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

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?

• By solar probe, you mean something in heliocentric orbit, not something investigating the sun, right? – Camille Goudeseune Jan 19 at 23:24
• @CamilleGoudeseune I refer to an object on a parabolic or hyperbolic orbit relative to Earth – Essan Verne Jan 20 at 2:47
• no, but you can determine the nature of it's current trajectory – JCRM Jan 21 at 8:58

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

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

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.

• And yet a satellite falling out of orbit might be indistinguishable from a high-altitude ballistic missile. I agree with your math but think the question is a bit vague. – Carl Witthoft Jan 20 at 12:52
• A ballistic missile will have been detected at launch. The US, Russia, and I assume many other countries all have extensive launch detection systems. The US systems are so good they detected the surface-to-air missile that shot down Malaysia Airlines Flight 17 in 2014. If you backtrack the object's path, an ICBM will have a launch event at its beginning, a falling satellite will not. – Ryan_L Jan 20 at 19:28