The Lindy effect and the stability of planetary systems

How can we predict how long something will stick around?  Flash back to 1992, when the Ronco pasta and sausage maker and Clif bars were both introduced. Clif bars are still around, in every convenience store and supermarket.  The Ronco pasta and sausage maker, not so much.  (Even though, in the early 2000s, my wife and I used ours several times a week and loved it — and you can still find them on ebay.)

The Lindy Effect provides a useful framework to predict something’s longevity. The effect is named after Lindy’s deli on the lower East side of Manhattan, which was the setting for discussions between comedians about the longevity of their careers (made famous in an article by Albert Goodman in 1964 entitled Lindy’s Law). The Lindy effect proposes that one can predict how long something will stick around based on how long it’s already been around. Its future will be a reflection of its past. It only applies to nonperishable things.

The Lindy effect has been studied by some thinkers I greatly admire, including Benoit Mandelbrot (who I once had the honor of having lunch with) and Nassim Nicholas Taleb. In his book Antifragile (one of my all-time favorites), Taleb explains how the Lindy effect implies that objects ‘age in reverse.’  That is, for every year something sticks around, its life expectancy lengthens by a year (or more) rather than shrinking by a year. 

In Antifragile, Taleb recounts an experiment performed by the physicist Richard Gott in 1993 that validates the concept of the Lindy effect.

Gott made a list of Broadway shows on a given day, May 17, 1993, and predicted that the longest-running ones would last longest, and vice versa. He was proven right with 95% accuracy. He had, as a child, visited both the Great Pyramid (more than fifty-seven hundred years old) and the Berlin wall (twelve years old), and correctly guessed that the former would outlive the latter.

In short, the Lindy effect predicts that the expected lifetime of a nonperishable item correlates with its age. This happens because the robustness of anything increases with age — long-lived (Lindy) objects survive the random fluctuations and instabilities that invariably happen, and may even be strengthened. These same fluctuations and instabilities weed out fragile items. That brand-new video game will probably no longer be popular in a year. Clif bars should stick around for decades to come. And spoons, which have been in use for thousands of years, are not going anywhere. 

The Lindy effect is a powerful heuristic but a statistical one — some new technologies will indeed last for a long time, and some old ones will die out.

Let’s explore how the Lindy effect applies to the stability of planetary systems. 

Imagine 100 newly-formed planetary systems, emerging from their cocoons, from the gas-rich protoplanetary disks in which they were born.  How many will survive for the next billion years?  All of them. The only way a planetary system can be completely destroyed is if another star passes extremely close by and gravitationally strips all the planets from the star. The odds of that happening are incredibly low (for a system like our own, there is a less than 1 in ten million chance in any billion-year time interval).

The question becomes: how many of those 100 planetary systems will survive intact for the next billion years?  The landscape of exoplanet systems is a wasteland: from an orbital point of view, the majority of systems look like the beat-up survivors of giant dynamical battles. Most giant exoplanets have very stretched-out orbits, a far cry from the near-circular orbits of our Solar System planets.  These distorted orbits are scars from their violent pasts: we think that their systems formed with many more giant planets than those we see today, but underwent dynamical instabilities that involved brutal gravitational kicks between the planets, leading to the ejection of one or more planets into interstellar space (to become free-floating planets).

The same pattern of violence appears to hold for systems of close-in “super-Earths” and “sub-Neptunes”.  Half of all stars have them – planets between Earth and Neptune in size on orbits much closer to their stars than Mercury is to the Sun. Their orbital distributions strongly suggest that they underwent a phase of dynamical instability similar to gas giant exoplanets. The difference is that for planets close to their stars, instability leads to giant impacts rather than to scattering and planetary ejection (as is the case for giant planets farther from their stars). The predominant model for the origins of super-Earths and sub-Neptunes is that they migrate inward and form a chain of orbital resonances (in which pairs of neighboring planets keep re-aligning in a stable way) near the inner edge of the disk. I like to say that the disk is like a (good) teacher, keeping the kids (planets) in place and well-behaved.  But when the teacher leaves the room (the disk dissipates after a few million years), the kids (planets) go bonkers and undergo an instability.  

Jupiter and Saturn’s orbits are more circular than most giant exoplanets’.  But they aren’t as well-behaved as you might expect, and there is compelling evidence that the Solar System’s giant planets underwent their own dynamical instability early in our system’s history. Signs to a relatively minor instability: the ice giants scattered off of the gas giants, but Jupiter and Saturn did not scatter off of each other – if they had, all the outer planets but Jupiter would have been ejected, Jupiter’s orbit would be much more stretched-out, and the rocky planets’ building blocks would have been tossed into the young Sun.  (And we wouldn’t be here thinking about the Lindy effect.)

Some exoplanet systems seem to have avoided the instability phase. These are the “good kids” that remained well-behaved even after the teacher (the gas disk) was long gone. The best examples are systems in which a whole cohort of planets are locked in long chains of orbital resonances.  In special cases such as the Trappist-1 system, all of the planets orbit their star in the same plane.

