- 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.
- 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.

**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.

- Earth size 6,371 km is not far from the gravitational Compton length of Sun GM/β
_{0}= r_{S}/2β_{0}which for β_{0}=v_{0}/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. - In the Nottale's model (see this), the value of the parameter β
_{0}=v_{0}/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 β_{0}and their distances like 1/β_{0}^{2}and one could speak of kinds of proto planets corresponding to some maximum value of β_{0}. - Using the gravitational Compton length Λ
_{gr}=GM/v_{0}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] . - 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 β_{0}/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.

- 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/v_{0}. 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. - 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.

- 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.
- 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.
- 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.

- Why the giant planets contain mostly helium?
- 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?

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

- 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.
- 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.

- 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.

- 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?

- Could the smaller value of β
_{0}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 β_{0} 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}.

- Assume that the Bohr model makes sense so that the radius of planetary orbits is given by
r

_{n}= n^{2}GM(star)4π/β_{0}^{2}. - The condition suggested by a standing wave in the radial direction
r

_{plan}= k Λ_{gr}= k GM(star)/β_{0}, 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.

- For the inner planets, one obtains the condition
r

_{orb}/r_{plan}= n^{2}4π/kβ_{0}.An appropriate generalization holds true for outer planets with different values of β

_{0}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 β_{0}<1 and k.For β

_{0}=1, this gives the lower boundr

_{orb}/r_{plan}≤ n^{2}4π/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). - 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
T

^{2}= 4π^{2}× r^{3}(orbit)/GMc^{2}.Using r

_{orb}= n^{2}(4π GM/β_{0}^{2}), one obtainsT= 8 π

^{5/2}(n^{3}/β_{0}^{3}) (r_{s}/c) .For a given period T and stellar mass M, this gives

β

_{0}= 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 β_{0}= 2.53×10^{-3}= 1.295× 2^{-9}to be compared with the estimate β_{0}= 2^{-11}for Sun. The result conforms with the idea that β_{0}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/β_{0}and the giant planet has the radius of Jupiter about 70,000 km, one has k= 2 r_{plan}β_{0}/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. - One might end up with problems with the idea of orbital expansion since the Bohr radius is given by r
_{n}=n^{2}GM(Sun)/β_{0}^{2}, 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 β_{0}.The Nottale model suggests that β

_{0}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 β_{0}is quantized as β_{0}=1/n, n integer. It must be emphasized however that outer and inner planets could also correspond to the same value of β_{0}if values of n for them come as multiples of 5. - The reduction β
_{0}→ β_{0}/5 appearing in Λ_{gr}=GM/β_{0}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. - 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 β

_{0}→ 3β_{0}for the Pluto-Sun pair at the time when Pluto originated if β_{0}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.

_{0}and k assumed to characterize planet-star pairs during the expansion.

- β
_{0}was reduced to β/5 at distance when it became impossible to realize circular Bohr orbits for β_{0}≈ 2^{-11}anymore. The radius of the planet was increased by a factor 5 and transformed an Earth-like planet to a giant planet. - 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. - 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 β_{0}→ β_{0}/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.

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