What would happen if a rouge neutron star were to pass through our solar system, specifically near earth?

Neutron star matter is only stable under neutron star gravity conditions, so a small bit of neuton star material would very rapidly 'explode' if it was removed from the neutron star - it would much prefer to be in the state of normal matter - it's only the extreme gravitation of the neutron star that keeps matter at nuclear density - so that as soon as you remove a chip of the neutron star it would very rapidly (say microsecond timescale or probably considerably faster) change to 'normal' matter. It's only the enormous gravitational force of the neutron matter that's keeping it in that form, and that will need a lot more than a golf ball sized bit to remain stable long enough to hit Earth.

Science is full of hypothetical scenarios, so I'll answer yours. The Solar System is in a delicate balance. The interactions of all the gravities of all the planets in the Solar System have acted in the same way for billions of years. The only known large-scale event like the one you are imagining that ever is known to have happened is when a Mars-sized planet smashed into the primordial Earth, the result being the Earth-Moon system. Now there are hints that other cataclysmic celestial events have also happened - mainly the axial tilt of Uranus being almost 90 degrees, which could have been caused by a transit of a large body, and by the slow retrograde motion of Venus, unique in the Solar System, but possibly due to gravitational influence from the Sun. So, that's the scientific basis for what might happen. If a planet sized object were to pass through the inner Solar System, it could impact a planet, causing a massive conflagration, creating a new moon for that planet, and completely rewriting the surface of that planet, at the same time shifting the gravitational balance of the whole Solar System. It would perhaps only transit the inner planets, in which case it could cause massive tides if it passed close enough to the Earth. Compare the size and distance of the Moon to possible sizes of this planet Nibiru. Gravity follows an inverse square law, remember, so if Nibiru passed twice as far away as the Moon, its gravitational effects would be one quarter of the Moon. This would lead to an extra tidal influence in Earth's oceans, and when the two tidal influences were added together, you'd get an exceptionally high tide, perhaps several meters higher than a normal high tide. The closer that Nibiru passed to the Earth, the worse the combined tidal effects would be. If Nibiru passed close enough to the Moon, gravitational forces could rip the whole thing apart, perhaps Nibiru too if it is of a similar size or passes closely enough. The same could be said for Earth. You would have to also consider how quickly Nibiru is moving. Or rather how slowly. The orbit of planets is quite slow, but a body from interstellar space might be travelling much faster. There might be all manner of interstellar bodies that have been ejected from their own Solar Systems for one reason or another. And if our Solar System's delicate balance were to be upset by a new planetary arrival, then one of our planets might be ejected as well. And if that planet was Earth, it would be quite bad, don't you think? But no one can know for sure until it is actually witnessed. Our models might be wrong, and we are only human after all. Fortunately, as you know, this is a hypothetical situation, because, as you know, no planet X or Nibiru has ever been located. It really ought to be visible by now if it was real...

1. How near, and how "big" (or massive is the neutron star)?

Sun gravitational effect is 1 ly away perhaps and should be longer than the (hypothetical) Oort Cloud location.

So if the neutron star is one solar mass, and it s in somewhere as "near" as Oort Cloud distance it will surely disrupt the earth (and other planets/asteroid/moon(s)) orbit.

If it s twice solar mass, it can disrupt our solar system even if it is as far as Alpha Centauri distance.

2. Neutron as a golf ball size?

"LAIN" has provided the best answer here. A golf ball size neutron star will explode if you remove it from its star.

But supposed an object as massive as a golf ball size neutron star collides with earth, what would its effect?

A golf ball size neutron star would weigh a half trillion tonnes. I ve lookup it up in Wikipedia and uses calculator.

Or a rock sphere with 7 Km diameter,

So if a 7 Km rock sphere hits the earth the effect would be...

Energy:

Energy before atmospheric entry: 1.70 x 1024 Joules = 4.05 x 108 MegaTons TNT

The average interval between impacts of this size somewhere on Earth during the last 4 billion years is 4.7 x 108years

Major Global Changes:

The Earth is not strongly disturbed by the impact and loses negligible mass.

The impact does not make a noticeable change in the tilt of Earth s axis (< 5 hundreths of a degree).

Depending on the direction and location of impact, the collision may cause a change in the length of the day of up to 1.87 milliseconds.

The impact does not shift the Earth s orbit noticeably.

Crater Dimensions:

Transient Crater Diameter: 93.3 km ( = 57.9 miles )

Transient Crater Depth: 33 km ( = 20.5 miles )

Final Crater Diameter: 169 km ( = 105 miles )

Final Crater Depth: 1.39 km ( = 0.861 miles )

The crater formed is a complex crater.

The volume of the target melted or vaporized is 10700 km3 = 2560 miles3

Roughly half the melt remains in the crater, where its average thickness is 1.56 km ( = 0.968 miles ).

source: http://impact.ese.ic.ac.uk/ImpactEffects/

A neutron star would likely be about 2 or 3 times the mass of the sun. Its gravity would disrupt the orbits of all the planets. Earth would be decimated.

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I think everyone would be quite surprised to find a "rouge" star anywhere, since pinkish stars are highly unusual.

On the other hand, if a "rogue" neutron star just happened to pass through our solar system, we would probably all be dead shortly after the earth's orbit was disrupted.

Can you specify what kind of neutron star?

If it was a magnatar it might simply just rip the iron from our body's.

Otherwise we would probably be destroyed by the super intense gamma ray bursts that could be 100AU away.

As for distance and direct effect on gravity I would say about 20 AU.

Neutron Golf Balls

If it were to pass near earth due to its high density first of all it would attract earth towards it gravitationally. If it would collide near america it would be a disaster as the one that caused extinction of dinosaurs on this earth.

I do not think about that ever.