How close can two solar systems safely be?

Imagine you have two Sun-mass stars, each with their own planetary system orbiting the galactic centre. How closely can they approach each other without disrupting each others’ planets?

Depends on what you mean by “disrupt”. Since Gravity falls off like the square of the distance, if X is the radius of the solar system, and if the distance between the two stars is 11X, that would put the furthest planet at 10X the distant from the other sun, and the gravitational pull of that other sun would be 1% of that planets own sun. That assumes the outer most planets are not overly massive.

Even without the influence of other stars, the solar system is not stable. Wikipedia has a short summary of the subject. An interesting bit:

Jacques Laskar and his colleague Mickaël Gastineau in 2009 took a more thorough approach by directly simulating 2500 possible futures. Each of the 2500 cases has slightly different initial conditions: Mercury’s position varies by about 1 metre between one simulation and the next.[13] In 20 cases, Mercury goes into a dangerous orbit and often ends up colliding with Venus or plunging into the Sun. Moving in such a warped orbit, Mercury’s gravity is more likely to shake other planets out of their settled paths: in one simulated case its perturbations send Mars heading towards Earth.

So there’s already some chance at wide-scale planetary disruption. A nearby passing star would obviously make things worse, but it’s not clear by how much.

The Oort Cloud goes out almost halfway to Alpha Centauri, and any disruption there would eventually have some effect on the inner solar system, so most likely there is already some external influence.

Other conditions would apply. If a really big – Jupiter-sized – planet is in each solar system, and the two have the same orbital period – resonance! Resonance is bad ™. This is why Saturn’s rings have gaps.

Stars routinely pass through the Oort Cloud. 70,000 years ago, Scholz’s Star passed less than a lightyear away. In less than a million and a half years, Gliese 710 will pass about a lightyear away. At most these passes disturb a few Oort Cloud objects and cause them to fall into the inner solar system, thus becoming long term comets. Unless one of those comets happen to hit one of the planets, there will be no significant effect on the inner solar system.

To put some numbers to it, in regards to Earth’s orbit, other than the Sun (and the moon, which doesn’t really count), Jupiter is the big gravitational influence. Jupiter’s orbit’s at about 5AU, which means it can come about 4AU from Earth. Assume that is an acceptable gravitational nudge.

The mass of the Sun is about 1000 times the mass of Jupiter. For another sun-mass star to give the same nudge as Jupiter, it would have to be sqrt(1000) ~= 32 times farther away, or 128AU, or about 0.2% of a light year.

Well, if you happen to be on one of those planets, that’s pretty significant.

That’s a pretty astonishing fact about Scholz’s star, though! 70ka is nothing… humans were around back then.

A very recent (astronomically speaking) but similar encounter is between Procyon and Luyten’s Star. They’re currently just over a lightyear apart and moving away from each other. According to Wikipedia, their closest approach was about 600 years ago at 1.12 ly.

As for Scholz’s Star, it’s too dim to have been visible to the naked eye even at closest approach, except perhaps when it flared. It also has a brown dwarf companion.

That’s less than a light-day. So perhaps a couple of light-days or a light-week for negligible effect?