A demonstration of the Titius–Bode law, the number of Saturn’s rings and the radius of Fine-Ring by Newtonian methods using the Kerr-Newman solution of the general relativity theory

Fumitaka Inuyama Senior Power Engineer (e-mail: [email protected])

Permanent address: 5-17-33 Torikai Jonan-ku, Fukuoka City, 814-0103, Japan

Thermal Power Dept., Kyushu Electric Power Co., Inc.(www.kyuden.co.jp) (Retired) Abstract The beautiful Titius–Bode law (휉 = 0.4 + 0.3 × 2n) , discovered 250 years ago, is considered to be a mathematical coincidence rather than an "exact" law, because it has not yet been physically proved. However, considering the disturbance restoration and the stability of the asteroid belt orbit, there must be some underlying logical necessity. Planetary orbits are often computed by Newtonian mechanics with the kinetic energy and the universal gravitation energy. However, applying the principle of energy-minimum to the Newtonian mechanics leads to the result that the stable orbital radius is only one value, which is totally incompatible with actual phenomena. This discrepancy must result from the shortage of elements which rule over the planetary orbits. Other elements to rule over the planetary orbits are the electric charge energy and the rotation energy, both of which are guided by the Kerr-Newman solution (discovered in 1965) of the general relativity theory. Here, I mathematically demonstrated the Titius–Bode law, and also calculated the number of Saturn’s rings, maximum 31 and the radius of Fine-Ring, by applying the principle of energy-minimum of Newtonian methods to the complicated energy equation which adopts mass, electric charge and rotation elements of the central core star and solving sole differential equation. Keywords Demonstration; Titius–Bode law; Saturn’s rings number; Fine-Ring radius; energy stable orbits; Kerr-Newman solution; relativity theory.

1. Introduction The Titius–Bode law, discovered 250 years ago, is considered to be a mathematical coincidence rather than an "exact" law [1], because it has not yet been physically proved. However, considering the disturbance restoration and the stability of the asteroid belt orbit, there must be some underlying logical necessity. Here, by Newtonian methods using the Kerr-Newman solution of the general relativity theory, I demonstrate the Titius-Bode law and apply its solution method to the calculation of the number of Saturn’s rings and the radius of Fine-Ring. This is a mathematical equation calculation. The detailed analysis process is provided in a separate sheet of paper [2].

2. Methods

The outline of the solution method and the key equation numbers in this article are as follows. 1) The equation for energy in the space-time field is obtained from the Kerr-Newman solution, a strict solution of the Einstein equations of the general relativity theory.

f 1(ρ, θ, dρ/dt, dθ/dt, dφ/dt, ε) = 0 (eq. 3） 2) This energy equation is partially differentiated by θ to the minimum energy. The result is θ=π/2, and the calculation below proceeds at θ=π/2, i.e., on the equatorial plane.

1 f 2(ρ, π/2, dρ/dt, 0, dφ/dt, ε) = 0 3)The angular momentum equivalent J is obtained by applying the variational principle to the Kerr- Newman solution to calculate dφ/dt . ξ(ρ, dφ/dt, J) = 0 (eq. 6) 4) Because an additional radius is dρ＝0 at the aphelion and perihelion distances R, the calculation below proceeds at distance R.

f 3(R, π/2, 0, 0, dφ/dt, ε) = 0 (eq. 7)

5) Substituting dφ/dt from ξ = 0 into f 3= 0 results in a relational expression of the radius, the angular momentum equivalent, and the energy.

f 4(R, π/2, 0, 0, J, ε) = 0 (eq. 9) 6) The orbital distance R is determined by the energy and the angular momentum equivalent, i.e.,

푅 = 푅(휀, 퐽). R is partially differentiated by ε , that is, f 4 is partially differentiated by ε. g (R, J, ε, 휕푅/휕휀) = 0 (eq. 10)

7) Derive the angular momentum equivalent J from f 4(R, π/2, 0, 0, J ,ε) = 0 and substitute it into g (R, J, ε, 휕푅/휕휀) = 0. Make this into an important differential equation which is just composed of the radius and the energy to analyze unique characteristics of orbits. h(R, ε, dε/dR) = 0 (eq. 11) 8) Solving the differential equation h results in a complicated set of arctan, log, power functions and an integration constant K. H(R, ε, K) = 0 (eq. 14) (eq. 15) 9) By using that the minimum energy is dε/dR=0 in h(R, ε, dε/dR) =0, following simultaneous equations are obtained and solved. ε

h(r, εmin , 0) = 0 ① H(r, εmin , K) = 0 ② (eq. 16) R 10) Because the integration constant K is common to all orbits, the Titius–Bode law is demonstrated

and also the number of Saturn’s rings and the radius of Fine-Ring are calculated. I (r, K) = 0 ( eq. 23) ( eq. 26) ( eq. 27)

2.1. The Energy Equation 2.1.1. Introduction to the Energy Equation There are two preconditions for the following analysis. 1) The analysis object must be sufficiently far from the center of mass. 2) The rotation speed of the center of mass must not be too fast. The characteristic Boyer-Lindquist coordinates in the Kerr solution are equal to general polar coordinates in the first-order term 푎/휌[3]. The strict Boyer-Lindquist metric of the Kerr-Newman geometry [4] is as follows.

