The Solar System's Manifestation of Planck Mass and Proton Mass

2.1.1 Let us present some well-known data on the Solar System. This data is concentrated in two tables—1 and 2 (data—NASA [2, 3]). The data in the tables, as well as all calculations, are presented (performed) in the CGS system ($\si{\cm}$, $\si{\gram}$, $\si{\sec}$).

2.1.2 Table 1 shows the diameters of the planets, the average distances of the planets from the Sun, and the average velocities of the planets around the Sun.

Table 2 shows the masses of the planets and the Sun, as well as the density of these space objects.

We will denote the orbital velocity of a planet (the average velocity of a planet as it orbits the Sun) by $V_{pl}$

2.2.1 If we examine quantization in the Solar System, then first of all it is necessary to point out the peculiarity of distances in this cosmic system. This feature is described by the well-known Titius–Bode law (calculation) [4, 5]. The Titius–Bode law (calculation) has existed for more than 200 years (it was established by Titius, and then slightly modified by Bode); it considered the planets that were known at that time (1766–1772).

2.2.2 According to this law (calculation), the average radii of the orbits along which the planets move, that is, the average distances of the planets from the Sun, are quantized by a certain macro-unit. This macro-unit is $1/8$ the distance from the Sun to Mercury; it is equal to $7.24 \cdot 10^{11} \si{\cm}$.

2.2.3 Table 3 shows the above-mentioned calculations—according to Titius–Bode—of the distance of a planet from the Sun (we denote it by $L_{{\!}_R}$)

2.2.4 Note that the actual distances from the Sun to the planets Neptune and Pluto do not correspond to the Titius–Bode law (calculation).

2.3.1 Let us explore the masses of the cosmic objects that make up the Solar System. Let us assume that quantization should manifest itself not only in the distances of the planets from the Sun, but also in the masses of the planets (as well as in the mass of the Sun).

Let us exclude the influence of the sizes (radii) of the planets and the Sun on the magnitude of the masses under study. We examine not the entire volume of each space object, but only a part of the volume—$1 \si{\km^3}$ (that is, the density of the planets and the Sun).

2.3.2 If the masses of these space objects are quantized, then there must be a corresponding macro-unit of mass. After analyzing the data presented in Table 2, we found that such a macro-unit is the mass of $1\si{\km^3}$ of the volume of Saturn. This is the smallest of the unit masses (namely, the masses of volume $1\si{\km^3}$) among all considered objects of the Solar System. It is equal to $0.687 \cdot 10^{15}\si{\gram}$.

2.3.3 Let us present the results of the calculations (see table 4)

2.3.4 We state: the masses of $1 \si{\km^3}$ of the volume of each planet and the Sun are quantized. Here are the indicated masses (expressed in whole macro-units) in order of increasing their magnitude: $1,2,2,2,2,3,6,8,8,8$

3.1.1 As noted in Section 1, Planck, considering together the three fundamental physical constants (the constant $\hbar$, the gravitational constant $G$ and the speed of light in vacuum $c$), obtained an expression describing some fundamental mass [1]. This mass was called the Planck mass and was denoted by $m_{{\!}_P}$ \begin{align*}\tag{1} m_{{\!}_P}=\left(\frac{\hbar c}{G} \right)^{\frac{1}{2}} \end{align*}

3.1.2 Let us introduce a nominal, hypothetical object that we will call a graton with the following definition (within the framework of this work):

a graton is a collection of molecules and/or atoms and/or elementary particles that has a total mass equal to the Planck mass. Each such collection can be interpreted as a certain physical object.

3.2.1 Let us explore the mass of the Sun, which we will denote by $\widetilde{M}$. Let us correlate the mass $\widetilde{M}$ with mass $m_{\pi}$ \begin{align*} \frac{\widetilde{M}}{ {m_{\pi}} } = \frac{1988.5\cdot 10^{30}}{2.1767\cdot 10^{-5}} = 0.914 \cdot 10^{38} \end{align*}

3.2.2 Let us transform the result we have obtained. Let us write it like this: \begin{align*}\tag{2} \widetilde{M}=(0.956\cdot 10^{19})^2 m_{\pi} \cong (10^{19})^2 m_{\pi} \end{align*}

According to the obtained expression, the Sun can (hypothetically) be divided into ${(10^{19})}^{2}$ gratons.

3.3.1 In Section 2, we showed the quantization of the following main parameters of the Solar System:

• the distances of planets (averaged) from the Sun (Table 3);
• the masses of $1 \si{\km}^3$ of each planet, as well as the Sun (Table 4).

