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# A Method for Empirical Estimation of Planck's Length, Mass, and Time through the Characteristics of an Electron. Improving the Accuracy of Some Physical Constants

Valery Timkov

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Valery Timkov. A Method for Empirical Estimation of Planck's Length, Mass, and Time through the Characteristics of an Electron. Improving the Accuracy of Some Physical Constants. 2023. hal-04124315

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# A Method for Empirical Estimation of Planck's Length, Mass, and Time through the Characteristics of an Electron. Improving the Accuracy of Some Physical Constants.

Valery F. Timkov

Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine, 03186, Kyiv, 25, Chokolivskiy bulv., ap.13, Ukraine.

valerytimkov@gmail.com

## ABSTRACT

A method for empirical estimation of Planck's length, mass, and time is proposed, which is based on the characteristics of the electron, Avogadro, and Euler numbers, and the fine structure constant. Basic physical constants can be expressed in terms of Planck's length, mass, and time. The disadvantage of this method is that Planck's elementary particle does not exist in nature. Planck's particle is presented in a hypothetical, virtual form, its characteristics are theoretically calculated through the reduced Planck's constant, Newton's gravitation constant, and the constant speed of light in a vacuum, and the accuracy of these characteristics is low. This is due to the low accuracy of the Newtonian constant of gravity. This shortcoming can be eliminated if the characteristics of the hypothetical Planck particle are associated with a real elementary particle, for example, with an electron, and through its characteristics with the characteristics of leptons and baryons. Since the characteristics of leptons and baryons are determined experimentally and are among the most accurate, for example, for a proton now is 11 decimal places, establishing a connection with them of a hypothetical Planck particle will increase the accuracy of the values: of Planck's length, mass and time, of Planck constant, of elementary electric charge, of the Newtonian gravitational constant, of electron mass, of Planck temperature up to the accuracy level of a proton.

**Keywords:** Planck's length, mass, and time, the accuracy of physical constants, Euler number, Planck's Universal Proportions, gravitational quantization step.

## Introduction

The structure of knowledge about the surrounding physical world consists of laws, constants, initial conditions, limits, restrictions, and thresholds. Physical constants play a special role in this structure. The laws of nature can be conditionally represented as a composition of their dynamic and static parts. The dynamic part of the laws is represented by the variables of these laws, and the static part is represented by physical constants. The static part of the laws of nature reflects the
structure of matter and the dynamic part of its movement. Consequently, the nature of physical constants consists in the numerical display of individual elements of the structure of matter at its structural level, at which the scenario of the action of a physical law unfolds. The totality of all existing physical constants forms a system of knowledge about the numerical values of the basic elements of the structure of matter. Physical constants have a set of specific characteristics. These characteristics of physical constants are discussed and described, for example, in [1, 2, 3, 4, 5, 6, 7, 8, 9]. It follows from these publications that the main characteristics of physical constants are: universality, invariance, stability, experimental verification and experimental confirmation of numerical values, and accuracy. When using physical constants in practice, the characteristics of stability, experimental measurements of their values, and accuracy are of the greatest interest. Stability and non-variability in time of the values of physical constants are directly related to their nature. A physical constant is stable to the extent that the structure of matter represented by this constant is stable. The problem of stability of physical constants is studied in publications [4, 9, 10, 11, 12, 13, 14, 15]. It follows from the publications that the greatest attention is paid to the study of the stability of constants: the speed of light in a vacuum, the fine structure constant, the ratio of the proton and electron masses, the elementary electric charge, and the gravitational constant. It is noted that there is a possibility of a certain instability of these constants in time. For a confident assessment of the instability of the constants, it is necessary to take into account the observation conditions in the experiments and increase the accuracy and stability of the measuring equipment. The measurement of physical constants, their dimension, and increasing the accuracy of values are studied in publications [16, 17, 18, 19, 20, 21, 22, 23, 24]. Almost all constants that are included in the CODATA listing [25] are measured. The greatest attention is paid to the experimental evaluation of numerical values: Newtonian constant of gravitation G, fine-structure constant α, elementary charge q, speed of light in vacuum c, Planck constant h, Boltzmann constant k, electron mass m_e, proton mass m_p, Compton wavelength λ_C. To improve the accuracy of physical constants, two directions are used. The main direction is realized through the experiment, with the help of improving its methodology, carefully taking into account the conditions of observation, and improving the measuring equipment. An auxiliary direction is analytical when the analytical formula of a physical constant with low accuracy is known, which is expressed in terms of constants with higher accuracy. For example, let's increase the accuracy of the value of the electron mass constant m_e from 10 to 11 decimal places. The following formula is presented in [21]: the moments of mass of a hypothetical Planck particle, as well as leptons and baryons, expressed in terms of their Compton wavelength (in particular, over 2pi), are equal to each other:

