Mystery or the "radius wall" for planets as evidence for the Bohr model of planetary system

Over 5,200 exoplanets have been confirmed hitherto. Exoplanets have posed several challenges for the existing models of the formation of planets (see this).

1. An expected finding is that giant exoplanets can have very small orbital radii. In some cases with orbital periods that last just a few days. The proposed explanation is that these planets have migrated to the vicinity of their stars.
2. The second mystery is that there is a mysterious size gap in the scale of exoplanets. Transit observations first by NASA's Kepler Space Telescope and now by TESS, the Transiting Exoplanet Survey Satellite, have found a puzzling absence of planets with radii between 1.4 and 2.4 times that of Earth. Astronomers call this the "radius valley" and although it seems to be telling us something fundamental about the nature, formation and evolution of planets, scientists have yet to ascertain what that something is. What comes in mind is quantization of orbit radii.

Helium could make up almost half the mass of the atmosphere of giant exoplanets that have migrated close to their star. A team led by PhD student Isaac Malsky of the University of Michigan and Leslie Rogers of the University of Chicago proposes a new approach to the radius valley problem (see this). Perhaps it could signal an increasing abundance of helium gas in the atmosphere of planets 2.4 times larger than Earth. Planets of this scale are often described as mini-Neptunes, and if they have a rocky core, it's deep beneath a thick atmosphere. But why the abundance of helium gas would be higher?

TGD view of the planetary system

Could TGD based quantum vision of planetary system (see this, this and this), and the closely related Expanding Earth hypothesis (see this, this, this, and this) provide some insights to this problem? One can start from some observations related to the planetary sizes in the solar system.

1. Earth size 6,371 km is not far from the gravitational Compton length of Sun GM/β₀= r_S/2β₀ which for β₀=v₀/c= 2^-11 is about Λ_gr=3,000 km, which is amazingly near to half radius of Earth about r_E=6371 km. Expanding Earth model in turn proposes that the Earth radius was r_E/2 before the Cambrian Expansion and therefore roughly the same as the radii of Mercury and Mars.
2. In the Nottale's model (see this), the value of the parameter β₀=v₀/c appearing in ℏ_gr is by a factor 1/5 smaller for outer planets than for inner, Earth-like planets, including Mars. This means that the value of the gravitational Compton length is scaled up by a factor 5: Λ_gr→ 5Λ_gr. If the radius is roughly equal to a multiple of Λ_gr. The radii of planets would scale like β₀ and their distances like 1/β₀² and one could speak of kinds of proto planets corresponding to some maximum value of β₀.
3. Using the gravitational Compton length Λ_gr=GM/v₀ for the Sun as a unit, Using Mkm as a unit, the radii of the planets (see this) are given by

   [r_E= 6.371, r_Ju= 69.911, r_Ur= 25.362, r_Me=2.4397, r_Ma= 3.3893, r_Ne= 24.622,r_Sa=58.232; r_Ve=6.0518] .

   If one uses 2Λ_gr=6000 km as a unit, the radii are given by

   [r_E= 1.0618, r_Ju= 11.6518, r_Ur= 4.2270, r_Me= 0.4066, r_Ma= 0.5649, r_Ne= 4.1037, r_Sa= 9.7053 , r_Ve= 1.0086] .

4. Giant planets of the solar system come in two varieties. Jupiter and Saturn, known also as gas giants, consist primarily of hydrogen and helium and have a radius of roughly 10r_E). Uranus and Neptune, also known as ice giants, consist of ice, rock, hydrogen, and helium and have a radius nearly to 4r_E not too far from 5r_E). Gas giants are also called failed stars because their composition resembles that of young stars consisting of light elements. Helium makes roughly one half of the mass of the atmosphere.

   Remarkably, the radii of giant planets are not very far from 2Λ_gr,β0/5 and 4Λ_gr,β0/5, and would very roughly correspond to first and second octaves of solar gravitational Compton length for β₀/5 in the model of Nottale (see this). In fact, the radii of inner planets radii are not far octaves for the radius of Mars. Does this mean that the expansion by a power of 2 proposed by Expanding Earth model (see this) has occurred for all planets except Mars and Mercury?

TGD view of planet formation

The following summarizes the TGD based model for the formation of planets by dark fusion and subsequent transformation of dark nuclei to ordinary nuclei.

1. In the TGD based model (see this and this), planets could have formed by dark fusion see this,this,this) as the dark matter at the magnetic flux tubes characterized by ℏ_gr=GMm/v₀. Dark matter would have consisted of dark proton (possibly nucleon with neutron as dark proton having charged color bond with the dark proton preceding it) sequences. These dark nuclei would have transformed to ordinary matter liberating almost all nuclear binding energy in this process. This would have induced an explosion.
2. First He and possibly also heavier elements would have formed by dark fusion. The process would have involved an explosion analogous to a supernova explosion, kind of a local Big Bang. The energy would have come from the liberation of nuclear binding energy. Due to the liberation of nuclear binding energy, the process would have led to a high temperature. Ordinary nuclear fusion starts if the temperature increases above the ignition temperature of ordinary fusion. In the proposed TGD based model, this would have led to a formation of a population II star.

