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So the stability of the Solar System is highly dependent on general relativity.

That seems a bit misleading. If you have a complex system that reaches a measure of stability over a large timescale and then change any of the underlying rules, it's going to become unstable. In this scenario there's nothing special about special relativity, it just happens to be the rules they messed with.

That's a fair criticism. If a dynamical system is stable under one model, it's not that surprising if suddenly swapping out the model causes an instability. Basically, different models have different stability islands, if you would.

But the question remains, how dependent on GR is the long term stability of an N-body system in general?

Here's an experimental idea: create a "random N-body system" generator, and compare what % of systems are stable under Newtonian vs Einsteinian physics models.

> in general

That's too broad to answer. Let's narrow to our solar system, or a close approximation, first. Below, I'll step up to more complicated arrangements, and binary black holes. These are only three or four families of (regions of) general curved spacetime; there is a very large infinity of other families one could reasonably associate with "in general".

In a (model) solar system, there is a large central mass and several other objects with extreme mass ratios to the sun. There are interesting secular resonances among Mercury, Venus, and Jupiter, and between Earth and Mars, where that means that from time to time (over many orbits), the precessions of the near-to-the-sun parts of the planets' orbits match. These resonances may increase the eccentricity of the smaller mass's orbit, which can in extremis yeet the small body out of the solar system, or cause a collision between planets (e.g. Mercury's orbit may near aphelion cross Venus's orbit).

For the Mercury-[Venus-]Jupiter the closeness of Mercury to the sun tends to break up the resonance enough to significantly stabilize the shape of Mercury's orbit.

Any sort of slow-down on the inner body when it and heavier outer bodies are relatively close together with the sun is liable to suffice, and that's what the authors of the link at the top argue what we get with (good approximations of) GR. (The approximations used by the authors are a post-Newtonian expansion <https://en.wikipedia.org/wiki/Post-Newtonian_expansion> and numerical methods). Very roughly, thanks to the large central mass and the very different distances of the planets involved, the outer body rushes ahead or equivalently the inner body lags behind, which avoids stabilizing the resonance.

Many -- possibly even most -- stars are in binaries, rather than singletons like our sun. Planetary orbits in a binary (or triple) star system are likely to be dramatically different. In binaries, the width and mass-ratio between the partners are relevant quantities. Triples can be arranged in all sorts of ways. PSR J0337+17 is one known arrangement <https://www.youtube.com/watch?v=oDgfqq_W_uM>. PSR B1620-26, a pulsar-white dwarf binary, has at least one planet. I have only weak intuitions about what Jupiter or Earth mass bodies would do in such systems, or even how to select a reasonably astrophysical set of initial orbital values. (There are known circumbinary planets. See <https://en.wikipedia.org/wiki/Circumbinary_planet> and <https://sci-hub.ru/10.1007/978-90-481-8687-7> in which Chapter 9 details how these can be simulated).

It is not too fanciful however to suppose that periastrons could be close enough that post-Newtonian effects are non-negligible, even if the stellar masses are non-compact. That doesn't say much about whether secular resonances (in the sense above) are likely in such systems. I have no idea, but I'd guess they aren't forbidden.

> compare what % of systems are stable under Newtonian vs Einsteinian [gravitation]

That's what the authors do in their paper <https://arxiv.org/abs/2303.05567>, although "systems" here is a simplified model of the solar system (they discuss several simplifications and their justifications: the oblateness of the sun and the presence of moons are probably interesting "future work" but are most likely not going to result in large corrections see their §3.2).

For non-isolated systems, interactions with the environment can magnify whether these properties drive dynamical stabilization (merger) or instability (fly apart). The authors in their other fresh paper even more freshly summarized at <https://astrobites.org/2023/03/14/chaos-planets-future-of-th...> discuss how "environmental events" (nearby stars and their potential to approach our solar system) might lead to e.g. collisions between Earh and Mars, or an ejection of one or more planets into deep space.

Astrobites has also freshly summarized another "environment can change orbits" (for 2-body + dusty environment) at <https://astrobites.org/2023/03/13/will-it-merge-investigatin...>. ("Dust" in the GR sense of <https://en.wikipedia.org/wiki/Dust_solution>, rather than in the sense of fine particles of e.g. chalk that allegedly accumulate on sedentary GR theorists).

> Here's an experimental idea

Been done (and improving) for many decades. See for example Fig. 1 in <https://arxiv.org/abs/1105.1082v1>.