$$m_p \cdot l_p = m_e \cdot \lambda_C = m_\mu \cdot \lambda_{C\mu} = m_\tau \cdot \lambda_{C\tau} = m_{pr} \cdot \lambda_{Cpr} = m_n \cdot \lambda_{Cn}, \quad (1)$$
where respectively: $m_p, m_e, m_\mu, m_\tau, m_{pr}, m_n$, – this is the Planck mass, as well as the masses:

electron, muon, tau, proton, neutron; $l_p, \lambda_C, \lambda_{C\mu}, \lambda_{C\tau}, \lambda_{Cpr}, \lambda_{Cn}$, – this is the Planck length, as well as the Compton wavelength over 2pi: electron, muon, tau, proton, neutron.

In formula (1), the proton has the highest accuracy and the smallest value of Uncertainty, therefore, for the condition of the task, formula (1) has the form:

$$m_e \cdot \lambda_C = m_{pr} \cdot \lambda_{Cpr}. \qquad (2)$$

From:

$$m_e = \frac{m_{pr} \cdot \lambda_{Cpr}}{\lambda_C}.$$

Then, according to CODATA 2018 [25] and taking into account the NIST Uncertainty Machine [26]:

$$m_e = \frac{m_{pr} \cdot \lambda_{Cpr}}{\lambda_C} = 9.10938370154 \cdot 10^{-31} (3.92 \cdot 10^{-40}) \text{kg}. \qquad (3)$$

The accuracy value of the Newtonian constant of gravitation $G$ is one of the lowest among all fundamental constants. The Newtonian constant of gravitation in 1969 [24] was:

$G = 6.6732(31) \cdot 10^{-11} m^3 kg^{-1} s^{-2}$. Now [25] Newtonian constant of gravitation has the value:

$G = 6.67430(31) \cdot 10^{-11} m^3 kg^{-1} s^{-2}$. In the process of measurements from 1969 to the present, the value of the Newtonian constant of gravitation has changed significantly, and the accuracy of its value has increased by only one order of magnitude. This is the basis for this constant to change the priority of methods for estimating its values, that is, the main method for estimating the Newtonian constant of gravitation should be considered analytical, which makes it possible to increase the accuracy of measuring Newtonian constant of gravitation from 5 to 11 decimal places.

Physical constants can be divided into dimensionless and having dimensions. Constants having dimensions are mainly represented in units of electromagnetism or units of gravity. Since the electric charge is a function of the moment of mass, and the Coulomb law can be represented in the gravitational form [27], the electromagnetic field is a special case of the gravitational field. This means that all the basic units of measurement of electromagnetism can be represented in units of gravity: meter, kilogram, and second. Based on the concepts: 1) electric charge is a function of the moment of mass, 2) the moments of mass of leptons and baryons, defined through their Compton wavelength over 2pi, are equal to each other and equal to the moment of mass of a hypothetical Planck particle, it is possible to improve the accuracy of some physical constants, for example, constant fine structure, elementary electric charge, electron mass, Bohr magneton.

This accuracy value corresponds to the accuracy of the characteristics of the electron and proton, that is, 11 decimal places. Improving the accuracy of measurements of the characteristics of the electron and proton will increase the accuracy of the above physical constants.
The idea of a connection between the characteristics of a hypothetical Planck particle and a real elementary particle is presented in [28]. The article proposes to connect the characteristics of the hypothetical Planck particle with the characteristics of the muon through the golden algebraic fractal.

