Abstract
 
Since the first discovery of an exoplanet system in 1992, it has become apparent that the structure of the solar Solar system as it exists today is by no means typical of planetary systems in general. The early discovery of hot Jupiters orbiting their stars close in, although a result of observational bias in the radial velocity method, has been confirmed by other detection methods and has led to the concept of planetary migration over timescales of millions to billions of years. Studying the likely history and possible future of the solar Solar system reveals a chaotic environment for the inner planets and a semi-stable environment for the giant planets. Long term orbital instability has profound implications in terms of planetary migration, collisions and ejections for both the solar Solar system and for exoplanet systems.

Introduction 

The evolution of planetary systems has been the subject of study for several hundred years, with Kepler (1571-1630), Newton (1643-1727), Laplace(1749-1827), Lagrange (1736-1813), Gauss (1777-1855) and Poincaré (1854-1912) all having made great contributions to the field. Also, as is well known, the precession of Mercury's orbit was first accurately explained by Einstein's, (1879-1955) theory of General Relatoivity.
Approximately 4.5 billion years ago, gravity pulled a cloud of dust and gas together to form the protoplanetary nebula from which the Sun and the rest of Solar system evolved. Subsequently, by processes of both acretionaccretion and condensation in the gas cloud, planetesimals formed and collisions between them led to the formation of planets. The present Solar system structure is like a flat disk, with objects within the disk orbiting in almost the same plane. The objects generally orbit in the same direction and, with the exceptions of Venus and Uranus, rotate in the same direction.
There are several general types of such objects:

• Planets of which there are two types:
  • Terrestrial (Earth-like) planets: Mercury, Venus, Earth, Mars. Mass < ~15 Earth masses
  • Jovian (Jupiter-like) planets: Jupiter, Saturn, Uranus, Neptune. Mass > ~15 Earth masses
    
• Minor planets: Ceres, Pluto
• Debris: Asteroids (of which some are relatively large; the current number of known asteroids is over 1 million), meteors, comets

All these objects are believed to be ~ 4.6 Billion years old.
The Sun contains >99% of the mass of the Solar system, whereas the major planets account for
 ~ 99% of the angular momentum of the Solar system.
Gravitational influence of the giant planets has a major effect on the smallest planets. Jupiter, with the greatest mass of all, perturbs the orbits of the two least massive planets so that Mercury has the most eccentric orbit  and Mars the second most eccentric orbit in the Solar system.

Discussion

Planetary systems 

Using the 305m Arecibo radio telescope (sadly, now defunct),  it was demonstrated that the 6.2ms pulsar PSR1257 +12 is orbited by two or more planet-sized bodies – the first detection of an exoplanet system (Wolszczan & Frail, 1992).  Perhaps more well-known, 51 Pegasi is a Sun-like star located in the constellation of Pegasus at a distance of approximately 15 parsecs from the Solar system. 51 Pegasi was the first main-sequence star (type G2IV) found to have an exoplanet - designated by convention as 51 Pegasi b (Mayor & Queloz, D et al, 1995). Until the early 1990s it was widely assumed most planetary systems would be essentially similar to the Solar system. That this is not the case may be explained by considering the whole time-domain of the evolution of all planetary systems. Each system we detect maybe at any stage in its development, and will therefore most likely have different characteristics to the present-day Solar system.
We begin by considering gravitational effects. Generally, a low mass object orbiting a much more massive body would be expected to be forced gravitationally into an almost circular (i.e. zero-eccentricity) orbit. For example, this is the case with the orbits of the Galilean moons of Jupiter, where each moon's mass is much less than Jupiter’s mass (see  Appendix: resonant and non-resonant orbits).
Considering just two gravitatonally-bound objects, Newton’s law of gravitation can be written as:

Where the respective forces are as shown in Figure 2.

This is mathematically called a 2-body problem, which as may be seen from the equation has a simple algebraic solution.
 
We should also recall Kepler's Laws, where he postulated that all orbits are elliptical. In particular, Kepler's Third Law tells us that

where k is a constant of proportionality given by

It turns out that if P is in Earth years and a is in Astronomical Units, (AU), k=1. The orbits of the Solar system’s major planets at the present time conform exactly to Kepler's Third Law as shown in Figure 3 below:

The preceding evidence gives the impression of a well-ordered and static system. We must however remember that what we are seeing at any instant in time is a snapshot in the history of the Solar system. As we shall see, planetary systems are generally much more complicated.


