An analytical method for increasing the accuracy of the value of the Newtonian constant of gravitation

Despite hundreds of measurements of the Newtonian constant of gravitation, its accuracy remains very low. Over the past 55 years, it has improved by only one order of magnitude - from four to five digits after the decimal point. In this study, a new analytical method for improving the accuracy of estimating the value of the Newtonian constant of gravitation is proposed. Using the proposed method, its accuracy is increased by 7 orders of magnitude relative to the CODATA 2022 data. The method is based on the analytical estimate of the Planck mass, length, and time, with an accuracy of values ​​that is 5 orders of magnitude higher than their accuracy according to CODATA 2022. Such a significant increase in the accuracy of the Planck mass, length, and time values ​​was made possible by the integrated use of: 1) precision formulas for the Planck momentum; 2) representation of the speed of light in a vacuum through the Planck length and time; 3) the De Broglie principle: the moments of the Planck mass, leptons, and baryons are equal to each other; 4) high-precision characteristics of the proton. The method of analytical evaluation of Planck mass, length, and time allowed us to connect the main characteristics of the hypothetical virtual Planck particle with the main characteristics of the proton. Increasing the accuracy of the proton characteristics will entail increasing the accuracy of Planck mass, length, and time. Accordingly, the accuracy of the value of the Newtonian gravitational constant and all physical constants that can be represented through Planck mass, length, and time will be increased, which is especially important in light of the decisions of the 26th General Conference on Weights and Measures.