The proposal is interesting from the TGD point of view because TGD raises the question whether stars and astrophysical objects in general could have a layered structure.

- One of the early "predictions" of TGD for stars coming from the study of what spherically symmetric metrics could look like, was that it corresponds to a spherical shell, possibly a hierarchical layered structure in which matter is condensed on shells. p-Adic length scale hierarchy suggests shells with radii coming as powers of 2
^{1/2}. - Nottale's model for planetary systems suggests Bohr orbitals for planets with gravitational Plack constant GMm/β
_{0}. The value of the velocity parameter β_{0}=v_{0}/c≤1 is from the model of Nottale about 2^{-11}for the inner planets and 1/5 times smaller for the outer planets. This might reflect the fact that originally the planets or what preceded them consisted of gravitationally dark matter or that the Sun itself consisted of gravitationally dark matter and perhaps still does so.

**1. Could harmonic oscillator model for stars and planets make sense?**

The Nottale model is especially interesting and one can look at what happens inside the Sun or planets, where the mass density is in a good approximation constant and gravitational potential is harmonic oscillator potential. Could particles be concentrated around the orbitals predicted by the Bohr model of harmonic oscillator with radii proportional to n^{1/2}, n=1,2,3,.. . The lowest state would correspond to S-wave concentrated around origin, which is not realized as Bohr orbit. The wave function has nodes and would give rise to spherical layers of matter.

One can perform the simple calculations to deduce the energy values and the radii of Bohr orbits in the gravitatational harmonic oscillator potential by using the Bohr orbit model.

- The gravitational potential energy for a particle with mass m associated with a spherical object with a constant density would be GmM(r)/r = GMmr
^{2}/R^{3}, where M is the mass of the Sun and R is the radius of the object. This is harmonic oscillator potential. - The oscillator frequency is
ω= (r

_{S}/R)^{3/2}/r_{S},where r

_{s}= 2GM is the Schwartschild radius of the object, about 3 km for the Sun and 1 cm for Earth. - The orbital radii for Bohr orbits are proportional to n
^{1/2}inside the star. By the Equivalence Principle, the radius does not depend on particle mass. One obtainsr

_{n}= n^{1/2}(2β_{0})^{-1/2}(r_{S}/R)^{1/4}× R.

Of course, one must remember that in the recent Sun and Earth ordinary matter is probably not gravitationally dark: only the particles associated with the U-shaped monopole flux tubes mediating gravitational interaction could be gravitationally dark and would play an important role in biology.

The situation could have been different when the planets formed. I have proposed a formation mechanism by an explosive generation of gravitationally dark magnetic bubbles ("mini big bangs"), which then condensed to planets (see this and this). This would explain why the value of β_0 for the Earth interior is the same as for the system formed by the interior planets and Sun. The simple calculations to be carried out that for the outer planets only the core region emerged in this way and the gravitational condensation gave rise to the layer above it. The core should have the properties of Mars in order that it could correspond to S-wave state.

The model of stars and planets as gravitational harmonic oscillators turns out to be surprisingly successful. It turns out that the radius of the core of Earth corresponds to the Bohr radius for the first orbital, which suggests that the core of Earth, and more generally of the inner planets and Mars corresponds to an S-wave ground state. For the Sun the $n=1$ S-wave orbital is 1.5 times the solar radius.

For the outer planets the first Bohr radius is larger than the radius of the planet, which suggests that they are formed by gravitational condensation of matter around the core. The wild guess is that the core has the radius of Mars. Also the rings of Jupiter (and probably also of Saturn) can be understood quantitatively, which gives strong support for the assumption that the core is Mars-like. This picture would suggest that at the fundamental level the planetary system is very simple.

** 2. Application of the oscillator model to solar system**

In this section the above simple model is applied to the solar system.

** 2.1 Oscillator model for the Sun and Earth**

Consider first the model for the Sun.

- For the Sun one has r
_{S}/R = 4.3×10^{-6}. For β_{0}=2^{-11}for the inner planets one obtains r_{1}= 1.45R so that this value of β_{0}is too small. For β_{0}=10^{-3}would give r^{1}≈ R. Solar interior would correspond to ground the S-wave concetrated around origin for β_{0}≤ 0^{-3}.β

_{0}=1 gives r_{1}=.032R, which is smaller than the radius of the solar core about .2R. β_{0}=0.026 would give r_{1}= .2R. r_{25}would be near to the solar radius. The set of the nodes of a harmonic oscillator wave function would be rather dense: at the surface of the Sun the distance between the nodes would be .1R. Note that the convective zone extends to .7R.

What about the Earth?