No, I don't think that follows. It is plausible that a system could remain stable under perturbations to the dynamical laws. In their case they find that there seems to be no critical threshold for GR on the stability of the system. Presumably if you strengthened GR somewhat rather than weakening it you would find increased stability. (Though I don't know that they ran that experiment.)

I should note that you could make these changes two ways: you could just suddenly change the dynamical rules or you could gradually change them (a so-called "adiabatic" change). Making a sudden change is more likely to result in an instability. In this paper they were looking at adiabatic changes.

Please forgive what I suspect is something of a rant.

I'm not comfortable with the wording "strengthened GR" because there's an implicit assumption that one can straightforwardly map the authors' time-dependent parameter on the perihelion procession (their eqs (1), (2), and (3) in <https://arxiv.org/abs/2303.05567>) into a relativistic theory at all. The experience of various alternatives to GR (and in particular MONDian alternatives) suggests that assumption may be a bad one.

What they are doing is essentially exploring a family of theories of gravitation (which may or may not solve the Einstein Field Equations, or even a comparable set of equations) where each theory in the family produces a different evolution of the orbit of a planet. General Relativity is one of the theories in that family. But how would you even get from eqns 1 & 2 with the parameter fixed at unity (and not varying in time) to the EFEs? What do the Einstein and stress-energy tensors look like?

I do sympathize with the implied (and even partially-stated in §3.1) idea that each theory in the family of theories maps to a different set of initial (stress-energy) values for the planet (ignoring backreaction onto the metric) in standard GR, but I wouldn't bet much on that being right.

The authors write about the (and in particular "some critical") general relativistic precession rate (or frequency), and I don't mind that wording.

> nothing special about special relativity

What's special about special relativity is that the spacetime is flat, i.e., there is no gravitation at all.

This work is about the sun's gravity, i.e. the (weakly) curved spacetime sourced by the central mass of our solar system, and so is about general relativity and not special relativity.

> change any of the underlying rules, it's going to become stable

The relevant thing here is chaos.

[italicizations below are mine]

§3.1 in the authors' preprint at <https://arxiv.org/abs/2303.05567> makes the point: "because the systems are chaotic and we use slightly different general relativistic corrections in each simulation, the simulations diverge quickly. This is effectively the same as varying the initial conditions of one planet by a tiny amount".

Quoting Wikipedia's Chaos Theory article: "Chaos theory is [...] focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions".

(The initial conditions in question are the orbital parameters--or more broadly the momentum--of Mercury. The equivalence is fairly obvious, but I think I prefer to look it at similar to your wording, "... change any of the underlying rules ...", although not just any rule, they change a specific rule (or closely related family of rules, more on that below).)

In the paper's §3.2 they identify the most prominent post-Newtonian contribution ("... the precession from GR (the gravito electric effect) .. follows closest behind Newtonian planet-planet interactions and is more than three orders of magnitude more influential than any additional effects"), and the paper in general focuses on the question of whether simulated planet-planet interactions' tendencies to eject Mercury from the solar system are dampened by GR's influence on the perihelion precession of Mercury. (Their answer: yes - Mercury sticks around for many billions of years even if the gravito-electric effect is small, but if it's zero, simulated Mercury is typically yeeted right out of simulated solar system in relatively few billions of years).

More specifically they look at the evolution of Mercury's orbital eccentricity and use it as a proxy for Mercury's ejection <https://www.urbandictionary.com/define.php?term=Yeet>. Mercury's orbit becomes more elliptical if over long periods of time its perihelion coincides with the perihelion of Jupiter. With practically any perihelion or ascending node precession of Mercury or Jupiter, Mercury's orbit remains roughly circular.

Even more specifically in their eqn. (2) they parameterize the general relativity corrections to the planets' perihelion precessions, and vary the parameter (eqn (3)). Eqns. (1) & (2), with the parameter \alpha set to unity, are straightforwardly extracted from an approximate solution to the Einstein Field Equations (of General Relativity; the EFEs), but going the other way from a modification of the parameter \alpha to a solution of the EFEs is something reviewers or a paper in reply may question, since it's not as straightforward. (One might compare how (Milgrom's) MOND can be seen as a parametrization in the same spirit eqns (1) & (2), "simples", but nobody has completely reverse-engineered the parametrized equation back into a set of relativistic field equations. The authors touch on this at the bottom of the first page's second column.)