R2Δ 2 ρ2 R4sin2θ a 2 ds2 = − (dt − asin2θdφ) + dr2 + ρ2dθ2 + (dφ − dt) ρ2 R2Δ ρ2 R2 At the large radius r, the Boyer-Lindquist metric is as follows.

2M 4aMsin2θ 2M ds2 → − (1 − ) dt2 − dtdφ + (1 + ) dr2 + r2(dθ2 + sin2θ dφ) r r r Symbols are changed from the Boyer-Lindquist metric to the general polar coordinate.

2 The Kerr-Newman solution of the general relativity theory is given by (eq.1). In this expression, 푚, 푎 and e are the mass, rotation and electric charge elements respectively. 2푚휌 − 푒2 휌2 + 푎2 cos2 휃 (1) 푑푠2 = (1 − ) (푐푑푡)2 − 푑휌2 − (휌2 + 푎2cos2휃)푑휃2 휌2 + 푎2cos2휃 휌2 + 푎2 − 2푚휌 + 푒2 (2푚휌−푒2)푎2sin2휃 2(2푚휌 − 푒2)푎 sin2휃 − [ (휌2 + 푎2) + ] sin2휃 푑휑2 − 푐푑푡 ∙ 푑휑 휌2 + 푎2 cos2 휃 휌2 + 푎2cos2휃 Γ is as follows when ds is divided by the time elements (c dt). 1 푑푠 2 = ( ) 훤2 푐푑푡 The Lorentz transformation factor γ (= c dt/ds) in the Minkowski space-time of the special relativity 2 2 theory is an important component of the energy E = M c = M0 γ c . Γ (= c dt/ds) of the Kerr-Newman solution of the general relativity theory is analogous to γ. On this occasion, by following the principle of minimum energy, the sign of 푚 is changed to −푚 , 푎 is changed to +푎, and e is changed to +e. Therefore, the energy equation is Ｅ＝Γ ( ρ, θ, φ, t, −푚, 푎, e). 1 2푚휌 + 푒2 휌2 + 푎2 cos2 휃 푑휌 2 푑휃 2 (2) = (1 + ) − ( ) − (휌2 + 푎2cos2휃) ( ) 퐸2 휌2 + 푎2cos2휃 휌2 + 푎2 + 2푚휌 + 푒2 푐푑푡 푐푑푡 (2푚휌+푒2)푎2sin2휃 푑휑 2 2(2푚휌 + 푒2)푎 sin2휃 푑휑 − [(휌2 + 푎2) − ] sin2휃 ( ) + ( ) 휌2 + 푎2 cos2 휃 푐푑푡 휌2 + 푎2cos2휃 푐푑푡

2 2 Since E has a decisive massive energy M0c , it is converted into ε 1/퐸 = 1 − 2휀 in (eq.3). 2푚휌 + 푒2 휌2 + 푎2cos2휃 푑휌 2 푑휃 2 (3) − 2ε = − ( ) − (휌2 + 푎2cos2휃) ( ) 휌2 + 푎2cos2휃 휌2 + 푎2 + 2푚휌 + 푒2 푐푑푡 푐푑푡 (2푚휌+푒2)푎2sin2휃 푑휑 2 2(2푚휌 + 푒2)푎 sin2휃 푑휑 − [(휌2 + 푎2) − ] sin2휃 ( ) + ( ) 휌2 + 푎2 cos2 휃 푐푑푡 휌2 + 푎2cos2휃 푐푑푡 Partial differentiation is used to minimize the energy ε(ρ，θ，φ，t) by using 휕휀/휕휃 = 0. (2푚휌 + 푒2)푎2 푎2 푑휌 2 푑휃 2 + ( ) + 푎2 ( ) (휌2 + 푎2cos2휃)2 휌2 + 푎2 + 2푚휌 + 푒2 푐푑푡 푐푑푡 (2푚휌 + 푒2)2푎2sin2휃 (2푚휌 + 푒2)푎4sin4휃 푑휑 2 − [(휌2 + 푎2) − − ] ( ) ・sin 2θ ＝ 0 휌2 + 푎2cos2휃 (휌2 + 푎2cos2휃)2 푐푑푡 2(2푚휌 + 푒2)푎 2(2푚휌 + 푒2)푎3sin2휃 푑휑 + [ + ] ( ) 휌2 + 푎2 cos2 휃 (휌2 + 푎2 cos2 휃)2 푐푑푡 That is, the energy E and ε are minimized at θ=π/2 and the planets gather on the equatorial plane where the energy is stable.