## A Method for Empirical Estimation of Planck's Length, Mass, and Time through the Characteristics of an Electron

Let us estimate the characteristics of one mole of the hypothetical Planck particles, and then compare them with the characteristics of an electron. The physical meaning of the proposed method will not change if in its algorithm instead of the Avogadro constant: $N_A = 6.02214076 \cdot 10^{23} mol^{-1}$ we will use the dimensionless Avogadro number: $N_A^d = 6.02214076 \cdot 10^{23}$, meaning by this number the number of structural elements of matter in one mole. Planck length of one mole of hypothetical Planck particles $l_p^m$, with [25]:

$$l_p^m = l_p \cdot N_A^d = 9.733315 \dots \cdot 10^{-12} m, \qquad (4)$$

where: $l_p = 1.616\,255(18) \cdot 10^{-35} m$ is the Planck length [25].

Let us estimate how many times A the Bohr radius $a_0$ [25] is greater than the Planck length of one mole of hypothetical Planck particles:

$$A = \frac{a_0}{l_p^m} = \frac{a_0}{l_p \cdot N_A^d} = \frac{5.29177210903 \cdot 10^{-11}}{9.733315 \cdot 10^{-12}} = 5.436762343582 \qquad (5)$$

The number A is close to twice Euler's number. The relative error of non-coincidence of their values is: 0.0037%. Let the number A be equal to twice the value of the Euler number, then:

$$A = 2 \cdot e = \frac{a_0}{l_p \cdot N_A^d} = 5,436563656918090 \dots \qquad (6)$$

Whence the Planck length, taking into account [26], is equal to:

$$l_p = \frac{a_0}{2 \cdot e \cdot N_A^d} = 1.61631406825 \cdot 10^{-35} (2.44 \cdot 10^{-45}) \, m. \qquad (7)$$

Let's estimate the mass of one mole of electrons $m_e^m$:

$$m_e^m = m_e \cdot N_A^d = 5.48579908873 \cdot 10^{-7} \, kg,$$

where $m_e = 9.1093837015(28) \cdot 10^{-31} kg$ is the electron mass [25]. Let us estimate how many times A1 the mass $m_e^m$ is greater than the Planck mass $m_p = 2.176434(24) \cdot 10^{-8} kg$ [25]:

$$A1 = \frac{m_e^m}{m_p} = \frac{m_e \cdot N_A^d}{m_p} = 25.20544656409650 \dots \qquad (8)$$
The number $A1$ is close to the value of the number: $\frac{1}{2 \cdot e \cdot \alpha}$. The relative error of non-coincidence of their values is: 0.0037%. Let:

$$A1 = \frac{1}{2 \cdot e \cdot \alpha}. \qquad (9)$$

Then, taking into account formula (8):

$$m_p = 2 \cdot e \cdot N_A^d \cdot \alpha \cdot m_e. \qquad (10)$$

Let's increase the accuracy of the fine-structure constant $\alpha$ from 10 to 11 decimal places. The value of the fine-structure constant [21] in terms of the Bohr radius $a_0$ and the Rydberg constant $R_\infty$ [25, 26]:

$$\alpha = 4 \cdot \pi \cdot a_0 \cdot R_\infty = 7.29735256929 \cdot 10^{-3} (1.1 \cdot 10^{-12}). \qquad (11)$$

Taking into account formulas (3) and (11), as well as [25, 26], the Planck mass has the value:

$$m_p = 2 \cdot e \cdot N_A^d \cdot \alpha \cdot m_e = 2.17635483774 \cdot 10^{-8} (9.92 \cdot 10^{-18}) \text{ kg}. \qquad (12)$$

Let us estimate the quantum $A2$ of the atomic unit of time $t_{au}$ [25] of one mole of electrons:

$$A2 = \frac{t_{au}}{N_A^d} = 4.0166519232700 \dots 10^{-41} \text{ s}. \qquad (13)$$

The value of quantum $A2$ is approximately in $(\frac{2 \cdot e}{\alpha})$ times the Planck time $t_p$ [25]. The relative error of non-coincidence of their values is: 0.0037%. Let's replace the attribute approximately with the attribute exactly, then, taking into account the formula (11) and [25, 26]:

$$t_p = \frac{t_{au} \cdot \alpha}{2 \cdot e \cdot N_A^d} = 5.39144339732 \cdot 10^{-44} (8.13 \cdot 10^{-54}) \text{ s}. \qquad (14)$$

We denote the dimensionless constant $(2 \cdot e \cdot N_A^d)$ as $N_{AE}$ and call it the Avogadro-Euler constant:

$$N_{AE} = 2 \cdot e \cdot N_A^d = 3.27397515926610 \dots 10^{24} \text{ (exact)}. \qquad (15)$$