The simplest assumption is that ordinary nuclear fusion has not started for planets although one cannot exclude this possibility in the case of the Earth-like planets with inner core.

1. If a spherical shell of dark matter was emitted, a gravitationally induced spontaneous breaking of spherical symmetry could be in question. The flow of the matter along magnetic flux tubes of the magnetic bubble to the spot, which became a planet, would have heated it. Also Moons could be these kinds of hot spots and planetary rings. The fact that largest exoplanet HD 100546 b (see this) is accompanied by a spherical shell supports this option.
2. The quantum option, which might be too readical, is that the dark planet would not have a spherical mass shell but a quantum version of a radial jet delocalized over angular degrees of freedom as, say, angular momentum eigenstate. The formation of a planet would have been a localization in momentum space so that the wave function would have been replaced by a time dependent wave function localized at a positing describing Kepler orbit. The mass would be concentrated at the slowly increasing orbital radius. This picture would conform with the Bohr orbit model.
3. An option, which is more in line with the standard view, is that the inner core is not due to planetary dark or nuclear fusion. Rather, the dark fusion at the spherical surface would have produced matter, whichwas gravitationally attracted by the pre-existing core region.

A rough sketch for the planetary evolution

Could one understand the differences between Earth like giant planets and giant exoplanets in this framework? One must answer at least the following questions.

1. Why the giant planets contain mostly helium?
2. How giant exoplanets can have very small orbital radii in contrast to the solar giant planets? Have the giant exoplanets migrated near their stars or could some other mechanism explain their small orbital radii?

Perhaps the following rough sketch could catch some elements of truth. Suppose that the formation of planets indeed involves a local Big-Bang throwing a layer or stellar surface outwards, which is induced by the liberation of nuclear binding energy in the transformation of dark nuclei to ordinary matter after dark fusion producing dark nuclei.

The fact that outer planets are older and thrown out of Sun earlier suggests a general view of the planetary evolution.

1. The outer planets are oldest and for them the dark fusion at the surface of Sun would not have had enough time to produce dark variants of heavier elements. As the transformation to ordinary nuclei occurred in the formation of planet, only relatively light elements were produced.
2. For the Earth-like planets, dark fusion occurring at the surface of the star would have had enough time to produce a spherical layer or pre-planetary spot of dark variants of heavier elements before the explosion accompanying the transformation of the dark nuclei to ordinary nuclei, occurred.

   What would be new as compared to the standard model would be that elements like Fe of planetary inner cores would have been generated by dark fusion following by an explosion of spherical shell rather than coming from decay proecuts of supernovas and thrown out in the formation of planets at the surface of the expanding magnetic bubbles.

3. Could ordinary nuclear fusion play any role? The temperature at the surface of Sun was certainly too low for the ordinary nuclear fusion to start. If the heating induced by the transformation to ordinary nuclei was not enough to initiate ordinary fusion in the planetary core, the planet would be a failed star. Even if the ordinary fusion was initiated, the increase of the planetary radius by a process analogous to what Expanding Earth model proposes, could have made the density of the fuel too small for nuclear fusion to continue.

One should understand also the sizes of planets.

1. Why should the solar giant planets have large orbital radii? Could the radius of the planet increase in discrete steps as the model for Expanding Earth suggests? If the size increases in discrete steps, the large size could be due to the fact that the explosion from them has reached a considerably later stage for the solar system as compared to the exoplanetary systems. Could giant exoplanets with small orbital radii accompany very young stars?

   Or does the size remain constant as the existence of giant planets with very small orbital radius suggests?

2. Could the smaller value of β₀ for outer planets imply a larger radius as is suggested by the fact that giant planets have radii, which are roughly 5 and 10 times the radius of Earth?

A concrete model

Since the orbital radius of the planet correlates with the duration of expansion, outer planets would have formed before the inner planets. Planets would been emitted as magnetic bubbles containing dark matter or as quantum jets described above. Planetary systems would tell the story of planetary evolution: an astrophysical variant of the phylogeny recapitulates ontogeny principle would be realized.

To build a more concrete model, assume that the value of the parameter β₀ characterizes the Sun-planet pair. Second parameter would be an integer k characterizing the radius of planet as multiple of Λ_gr. This assumption is inspired by the observation that the planetary radii are multiples of Λ_gr≈ r_Mars.

1. Assume that the Bohr model makes sense so that the radius of planetary orbits is given by

   r_n= n²GM(star)4π/β₀² .

2. The condition suggested by a standing wave in the radial direction

   r_plan= k Λ_gr = k GM(star)/β₀ , k=1,2...

   is certainly approximate but would conform roughly with the radii of solar giants planets for k=2,4 suggesting that k is power of two as Expanding Earth model assumes. All planets except Mercury and Mars would have experienced the transition k=1→ 2.

3. For the inner planets, one obtains the condition

   r_orb/r_plan= n² 4π/kβ₀ .

