Q:

How likely is the flotsam of a space battle to be captured by our sun's gravity well?

A light-year distant from Sol saw the collision of a massive space-born battle fleet against an awe-inspiring dreadnought.  At one point in the battle, the flotilla fired rail-guns with 100 metric ton slugs and nuclear missiles.
...And a lot of them missed their target
...And kept on sailing through space, generally toward our beloved Sol.
Eons later, while eating an Italian custard at a French bistro whilst admiring the visas in my American passport, I look up to see an odd reflection in the early afternoon sun.  Hours later, Earth becomes collateral damage in a battle fought long before Gelato ice cream was invented.

Ignore the scattering effect of objects sailing through space.  Assume that the rail-gun slugs and missiles (their fuel long since exhausted) are lined up to pass through our system, parallel with the ecliptic, 15 degrees above the ecliptic referenced from the sun, and 2 AU from the sun.
Assume the missiles are similar in size and mass to a Minuteman-III missile.
Rail-gun slug: 100 metric tons, muzzle velocity of 2.5 Km/s.

Question: Is it possible for these munitions to become captured in our sun's gravity well, thereby making them a believable catastrophe-in-the-making?

A:

Virtually none of them would be captured and it's unlikely we'd notice anything whizzing past.
Basically, an object that falls into a gravity well from infinity (and a light year off is pretty close to that) will, unless it hits something while passing through, fall right back out again.  By definition it picks up escape velocity as it falls in and loses it as it falls back out. So it passes through the solar system on a hyperbolic trajectory.  And if the projectiles were moving at high speed before falling into the solar system, it will be a very flat hyperbola -- i.e., they'll hardly be deflected at all by the Sun's gravity.
If the slugs are aimed to pass through at 2AU, there's basically nothing there to hit and all they do is think to themselves, "Here comes the Sun...there goes the Sun...hello interstellar darkness..."
(I'm not quite sure what you mean by "to pass through our system 15 degrees above the ecliptic."  It you mean they are aimed to pass well above the plane of the planetary orbits, then there's still less likely to hit anything or even be noticed.)
The planets -- even Jupiter -- would have little effect on the projectiles since most of them would not come near enough to be affected, and the rest would only be deflected a bit.  (Remember, they're passing by at greater than escape velocity.  The deflection a body can provide depends on the projectile's velocity (the slower it's moving relative, the bigger the defection) and on the impact parameter (the closer the projectile passes the bigger the deflection.)
Hitting the Earth accidentally is hard.  The Earth occupies .25(r/R)2 of a 2AU circle, where r is the Earth's radius and R is the Earth's orbital radius (the AU). That fraction is about 5*10-10. So less than one in a billion of the projectiles would hit Earth even if they were aimed to all pass through the ecliptic within 2AU of the Sun.
Finally, if they did hit the Earth we'd probably not notice them.  A 100 metric ton slug of metal would be about 10 cubic meters if it's iron, less if it's something denser.  That's a sphere less then ten feet in diameter.  It would produce a nice bolide, but would come down pretty much intact, making a 50 meter crater with about a 4 kiloton explosion.  Nothing you'd want to be next to, of course, but basically unnoticeable more than twenty miles away.  Tunguska was close to 100 times bigger (though probably mostly ice) and its explosion was around four thousand times greater, and it was nearly missed.  Had it hit water, it would have been missed.

A:

To be captured by the sun's gravity well means to end up orbiting the sun.
If a projectile is fired from outside the solar system, it may do a flyby around the sun. Its speed at any altitude (relative to the sun) will be greater than the escape velocity at those points.
Scientists believe this is exactly what the Oumuamua object did.
For the flotsam to be captured, it must be decelerated by a planet. Jupiter is the best candidate to do so, but the odds are against it. Even if the flotsam does a flyby by Jupiter, it is more likely to either hit the planet or be accelerated even further, causing it to exit the solar system earlier.
If it does get decelerated by a planet, the flotsam will be locked into an orbit that is unlikely to cross the Earth's own. It will cross Jupiter's orbit during its perihelion, which means it may crash into Jupiter eobs later, and it has an astronomical probability of having an orbit even more inclined relative to the Earth's than Pluto. All this makes an encounter with Earth very unlikely.

A:

Short answer is a space weapon fired at any sort of realistic speed will never be captured by the Sun's gravity well. Consider that mere human technology has created nuclear pumped weapons like project Prometheus, which captured a small fraction of a nuclear explosion to fire a stream of pellets at 100km/sec. Changing some elements allows you to use similar equipment to make shaped charges, Explosively formed projectiles and even streams of star hot plasma moving at speeds measured in percentages of the speed of light. In short, your space armada would be sliced to pieces before they get a shot in using weapons with velocities of 2.5km/sec. In fact, orbital velocity around the Earth is already 7km/sec.
Realistic space weapons will be moving so fast that they will never be captured, while extremely slow moving bodies like you describe will likely not reach Earth at all, but either drift harmlessly past, or be captured by some of the Gas Giant planets in the outer Solar System.