Then formulas (7, 12, 14) can be represented as:

$$l_p = \frac{a_0}{N_{AE}} = 1.61631406825 \cdot 10^{-35} (2.44 \cdot 10^{-45}) \text{ m}. \qquad (16)$$

$$m_p = N_{AE} \cdot \alpha \cdot m_e = 2.17635483774 \cdot 10^{-8} (9.92 \cdot 10^{-18}) \text{ kg}. \qquad (17)$$

$$t_p = \frac{t_{au} \cdot \alpha}{N_{AE}} = 5.39144339732 \cdot 10^{-44} (8.13 \cdot 10^{-54}) \text{ s}. \qquad (18)$$

Partial verification of formulas (16, 17, 18). In accordance with formula (1) and taking into account the most accurate characteristics of a proton, it is true:

$$m_p \cdot l_p = m_{pr} \cdot \lambda_{Cpr}. \qquad (19)$$

According to CODATA 2018 [25]:
$$m_p \cdot l_p = 3.51767233467 \cdot 10^{-43} \text{ kg} \cdot \text{m}^1. \qquad (20)$$

$$m_{pr} \cdot \lambda_{Cpr} = 3.51767294175 \cdot 10^{-43} \text{ kg} \cdot \text{m}^1. \qquad (21)$$

The relative error of non-coincidence of the result of formulas (20) and (21) is: 0.000017%.

The moment of mass of a hypothetical Planck particle according to formulas (16, 17):

$$m_p \cdot l_p = \left( \frac{a_0}{N_{AE}} \right) \cdot (N_{AE} \cdot \alpha \cdot m_e) = m_e \cdot \lambda_C = 3.51767294176 \cdot 10^{-43} \text{ kg} \cdot \text{m}^1. \qquad (22)$$

The relative error of non-coincidence of the result of formulas (21) and (22) is: 0.0000000002%.

Speed of light in a vacuum according to CODATA 2018 [25]: $c = 299792458 \text{ m}^1 \text{s}^{-1}$.

Speed of light in a vacuum in terms of Planck length and Planck time according to CODATA 2018:

$$c = \frac{l_p}{t_p} = 299792423 \text{ m}^1 \text{s}^{-1}. \qquad (23)$$

The relative error of non-coincidence of the result is: 0.000012%. The speed of light in a vacuum according to formulas (16) and (18):

$$c = \frac{l_p}{t_p} = \left( \frac{a_0}{N_{AE}} \right) \cdot \left( \frac{t_{au} \cdot \alpha}{N_{AE}} \right) = \frac{a_0}{t_{au} \cdot \alpha} = 299792458 \text{ m}^1 \text{s}^{-1}. \qquad (24)$$

The match of the result is 100%.

Tests have shown that the proposed estimates for Planck's length, mass, and time are more efficient than the estimates provided in CODATA 2018.

## Improving the Accuracy of Some Physical Constants

It is known [27] that the electric charge $q$ is a function of the moment of mass:

$$q^2 = 10^7 \cdot \alpha \cdot m_p \cdot l_p. \qquad (25)$$

Then, taking into account formulas (11) and (19), as well as [25, 26]:

$$q = \sqrt{10^7 \cdot \alpha \cdot m_{pr} \cdot \lambda_{Cpr}} = 1.60217663444 \cdot 10^{-19} (2.51 \cdot 10^{-29}) C, [\text{kg} \cdot \text{m}^2]. \qquad (26)$$

Elementary electric charge according to CODATA 2018 [25]: $q = 1.602176634 \cdot 10^{-19} C$ (exact).

The accuracy of the value of the elementary electric charge has been increased by 2 orders of magnitude, and the exact attribute should be removed. Planck constant $h$, taking into account formula (19), as well as [25, 26]:

$$h = 2 \cdot \pi \cdot m_p \cdot l_p \cdot c = 2 \cdot \pi \cdot m_{pr} \cdot \lambda_{Cpr} \cdot c = 6.62607015001 \cdot 10^{-34} (2.85 \cdot 10^{-43}) \text{ J}^1 \text{s}^1. \qquad (27)$$