- One has r
_{S}= 1 cm and R= 6,378 km. At the surface of Earth β_{0}=1 is the favoured value and would give r_{1}= ≈ 151.6 km. The radius of the inner inner core is between 300 km and 400 km. n=4 would correspond to 300 km and n=7 to 400 km. β_{0}scales like (r_{1}/R_{E})^{2}. At the surface of Earth one would have n = (R_{E}/r_{1})^{2}≈1784 and the distance between two nodes would be R_{E}/2n≈1.8 km. - One can write β
_{0}(r_{1}) as β_{0}(r_{1})= (151.6/r_{1})^{2}.- For r
_{1}=3471 km, the core radius, this gives β_{0}≈1.9× 10^{-3}. - The gravitational Compton length of the Sun is one half of Earth's radius, which conforms with the Expanding Earth hypothesis, and is not far from the radius of the core. This gives β
_{0}= 2.2× 10^{-3}. - For r
_{1}=R_{E}, one has β_{0}≤5.6× 10^{-3}, which is quite near to the nominal value of β_{0}=2^{-11}for the magnetic body of the Sun, the Earth interior would correspond to the ground state S-wave concentrated around origin.β

_{0}≈ 1 should hold true above the surface of the Earth, which suggests that it characterizes the gravitational magnetic body of Earth.

- For r

**2.2 The radii of first Bohr orbits for planets modelled as gravitational harmonic oscillators?**

The above observations raise the question whether the value of β_{0} for Sun and inner/outer planets is such that both the entire Sun or its core and the cores of at least some rocky planets correspond to the ground state S-waves for the value of the gravitational Planck constant assigned with the planet. The allowed n ≥ 1 states could correspond to layers above the core.

Note that the Bohr orbital in plane corresponds to a wave function for Schrödinger equation localized to an orbital located near the orbital plane and that there are several orbitals for a given value of n. This state could have been the primordial dark matter state and the recent state could carry some information about this state.

The condition r_{1} ≤ R_{p} requires

r_{S,P}/R_{P} ≤ 4β^{2}_{0}(Sun,P) .

Using M_{e} and R_{e} as units, this condition reads for inner planets as

r_{S,P}/R_{P} < 1

and for outer planets as

r_{S,P}/R_{P} < K^{2} ,

where one has K = 1 or K = 1/5 depending on what option is assumed.

- The first option giving K = 1 assumes that the principal quantum numbers n are of the form n = 5k, k = 1,2,.. for the outer planets. This is possible although it looks somewhat un-natural.
- The second option, proposed originally by Nottale [Nottale], is β(outer) = Kβ(inner), K = 1/5.

Recall that the prediction for the radius of the first Bohr orbital is

r_{1}/R_{P} = (2β_{0})^{-1/2} < (r_s/R_{P})^{1/4} .

It is interesting to see whether the condition holds true (see this).

**2.3.1 Rocky planets**

Consider first the rocky planets, which include inner planets and Mars. For Mercury the ratio r_{1}/R_{Mars} is (R_{E}/R_{Mars})(M_{Mars}/M_{E})^{1/4}) ( r_{1}(E)/R_{E}) ≈ .388 . For Venus and Earth with nearly equal masses, which suggests that Venus has also a core of nearly the same radius, which corresponds to r_{1}≈ .36R.

For Mars, which is also a rocky outer planet, the condition for the K=1/5 option gives the value of r_{1}/R} for Mars by a scaling the value .36 for the Earth by the factor (1/K)^{1/2}× (R_{E}/R_{Mars})(M_{Mars}/M_{E})^{1/4} ≈ .931 so that one r_{1}= .33R_{Mars}. The situation for the mantle region would be very similar to that for the Earth. Note that the values of r_{1}(P)/r_{P} are rather near to each other, which suggests that all are formed by the condensation of the mantle on top of the core.

Planet | M_{P}/M_{E} |
R_{P}/R_{E} |
r_{1}/R_{P} |
---|---|---|---|

Mercury | 0.0553 | 0.383 | .39 |

Venus | 0.815 | 0.949 | .35 |

Earth | 1 | 1 | 0.36 |

Mars | 0.107 | 0.532 | .54 |

What is truly remarkable and raises hope that the proposed model has something to do with reality, that in the case of Earth r_{1} is identifiable as the core radius.

**2.3.2Giant planets**

The outer planets are gas giants apart from Mars and apart from Neptune, which is an ice giant. The following table gives the values of the radius r_{1} for the first oscillator orbit assuming K=1/5.

Planet | M_{P}/M_{E} |
R_{P}/R_{E} |
r_{1}/R_{P} |
---|---|---|---|

Jupiter | 317.8 | 11.21 | 5.16 |

Saturn | 95.2 | 9.45 | 4.0 |

Uranus | 14.5 | 4.01 | 3.1 |

Neptune | 17.1 | 3.88 | 3.2 |

For K=1/5 the values of r_{1} for the giant planets are systematically larger than the orbital radius. The reason for this is that the large value of the mass of the planet increases like R_{P}^{3} and makes ℏ_{gr} ∝ r_s/R_{P} large. For K=1, also allowed by the Nottale model,r_{1} would be replaced by .45 r_{1}. Also now r_{1}/R_{P} > 1 would be true.

What is interesting is that r_{1}/R_{P} >1 is true also for the Sun.