Resonant chain systems were “born Lindy”. They emerged from their disks in a configuration that was already beautifully stable for billions of years — like spoons, destined to last for a long time.  These systems are sometimes in surprisingly fragile configurations – a small perturbation from a leftover planetary embryo would have destabilized them. But, left to their own devices, they remain stable indefinitely.

What about the other 90-95% of planetary systems that DID go unstable?  The systems themselves were not weeded out, but their orbital setup has changed. So, we can think of them as the market itself rather than a product.  Their original orbital configurations were like the Ronco pasta maker (or, even shorter-lived, like the infamous musical Spiderman: Turn off the dark). Their phases of instability were a necessary step in the to reach a more robust, longer-lived configuration.  And all planetary systems undergo a multitude of phases of instability on the pathway to growing the planets themselves, as smaller bodies repeatedly collide and grow, each a small-scale instability. The time between successive instabilities becomes longer as the planets get more massive, following a roughly logarithmic distribution (with collisions happening at 1, 10, 100 thousand years rather than at 10, 20 30 thousand years). This is, in some sense, the Lindy effect in action: the system is converging toward a longer- and longer-lived configuration.

What does the Lindy effect imply for planetary systems in a general sense?  First of all, most systems that will go unstable do so early. Systems that are fragile break, and most breaks will happen fast. The instability phase that accompanies a “break” is chaotic, and randomly samples the parameter space of possibilities until a new solution is found. The next orbital configuration will generally last longer than the previous one — physically-speaking, this is because instabilities remove planets (by collision or ejection), and tend to spread out their orbits. Wider spacings with fewer planets are naturally more stable.

Exoplanets are found around stars with a wide range of ages. According to the Lindy effect, each system system is likely to be stable in the future for a time similar to its current age. And, we can see some evolution in exoplanet systems’ orbital configurations. For instance, the fraction of systems in resonance seems to drop quickly in time, perhaps as systems “break the chains“.

This drop in the fraction of resonances is basically the number of “good kids” dropping off in time. This makes sense — the longer the teacher is gone, the higher the chance of kids misbehaving. It also reinforces that the long-term (Lindy) trend is for systems to leave resonance and to look beat-up (by life or your planetary system going unstable).

Planetary systems are perturbed from the inside and out, by perturbations between the planets themselves and also by passing stars. In the same way, products can be rendered obsolete by internal (functional) or external (lack of demand) issues. There is a roughly 1% chance that the Solar System will become unstable in the next 5 billion years due to internal chaos, and an additional 0.5% chance from passing stars (see here for details). The characteristic instability timescale is also about the same as the Solar System’s current age, roughly matching the Lindy expectation (although flybys are likely to trigger earlier instability).

Let’s apply this Lindy way of thinking to the young Solar System, and see what it has to say about when our own system’s giant planet dynamical instability took place. The predominant view after the return of the Apollo samples was that there was a delayed, cataclysmic event that generated a giant bombardment on the Moon — and also on the other planets, but only the Moon’s craters survived. The original “Nice model” proposed that the Solar System’s dynamical instability took place about five hundred million years after Earth had finished forming and caused this cataclysmic bombardment.

For more than a decade, the idea of a delayed giant planet instability was in vogue (the “Nice model“). But it was a delicate balancing act. One colleague described it as being like placing a piece of paper on the edge of a table that finally falls off a year later. Of course, some systems do go unstable late, like a technology that existed for hundreds of years and then became obsolete. It happens, but it’s rare — and the longer the delay, the rarer it is. Our Solar System may be out of the ordinary in some ways (such as in its orbital architecture), but a very late instability would be a pretty extreme cases of the unusuals. The Lindy argument, of course, is that things that break usually do so quickly.

The Lindy viewpoint proved itself again. Recent reanalyses concluded that there was no cataclysmic spike in bombardment on the Moon. The spike was an illusion caused by sampling bias, the fact that the last big impact (that created the Imbrium basin) likely sprayed rocks across the Moon’s surface, to a number of different Apollo landing sites.

Current thinking is that the instability did indeed take place, but much earlier than previously thought — no later than 100 million years after the start of planet formation. The exact timing remains a matter of debate, although there is a contingent of scientists still arguing for a late instability within that time window. Lindy would suggest otherwise (although, we must always keep in mind that it is only a statistical heuristic, and rare events do happen all the time).

Nothing lasts forever (even cold November rain). No planetary system is protected from everything that could go wrong, just like no Broadway show is protected from a nuclear war. Our Solar System will eventually disintegrate, the planets either swallowed by the red giant Sun or set free by passing stars. The Ronco pasta and sausaker maker is already gone. Ultra-compact, stable systems like Trappist-1 will eventually be destroyed. Even Lindy’s deli closed in 2018.

Luckily, we still have the Lindy effect to guide our thinking.

Questions? Comments? Words of wisdom?

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Additional Resources

• The Lindy Effect (Wikipedia and sketchyideas.co)
• Antifragile by Nicholas Nassim Taleb
• How planets die: when good Jupiters go bad
• Super-Earths: breaking the (resonant) chains
• The Trappist-1 system: a poetic introduction and bombardment
• Amazon page for the Ronco pasta and sausage maker. Sadly obsolete…