2.1.2. Time component from the variational principle When the rotation speed of the center of mass is not too fast, the Kerr-Newman solution expanded in the first order of 푎/휌 takes the form given in (eq.4): 푑푠 2 2푚 푒2 푐푑푡 2 1 푑휌 2 푑휃 2 푑휑 2 (4) ( ) = 1 = (1 − + ) ( ) − ( ) − 휌2 ( ) − 휌2sin2휃 ( ) 푑푠 휌 휌2 푑푠 2푚 푒2 푑푠 푑푠 푑푠 1 − + 휌 휌2 2푎 푒2 푐푑푡 푑휑 − (2푚 − ) sin2휃 ( ) ( ) 휌 휌 푑푠 푑푠

3 The Euler–Lagrange equation [5] is adopted by applying the variational principle to the Kerr-Newman solution. 2푚 푒2 푐푑푡 2 1 푑휌 2 푑휃 2 푑휑 2 δ ∫ (1 − + ) ( ) − ( ) − 휌2 {( ) + sin2휃 ( ) } 휌 휌2 푑푠 2푚 푒2 푑푠 푑푠 푑푠 1 − + 휌 휌2 2푎 푒2 푐푑푡 푑휑 − (2푚 − ) sin2휃 ( ) ( ) 푑푠 = 0 휌 휌 푑푠 푑푠 Eventually, (eq.5) is obtained at the equatorial plane of the rotating center of mass where the energy is stable. Hereafter, I perform the calculation at the equatorial plane (θ＝π/2) of the rotating center of mass. 2 2 푑 2푚2푚 푒 푐푑푡 푎 푒 푑휑 [(1 − + 2) ( ) − (2푚 − ) ( )] = 0 time component 푑푠 ( 휌 휌) 휌 푑푠 휌 휌 푑푠 (5) 푑 푑휑 푎 푒2 푐푑푡 [휌2 (2푚푎) + (2푚 − ) ( )] = 0 휑 component 푑푠 ( 푑푠 ) 휌 휌 푑푠 2푚푎 The two equations in (eq.5) are integrated over ds. 푑휑/푑푡 (eq. 6) with an integration variable J is obtained by using the resulting pair of simultaneous equations, 푑휑 푒2 푒2 ( ) 퐽 (휌 − 2푚 + ) + 푎 ( − 2푚) 푑휑 푑푠 휌 휌 퐽 ∶ the angular momentum equivalent (6) = = ∙ 푐 푑푡 푒2 （ ） 푑푡 ( ) 3 a kind of Carter constant in relativity theory 푑푠 휌 + 퐽푎 (2푚 − 휌 ) The distance variables are defined as follows: ρ：An arbitrary orbital distance in two- or three-dimensional coordinates. R：The aphelion and perihelion distances at the equatorial plane of the rotating center of mass. r：The aphelion and perihelion distances, both of which are energetically stable at the equatorial plane.

2.1.3. Introduction of the angular momentum equivalent Because additional ρ at the aphelion and perihelion distances is dρ＝0, the energy equation is given by (eq.7). 2푚 푒2 푑휑 2 4푎 푒2 푑휑 (7) 0 = 2휀 + + − 푅2 ( ) + (푚 + ) ( ) 푅 푅2 푐푑푡 푅 2푅 푐푑푡 dφ/cdt (eq. 6) is composed of the angular momentum equivalent and is substituted into (eq.7). J is obtained as in (eq.8) by adopting the secondary order R. 4푎푚 + 푅훿√푅(2휀푅 + 2푚 + 퐶) (8) 퐽 = 푅2 푅2(푅 − 2푚 + 퐶) − 푎(2푚 − 퐶)훿√푅(2휀푅 + 2푚 + 퐶) Here, 훿 = ±1 and 퐶 = 푒2/푅 . δ is related to the orbital rotation direction.

