Why are the planets spaced just so?

Why are the planets spaced just so?

© 2019 Michael Clarage
Why is the Earth not just a little bit closer to the Sun? Why is Jupiter not just a little bit further away? Would that matter? Can planets be placed in a variety of orbits, or is there only one arrangement of planet orbit sizes and orbital periods possible for a given star system? Truth is we do not yet know.

Thanks to the Kepler and TESS telescopes, we have the orbital sizes and periods for dozens of other star systems. They are not exactly like our Solar System, but the similarities are obvious. With multiple systems to study, astrophysics should be able to go at this problem afresh. I follow several guiding principles

• The initial formation of star systems is governed mostly by electric and magnetic forces

• Star systems grow: they start very small and get bigger with time

• The arrangement of planets is for the solar system as a whole. It is not sufficient to look only at the relationships between pairs of planets.

A good paper that contains references to a very large number of other papers on this topic, as well as presenting a very good model of for planetary spacing where the times of orbits must be in simple ratios: https://arxiv.org/pdf/1701.08181.pdf

Relations in space and time between two planets

If we map the relationship between two planets over several cycles around the Sun,

Figure 1 The relative positions of Venus and Earth over the course of 7 years.

Imagine looking down upon our solar system, and mapping out the line connecting Venus to Earth. The length and direction of that line would constantly be changing.Figure 1is a plot of the relationship between Venus and Earth over a 7 year period. When the two planets are close together the curve is close to the origin, if the Earth is to the right of Venus the curve moves to the right. This is an analysis method pioneered by John Martineau, which can be found in the book entitled ‘A Little Book of Coincidence’.  The two planets get closer to each other, then farther away, and over 7 years time the pattern of their "dance" traces a 5-fold symmetric figure. To see all the "dances" between all the planets of our solar system (and a couple of other star systems) see this post: https://mclarage.blogspot.com/2019/06/other-solar-systems-as-electrical_8.html

Small variations in the sizes of orbits

What would the pattern look like if the orbit of Venus were just 1.0% larger?Figure 2shows result. A small change in the size of Venus' orbit completely changes the space-time symmetry between the two planets.

Figure 2 The dance between Venus and Earth if Venus' orbit were 1.0% larger.

What if we make Venus' orbit only 0.5% larger, will it be somewhere between a 5-fold and 3-fold symmetric figure?Figure 3shows the interplanetary correlations for this smaller change to the orbit size of Venus.

Figure 3 The dance between Venus and Earth if the orbit of Venus is 0.5% larger than what we actually see in our solar system.

In other words, the relation between Venus and Earth would be completely different if their orbits were fractions of a percent different than what we actually observe. Since all existing gravitational models for planet spacing only get the relative orbit spacing correct to within a few percent, and since the interplanetary relationship changes completely when varying only by fractions of a percent, it seems clear the gravitational models are missing something. 


I had my student Neha manually examine the symmetries of this Venus-Earth relation while slowly varying the orbit size of Venus, resulting inFigure 4. The horizontal axis represents various values for the orbit size of Venus. The vertical axis represents the symmetry of the resulting figure between Earth and Venus. The red marks show the actual orbit size of Venus, 0.7233 times the orbit size of Earth. The red circle surrounds the dot representing the 5-fold pattern that Venus and Earth actually make with each other. All the other blue dots are the symmetries of the patterns produced when selecting a certain value for the size of Venus' orbit around the Sun.

Figure 4 Blue dots show the rotational symmetry of the figure traced by the spatial relation between Venus and Earth by assuming a variety of different possible values for the size of the orbit of Venus, ranging from 0.6 to 0.8 times the size of Earth's orbit. The red marks show what is the actual orbit size of Venus, 0.7233 times the orbit size of Earth. The red circle surrounds the dot representing the 5-fold pattern that Venus and Earth actually make with each other.

 To make the further work faster, the patterns of orbital relations were digitize, and the Fourier transform (FFT) taken in both the angular and radial directions. If we take the FFT of the angular pattern, we will see larger intensities at the values of the symmetries - in other words, if the angular pattern has a 5-fold symmetry the FFT in the angular direction will have a peak at 5.Figure 5shows just such a FFT, in the angular direction, of the "dance" between Venus-Earth, for one particular value for the Earth and Venus orbit sizes.

Figure 5 The Fourier transform, in the rotational direction, of the orbital pattern described between Venus and Earth. This is for one value of Venus and Earth orbit size. Many of these plots are combined to visualize how the rotational symmetries depend upon the orbit size of Venus and Earth.

 We can combine many such plots, one line for each value of the orbit sizes. This will allow us to visualize how variations in orbit sizes affect the symmetries of the relations between the planets.Figure 6shows an example. In this case, the orbital size of Earth was slowly varied from .98 to 1.02 times its actual size. The "dance" pattern between Venus and Earth was computed for each value of Earth radius. The FFT was computed for the angular direction, and placed along the vertical direction. For each chosen value of Earth's radius, the bright spots above that radius value show the strong symmetries of the interplanetary figure. A red-dashed line is placed at Earth radius 1.0 (the actual value), this line crosses a bright spot at a vertical value of 5, indicating that for this value of Earth's radius, the relation between Venus-Earth has a strong 5-fold symmetry. It is surprising how many different symmetries exist for such a small variation in the orbital distance of Earth. Note the continuous vertical line at R=1.15. This means there is no repeated pattern between Venus and Earth if the Earth has this distance from the Sun.