Complexity of multi-body systems 

Kepler's Laws are regarded as fundamental Laws of planetary systems, so Kepler's Third Law is assumed to apply throughout the life of the Solar system and that of any exoplanet system. However, this does not mean the configuration over the lifetime of the system is static. The orbits of the system will always be elliptical (and hence conform to Kepler’s Laws), but the parameters eccentricity, major and minor axes of the ellipses will change as the planets interact with each other. The only case where the orbit of a planet will acquire an orbital eccentricity e>1 is in the case of a planetary ejection from the system (Laskar, 1994)
The plot in Figure 4 shows the eccentricities of the major Solar system planets. We can see from this plot that whereas the orbits of Venus, Earth, Jupiter, Saturn, Uranus and Neptune have similar eccentricities, the orbits of Mercury and Mars are clearly outliers. Mercury has a particularly  eccentric orbit Ε = 0.2056.

Exactly as one would expect, the Left-to-Right order of the plots is the order outwards from the Sun at which the planets orbit. Of all the major planets in the solar system,. Mars has the second most eccentric orbit with Ε =0.0935.
 
In Figure 5, we plot orbital eccentricity, Ε against planetary mass, M for each major planet.  Here the plots are not in order of distance from the Sun - the rightmost plot is the most massive planet, Jupiter which has therefore the most domiantdominant influence on the rest of the Solar system planets (Hayes, 2010).

Mercury has been described as having the most unstable orbit in the Solar system (Lithwick & Wu, 2014). The root cause of this appears to be that at least the inner Solar system is chaotic, and the outer Solar system is borderline stable (Lithwick & Wu, 2014). What we see here is the situation in the Solar system's time-domain as of today. Beyond a few tens of Myr into the future the motion of the planets based on what we observe today cannot be accurately predicted (Woillez and Bouchet, 2020).
On longer timescales, planetary trajectories can only be studied probabilistically with software using numerical algorithms running on supercomputers (Budrikis, 2020). Simulations using these methods predict that over the Sun’s remaining lifetime around 1% of possible trajectories of the planets show Mercury’s orbit becoming so eccentric it may be involved in a collision with Venus or the Sun (Woillez and Bouchet, 2020). Early studies have discussed the possibility of Mercury being ejected from the Solar system altogether or its collision with Venus (Laskar, 1994). This is also indicated in more recent work which shows the strong influence of the outer planets inducing chaotic variations by the inner planets (Hayes, 2010) . In case these scenaria sound improbable, consider that the most widely accepted theory of the origin of Earth's Moon is that of a proto-Earth colliding with a Mars-size planetesimal.
To better understand this, consider a simple thought-experiment where a planetary system consists of just the Sun, Mercury and Jupiter. The gravitational forces acting on Mercury due to Jupiter and the Sun respectively will be the vector:

The vector components on the right-hand side will be, according to the Inverse Square Law, inversely proportional to the respective vector distances of the Sun and Jupiter respectively, rSUN and rJUPITER:

When Mercury and Jupiter are on the same side of the Sun, these vector components will be at their highest. When Mercury and Jupiter are on opposite sides of the Sun, these vector components will be at their lowest.
As we can see, the two vector sums change continuously as the two planets proceed along their respective orbits, and the calculation of the vectors is relatively tedious..
Now, aside from this thought-experiment, in the real Solar system, we must recall several things, namely:

1. During the time that Jupiter makes one orbit, Mercury makes just over 49 orbits; during each of Mercury’s orbits, the resultants of the vector sum above will be different;
   
2. While this is happening the four terrestrial planets and two minor planets (Ceres and Pluto), plus numerous asteroids and comets, each with differing masses, orbital periods and eccentricity will influence the vectors.
3. The spin axis of each planet is precessing (see September 2020 AstroBlog).
   
4. The orbit of each planet is precessing (Ibid)
5. The largest mass in the system, the Sun, is also orbiting the common centre of mass in the Solar system ( called the barycentre) - the so called "wobble" by which exoplanet systems are detected using the radial velocity method.

 
We can now see that the calculations are of very high complexity, even for a single instant in time, and vastly more complex if we look at the distant past or distant future. This is known as the “N-body problem”, the problem of predicting the individual motions of a group of celestial objects which gravitationally interact. Though analytic solutions have been proven up to N=3 (at least in cases where M1 >> M2) , there is no analytic solution to the N-body problem where N>3 (Heggie, 2005), and instead numerical solutions must be run on computers.
At earlier times in the history of the Solar system, it has been suggested that Jupiter and Saturn were likely in the 3:2 resonance, defined as PSATURN/PJUPITER = 1.5, where PJUPITER and PSATURN are the respective orbital periods of Jupiter and Saturn. Today, this ratio has become 2.49, the resonance having been disrupted by gravitational interactions (Nesvorny, D, 2011)