Planck constant according to CODATA 2018 [25]: $h = 6.62607015 \cdot 10^{-34} \text{ J}^1 \text{s}^1$ (exact).
The accuracy of the value of Planck's constant has been increased by 3 orders of magnitude, and the exact attribute must be removed. Let us estimate the auxiliary constants. Estimation of the linear density of the Planck mass $\rho_{Pl}$, taking into account formulas (16) and (17):

$$ \rho_{Pl} = \frac{m_p}{l_p} = 1.34649254157 \cdot 10^{27} (6.14 \cdot 10^{17}) \text{ kg} \cdot \text{m}^{-1}. \quad (28) $$

Estimation of the linear rarefaction of the Planck mass $\rho_{Pl}^{-1}$, taking into account formulas (16) and (17):

$$ \rho_{Pl}^{-1} = \frac{1}{\rho_{Pl}} = \frac{l_p}{m_p} = 7.42670285296 \cdot 10^{-28} (1.12 \cdot 10^{-37}) \text{ kg} \cdot \text{m}^{-1}. \quad (29) $$

Estimation of the Planck charge $q_p$, taking into account formula (25):

$$ q_p = \frac{q}{\sqrt{\alpha}} = 1.87554603829 \cdot 10^{-18} (3.26 \cdot 10^{-28}) C, \quad [\text{kg} \cdot \text{m}^2]. \quad (30) $$

Estimation of the Planck acceleration $a_p$, taking into account formulas (16) and (18):

$$ a_p = \frac{l_p}{t_p^2} = 5.56052314579 \cdot 10^{51} (1.88 \cdot 10^{42}) \text{ m} \cdot \text{s}^{-2}. \quad (31) $$

Estimation of the static moment of the Planck mass $J_{Ps}$, taking into account formula (19):

$$ J_{Ps} = m_p \cdot l_p = m_{pr} \cdot \lambda_{Cpr} = 3.51767294175 \cdot 10^{-43} (1.07 \cdot 10^{-52}) \text{kg} \cdot \text{m}^{-1}. \quad (32) $$

Estimation of the dynamic moment of the Planck mass, that is, the moment of motion of the Planck mass $J_{Pm}$, taking into account formula (17):

$$ J_{Pm} = m_p \cdot c = 6.52454766286 (2.97 \cdot 10^{-9}) \text{kg} \cdot \text{m} \cdot \text{s}^{-1}. \quad (33) $$

Estimation of the Planck force $F_p$:

$$ F_p = m_p \cdot \frac{l_p}{t_p^2} = m_p \cdot a_p = 1.21016714487 \cdot 10^{44} (6.87 \cdot 10^{34}) \text{ kg} \cdot \text{m} \cdot \text{s}^{-2}. \quad (34) $$

Estimation of the Planck energy $E_p$, taking into account formulas (16, 17, 18, 26):

$$ E_p = m_p \cdot c^2 = 1956010181.19(0.892)J, \quad (35) $$

$$ E_p = \frac{\hbar}{t_p} = 1956010181.19(0.595)J, \quad (36) $$

$$ E_p = \frac{q_p^2 \cdot c}{10^7 \cdot t_p} = 1956010181.19(0.679)J, \quad (37) $$

where $\hbar$ is the reduced Planck constant, that is, is the Dirac constant.

Taking into account formulas (35-37), as well as [25,26], we estimate the value of the Planck temperature $T_p$:
$$T_p = \frac{E_p}{k} = 1.41673240714 \cdot 10^{32} (4.31 \cdot 10^{22}) K, \qquad (38)$$

where k is the Boltzmann constant.

Planck temperature according to CODATA 2018 [25]: $T_p = 1.416784 \cdot 10^{32} K$.

The relative error of non-coincidence of values with formula (38) is: 0.0036%.

Let's estimate the value of the Newtonian constant of gravitation G:

$$G = \frac{\hbar \cdot c}{m_p^2} = 6,67478765001 \cdot 10^{-11} (2.87 \cdot 10^{-20}) \text{ kg}^{-1} \text{m}^3 \text{s}^{-2}. \qquad (39)$$

$$G = \frac{l_p^2 \cdot c^3}{\hbar} = 6.67478765001 \cdot 10^{-11} (3.51 \cdot 10^{-20}) \text{ kg}^{-1} \text{m}^3 \text{s}^{-2}, \qquad (40)$$

$$G = \frac{t_p^2 \cdot c^5}{\hbar} = 6,67478765001 \cdot 10^{-11} (3.5 \cdot 10^{-20}) \text{ kg}^{-1} \text{m}^3 \text{s}^{-2}, \qquad (41)$$

The Newtonian constant of gravity according to CODATA 2018 [25]:

$$G = 6.67430 \cdot 10^{-11} \text{ m}^3 \text{kg}^{-1} \text{s}^{-2}.$$

The accuracy of the Newtonian constant of gravitation increased by 6 orders of magnitude.