Figure 6 Showing the rotational symmetries of the pattern created between Venus and Earth created by small variations in the size of Earth's orbit. The red-dashed line is at the actual value of Earth's radius from the Sun. This line crosses a bright spot at vertical value 5, indicating that the actual pattern between Venus-Earth has a strong 5-fold symmetry.

 Following the principle that the arrangement of planets is for the solar system as a whole, we look next at the mutual relations between three planets. Are there any combinations of spacings that produce particularly simple relationships? Looking at Jupiter-Saturn-Uranus, the patterns that exist now in our solar system are these shown inFigure 7.

Figure 7 The interplanetary orbital relations that exist in our solar system between Jupiter-Saturn, Jupiter-Uranus, and Saturn-Uranus.

We then make small variations in the orbital sizes of Jupiter and Uranus, and recompute the patterns between all three planets. This generates a large number of images, which can be put into a movie.

[I cannot get the movies to carry over from Blogger :( ]

The bottom frames are the FFT in angle (horizontal) and separation distance (vertical). If the interplanetary pattern has a strong angular symmetry, the FFT will show strong vertical lines. For reference, the actual orbit sizes are Jupiter=5.2012, Saturn=9.5856, Uranus=19.1912.

The three FFT patterns can be combined into one image, using Red for Jupiter-Saturn, Green for Saturn-Uranus, and Blue for Jupiter-Uranus. Of interest are the frames that show simultaneous strong lines in the pure RGB colors, meaning there is one particular arrangement of orbit sizes that gives all three planets simple relations. Seeing Yellow (Red+Green) means that Jupiter-Saturn and Saturn-Uranus are simply related while Jupiter-Uranus is not. Purple=Red+Blue. Cyan=Green+Blue.

Summary on small variations of orbit sizes

Small variations of orbit sizes, even less than 1%, drastically affects the symmetry of the interplanetary relationship. This has strong implications for any theory of planet spacings that looks for harmonic relations.

Ideal Spacing

An "Ideal" set of planetary spacing is specified by any particular theoretical model. For example, it is well known that an exponential spiral is a good approximation for planet spacing.

           R(n) = A * exp( B*n / TwoPi )

Where A and B are numeric constants that are different for every solar system, and n is the planet number. In our solar system, the exponential spiral

            R(n) = 0.176 * exp( 0.574*n / TwoPi )

gives a very close prediction for the orbit size for planet number n. All other solar systems studied so far also fit very close to exponential spirals, with different numeric values for A and B.

Figure 8 All solar systems have their planets' orbits approximately spaced by an exponential spiral. 

The best model for planetary spacing is theHarmonic Resonance Modelfrom Aschwanden, where the orbital times of neighboring planets are required to be in simple ratios. This elegant and quite reasonable assumption gives a very close fit to the observed orbit sizes.

Planet Pair  ( ratio of the times of orbit )
Mercury-Venus (5:2)
Venus-Earth (5:3)
Earth-Mars (2:1)
Mars-Ceres (5:2)
Ceres-Jupiter (5:2)
Jupiter-Saturn (5:2)
Saturn-Uranus (3:1)
Uranus-Neptune (2:1)
Neptune-Pluto (3:2)

However, even this best of model predicts orbital sizes that are on the order of 1% different that actual. For example, Venus is placed at 0.71 AU compared to the observed 0.723, or 2% off. We saw above that this will produce an entirely different symmetry between Earth-Venus than what is actually observed. Looking at the Earth-Venus relation, and using Aschwanden's 5:3 ratio between a year on Earth compared to a year on Venus, if the radius of Earth's orbit is 1.0, then the radius of Venus' orbit is 0.711. The figure below shows the spatio-temporal pattern created between the two. Very different from the actual solar system.

Figure 9 The pattern traced between Venus-Earth if the ratio of their periods is 5:3

Summary of Ideal Spacing

While Aschwanden's model does the best so far to predict planet spacing, even that model does not predict the actual interplanetary symmetries.

Summary and Next Steps

All known solar systems show simple symmetries of the "dances" between their planets. So far, no theoretical model predicts the orbit sizes needed to reproduce these observed simple symmetries. It is as if there is a rough rule, such as theexponential spiralor theharmonic relation of periods, but in the real world small adjustments are made to this ideal so that the symmetries between planets become simpler.

Next step is to calculate a quantity, call itharmonic measure, which will decrease when the symmetries in space and time are simpler and well defined. If two planets have no simple symmetry between their orbits, then the value ofharmonic measurewill be larger. Then, starting with one model (exponential spiral or harmonic periods) and make small variations in all the planets' orbits. Find the arrangement that will give the smallest value ofharmonic measure. 

Another next step is to pursue the growth of solar systems. The smallest protostellar cores we see have linearly spaced rings. How do these grow into the exponentially spaced ones we see.