Planetary migration 

As planets evolve, their mass is subject to changes. Collisions and absorption of dust small objects increase a planet’s mass, and collisions may either increase or decrease mass. This results in an exchange of angular momentum between an evolving planet and the protoplanetary disc, which causes the planet to migrate through the disc. Until the long-awaited discovery of exoplanets, when planetary formation models were based on the Solar system, planetary migrations were usually considered unlikely.
More recently, data from exoplanet systems has provided strong evidence of planetary migration. For example, WASP-107b, a super-Neptune discovered in 2017, is estimated to have a rocky core of ~ 10 Earth masses, and a large gaseous envelope consisting mainly of H and He. This means WASP-107b’s most probably formed several AU from the host star where the protoplanetary disk is rich in gas, ices and dust (Piaulet et al, 2020). However, the current orbital semi-major axis of WASP-107b is only 0.055±0.001 AU (NASA Exoplanet Archive). This leads to the conclusion that WASP-107b has most likely undergone inwards migration (Piaulet et al, 2020).


Decaying planetary orbits 

Have any decaying planetary orbits been observed? Yes, but not that many so far.
The orbital period of exoplanet TrES-1 b, discovered in 2004 (Alonso et al 2004) is getting shorter by around 11 milliseconds per year. This may not sound much, but over an astronomically short time of ~300,000 yr, that's approximately 3 days. The orbital period of TrES-1 b is very close to 3 days according to recognized sources (Exoplanet.eu: P~3.0300722d; NASA Exoplanet Archive: P~3.030070±0.000008d; (see NASA Exoplanet Archive).

Ejection of planets from systems 

As mentioned earlier, ejection of small terrestrial planets is plausible (Laskar, 1994). But what about giant planets? Studies of giant planets’ interaction within the protoplanetary gas disk indicate that planetary migration is usual. Moreover, planets emerging from mergers of planetesimals are expected to be in orbital resonance. Planetary systems formed from protoplanetary disks can become dynamically unstable after the gas disappears, since the gas exerts a stabilising influence. This leads to a phase when planets scatter off of each other. According to this model, Jupiter and Saturn were most likely trapped in the 3:2 resonance (Nesvorny, 2011). We can see from Figure 7 that this is certainly not the case today.

Using 6000 scattering simulations, Nesvorny et al evaluated a historic Solar system with both four and five gas giant planets. With four gas giants (i.e. as in the current epoch) the best results were obtained with disk masses between 35 and 50 Earth masses. The fraction of simulations with an initial four outer planets producing a final system also having four outer planets was only between 10% and 13%, showing an unlikelihood that the Solar system evolved from a four giant planet system. When run with a five outer planet system as the initial configuration, simulations showed it roughly 10 times more likely to obtain a Solar system analog.
The conclusion reached by Nesvorny, postulating a "jumping Jupiter" scenario, is that the fifth giant planet was ejected from the early Solar system about 3Myr ago. It should be noted that to explain the current orbital parameters of Jupiter, there is a dependency on the 3:2 resonance between Jupiter and Saturn mentioned earlier. More recently, investigation into the framework of this jumping-Jupiter model assessed the possibility that the high eccentricity and inclination of Mercury originated during the instability and concluded instability can produce the presently large values of eccentricity and inclination of Mercury. (Roig et al, 2021)
How plausible is it that a planet - in the case of the 2011 study by Nesvorny study, a giant planet - could be ejected from a planetary system, becoming a lone planet? An observational detection of a giant-mass lone planet using gravitational micro-lensing has been made (Mroz et al, 2018). In the case of this study, there are some major uncertainties as to where the lens is located. If it is in the Galactic disk, the lone planet should be of Neptune-mass; if the lens belongs to the Galactic bulge population, the lone planet should be a Saturn-mass. It is interesting to note that today’s micro-lensing surveys are able to detect lone planets as small as a single Earth-mass.


Conclusion 

The configuration of the Solar system as it is today is atypical of planetary systems we have discovered to date. The current configuration would not have existed throughout the history to date of the Solar system, neither will it remain static in the future. The same reasoning apples to exoplanet systems. Since we are examining exoplanet systems as they exist now may explain why most exoplaenet systems discovered to date show very different characteristics and structure to those of the current Solar system structure.
When we are observing exoplanetary systems, we are observing evolving systems at an instant in time. These systems will also have changed over time and will continue to do so in future. The same reasoning applies in the case of the Solar system.  Modeling these changes throughout the expected time-domain is only possible by using numerical methods which require large amounts of computer time.
In planetary systems, events such as collisions, both between orbiting objects with other orbiting objects, and between orbiting objects and the parent star are regarded as normal rather than exceptional. Migration of planets from their current orbits is also not unusual. Planetary ejections also occur, which may be one reason for the existence of some lone planets.

