow{v}}\| = rv\sin\left(\frac{\pi}{2}+ \gamma\right) =rv\cos\left( \gamma\right)$$

Then the orbital eccentricity is

$$e=\sqrt{1+\frac{2\epsilon h^2}{\mu^2}}$$

and the semimajor axis is

$$a = -\frac{\mu}{2\epsilon}$$

That yields the periapsis distance $q = a(1-e)$.

If $q$ is less than the radius of the planet being orbited, the object is on a suborbital path, and given the constraints of objects in the original question, would be an ICBM.