Golden ratio, Concentric Circumferences and Planetary Distances

In this work we will show some important relations that includes the study of the golden ratio between two concentric circles, showing that there is a linear combination between the radii of the circumferences. We derive a constant K which corresponds to the perfect number representing the largest root of the golden ratio which is a function of the radii of two concentric circles C_1andC_2, respectively. This relation makes it possible to find the values of the radii r_1of C_1and r_2of C_2 or vice versa. We will apply the results obtained in a problem related to the uranium and Neptune planets where we will use the known astronomical distances of said planets with respect to the sun to calculate the minimum and maximum distance comparing the percentage of the relative error with the known astronomical values in the scientific literature. Plot the graphs comparing the planetary distances in relation to the distances of Kepler and Titus as well as the margin of error.