I understand the point made about effectively the same as varying the initial conditions thanks to spending some student time on Smale and non linear dynamics, and I appreciate this paper is exploring the sensitivity of gravitational N-body systems with respect to GR perturbations, however this caught my eye:

> and is more than three orders of magnitude more influential than any additional effects

Mercury is just so damn close to the sun, how much force does the solar wind pressure exert on the planet (directly | via magnetic field interactions)?

I imagine it must be non trivial and subject to variation.

Hey, I'm the first author on this paper. We don't expect solar wind to have a strong effect. I focus on dynamics and not electromagnetic effects, but we took our values for the effects contributing to the perihelion precession of Mercury from NASA by Park et al. (2017) Precession of Mercury's Perihelion from Ranging to the MESSENGER Spacecraft <https://iopscience.iop.org/article/10.3847/1538-3881/aa5be2>.

However, I don't have an order of magnitude estimate for the effect and like you said, the effect is certainly subject to variation, especially due to the Sun's solar cycle <https://en.wikipedia.org/wiki/Solar_cycle>. My guess is that since the effect of solar wind isn't included in the calculations and measurements by Park et al. (2017), then it's likely around a thousandth of an arcsecond per century or less.

Thanks for the reply, I was in the ground based geophysics game for a decade or two and I'm fine with the envelope guess work and principle focus on gravity + GR dynamics .. but I am now piqued by this, I'm grabbing Park (thanks for the link) and will poke about further.

There's an orbiting-body size and shape (and ultimately mass) dependence on solar radiation pressure's effects on the evolutions of orbits in the solar system that dominates other factors for bigger solar system bodies (planets, dwarf planets, round moons).

For planetary orbits, perturbations from radiation pressure while the sun is on the main sequence will be so small as to be negligible. Small perturbations may matter over billions of years, but perhaps more billions than the sun has life.

Small bodies which are not in hydrostatic equilibrium (i.e., not massive enough to be round through gravitational collapse) can experience many orders of magnitude larger perturbations. However, for lumpy asteroids, radiation pressure gives net accelerations orders of magnitude below 10^-10 m/s^2. A literature search on the Yarkovksy effect might take you to radiation-pressure work on larger, rounder bodies (or more likely their outermost layers' chemistry and density), possibly even planets. Effects like these are generally proportional to ellipticy of the body itself and its rotation, and inversely proportional the body's size, with insolation and albedo as surprisingly minor factors. The Yarkovsky effect might cause a small, lumpy, quickly-spinning asteroid-belt body to dive into the inner solar system. https://arxiv.org/abs/1708.05513 has an interesting overview. At a more pop-sci level, https://skyandtelescope.org/astronomy-news/the-asteroid-and-...

The MESSENGER programme did some science on Mercury's exosphere <https://en.wikipedia.org/wiki/Atmosphere_of_Mercury> (and effects like ion sputtering photon-stimulated desorption, each of which kicks mainly sodium atoms off Mercury's surface into its long cometary tail) that is likely to be interesting, but is likely to undermine the idea that electromagnetic forces are significant with respect to the planet's orbital evolution.

There is a substantial body of literature on radiation pressure's effects on artificial Earth satellites, but such devices are certainly not in hydrostatic equilibrium and rarely anything close to spherical (extended solar panels, etc.) Putting "solar radiation pressure satellites" into your favourite Internet search engine will be fruitful.

There is also a body of literature where the radial solar radiation force is important in calculating the trajectories of space probes sent to various parts of our solar system. Again, these are low-mass objects on many-year flights. The Mariner programme generated papers relevant to Mercury e.g. the 160 pages of <https://ntrs.nasa.gov/citations/19750009204>.

Cheers.

On a related note, are there any examples of magnetic field coupling having a significant effect (for some time scale) on orbiting masses? Mercury, again, might be good case for such a thing.

In a similar vein as

> (searching) "solar radiation pressure satellites"

is "Magnetorquer satellites" - small effects for attitude adjustment with a complicated "surfing" twist as the field fluxes so much on a daily basis.

I'm under no illusions that these (solar particle pressure, magnetic coupling) would be significant - but theis is a thread about a paper on the evolution of small perturbations :-)

We are straying far from my area of expertise. :-) (except at the end).

> Mercury, again, might be a good case

In our solar system I'd look to Jupiter, whose magnetic strength is tens of thousands of times stronger than Earth's, and which has lots of bodies of various sizes orbiting it.

There are magnetic interactions with several of the moons explored by recent space probes, but they are very small. Ganymede is probably the most interesting case. I would be surprised if there were significant orbital perturbations from magnetics on the Galilean moons, or even the mid-size ones.