2.2. The Space Fantasy Differential Equation 2.2.1. Introduction of the Space Fantasy differential equation It leads not to the numerical analysis but to the analytical unique characteristics. The relation of R , ε, and J are given as (eq.9) at the aphelion and perihelion distances R by changing the angular momentum equivalent J (eq. 8). (eq.9) is far more complicated than the Kepler-Newton equation 2휀푅2 + 2푚푅 − 퐽2 = 0

4 2 푒2 푒2 2 퐽 (푅 − 2푚 + ) + 푎 ( − 2푚) 2푚 푒 2 푅 푅 (9) 0 = 2휀 + + 2 − 푅 [ 2 ] 푅 푅 3 푒 푅 + 퐽푎 (2푚 − 푅 ) 푒2 푒2 4푎 푒2 퐽 (푅 − 2푚 + 푅 ) + 푎 ( 푅 − 2푚) + (푚 + ) [ 2 ] 푅 2푅 3 푒 푅 + 퐽푎 (2푚 − 푅 ) Since the orbital distance R is determined by the energy ε and the angular momentum equivalent J, 푅 = 푅(휀, 퐽). A new differential equation is given as (eq.10) by partially differentiating R by ε, then substituting J into this, and adopting the reciprocal of 휕푅/휕휀. 휕휀 (10) [푅3 + 퐽푎(2푚 − 퐶)]2 휕푅

(푚 + 퐶)[푅3 + 퐽푎(2푚 − 퐶)]2 [퐽(푅 − 2푚 + 퐶) − 2푎푚 + 푎퐶] [퐽(푅 − 2푚 + 퐶) + 3푎퐶] ∙ 푅 = + 푅2 1 2푅2[퐽(푅 − 2푚 + 퐶) − 4푎푚] [퐽2푎(푚 − 퐶) − 퐽푅2(푅 − 3푚 + 2퐶) + 푎푅2(3푚 − 2퐶)] + 푅3 + 퐽푎(2푚 − 퐶) Here, by substituting J of (eq. 8) into (eq.10), the second order R is obtained. Through all these extensive calculation processes, the relation between ε and R is summarized as (eq.11). 풅휺 (11) 푹ퟒ(푹ퟐ − ퟒ풎푹 + ퟐ푪푹 + ퟒ풎ퟐ) 풅푹 = 풎푹ퟐ(−푹ퟐ + ퟖ풎푹 − ퟒ푪푹 − ퟏퟐ풎ퟐ) + 휺 ∙ ퟐ푹ퟑ(−푹ퟐ + ퟔ풎푹 − ퟒ푪푹 − ퟖ풎ퟐ) +ퟐ풂풎(ퟐ푹ퟐ + ퟐ풎푹 − 푪푹 − ퟏퟐ풎ퟐ)휹√푹(ퟐ휺푹 + ퟐ풎 + 푪) +휺 ∙ ퟒ풂푹(ퟑ풎푹 − ퟐ푪푹 − ퟔ풎ퟐ + ퟕ푪풎)휹√푹(ퟐ휺푹 + ퟐ풎 + 푪) 퐶 = 푒2/푅 (푅 2ry order)

I call this second order equation (eq. 11) “the Space Fantasy (SF) differential equation”. The change of variables is performed to solve the SF differential equation for S. The result is (eq.12).

푆 = 푅√푅(2휀푅 + 2푚 + 퐶) 풅푺 ퟐ풆ퟐ(풆ퟐ + ퟐ풎ퟐ) ퟒ풂휹풎 + 푺 ퟔ풂휹풎푺ퟐ (12) = + + ( 푅 0 order) 풅푹 푺푹 푹 푹ퟓ The form of the differential equation in (eq.12) is more complicated than the Riccati's differential equation, which never has an exact general solution [6]. Since 6푎훿푚푆2/푅5 is much smaller than S/R and 4푎훿푚/푅, it can be treated as a constant θ. Also, (eq.12) can be reduced to the problem of an approximate differential equation, and it is given as (eq.13). 푑푆 1 2퐸4 4푎훿푚푆 6푆2 푆2 = [ + (1 + ) + ] 퐸4 = 푒2(푒2 + 2푚2) 푑푅 푆 푅 푅 4푅4 푅

4 2 2 1 2퐸 4푎훿푚푆 푆 3푆0 2 4 2 4 ≒ [ + (1 + 휃) + ] 휃 = 4 (푆0 , 푅0 are centroids 푆 /3 , 푅 /5) 푆 푅 푅 푅 2푅0 푆푑푆 푑푅 (13) = 푆2 + 4푎훿푚푆(1 + 휃) + 2퐸4 푅 Solving (eq.13) by the quadrature formulae [7] leads to (eq.14) and (eq.15).