## Estimation of the Newtonian Constant of Gravitation Based on Experimental Data

Let's check the Newtonian constant of gravitation estimate based on experimental data. It was shown in [29] that the gravitational field has a quantum wave character. Any gravitating object of the observable Universe, like the Hubble spheres, including the Hubble sphere itself, creates a standing gravitational wave in space-time. If we represent this gravitating object as a material point, then the structure of the gravitational field around this point can be represented as a set of nested spherical equipotential surfaces centered at this material point and separated from each other by the length of a standing gravitational wave, which is the quantization step. The length of the quantization step is proportional to the mass of the gravitating object, and the coefficient of proportionality is equal to the linear rarefaction of the Planck mass. The quantum of the gravitational field is the square of the speed of light in a vacuum, and the physical nature of the quantum of the gravitational field is the kinematic gravitational viscosity of the gravitational field of the observable Universe. The gravitational field has two fundamental properties: 1) it bends space-time around any gravitating object, 2) it has a kinematic gravitational viscosity, due to which there is a finite rate of transmission of physical interaction in nature - the speed of light in vacuum. The main proof of the quantum-wave nature of the gravitational field is the presence in nature of a gravitational-electromagnetic resonance at the frequency of a standing gravitational wave, which is shown in the experiments presented in [30, 31, 32]. In accordance with the concept of the quantum-wave nature of the gravitational field, any gravitating object of the observable Universe, like the
Hubble spheres, including the Universe itself, has three self-sufficient and interrelated characteristics: mass *m*, quantization step *S<sub>q</sub>*<sub>q</sub>, delay time for the propagation of a light signal over a distance equal to the quantization step *t<sub>d</sub>*<sub>d</sub>. Then, if *M<sub>U</sub>*<sub>U</sub> this is the mass of the baryonic matter of the observable Universe (the Hubble sphere), *R<sub>U</sub>*<sub>U</sub> this is the radius of the Hubble sphere, *T<sub>U</sub>*<sub>U</sub> this is the Hubble time, that is: *T<sub>U</sub>*<sub>U</sub> = 1/H<sub>0</sub>, where *H<sub>0</sub>*<sub>0</sub> is the Hubble constant, then, in accordance with [30], the following formulas are true:

$$E = m \cdot c^2; E = F_p \cdot S_q; E = \frac{E_p}{t_p} \cdot t_d; E_U = M_U \cdot c^2; E_U = F_p \cdot R_U; E_U = \frac{E_p}{t_p} \cdot T_U, \quad (42)$$

where *E* is the energy of any gravitating object, *E<sub>U</sub>*<sub>U</sub> is the energy of the baryonic matter of the observable Universe.

Based on the concept of Planck's Universal Proportions [30], the following formulas are true:

$$\rho_{Pl} = \frac{m_p}{l_p} = \frac{m}{S_q} = \frac{M_U}{R_U}; c = \frac{l_p}{t_p} = \frac{S_q}{t_d} = \frac{R_U}{T_U}; \frac{m_p}{t_p} = \frac{m}{t_d} = \frac{M_U}{T_U} = M_U \cdot H_0. \quad (43)$$

It follows from formulas (43) that by knowing the values of the proportional ratios of Planck's length, mass, and time and experimentally determining one of the three characteristics of a gravitating object, one can estimate the other two unknown characteristics.

Taking into account formula (43), the formulas are true:

$$F_p = m_p \cdot \frac{l_p}{t_p^2} = m \cdot \frac{S_q}{t_d^2} = M_U \cdot \frac{R_U}{T_U^2} = \left( m_p \cdot \frac{l_p}{t_p^2} \right) \cdot \frac{l_p}{l_p} = \rho_{Pl} \cdot c^2, \quad (44)$$

$$G = \frac{l_p^3}{m_p \cdot t_p^3} = \frac{S_q^3}{m \cdot t_d^2} = \frac{R_U^3}{M_U \cdot T_U^2} = \rho_{Pl}^{-1} \cdot c^2. \quad (45)$$

In [30], the results of an experiment on estimating the parameters of the Earth's gravitational-electromagnetic resonance, namely, the quantization step of the Earth's gravitational field are given: *S<sub>q⊕</sub>*<sub>q⊕</sub> = 0.00443474*m*. Using the estimation of the value of the Earth quantization step, let's estimate its effective radius *R<sub>e⊕</sub>*<sub>e⊕</sub>, corresponding to the standard acceleration of gravity on the Earth *g<sub>⊕</sub>*<sub>⊕</sub>.

