Summary - This is just exceptionally unlikely. Theoretically it is possible but as far as worrying about it, forget about it.

The chances of getting closer than Neptune in any one million year period are

• 1 in 48 million for a neutron star

The chances of getting as close as Earth, hitting Earth or hitting the Sun are vanishingly small. .

IN DETAIL - BROAD PICTURE

There are several hundred million neutron stars in the galaxy as a rough estimate (see Compact Objects in Astrophysics) . Most of them would be very old and cold, due to supernova explosions billions of years ago and an old cold neutron star would be very hard to detect, that's true. You might think that with so many of them, that we’d have a chance that one of them would hit us. But the galaxy is also very huge, as worked out by Phil Plait The volume of the Milky Way, our galaxy, is roughly 8 trillion cubic light years

HOW MANY OF THEM ARE THERE WITHIN TWELVE LIGHT YEARS?

With the volume of the galaxy 8 trillion cubic light years, the calculations are simplified if we look at a volume of 8 thousand cubic light years, i.e. in a sphere of radius twelve light years.

So those several hundred million neutron stars correspond to about ten neutron stars within twelve light years. By comparison our galaxy has 400 billion stars (you get various estimates here, some say 100 billion, I’m going by the higher estimate) which makes it around 200 normal stars in that same 8,000 cubic light years. T

So in short we have

• Ten neutron stars (max)

within that twelve light years radius.

Stars are from several hundred to a thousand times more common than neutron stars. That might surprise you given that many stars have short lifetimes and of those many end in neutron stars - so what has happened to all those stars that reached the end of their lifetime billions of years ago?

The thing is that those bright stars are far outnumbered by the numerous fainter stars. Most stars are red dwarfs with trillions of years long lifetimes - Red Dwarfs: The fascinating stars that live for trillions of years ).

SO HOW LIKELY IS IT THAT A STAR DOES A FLYBY?

Stars, even though they are much more common, are also very unlikely to do close passes of the solar system. Never mind hitting the Earth, or the sun, they are extremely unlikely to get as close as Pluto. The closest flyby of a star in recent past is Scholtz's star, which passed 0.8 light years (around 9.6 light months) away 70,000 years ago. https://www.sciencedaily.com/rel...

By comparison Pluto is 5.5 light hours away. So - when a very rare close encounter may take a star 9-10 light months away - how likely is it that a star would pass as close as 5.5. light hours away? With neutron stars a thousand times less common than stars, how likely that a neutron star would come anywhere near our solar system?

To get close enough to pass between Earth and the Moon it would need to pass just one light second away. To hit Earth it would need to hit us accurately to the nearest 0.02 light seconds - within two hundredths of a light second. The sun’s diameter is 4.64 light seconds, so to hit the Sun it would need to get within a few light seconds of it.

I think you can see that all these things are so unlikely it will surely never happen.

IMPOSSIBILITY OF CAPTURE

And it is just a single pass, because if any rogue planet, or neutron star, or black hole or anything were to pass through our solar system - it would be going too fast to do anything except just fly out again.

Capture into solar system orbit by Jupiter - tracing the path of Voyager 2 or Voyager 1 backwards - is so very improbable you can forget about it. It is easy for a solar system to eject a planet, and very hard for it to capture one. That’s a bit like the way it is easy for a cup to break but very hard for a cup to spontaneously assemble from the broken pieces on the floor.

Another analogy - it would be theoretically possible to drop a pin on a hard polished floor and for it to land point down balanced exactly. And if all the tiny drafts of air pushed in the right way to keep it balanced, it could stay balanced like that for an hour or more. It's possible but surely even with the thousands of pins dropped on polished floors, it's surely never happened in the history of humanity!

EXACT CALCULATION OF THE CHANCES

So now, what is the chance that Earth or the Sun is hit by any of these, or that they come into our solar system?

I’ve found a way to do an exact calculation

There's a formula, we can use here, from Perturbation of the Oort Cloud by Close Stellar Approaches. Our sun has approximately 4.2*D^2 encounters with other stars every million years.

There D is the diameter in parsecs of the spherical region around the star.

Neptune's semi major axis is 4.49506 billion kilometers so it's diameter is around 0.00029135 parsecs. So substituting that for D, every million years there is 1 chance in 2.8 million (calculated as 1/(4.2*0.0002913^2)) of a star passing closer to the sun than Neptune.

So then back to our neutron stars, then there is less than one of those to every 20 normal stars, so that makes it one chance in 48 million of a neutron star passing closer to the sun than Neptune in a million years.

So to summarize, chances of getting closer than Neptune are

• 1 in 48 million for a neutron star

These are almost vanishingly small chances already.

Now for the chance of a star hitting the Earth. Now D is 12,742 km, which is 4.129401e-10 parsecs. So now the calculation is one in 1/(4.2*(4.129401e-10)^2).

So the chance of a star hitting Earth in the next million years is about 1 in 1.3962931 * 10^18. Or about 1 in 1,400,000,000,000,000,000

We can also look at the chance of a star hitting the Sun. Now D is 1.3914 million km, or 4.5092203 × 10^-8 Parsecs 1/(4.2*(4.5092203e-8)^2) makes it 1 chance in 117,000,000,000,000.

So to summarize, every million years there is a

• 1 in 1,400,000,000 billion of a star hitting Earth
• 1 in 117,000 billion chance of a star hitting the Sun

So about one chance in 117,000 billion of it happening every million years. With 400 billion stars in the galaxy, we get one chance in 117,000/400 or about one chance in 300 that we get a collision between two stars somewhere in the galaxy every million years. With the galaxy 13.21 billion years old, then it may have happened 13,210/300 times or about 44 times since the galaxy formed, that one star has hit another star.

For neutron stars, divide by 20, so we might have had a couple of collisions of neutron stars with another star.

So we have the figures for anywhere in our galaxy since it formed:

• 2 collisions of a neutron star with a star

Those are averages though. The stars are much more densely packed in the center of the galaxy, so stellar collisions there should be more common. The galaxy also has a giant black hole in its core, and stars must collide with it quite often. I’ve also assumed that there is not enough gas or dust to cause significant drag on the approaching object. That’s true for our sun but not true for newly born stars or stars that hit the accretion disk of a black hole.

However, we orbit far from the galactic center, and are at no risk of collision, no more risk than Earth is at risk hitting the sun, because we orbit the galactic center in a long term stable orbit.

In the universe as a whole, there are so many stars that such collisions are common. Collisions of a neutron star with a black hole cause some of the enigmatic gamma ray bursts - the shorter ones.

Artist’s impression of a neutron star captured by a black hole. In a Flash NASA Helps Solve 35-year-old Cosmic Mystery

This is of no danger to Earth whatsoever