5 In the case that the discriminant is 훥 ＝퐸4 − 2푎2푚2(1 + 휃)2 ＞0: 1 4푎훿푚(1 + 휃) 2S + 4푎훿푚(1 + 휃) log[푆2 + 4푎훿푚(1 + 휃)푆 + 2퐸4] − arctan ( ) 2 2√2퐸4 − 4푎2푚2(1 + 휃)2 2√2퐸4 − 4푎2푚2(1 + 휃)2 = log 푅 + 퐾 푆2 + 4푎훿푚(1 + 휃)푆 + 2퐸4 −4푎훿푚(1 + 휃) 푆 + 2푎훿푚(1 + 휃) (14) 퐾 = ∙ EXP [ arctan ( )] 푅2 √2퐸4 − 4푎2푚2(1 + 휃)2 √2퐸4 − 4푎2푚2(1 + 휃)2

In the case that the discriminant is 훥 ＝퐸4 − 2푎2푚2(1 + 휃)2 ＜0: log[푆2 + 4푎훿푚푆(1 + 휃) + 2퐸4] 2푎훿푚(1 + 휃) 푆 + 2푎훿푚(1 + 휃) − √4푎2푚2(1 + 휃)2 − 2퐸4 − ∙ log [ ] = 2 log 푅 + 퐾 √4푎2푚2(1 + 휃)2 − 2퐸4 푆 + 2푎훿푚(1 + 휃) + √4푎2푚2(1 + 휃)2 − 2퐸4 푆2 + 4푎훿푚푆(1 + 휃) + 2퐸4 2 ( ) 푅 15 퐾 = log 2푎훿푚(1+휃)

푆 + 2푎훿푚(1 + 휃) − √4푎2푚2(1 + 휃)2 − 2퐸4 √4푎2푚2(1+휃)2−2퐸4 [ ] [ 푆 + 2푎훿푚(1 + 휃) + √4푎2푚2(1 + 휃)2 − 2퐸4 ]

In the case that the discriminant is 훥 ＝퐸4 − 2푎2푚2(1 + 휃)2 = 0: 푑푆 1 (푆 + √2퐸2)2 = ∙ Solving this equation by the quadrature formulae leads 푑푅 푆 푅 푆 + √2퐸2 √2퐸2 (16) 퐾 = 퐸푋푃〔 ］ 푅 푆 + √2퐸2

2.2.2. Conditions of the energy minimum orbit Since the minimum energy is 푑휀/푑푅 = 0 in the SF differential equation (eq. 11), it is a cubic equation in ε. 0 = 휀3 ∙ 32푎2푟3(3푚푟 − 2퐶푟 − 6푚2 + 7퐶푚)2 + 휀2 ∙ 푟2 16푎2(3푚푟 − 2퐶푟 − 6푚2 + 7퐶푚)2(2푚 + 퐶) +32푎2푚(2푟2 + 2푚푟 − 퐶푟 − 12푚2)(3푚푟 − 2퐶푟 − 6푚2 + 7퐶푚) −4푟3(−푟2 + 6푚푟 − 4퐶푟 − 8푚2)2

+ 휀 ∙ 4푚푟 2푎2푚(2푟2 + 2푚푟 − 퐶푟 − 12푚2)2 +4푎2(2푟2 + 2푚푟 − 퐶푟 − 12푚2)(3푚푟 − 2퐶푟 − 6푚2 + 7퐶푚)(2푚 + 퐶) −푟3(−푟2 + 8푚푟 − 4퐶푟 − 12푚2)(−푟2 + 6푚푟 − 4퐶푟 − 8푚2)

+푚2[4푎2(2푟2 + 2푚푟 − 퐶푟 − 12푚2)2(2푚 + 퐶) − 푟3(−푟2 + 8푚푟 − 4퐶푟 − 12푚2)2]

Solve this cubic equation. A solution 휀푚푖푛 (eq. 16) very close to 0 is adopted in accordance with the principle of the energy minimum. −푚 푟3(푟2 − 8푚푟 + 4퐶푟 + 12푚2)2 − 4푎2(2푚 + 퐶)(2푟2 + 2푚푟 − 퐶푟 − 12푚2)2 (17) εmin = ・ 4푟 푟3(푟2 − 8푚푟 + 4퐶푟 + 12푚2)(푟2 − 6푚푟 + 4퐶푟 + 8푚2)

−4푎2(2푚 + 퐶)(2푟2 + 2푚푟 − 퐶푟 − 12푚2)(3푚푟 − 2퐶푟 − 6푚2 + 7퐶푚) −2푎2푚(2푟2 + 2푚푟 − 퐶푟 − 12푚2)2

−푚 ≒ ( 푟 0 order) 4푟

6 ( ) ( ) ( ) 휀푚푖푛 eq. 17 is substituted into the change of variables S = 푟√푟 2휀푟 + 2푚 + 퐶 eq. 12 . −푚푟4 푟4(푟2 − 8푚푟 + 4푒2 + 12푚2)2 − 4푎2(2푚푟 + 푒2)(2푟2 + 2푚푟 − 푒2 − 12푚2)2 S2 = ・ 2 푟5(푟2 − 8푚푟 + 4푒2 + 12푚2)(푟2 − 6푚푟 + 4푒2 + 8푚2)