Smaller, especially grain-sized bodies, I don't know. I know next to nothing about Jupiter's rings, but on general chemistry grounds there will be electromagnetic interactions with the planet's magnetosphere. (I suspect these are dwarfed by radiation pressures though; the biggest forcing radiation will be tangential to the ring-components' orbits around Jupiter, and as those objects are small the effect will be relatively large, causing illuminated grains to tend to spiral inwards to Jupiter, but how long does a ring particle stay illuminated rather than in shadow? And there are likely forces that push grains outward, too). None of this likely needs corrections to classical Newtonian gravitation. (The authors of the paper linked at the top do not consider any of the solar system's moons or smaller objects).

I'd start here <https://en.wikipedia.org/wiki/Magnetosphere_of_Jupiter#Inter...>, notably the paragraph starting, "The icy Galilean moons, Europa, Ganymede and Callisto, all generate induced magnetic moments in response to changes in Jupiter's magnetic field", and visit the various citations in that paragraph and the following two.

> evolution of small parameters

You've hit on the right word, "evolution", which is much more important than "small". Can we apply perturbation theory? Do we get a nice Taylor-series-like expansion where higher order terms can be treated as marginal or negligible, in the style of Wilsonian effective field theory? See Clifford Will's article, <https://www.pnas.org/doi/pdf/10.1073/pnas.1103127108>, which is probably close to accessible. It is mainly interested in "quite relativistic" systems, however, and we do not have those in our solar system. Almost nothing in the solar system needs even 1PN (see what that means in the article, or in e.g. the wikipedia article on the post-Newtonian expansion) correction. The work linked at the top is neat because PN effects may actually matter for the ultimate fate of Mercury.

Finally, this beautiful paper by Soffel (Past President of IAU Commission 52 Relativity in Fundamental Astronomy) discusses the (ir)relevance of higher-order post-Newtonian corrections to the Earth's gravitational field. <https://link.springer.com/article/10.1007/s00190-016-0927-4> (find it on sci-hub.ru). ("Above, we have demonstrated that the first post-Newtonian approximation is completely sufficient for a description of the Earth's gravitational field and also for a description of satellite orbits."). Table 3 lists the various accelerations on the LAGEOS satellite. Solar radiation pressure, thermal re-radiation, and even albedo pressure (at 3.5e-19 m/s^2) are in the table; nothing like like "magnetotorquer" effects even appears.

This article by Lhotka & Gale (2019) <https://arxiv.org/abs/1911.02778> is a little beyond my patience for understanding, but they have an excellent parameter space plot in Figure 1 relating a bunch of different effects with respect to dust grain size. Their figure only goes up to grains 1 cm in size though. However, it's a detailed paper on the relative "importance of the interplanetary magnetic field on the orbital evolution of dust". I met Lhotka at a conference where they presented this work.

I think this discussion is really fascinating, and it's certainly something that I didn't consider when working on the paper. However, I think the results that we found can be generalized to better understand how a small effect might impact the stability of the solar system on a statistical level, even if we can't say much about the effect in specific cases.

For example, I expect that as the total mass of the sun decreases due to solar wind the secular frequencies would all change by the same amount, proportional to the decrease in mass <https://arxiv.org/abs/1811.07135>. This would mean that the difference between the g1 and g5 modes would be closer by the same amount. However, we wouldn't expect the secular diffusion to change so it would be more akin to dashed line in Figure 2 of our paper. So stellar mass loss could cause Mercury to go unstable sooner than without mass loss, but we wouldn't expect the effect to be significant.

"In the planetary system, solar gravity determines the orbits of all bodies larger than dust particles for which other forces become important. But even for dust, gravity is an important factor. Near planets, planetary gravity takes over", §3.1 of Krüger & Grün's Chapter 29 in Encyclopedia of the Solar System (2014) <https://doi.org/10.1016/B978-0-12-415845-0.00029-3> (it's on scy hob :) ). See Table 29.4.

Where dust is gravitationally interesting at the present age of the solar system is probably in ring/moon interactions, but mostly with the moon acting on the ring dusts as with Saturn's Daphnis and Pan (Cassini).