3.3.2 There is one more main parameter—the orbital velocity of the planet (averaged). This parameter was denoted by $V_{pl}$ in Section 2; we will study it not separately, but as part of the following complex parameter: angular momentum (corresponding to each planet)—$\bar{m}V_{pl} L_{{\!}_R}$.

3.3.3 The values of the parameters $V_{pl}$ and $L_{{\!}_R}$ are given in Table 1 ($L_{{\!}_R}$ is the average distance of the planets from the Sun). Parameter $\bar{m}$ must be selected; the mass $\bar{m}$ can be selected as:

• the mass of the entire planet (according to Table 2)—option I;
• the mass of $1 \si{\km}^3$ of the planet's volume (according to Table 4)—option II;
• the mass of a certain group of $S$ gratons—option III.

3.3.4 Options I and II cannot be accepted. The mass of the entire planet (option I) and the mass of $1 \si{\km}^3$ of the volume of each planet (option II) have different values for different planets. This means that the angular momenta of the planets will be incomparable (due to different values $\bar{m}$). Consequently, it will be impossible to establish the quantization (or lack of quantization) of the orbital velocities of the planets.

3.3.5 Let us move on to option III. A group of $S$ gratons will have the same mass for any planet (assuming $S = const$). For a fundamental solution to the problem posed, it is necessary and sufficient to consider only one graton (namely, we can assume that $S = 1$). We will examine the angular momenta of the unit corresponding to one or another planet—according to the formula \begin{align*}\tag{3} m_{\pi}V_{pl} L_{{\!}_R}; \end{align*} moreover, the nominal, hypothetical object (graton) under consideration must be located in the region of the center of mass of each planet.

3.3.6 The above calculations are presented in Table 5.

3.4.1 Let us present the numerical results written above in square brackets in an approximate form (without the repeating factor $10^{19}$): \begin{align*} 2.7; \: 3.8; \: 4.5; \: 5.5;\: 10.2;\: 13.9;\: 19.5;\: 24.3;\: 27.8 \end{align*}

3.4.2 After analyzed these approximate results, we come to the conclusion that an additional dimensionless factor—coefficient $\bm{1.08}$, should be introduced into the calculations presented in Table 5. This coefficient takes into account what is associated with the ellipticity of the orbits along which the planets move around the Sun.

3.4.3 New (refined) values of the angular momenta of the unit—using the factor $1.08$, but without specifying units, are given in Table 6.

The obtained results can be described as follows (taking into account formula (3)) \begin{align*}\tag{4} m_{\pi}(V_{pl} L_{{\!}_R})=m_{\pi}(k\cdot 10^{19}\mathrm{cm^{2}\cdot \sec^{-1}}); \end{align*} while $k$ has the following values: \begin{align*} k_{{\!}_{A}} = 3, 4, 5, 6; \quad k_{{\!}_{B}} = 11, 15, 21, 26, 30 \end{align*} We come to the following conclusion: the product $\boldsymbol{V_{pl} L_{{\!}_R}}$ is quantized.

3.4.4 We additionally state: the group of planets A is radically different from the group of planets B.

It is well known that the density of planets of group A is several times higher than the density of planets of group B (see Table 2). Table 6 shows that there is another fundamental difference—in the values of the quantized product $\bm{V_{pl} L_{{\!}_R}}$, corresponding to the planets of group A and the planets of group B.

3.5.1 It is necessary to answer the question—why is there a constant factor $10^{19}$ in all the data in Table 6? This is because the factor expresses the ratio between the mass of a graton $\bm{m_{\pi}}$ (that is, Planck mass $\boldsymbol{m_{\!_P}}$) and the mass of a proton $\bm{m_{p}}$. \begin{align*} \frac{m_{\pi}}{m_{p}} = \frac{2.1767\cdot10^{-5}}{1.6726 \cdot 10^{-24}} = 1.301\cdot 10^{19} \end{align*} \begin{align*}\tag{5} m_{\pi} \cong 1.3 \cdot 10^{19} m_{p} \end{align*}

3.5.2 Further we will examine only the group of planets A Let us introduce the relation between the masses of a graton and a proton in the data of Table 6; we obtain

A new constant factor has appeared—$\boldsymbol{{(10^{19})}^{2}}$. The same factor characterizes the ratio between the mass of the Sun and the mass of a graton (see §3.2.1) \begin{align*} \widetilde{M} / m_{\pi} \cong (10^{19})^{2} \end{align*}