−4푎2(2푚푟 + 푒2)(2푟2 + 2푚푟 − 푒2 − 12푚2)(3푚푟2 − 2푒2푟 − 6푚2푟 + 7푚푒2) −2푎2푚푟2(2푟2 + 2푚푟 − 푒2 − 12푚2)2

+ 푟2(2푚푟 + 푒2) 푟4 × [푟8 polynomial ] + 푟2(2푚푟 + 푒2) × [푟9 polynomial ] 푟2 × 푃 = = [푟9polynomial ] 푄 3푚 ≒ 푟3 ( 푟 0 order) 2 Here, P and Q are given by (eq.18) and (eq.19). (18) 푃 = −푚푟2/2 〔 푟4(푟2 − 8푚푟 + 4푒2 + 12푚2)2 − 4푎2(2푚푟 + 푒2)(2푟2 + 2푚푟 − 푒2 − 12푚2)2 〕 +(2푚푟 + 푒2) × 푄 [푟10 polynomial ] (19) 푄 = 푟5(푟2 − 8푚푟 + 4푒2 + 12푚2)(푟2 − 6푚푟 + 4푒2 + 8푚2) −4푎2(2푚푟 + 푒2)(2푟2 + 2푚푟 − 푒2 − 12푚2)(3푚푟2 − 2푒2푟 − 6푚2푟 + 7푚푒2) −2푎2푚푟2(2푟2 + 2푚푟 − 푒2 − 12푚2)2 [푟9 polynomial ] And for 휃, 2 2 3푆0 5푆 5푃 15푚 휃 = 4 = 4 = 2 ≒ ( 푟 at 0 order) 2푅0 2푟 2푄푟 4푟

2.3. The Titius–Bode Law In the case that the discriminant is 훥 ＝퐸4 − 2푎2푚2(1 + 휃)2 ＞0 of the SF differential equation, the function 푓(θ) is given in (eq.14) and is subjected to a Maclaurin series expansion. Terms above 휃2 are neglected. The result is given in (eq.20). 푆2 + 4푎훿푚(1 + 휃)푆 + 2퐸4 −4푎훿푚(1 + 휃) 푆 + 2푎훿푚(1 + 휃) 푓(θ) = EXP [ arctan ( )] 푅2 √2퐸4 − 4푎2푚2(1 + 휃)2 √2퐸4 − 4푎2푚2(1 + 휃)2 −퐾 = 0 1 휕푓(0) 1 휕2푓(0) 푓(θ) = 푓(0) + ∙ 휃 + ∙ 휃2 + ⋯ = 0 1! 휕휃 2! (휕휃)2

3푚푟 −4푎훿푚 푟√3푚푟 (20) 푓(θ) = EXP [ arctan ( )] × 2 √2퐸4 − 4푎2푚2 2√퐸4 − 2푎2푚2 30푎훿푚2퐸4 푟√3푚푟 × [1 − × arctan ( )] − 퐾 = 0 3 4 2 2 푟[2퐸4 − 4푎2푚2]2 2√퐸 − 2푎 푚

푟 3푚푟 2√퐸4−2푎2푚2 Since r is very large, it is given as arctan ( √ ) = 휋/2 + 휋푁 − . This is substituted 2√퐸4−2푎2푚2 푟√3푚푟

into (eq.20). 3mr −2푎훿푚휋(1 + 2푁) 30푎훿푚2퐸4 휋(1 + 2푁) 퐾 = EXP [ ] ∙ [1 − ∙ ] 2 4 2 2 3 2 √2퐸 − 4푎 푚 푟[2퐸4 − 4푎2푚2]2 Since the integration constant K is common to all planets that orbit the center of mass, the base planet

and the distance ratio to the base planet can be set as follows. r1 , N1 , N－N1＝푛 − 1, and 휉 = 푟/푟1. The result is given in (eq.21).

7 2 4 15푎훿푚 퐸 휋(2푁1 + 2푛 − 1) 휉 − 3 4 2 2 4 2 2 √2퐸 − 4푎 푚 푟1[2퐸 − 4푎 푚 ]2 (21) 푛 − 1 = ∙ log 2 4 4푎훿푚휋 15푎훿푚 퐸 휋(2푁1 + 1) 1 − 3 4 2 2 [ 푟1[2퐸 − 4푎 푚 ]2 ] On the other hand, the Titius-Bode law is changed into (eq.22). ＝ 푛 ＝ 푛−1 ： 휉퐸푎푟푡ℎ 0.4 + 0.3 × 2 0.4 + 0.6 × 2 ( 휉퐸푎푟푡ℎ the Earth basis 흃) 1 휉 − 0.4 (22) 푛 − 1 = ∙ log 퐸푎푟푡ℎ log 2 1 − 0.4 The Titius-Bode law (eq. 22) is remarkably similar to the solution (eq. 21) of the approximate SF differential equation. If the two coefficients are the same, the two equations are almost equal. (The Earth is the base planet, 푛=1.) 4 2 2 2 4 1 √2퐸 − 4푎 푚 15푎훿푚 퐸 휋(2푁1 + 1) = 0.4 = 3 log 2 4푎훿푚휋 4 2 2 푟1[2퐸 − 4푎 푚 ]2 8 Since 푟1 = 1.5 × 10 푘푚 for the Earth, and, 푚＝1.476푘푚 and 푎＝0.32푘푚 [8] for the Sun, it is calculated 7 that e = 2.1푘푚, and 푁1 = 1.5 × 10 . The 2n on the right side of (eq.21) is neglected because of the very