Of course there was a lot more dust doing a lot more interesting things gravitationally much earlier in the solar system's history, which might be interesting if you decide to use your approach to simulate the solar system's past (some of that dust could be related to your link to the massive early sun hypothesis paper). :-)

Your other paper on stellar flybys provokes an interesting question: could a fly-by deposit a significant mass of trans-Neptunian dust? I'd wager a lot of fresh dust captured out there could increase the eccentricity of planetary orbits through your 0.9 threshold.

Although not plausible for the solar system, relatively high mass-density dust around compact binaries can drive the compact binary's orbital evolution. See for example the very fresh Siwek, Weinberger & Hernquist (2023) <https://arxiv.org/abs/2302.01785>, reasonably summarized at <https://astrobites.org/2023/03/13/will-it-merge-investigatin...>.

This is not terribly surprising because it turns out that nearly every dynamical system with more than three bodies is only marginally stable on a timescale of order its own lifetime.

Does this mean that it's highly likely that our solar system had more planets a few billion years ago?

It's 100% certain, modulo your definition of "planet". Our solar system began as a cloud of dust, which over time accreted into larger and larger objects. During that accretion process, many objects large enough to be called planets accreted by colliding with each other. That's how earth's moon was formed, and it's very likely the cause of Uranus's extreme axial tilt and Venus's retrograde rotation.

Ok, I'm thinking of the "cleared its orbital neighborhood" definition. So not the initial accretion process but after it reached a more modern-looking system of planets separated by mostly empty space.

Well, like I said, it all turns on how you define "planet". Comet Shoemaker-Levy impacted Jupiter in 1994, so before that there was one more body orbiting the sun than after. But I'm guessing that doesn't count because it was a "comet" and not a "planet". Meteors hit the earth and moon constantly, but again, not "planets".

Orbiting bodies are eliminated regularly, either by impacts or by being ejected, the former happening much more often than the latter because the ejection window is smaller than the impact window. And the bigger the body, the less likely it is to happen because to be eliminated it has to hit or do a flyby of something bigger, and the bigger you get the fewer things there are that will do the job. Jupiter, for example, ain't goin' nowhere.

At some point in the past we had a lot more Mercury-to-Mars-sized things floating around than we do now. How long ago the last one got eliminated is anyone's guess, but it was almost certainly a few billion years.

> At some point in the past we had a lot more Mercury-to-Mars-sized things floating around than we do now. How long ago the last one got eliminated is anyone's guess, but it was almost certainly a few billion years.

The last one hasn't been eliminated - we still have Mercury and Mars. You probably don't think of it that way because we think of the existing planets as constants, not variables.

I meant: no one knows how long the last one that was eliminated was eliminated. But that sounded awkward (and I thought it would be obvious).

You might be interested in the "Five Planet Nice Model": https://en.wikipedia.org/wiki/Five-planet_Nice_model

The most popular theory about the Moon's formation is that Earth collided with a Mars-sized planet, so if that's true there definitely used to be at least one more planet than what we have now.

Hanno Rein writes some of the prettiest C code I've seen. He also has a cool app, Exoplanets. Big fan!

>What they find is that after switching off general relativity the Solar System goes unstable shortly thereafter.

Not my field, and I only glanced at the figures, but I think "shortly thereafter" is something only a cosmologist could say without blushing. Unless I completely bungled the meaning (very plausible), the paper is using time on the scale of billions of years.

I guess the timescale was picked up to match the expected end of Sol's main sequence (in ~5 billion years) and its growth into a planet-engulfing red giant, in which case continuing with the model beyond the next order of magnitude becomes unrealistic.

IIUC the instability time scatter plot shows that GR basically stabilises our local N body problem by one to two orders of magnitude in time scale (10^8~10^8.5 vs 10^9~10^10), IOW giving a 900~9000Myr extra vs being derailed by a Mercury-Venus collision in ~100Myr, which would indeed comparatively be "shortly thereafter" at these timescales.

I did chuckle at the phrasing though.

(Disclaimer: not my field either, so I may be 10^10 wrong)

Hah, fair. As an astronomer I don't really blink when someone says that 100 million years is a short time period.

Since Mercury is the closest planet to the sun, wouldn’t it need the most delta-v to escape the solar system? What about it makes it the most likely planet to be ejected?

It is less massive than the other planets. Momentum is conserved, so any escape scenario would see at least one other planet with a smaller orbit.

The solar system is a chaotic, dynamical, and n-body system though, so the 1 or 2-body problems involved in delta-v calculations are much more simple. In addition, Mercury's mass is low, relatively speaking.

Neat.

I was half-disappointed that they didn't use scmutils (like Sussman & Wisdom) : (