References 

Alonso, R et al (2004). TrES-1: The transiting planet of a bright k0 v star. ApJ, 613:L153–L156, 2004 October 1. https://iopscience.iop.org/article/10.1086/425256/pdf  Accessed September 23, 2021
Budrikis, Z (2020). Nature Reviews Physics. The path to the Solar system’s destabilization.
DOI:https://doi.org/10.1038/s42254-020-0218-0https://doi.org/10.1038/s42254-020-0218-0   Accessed Sep-8-2021
Hayes, W et al 2010). The interplay of chaos between the terrestrial and giant planets. MNRAS 407, 1859–1865 (2010). DOI: 10.1111/j.1365-2966.2010.17027.x Accessed Sep 26 2021.
Heggie, D (2005). The Classical Gravitational N-Body Problem.
https://arxiv.org/pdf/astro-ph/0503600.pdf  Accessed September 14 2021
Kokori, A; Tsiaras, A; Edwards, B et al. (2020) ExoClock project: an open platform for monitoring the ephemerides of Ariel targets with contributions from the public. https://www.exoclock.space/project Accessed September 25, 2021
Laskar, J (1994). Large scale chaos in the solar Solar system. A&A 287 L8-12(1994)
Lithwick, Y; Wu, Y (2014). Secular chaos and its application to Mercury, hot Jupiters, and the organization of planetary systems. Proceedings of the National Academy of Sciences, Volume 111, Issue 35, 2014, pp.12610-12615. https://arxiv.org/pdf/1311.1214.pdf Accessed Sep-8-2021.
Mayor, M.; Queloz, D; et al (1995).  51 Pegasi.   IAU Circ., No. 6251, #1 (1995)
DOI: https://doi.org/10.1038/355145a0   Accessed Oct 15 2021
Morbidelli, A et al (2010) Evidence from the asteroid belt for a violent past evolution of Jupiter’s orbit. ApJ, 140:1391–1401, 2010 November. https://iopscience.iop.org/article/10.1088/0004-6256/140/5/1391/pdf Accessed Sep 25 2021.
Mroz, P (2018). A Neptune-mass Free-floating Planet Candidate Discovered by Microlensing Surveys.  ApJ , 155:121 (6pp), 2018 March https://iopscience.iop.org/article/10.3847/1538-3881/aaaae9/pdf  Accessed Sep 22 2021. Submitted to ApJ
Nesvorny, D (2011). Young Solar System’s Fifth Giant Planet? https://arxiv.org/pdf/1109.2949.pdf Accessed Sep 22 2021.
Piaulet, C et al (2020). WASP-107b’s density is even lower: a case study for the physics of planetary gas envelope accretion and orbital migration. https://arxiv.org/pdf/2011.13444.pdf Accessed Sep-14 2021
Roig, F (2016). Jumping Jupiter can explain Mercury’s orbit. ApJ 820:L30 (5pp), 2016 April 1. https://iopscience.iop.org/article/10.3847/2041-8205/820/2/L30/pdf Accessed Oct-2-2021
Saillenfest, M; Laskar, J; Boué, G (2019) Secular spin-axis dynamics of exoplanets A&A Volume 623, March2019. https://www.aanda.org/articles/aa/pdf/2019/03/aa34344-18.pdf Accessed Sep-8-2021.
Wolszczan, A & Frail, D (1992). A planetary system around the millisecond pulsar PSR1257 + 12.  Nature volume 355, pages 145–147 (1992).  DOI: https://doi.org/10.1038/355145a0   Accessed Oct 15 2021

Acknowledgments

The author gratefully acknowledges two anonymous referees, whose critiques have resulted in improvements to this paper.


Data sources used in this work

Exoplanet.eu
NASA Exoplanet Archive
NASA Index of Planetary Fact Sheets

Appendix: resonant and  non-resonant orbits 

In a system where a low mass satellite is orbiting a much more massive body, we would expect the satellite’s orbit to be forced gravitationally into an almost zero-eccentricity (i.e. very nearly circular) orbit - a process called circularization.
Consider the case of the Galilean moons of Jupiter, where this is indeed the situation. Although there are 79 known Jovian moons (as of September 2021), the Galileans account for 99.007% of the orbiting system’s mass. In this system, each of the four moons’ mass is much less than Jupiter’s mass, and as Table 2 shows, all the respective orbits are of very low eccentricity.
The reason this system’s orbital configuration is not like that of the Solar system is because in the case of the Galilean system, there are no high mass Jovian satellites beyond Callisto, the outermost Galilean moon

A mean-motion orbital resonance occurs when two bodies orbiting a larger primary have periods of revolution that are an in an integer ratio. An example is the resonance between the Galilean moons of Jupiter Io, Europa, and Ganymede, which are in 1:2:4 resonance. That is to say, the furthest orbiting moon, Ganymede makes one orbit, Europa makes two orbits and Io makes four orbits.  The fourth Galilean moon Callisto does not cross the zero point simultaneously with the others and is therefore not in orbital resonance with the other three.