It is known that the mass of the Earth can be estimated through the known values of the standard acceleration of gravity and the effective radius:

$$M_{\oplus} = \frac{g_{\oplus} \cdot R_{e\oplus}^2}{G} = \frac{g_{\oplus} \cdot R_{e\oplus}^2 \cdot M_{\oplus}}{S_{q\oplus} \cdot c^2}. \quad (46)$$

From:

$$R_{e\oplus} = \sqrt{\frac{S_{q\oplus} \cdot c^2}{g_{\oplus}}}. \quad (47)$$
It follows from formula (47) that the effective radius of the Earth and the standard acceleration of gravity do not directly depend on the mass of the Earth, but depend on the quantization step of its gravitational field. According to CODATA 2018: $g_⊕ = 9.80665 m^1 s^{-2}$ (exact). Then:

$$R_{e⊕} = \sqrt{\frac{S_{q⊕} \cdot c^2}{g_⊕}} = 6.375209290... \cdot 10^6 m. \quad (48)$$

The value of the effective radius of the Earth was obtained by calculation on the basis of experimental data on the standard acceleration of gravity and the quantization step of the Earth's gravitational field. In [33], the value of the Earth's mass is given: $M_⊕ = 5.9722 \cdot 10^{24} kg$.

Then the estimate of the value of the Newtonian constant of gravitation is:

$$G = \frac{g_⊕ \cdot R_{e⊕}^2}{M_⊕} = 6.673831320 \cdot 10^{-11} m^3 kg^{-1} s^{-2}. \quad (49)$$

The values of the standard acceleration of gravity and the effective radius of the Earth were established through the experiment, and the mass of the Earth was determined by calculation through the Newtonian constant of gravitation, which means that in the formula (49) there should be a result in accordance with CODATA 2018: $G = 6.67430 \cdot 10^{-11} m^3 kg^{-1} s^{-2}$.

But this result does not exist. Therefore, both the mass of the Earth according to [33] and the value of the Newtonian constant of gravitation according to CODATA 2018 have erroneous estimates. Estimating the mass of the Earth through Planck's Universal Proportions:

$$M_⊕ = \rho_{Pl} \cdot S_{q⊕} = 5.971344 \cdot 10^{24} kg. \quad (50)$$

To estimate the mass of the Earth using formula (50), estimate the value of the Newtonian constant of gravitation:

$$G = \frac{g_⊕ \cdot R_{e⊕}^2}{M_⊕} = 6,67478765003 \cdot 10^{-11} m^3 kg^{-1} s^{-2}. \quad (51)$$

The value of the Newtonian constant of gravitation according to formula (51) is close to its value according to formulas (39 - 41).

## Conclusions

A method for empirical estimation of the Planck length, mass, and time based on the characteristics of an electron is proposed. This method relates the characteristics of a hypothetical Planck particle with the characteristics of leptons and baryons, which makes it possible to improve the accuracy of the following physical constants: Planck's length, mass, and time, Newtonian constant of gravitation, Planck's constant, fine structure constant, elementary electric charge, electron mass, temperature Planck, as well as all physical constants that can be expressed analytically in terms of Planck's length, mass, and time, for example, the Hartree energy, the Bohr magneton, the von
Klitzing constant. With an increase in the accuracy of the characteristics of the electron and proton, the accuracy of all the above physical constants will be increased accordingly.

**CRediT authorship contribution statement**

The author confirms that all stages of preparation of this paper: Idea, Conceptualization, Methodology, Supervision, Writing & Review & Editing Original Draft, Data Curation, Software, Investigation, Formal Analysis, Validation, Project Administration are performed by the author alone.

**Declaration of competing interest**

The author declares that he has no conflict of interest that could affect the work presented in this paper.

**Data Availability**

Data will be made available on request.

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