large 푁1. Thus,

ퟑퟎ풂휹풎ퟐ푬ퟒ흅푵 ퟒ풂풎흅(풏 − ퟏ) ퟑퟎ풂휹풎ퟐ푬ퟒ흅푵 (23) 흃 = [ퟏ − ퟏ ] ∙ 퐄퐗퐏 [ ] + ퟏ ퟑ ퟒ ퟐ ퟐ ퟑ ퟒ ퟐ ퟐ √ퟐ푬 − ퟒ풂 풎 ퟒ ퟐ ퟐ 풓ퟏ[ퟐ푬 − ퟒ풂 풎 ]ퟐ 풓ퟏ[ퟐ푬 − ퟒ풂 풎 ]ퟐ

풏−ퟏ 휉퐸푎푟푡ℎ = (ퟏ − ퟎ. ퟒ) ∙ ퟐ + ퟎ. ퟒ 훿 = ±1 is related to the orbital rotation direction. (Eq.23) is now exactly equal to (eq.22). The Titius-Bode law has therefore been demonstrated.

2.4. The Saturn’s Rings Since the autorotation of the Saturn is fast, the discriminant is 훥 ＝퐸4 − 2푎2푚2(1 + 휃)2 ＞0. The solution (eq. 15) of the SF differential equation is as follows. 푆2 + 4푎훿푚푆(1 + 휃) + 2퐸4

푅2 퐾 = log 2푎훿푚(1+휃)

푆 + 2푎훿푚(1 + 휃) − √4푎2푚2(1 + 휃)2 − 2퐸4 √4푎2푚2(1+휃)2−2퐸4 [ ] [ 푆 + 2푎훿푚(1 + 휃) + √4푎2푚2(1 + 휃)2 − 2퐸4 ]

2푎훿푚(1+휃) Since the power number [ ] is nearly 1, the denominator is expressed as ( 1 − 휆). 휆 is √4푎2푚2(1+휃)2−2퐸4 extremely small, but not zero. The solution of the SF differential equation is (eq.24).

2푎훿푚(1+휃) 푆 + 2푎훿푚(1 + 휃) − √4푎2푚2(1 + 휃)2 − 2퐸4 √4푎2푚2(1+휃)2−2퐸4 1 − 휆 = [ ] 푆 + 2푎훿푚(1 + 휃) + √4푎2푚2(1 + 휃)2 − 2퐸4 푆2 + 4푎훿푚푆(1 + 휃) + 2퐸4 1 (24) 퐾 = ∙ 푟2 (1 − 휆) The integration constant K is common to all the rings that belong to the Saturn. For the base ring, the

variables are r1 and F =K, and the polynomial of S is (eq.25). (25) 푆4 − 2푆2[퐹(1 − 휆)푟2 − 2퐸4 + 8푎2푚2(1 + 휃)2] + [퐹(1 − 휆)푟2 − 2퐸4]2 = 0

8 푃 (eq. 18) and 푄 (eq. 19) are substituted into (eq.25) to give S and 휃. Finally, the polynomial of r is (eq.26). (26) 푸풓ퟐ ( 푷풓ퟐ − 푸 〔 푭(ퟏ − 흀)풓ퟐ − ퟐ푬ퟒ ] )ퟐ − ퟒ풂ퟐ풎ퟐ푷 ( ퟐ푸풓ퟐ + ퟓ푷 )ퟐ = ퟎ The degree of (eq.26) is the highest at the first term 푃2푄푟6, and is 푟 to the power of 35〔10×2+9+6〕． That is, (eq.26) is a polynomial of 푟35 with high degree coefficient 훌 and has four micro roots. Thus, planets with rings such as the Saturn have a maximum of 31 rings. The real number of rings decreases because of roots of complex number, minus roots, equal roots and the swelling of the center core. It is expected to observe and determine the rotation element 푎 and the electric charge element e .