Well the autumn equinox is past now and we should begin to enjoy some dark skies again. Remember the clocks go back on the 31st of this month. The Harvest Moon on the 21st September looked quite spectacular as I caught it rising in the east and just fitting between two nearby houses.
 
Observing
The following chart represents the night sky at 11.00pm BST on the 8th of October and at 10.00pm BST on the 23rd October. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

The chart this month should look familiar because facing south and just above your zenith will be the magnitude 2.3 star ,Caph, a member of five bright stars forming the readily recognisable ‘W’ shaped asterism in the constellation Cassiopeia. Being circumpolar it is visible all year round and six months ago it was low over the northern horizon but now we have a chance to admire it high in the sky.
            Below Cassiopeia and to the right is the Great Square of Pegasus which we looked at last month. However, our focus this month is on the constellation Andromeda- the princess and daughter of the mythological Queen Cassiopeia and King Cepheus. Remember that the brightest of the stars in the Great Square of Pegasus, Alpheratz, is actually in the constellation Andromeda so we shall start from there. The main features of Andromeda are two curved strings of relatively faint stars starting at Alpheratz and lying above it and to the left, below Cassiopeia. The bright stars Algol and Mirphak in the constellation Perseus to the east can also help with navigation. The lower string of stars is fairly easily followed by star hopping from Alpheratz: delta Andromeda, Mirach and Almach have magnitudes of 3.3, 2.1 and 2.2 so no problem. The higher string of stars is fainter and poor viewing conditions and light pollution will make them difficult to see with the unaided eye as they are typically of magnitude 3.5 to 4.5. The constellation Andromeda is home to one of the most famous objects in the night sky- the Andromeda galaxy also known as M31 and shown on the chart by a red cross. The Andromeda galaxy is the nearest large galaxy to Earth and is similar in many ways to our Milky Way galaxy and is the only one visible to the unaided eye in the northern hemisphere. To locate it for observing, (your eyesight needs to be pretty good as its magnitude is 3.4), start at Alpheratz and by star hopping, jump to the second pair of stars along the curved strings and extend a line from Mirach through the second star and the Andromeda galaxy will be at a distance approximately equal to the distance between the two stars.  Now some mind boggling statistics- the distance to the Andromeda galaxy is about two and a half million light years which means that the light entering your eyes from Andromeda set out two and a half million years ago. Andromeda is the most distant object you can see with the unaided eye but you will need a dark site with no light pollution and clear skies. Don’t expect to see something like the images shown in the gallery of the WMA website, you will have to settle for something which might be described as a smudge or fuzzy star but that doesn’t detract from the sense of achievement. Perhaps we will be able to do it together when we meet at Oakhill on the 9th October. Let’s hope for clear skies.
            Only two stars in the constellation Pisces- The Fishes, are brighter than magnitude 4 so it offers little to the unaided eye. The constellation, Triangulum- The Triangle, is equally insignificant but because of its compact size, and a shape matching its name, it is relatively easy to spot.
            We’ll finish this month with another zodiacal constellation Aries- The Ram. In Greek mythology it represents the golden ram whose fleece was sought after by Jason and the Argonauts. I can see no resemblance to a ram but its brightest star, Hamal, shines brightly at magnitude 2.0 and can be readily picked out.
                         
Something to look out for
There will be a close approach of the Moon with Saturn and Jupiter on the 14th October and 15th October respectively. Perhaps I should explain that the chart in this blog is selected because it gives the view of the stars shown as they are crossing our meridian and therefore at their highest in the sky and as far away as possible from atmospheric disturbance closer to the horizon. However it is good to take a wider view from time to time and if you look west you can see the Summer Triangle with Altair getting closer to the horizon before it disappears from view later on. Similarly if you look towards your eastern horizon you will catch the beautiful star cluster known as the Pleiades or Seven Sisters. And don’t forget to cross from Cassiopeia through the pole star, Polaris, to The Plough which is currently close to the northern horizon and looking like a plough! Clear skies.