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?

  • $\begingroup$ By solar probe, you mean something in heliocentric orbit, not something investigating the sun, right? $\endgroup$ – Camille Goudeseune Jan 19 '20 at 23:24
  • $\begingroup$ @CamilleGoudeseune I refer to an object on a parabolic or hyperbolic orbit relative to Earth $\endgroup$ – Essan Verne Jan 20 '20 at 2:47
  • $\begingroup$ no, but you can determine the nature of it's current trajectory $\endgroup$ – user20636 Jan 21 '20 at 8:58

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.

  • $\begingroup$ 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. $\endgroup$ – Carl Witthoft Jan 20 '20 at 12:52
  • 1
    $\begingroup$ 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. $\endgroup$ – Ryan_L Jan 20 '20 at 19:28

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