3.6.1 Let us examine this factor. We present the following product \begin{align*}\tag{6} m_{\pi} \cdot m_{\pi} = (1.3\cdot 10^{19})^2 m^2_{p} \end{align*} Let us transform the obtained expression \begin{align*}\tag{7} m_{\pi}^2 = ( 10^{19})^2 (1.3\cdot m_{p})^{2} \end{align*}

3.6.2 According to Planck's formula (see §3.1.1), taking into account that $m_{\pi} \equiv m_{\!_P}$, we come to the following expressions \begin{align*}\tag{8} m_{\pi}^2 = \frac{\hbar c}{G} \end{align*} \begin{align*}\tag{9} (10^{19})^2 = \frac{\hbar c}{ (1.3 \cdot m_{p})^2 G} \end{align*}

3.7.1 By introducing formula (9) into the expressions given in §3.2.1 and §3.5.2, we obtain the following \begin{align*}\tag{10} \widetilde{M} = \frac{\hbar c}{ (1.3 \cdot m_{p})^2 G} m_{\pi} \end{align*}

The angular momentum of the unit for the group of planets A (units are not indicated on the left side):

3.7.2 We indicate the following relations (formulas) \begin{align*} \widetilde{M}=(10^{19})^2 m_{\pi}; \quad (10^{19})^2 = \frac{\widetilde{M}}{m_{\pi}};\quad m_{p}(10^{19})^2=\widetilde{M} \frac{m_{p}}{m_{\pi}} \: \end{align*}

3.7.3 All angular momenta of the unit, which are considered in §3.3.4–§3.7.1, can be written as follows (without specifying the dimensions)

3.7.4 Summarizing what has been said in this section, we come to the conclusion that the Solar System is such a collection of the star and the planets moving around this star, which is one whole cosmic object.

3.8.1 Suppose that in §3.3.4, §3.3.5 instead of $S =1$ we would take $S = 10^{8}$. In this case, formula (3) \begin{align*}\tag{11} (10^8 m_{\pi})V_{pl} L_{{\!}_R}, \end{align*} and also those expressions presented in §3.5.2 would change accordingly. In this case, the expressions given in §3.7.1 would have the following form

3.8.2 We come to the following conclusion: for any value of the number $\bm{S}$ no fundamental changes occur—provided that $\bm{S < 10^{16}}$ (instead of the factor 1 in formula (3) another factor appears; in the example above—the factor $10^{8}$).

3.9.1 Results of parameter calculation $V_{pl} L_{{\!}_R}$ (of the angular momenta of the unit) corresponding to each planet (see Table 5) are quantities having the dimension $\si{\cm^2}\cdot \si{\sec^{-1}}$. Let us move on to other dimensions, for example, express the indicated results in $\si{\meter^2}\cdot \si{\sec^{-1}}$.

Because $1\si{\meter^2}=10^4 \si{\cm^2}$, then an additional factor $10^{-4}$ will appear. Therefore, the expressions given in §3.7.1 would take the form

3.9.2 Specifying linear parameters in centimeters or meters, specifying time in seconds, means using in our calculations such systems of units that are introduced according to well-known international agreements. However, suppose the following (as an assumption):

• all linear dimensions and distances are measured, for example, in yards;
• The unit of time is taken to be the time required for photons to travel a path equal to the length of the Earth's equator.

This means that another additional factor would appear in our calculations, and not $10^{-4}$.

However, neither in the case described in §3.9.1 nor in the case described in §3.9.2 can additional factors lead to any fundamental changes in the record of calculation results. We have chosen the CGS system of units ($\si{\cm, \gram, \sec}$), so that certain (possible) additional factors are equal to 1.

4.1.1 Consider the planetary model of a hydrogen atom—proposed by E. Rutherford and developed in depth by N. Bohr [6]. In accordance with this model, we will write down the parameters of an electron, which—as Bohr assumed—moves in the atom along one of the stationary (allowed) orbits; let us call them Bohr orbits. We present the well-known Bohr formula, which singles out the indicated trajectories (orbits) from all possible trajectories of an electron's motion in a hydrogen atom \begin{align*}\tag{12} m_{e}v_{n}R_{n}=n\frac{h}{2\pi} \end{align*} Here:

• $m_{e}$ is the rest mass of an electron;
• $v_{n}$ is the speed of movement of an electron around a proton;
• $R_{n}$ is radius of the stationary (permitted) orbit;
• $n$ is principal quantum number ($n = 1, 2, 3, 4, 5, 6 \dots$).

4.1.2 Let $n = 1$; in this case, formula (12) will have the form

• $v_{{}_1} = c/137$ ($c$ is the speed of light in vacuum);
• $R_{1} \cong 5.29\cdot 10^{-9}\mathrm{cm}$ (radius of the first stationary orbit).