2.5. Fine Ring star In the case that the discriminant is 훥 ＝퐸4 − 2푎2푚2(1 + 휃)2 = 0 because the balance of the mass, rotation and electric charge elements is exquisite. 푑푆 1 2퐸4 4푎훿푚푆 3푆2 푆2 = [ + (1 + ) + ] 퐸4 = 푒2(푒2 + 2푚2) 푑푅 푆 푅 푅 2푅4 푅

2 3푆2 the discriminan 훥＝2퐸4 − 4푎2푚2 (1 + ) = 0 ① 2푅4 2 푑푆 1 (푆 + √2퐸2) the differenntial equation = ∙ solving this equation 푑푅 푆 푅 푆 + √2퐸2 √2퐸2 퐾 = 퐸푋푃( ) ② 푅 푆 + √2퐸2 the energy stable equation S = 푟√3푚푟/2 ③ 푟 , 푆 and 퐾 are unknown, then solve simultaneous equations ①, ②, ③ ퟗ풂휹풎ퟐ (27) 풓 = ퟐ√ퟐ(푬ퟐ − √ퟐ푎훿풎) 푚 , 푒 and 푎 are small constant value. But if it is 퐸2 − √2푎훿푚 ≃ 0 generally, 푟 grows larger. That is, in the case of exquisite balance푠 푎2 ≃ 푒2(1 + 푒2/2 푚2), Super Fine-Ring is formed. In the case that the discriminant is 훥 ≠ 0 slightly out of alignment, Fine-Ring is formed and has some components of the Saturn’s rings or Solar System planets.

3. Discussion The Titius-Bode law, discovered 250 years ago, is considered to be a mathematical coincidence rather than an "exact" law. However, I successfully proved the law and showed here that the Saturn can physically have a maximum of 31 rings. 250 years of astronomical mystery is now solved not by the computer analysis but by the theoretical analysis. The Kerr-Newman solution of the Einstein’s equation is considered as follows. The no-hair theorem postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum. [9], [10]

9 In this manner, since this theory is based on the steady state of Kerr-Newman solution in the mature galaxies, it cannot be applied to the galaxies which are still young, unstable and transitional. Three important equations can be summarized as follows. (eq.11) is a fundamental differential equation based on the steady state, and it can be applied to the Solar system, other planets and rings in the galaxies. There must be many solutions of (eq.11). (eq.23) is one of the approximate solutions of (eq.11). Since this is energetically stable, it is applicable to the Solar system planets and many of the around 4000 extrasolar planets in the galaxies. However, it is not applicable to still young, unstable and transitional planets like comets. (eq.26) is one of the approximate solutions of (eq.11). This is also energetically stable and applicable to Saturn’s rings and some other extrasolar planets’ rings. 푚 , 푒 and 푎 are small constant value. But in the case of exquisite balance푠 푎2 ≃ 푒2(1 + 푒2/2 푚2), Super Fine Ring is formed (eq.27). This theory is applied to planets which belong to the center of mass in galaxies, but not available to the bulge space near to the center of mass.

Acknowledgments Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. I thank Dr. Yuko Masaki and Edanz Group (www.edanzediting.com/ac) for editing a draft of this manuscript.

Author Contributions F. I. developed the theory and wrote the manuscript.

Competing Interests The author declares no competing interests including financial and non-financial interests.

References [1] Internet Titius–Bode law - Wikipedia https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law, accessed in Jan 2018. [2] Internet A demonstration of the Titius –Bode law and the number of the Saturn’s rings, based on the theory of relativity http://sayuri-fumitaka.icurus.jp. accessed in Oct 2017 [3] Internet Boyer–Lindquist coordinates - Wikipedia https://en.wikipedia.org/wiki/Boyer%E2%80%93Lindquist_coordinates, accessed in Jan 2018. [4] Internet General Relativity, Black Holes and Cosmology, Andrew J S. Hamilton http://jila.colorado.edu/~ajsh/astr5770_14/grbook.pdf#search=%27general+relativity% 2C+black+hole+and+cosmology%27, accessed in Jan 2018. [5] Internet Euler-Lagrange Differential Equation http://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html, accessed in Jan 2018.

10 [6] Internet Riccati equation - Wikipedia (similar to Japanese) https://en.wikipedia.org/wiki/Riccati_equation, accessed in Jan 2018. [7] Formeln+Hilfen Höhere Mathematik, 2013 (translated into Japanese) Gerhard Merziger, Günter Mühlbach, Detlef Wille, Thomas Wirth. [8] Exploring Black Holes: Introduction to General Relativity, 2000, Edwin F. Taylor, John Archibald Wheeler (p272, translated into Japanese by Nobuyoshi Makino) [ퟗ] Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. pp. 875–876. ISBN 0716703343. Retrieved 24 January 2013. [ퟏퟎ] Internet No-hair theorem - Wikipediaen. https://en.wikipedia.org/wiki/No-hair_theorem, accessed in Jan 2018.

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