Are planets and stars gravitational harmonic oscillators?

I learned (thanks to Mark McWilliams and Grigol Asatiani) about a proposal that black-hole like stars, gravatars, could develop Russian doll-like nested structures, nestars (see this). Gravastar is a star proposed to replace blackhole. It has a thing layer of matter at horizon and de-Sitter metric in the interior. Nestar would consist of nested gravastars.

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.

1. 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.
2. Nottale’s model for planetary systems suggests Bohr orbitals for planets with gravitational Plack constant GMm/β₀. The value of the velocity parameter β₀=v₀/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.

1. The gravitational potential energy for a particle with mass m associated with a spherical object with a constant density would be GmM(r)/r = GMmr²/R³, where M is the mass of the Sun and R is the radius of the object. This is harmonic oscillator potential.
2. 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.

r_n = n^1/2 (2β₀)^-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 turns out to be surprisingly successful. The condition that the interior of the planet corresponds to an S-wave ground state with a maximal radius is satisfied for 3 inner planets and Mars. Also Mercury satisfies the inequality. For the Sun the n=1 S-wave orbital is 1.5 times the solar radius . For outer planets the conditions are not satisfied, which suggests that they are formed by gravitational condensation of matter around the core which must have the size and mass of Mars to satisfy the ground state S-wave orbital condition. 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 suggests 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.

1. For the Sun one has r_S/R = 4.3×10^-6. For β₀=2^-11 for the inner planets one obtains r₁= 1.45R so that this value of β₀ is too small. For β₀=10^-3 would give r¹≈ R. Solar interior would correspond to ground the S-wave concetrated around origin for β₀≤ 0^-3.

β₀=1 gives r₁=.032R, which is smaller than the radius of the solar core about .2R. β₀=0.026 would give r₁= .2R. r₂₅ 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?

1. One has r_S= 1 cm and R= 6,378 km. At the surface of Earth β₀=1 is the favoured value and would give r₁= ≈ 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. β₀ scales like (r₁/R_E)². At the surface of Earth one would have n = (R_E/r₁)²≈1784 and the distance between two nodes would be R_E/2n≈1.8 km.
2. One can write β₀(r₁) as β₀(r₁)= (151.6/r₁)².
   1. For r₁=3471 km, the core radius, this gives β₀≈1.9× 10^-3.
   2. 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 β₀= 2.2× 10^-3.
   3. For r₁=R_E, one has β₀≤5.6× 10^-3, which is quite near to the nominal value of β₀=2^-11 for the magnetic body of the Sun, the Earth interior would correspond to the ground state S-wave concentrated around origin.

β₀≈ 1 should hold true above the surface of the Earth, which suggests that it characterizes the gravitational magnetic body of Earth. 2.2 Do inner planets correspond to S wave ground states for gravitational harmonic oscillator?

The above observations suggest that the value of β₀ for Sun and inner is such that both the entire Sun and planets correspond to ground state S-wave states. This would indeed mean that the n=1 state corresponds to a thin layer at the surface (note that the orbital in plane is replaced by a wave function in Schrödinger equation). The large size of giant planets does not favor this hypothesis them. In any case, the condition r₁≤ R_P would predict

r_S,P/R_P≤ 4β²₀(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≤ K².

where the value of K= 1 or 1/5 depending on what option is assumed.

1. The first option assumes that the principal quantum numbers n are of the form n= 5k, k=1,2,.. for the outer planets and n= 3,4,5 for inner planets. This gives K=1. This is possible although it looks somewhat un-natural.
2. The second option, proposed originally by Nottale, is β₀(outer)= Kβ₀(inner)/5, K=1/5.

It is interesting to see whether the condition holds true (for the tables providing among other things masses and radii of planets see this).

1. Consider first the rocky planets, which include inner planets and Mars. For Mercury the ratio of this factor to that for Earth is .145 which conforms with the inequality hypothesis. For Venus and Earth with nearly equal masses and sizes the equality is true. For Mars, which is a rocky outer planet, the condition is also true for K=1/5 option. Therefore the equality holds true for 3 rocky planets.
2. The outer planets are gas giants apart from Neptune, which is an ice giant. For Jupiter , Saturn, Uranus, and Neptune, the ratios using M_E and R_E as units are 28.4, 10.4, 3.6, 4.6.The values of the core radii would be by a factor x/K too large, where x is 28.4, 10.4, 3.6, 4.6, where K is 1 or 1/5 depending on the option.

2.3 Do giant planets have a shell structure for gravitational harmonic oscil- lator in some sense?

Do the above observations imply that the giant planets have a layered structure predicted 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?

1. A natural mechanism for the formation of the giant planet would be gravitational condensation of matter from the environment around the core region.
2. The first 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. Mars with K=1/5 remains the only option.
3. 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^1/2R_core(P).

One can develop a more detailed model as follows.

1. 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.

v²/R= dV(R)/dR = -d(GM(R)/R)/dR = GM(R)/R²-G(dM/dR)/R,

where one has

dM/dR= 3R^2/R_P³ .

One obtains

v(R)²= (1/2)× (r_S(core)/R- 3r_S(layer)× (R/R_P)³).

vR= GM(core)/β₀

gives for R the equation

R/R_E= (r_S(core)/R_E)/β₀v(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^1/2R_M. For large enough radii one would obtain harmonic oscillator potential. Jupiter serves as a representative example. One has M_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 a₀= ℏ/2α m_e, α= e²/4πℏ is replaced with a_gr= ℏ_gr/2α_grm, ℏ_gr= GM_Marsm/β₀, α_gr= GM_Mm/4πℏ_gr= GMm β₀/4π. This gives

a_gr =(2π/β₀²)× (r_S,Mars²/r_S,J) .

Consider Jupiter as an example. By using M_J/M_Mars≈ 3178 and β₀≈2^-11/5, one obtains the estimate a_gr= (π/3.178)/× 10⁴ ≈ 10⁴ km. The radius of Jupiter is 7.4× 10⁴ 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 Are planets and stars gravitational harmonic oscillators?

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.