**2.3.3 Dwarf planets, Pluto, and some moons**

One can also estimate the values of r_{1} for some dwarf planets (Table 3) known to be promising places for the evolution of organic life and the Moon and some moons of Jupiter and Saturn.

Object | M/M_{E} |
R/R_{E} |
r_{1}/R |
---|---|---|---|

Pluto | .00218 | 0.1818 | .27 |

Eris | .0028 | .182 | .28 |

Ceres | 1.57× 10^{4} |
.2725 | .17 |

Moon | .0123 | .074 | .17 |

**2.4 Do giant planets have a shell structure for gravitational harmonic oscillator in some sense?**

The above observations give r_{1}/R_{P} >1 for the outer planets. The reason is that the large radius of the plane implies large mass and this in turn makes the gravitational Planck constant large. Should one accept that the giant planes are analogous to the ground state S-waves of the harmonic oscillator (whatever this means!) or have a layered structure suggested by the gravitational harmonic oscillator potential and they have a rocky core as an analog of the S-wave state with a size predicted by the equality? There are suggestions that giant planets could have a rocky core containing metals for which there is evidence (see this) with smaller mass.

- A natural mechanism for the formation of the giant planet would be gravitational condensation of matter from the environment around the core region.
The crucial assumption would be that the gravitational Planck constant GMm/\beta_0 is determined by the mass M

_{core}of the core region rather than the mass of the entire planet. This would reduce the value of R_{1}. - The first wild guess for the core region is as a rocky planet, either Mars or Earth. This determines the mass and radius of the core and it would correspond to the S-wave state of a gravitational harmonic oscillator with gravitational Planck constant proportional to M
_{E}or M_{M}. The n=1 harmonic oscillator orbital corresponds to the radius of the core. For definiteness let us consider Mars with K=1/5 as a guess for the core region. - The region outside the core could correspond in the first approximation to harmonic oscillator orbitals determined by the average density with radii given as r
_{n}= n^{1/2}R_{core}(P).

One can develop a more detailed model as follows.

- Newton's law for circular Bohr orbits and quantization condition for angular momentum in the gravitational potential V(R)= GmM(R)/R, where M(R) is
M(R) = M(core) + M(layer)×[(R/R

_{P})^{3}-(R_{core}/R_{P})^{3}) .Slightly below R(core) the force is harmonic force the interior R increases, the gravitational potential approaches to harmonic oscillator potential determined by M

_{P}. For outer planets the average density is considerably smaller than the density of the core. - The first condition is
v

^{2}/R= dV(R)/dR = -d(GM(R)/R)/dR = GM(R)/R^{2}-G(dM/dR)/R,where one has

dM/dR= 3R^2/R

_{P}^{3}.One obtains

v(R)

^{2}= (1/2)× (r_{S}(core)/R- 3r_{S}(layer)× (R/R_{P})^{3}). - The second condition corresponds to the quantization of the angular momentum
vR= GM(core)/β

_{0}gives for R the equation

R/R

_{E}= (r_{S}(core)/R_{E})/β_{0}v(R) .Mars is the natural choice for the core. From these data the radii of the Bohr orbits can be calculated. Near the boundary of the core the radii would go like n

^{1/2}R_{M}. For large enough radii one would obtain harmonic oscillator potential.

_{J}= 317.8M

_{E}and R

_{J}= 11.2R

_{E}≈22.4R

_{M}. The density of Jupiter is fraction .22 of the density of Earth. Most of the mass of Jupiter would be generated by the gravitational condensation of gas from the atmosphere. At least the dark matter at the gravitational magnetic body would be at the harmonic oscillator orbitals.

**2.4. Could one understand the rings of Jupiter and Saturn in terms of a gravitational analog of a hydrogen atom?**

Could one understand the rings of Saturn and Jupiter in terms of Bohr orbits with a small principal quantum number n for the gravitational analog of a hydrogen atom assuming the same gravitational Planck constant as for the interior of the planet and determined by the mass of the core?

The basic formulas for hydrogen atom generalize and one obtains that the radius of hydrogen atom as
a_{0}= ℏ/2α m_{e}, α= e^{2}/4πℏ is replaced with a_{gr}= ℏ_{gr}/2α_{gr}m, ℏ_{gr}= GM_{Mars}m/β_{0}, α_{gr}= GM_{Mm}/4πℏ_{gr}= GMm β_{0}/4π. This gives

a_{gr} =(2π/β_{0}^{2})× (r_{S,Mars}^{2}/r_{S,J}) .

Consider Jupiter as an example. By using M_{J}/M_{Mars}≈ 3178 and β_{0}≈2^{-11}/5, one obtains the estimate a_{gr}= (π/3.178)/× 10^{4} ≈ 10^{4} km. The radius of Jupiter is 7.4× 10^{4} km. a_{gr} is proportional to the square of the mass of the core. That orders of magnitude are correct, is highly encouraging. The radii of Bohr orbits are given by r_{n}=n^2a_{gr}. Could the radii for the rings correspond to n=3 Bohr orbit?

See the article A model for planets or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

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