4.2.1 We want to show the following.

In the Solar System there is a special group of planets; in this group of planets the unity of the main structural principles is manifested, according to which two natural objects are built, radically different in their scale and in their properties:

• a physical object consisting of a proton (nucleus) and an electron, which—according to the Rutherford–Bohr model—moves around a proton (a hydrogen atom);
• a physical object consisting of a star and a special group of planets moving around the star (we will call this special group of planets a planetary atom).

4.2.2 A hydrogen atom is the main element of the microcosm. The second object—a planetary atom in the Solar System—belongs to space, to the world of stars and galaxies.

4.2.3 A rigorous description of a hydrogen atom is possible only on the basis of quantum mechanics—wave or matrix. The Rutherford–Bohr model can be considered only as a first approximation, which gives a general representation of a hydrogen atom.

4.2.4 However, in the theoretical task that we have set—to show the unity of the main structural principles of constructing two such different natural objects—it is possible (and sufficient) to use an approximate representation of a hydrogen atom.

4.3.1 As is well known, the number of manifested energy levels in atoms of chemical elements is seven: $1, 2, 3, 4, 5, 6, 7$ (these numbers are the values of the first quantum number $n$).

4.3.2 Consequently, the planetary atom must consist of a group of such planets: Mercury, Venus, Earth and Mars. The following sequence of values of the product $V_{pl}L{\!_R}$ corresponds to these planets: $3, 4, 5, 6$ (see Section 3, §3.3.5, §3.3.6, as well as tables 5 and 6).

4.4.1 Let us return to formula (14) and perform the necessary calculations for $n=1$ \begin{align*} v_1 = \frac{c}{137} = 21.898 \cdot 10^7 \si{\cm}/\sec; \quad R_1 = 5.292 \cdot 10^{-9} \si{\cm} \end{align*}

4.4.2 It follows that \begin{align*} v_1 R_1 = \frac{h}{2\pi} \frac{1}{m_e} = (21.898 \cdot 10^7) \cdot (5.292 \cdot 10^{-9}) = 1.159 (\si{\cm}^2/\sec) \end{align*}

4.4.3 Let us move on to the planetary atom}. We will describe it with the following formula \begin{align*}\tag{15} V_{pl} L_{\!_R} = v_n R_n (\frac{3}{5} \frac{m_{\pi}}{m_p}) \end{align*} Here:

• $m_\pi \equiv m_{\!_P}; \quad m_\pi / m_p = 1.301 \cdot 10^{19};$
• $3/5$ is the coefficient taking into account the gravitational interaction between the planets.

4.4.4 The results of calculating the quantity $V_{pl} L_{\!_R}$ according to formula (15) are given in Table 7

(We assume that the significant difference between the actual value of $V_{pl} L_R$ for Venus and the calculated value of $V_{pl} L_R$ is due to the fact that Venus revolves (moves around the Sun) in the direction opposite to the direction of revolution of most other planets.)

• 1. Let us present the following data.
  • the minimum possible radius of a hydrogen atom (that is, the radius of the Bohr orbit at $n = 1$); we will denote this radius by $R_{\!_H}$ \begin{align*} R_{\!_H}=5.292\cdot 10^{-9} \mathrm{cm} \end{align*}
  • The ratio of Planck mass $m_{\!_P}$ and the mass of a proton $m_{p}$ (in this case: $m_{\!_P} \equiv m_{\pi}$—see §3.1.2); according to §3.5.1 \begin{align*} m_{\pi}=1.301\cdot 10^{19}m_{\!_p}\: \end{align*}
• 2. Let us calculate the radius of the Sun, which we will denote by $R^{*}_{\odot}$. Let us calculate this radius using the following formula \begin{align*} R^{*}_{\odot}=\frac{m_{\pi}}{m_p} R_{\!_H} \end{align*} Next we get \begin{align*} R^{*}_{\odot}=1.301\cdot 10^{19} (5.292\cdot 10^{-9})=6.885 \cdot 10^{10} \si{\cm} \end{align*}
• 3. Radius of the Sun (along the equator) according to NASA [3] \begin{align*} R_{\odot}=6.957 \cdot 10^{10} \si{\cm} \end{align*} Deviation of the calculated result from the observed data \begin{align*} \Delta = \frac{R_{\odot} - R^{*}_{\odot}}{R_{\odot}}=0.0103 \end{align*}
• 4. The following should be noted:

  the dimensionless parameter $\boldsymbol{m_{\pi}/m_p}$, as well as the parameter $\boldsymbol{m_p / m_{\pi}}$, are special quantitative parameters of the Solar System.