   An appropriate generalization holds true for outer planets with different values of β₀ and n. The small value of r_orb and large value of r_plan for the giants with small orbital period, favors small values of n, and large values of β₀<1 and k.

   For β₀=1, this gives the lower bound

   r_orb/r_plan≤ n²4π/k .

   Note that the solar radius is r(Sun)=696.340 Mm and roughly 10 times the radius r_Ju= 69.911 Mm of Jupiter. The largest known exoplanet HD 100546 b has radius about 6.9 r_Ju and is probably a brown dwarf (this).

4. The empirical input from the very short periods of giant planets, which are a few days (see this), gives an additional condition. For a circular orbit, the period T relates to the orbital radius via Kepler's law

   T²= 4π²× r³(orbit)/GMc² .

   Using r_orb= n²(4π GM/β₀²), one obtains

   T= 8 π^5/2 (n³/β₀³) (r_s/c) .

   For a given period T and stellar mass M, this gives

   β₀= 8× 2^1/3 π^5/6 (1/n)

   n=1 is natural for the lowest Bohr orbit. For solar mass one has r_S=3 km. For T= 24 hours this would give β₀= 2.53×10^-3= 1.295× 2^-9 to be compared with the estimate β₀= 2^-11 for Sun. The result conforms with the idea that β₀ decreases gradually during the evolution of the planetary system, perhaps in powers of 1/2.

   If the radius of the planet is given by r_plan= k GM/β₀ and the giant planet has the radius of Jupiter about 70,000 km, one has k= 2 r_planβ₀/r_S ≈ 59. In this case the planet could be regarded as a brown dwarf (see this), which had too low mass to reach the temperature making possible nuclear fusion.

5. One might end up with problems with the idea of orbital expansion since the Bohr radius is given by r_n=n²GM(Sun)/β₀², where n is the principal quantum number n. n should be small for a giant exoplanet with very small orbital radius. Too small orbital radii are not however possible for a given value of β₀.

   The Nottale model suggests that β₀ is dynamical, quantized, and decreases in discrete steps during the expansion for some critical values orbital radius so that also r_plan increases for certain critical values of r_orb. I have earlier developed an argument that β₀ is quantized as β₀=1/n, n integer. It must be emphasized however that outer and inner planets could also correspond to the same value of β₀ if values of n for them come as multiples of 5.

6. The reduction β₀→ β₀/5 appearing in Λ_gr=GM/β₀ appearing in the formula for r_plan would induce the increase of the planetary radius.

   Does value of the parameter k need change during the orbital expansion? The existence of giant planets with very small orbital radii would conform with the assumption that the value of k does not change during evolution. On the other hand, the idea that planets should participate cosmic expansion in discrete jerks and the observation that the radii of planets are roughly power of 2 multiples of Λ_gr≈ r_Mars, suggest that k can increase in discrete steps coming as power of 2.

7. The former planet Pluto (see this) is the largest object in the Kuiper belt, which has a torus-like shape. The radius of Pluto is 1,191 km to be compared with Λ_gr= 3,000 km and to the radius 2,439 km of Mercury.

   The assumption that Pluto is a planet of solar origin requires β₀ → 3β₀ for the Pluto-Sun pair at the time when Pluto originated if β₀ has remained unchanged during its evolution. This does not conform with the proposed model.

   Could the Kuiper belt (see this), composed of miniplanets, be analogous to a planetary ring, and be the oldest structure emanating from the Sun by the proposed mechanism? The total mass of Kuiper belt is recently about 10 per cent of the mass of Earth but there are reasons to believe that the original material has been 7 to 10 Earth masses so that Kuiper belt could be perhaps seen as a failed Jupiter sized giant planet for which the transformation of dark matter to ordinary matter did not lead to a single planet but to a large number of smaller objects.

These considerations suggest a simple model for the evolution of the parameters β₀ and k assumed to characterize planet-star pairs during the expansion.

1. β₀ was reduced to β/5 at distance when it became impossible to realize circular Bohr orbits for β₀≈ 2^-11 anymore. The radius of the planet was increased by a factor 5 and transformed an Earth-like planet to a giant planet.
2. The radii of Jupiter and Saturn would have been roughly 2r_E before this and the radii of Uranus and Neptune would have been roughly r_E. Mercurius and Mars would have had a radius not far from r_E/2. p-Adic length scale hypothesis is suggestive.
3. The increase of k is consistent with the Expanding Earth model involving the increase of Earth radius by a factor k=2.

   Expanding Earth model (see this) and the fact that Λ_gr is roughly r_E/2 ≈ r_Mars suggests an even simpler model. Outer planets have suffered the transition β₀→ β₀/5. Jupiter and Saturn with a radius about 20Λ_gr have also suffered two scalings k=1→ 2→ 4. The remaining planets except Mars and Mercury have suffered the scaling k=1→ 2. In the simplest model, the solar proto planet would have a radius roughly that of Mars and Mercury.

See the article Magnetic Bubbles in TGD Universe: Part I or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.