Titius-Bode Modified 2020-08-29
The Titius-Bode Rule or Law is a slope-intercept formula,
y = .3*x+.4, for the orbital radius of the planets.

The planets are the little circles, left to right, Mercury, Venus, Earth, Mars, Asteroids, Jupiter and Saturn at an x value of 32. Not shown are Uranus at 64, Neptune at 100 or Pluto at 128. The bottom line could be called the Sun line. The radial distance from the Sun to each planet is the "y" value, the vertical distance from the bottom line to the center of the little circles. The horizontal double lines are 0.4 astronomical units au, apart for Mercury. One "au" is the average distance from the Sun to the Earth. The graph and table show it is not perfect but close enough to be interesting. What a wonderful mystery. How does nature work to provide planetary spacings so easy to graph?

 Table 1 Titius-Bode Rule orbit radius mass orbit radius orbit period .3*x+.4 actual 1E24 orbit 1E9 1E6 Planet x au au kg years meters ln ln/ln seconds ln ln/ln Mercury 0 0.4 0.387 0.33 .2435 57.9 24.78 7.598 15.843 Venus 1 0.7 0.723 4.87 .6109 108.2 25.41 1.025 19.401 16.781 1.059 Earth 2 1.0 1.000 5.98 1.000 149.6 25.73 1.013 31.559 17.267 1.029 Mars 4 1.6 1.524 0.642 1.874 227.9 26.15 1.016 59.374 17.899 1.037 Asteroids 8 2.8 2.7 4.436 403.9 26.72 1.022 140.01 18.757 1.048 Jupiter 16 5.2 5.203 1900 11.86 778.3 27.38 1.025 374.55 19.741 1.052 Saturn 32 10 9.539 570 29.46 1427 27.99 1.022 929.77 20.650 1.046 Uranus 64 19.6 19.18 87 84.07 2871 28.69 1.025 2650.9 21.698 1.051 Neptune 100 30.4 30.06 102 164.8 4497 29.13 1.016 5201.2 22.372 1.031 Pluto 128 38.8 39.53 0.7 247.7 5913 29.41 1.009 7843.6 22.783 1.018

The rule: y = 0.3*x +.4, where y is the vertical distance to the bottom Sun line, in astronomical units, and x = 0,1,2,4,8,16,32,64,100,128.
The most interesting feature is the "x" values, doubling as planets are added, to the right except for Neptune. The rule works just as well using the same x and 1E9 meters,
45.83*x +57.9 = r, in 1E9_meters. Seen in this way each unit of x is 45.83E9_meters and the distance from the start, the Sun to 0, Mercury is 57.9E9_meters. The straight slope line shows the deviation of the radius from the rule.

Orbital resonance:
Neptune at x=100 kills the pattern of the doubling of x values. Pluto and Neptune are in, 2:3 orbital resonance. Pluto orbits twice around the sun while Neptune orbits three times. Pluto's orbit has a high inclination, to the other planets, and a high orbital eccentricity, which allows it to occasionally be closer to the sun than Neptune. Jupiters moon's; Ganemede, Europa and Io exhibit, a 1:2:4 orbital resonance around Jupiter. Orbital resonance is based on the periods of the planets and has been suggested as the cause of Titius-Bode but there are so many near resonances that it is hard to see the relevant pattern.
Mercury and Venus, 5:2
Mercury and Earth, 4:1. We are closest to Mercury four times per year.
Venus and Earth, 5:3. We are closest to Venus every three years.
Earth and Moon, 1:1. A binary system with the same periods.
Earth and Mars, 15:8. We are closest to Mars every fifteen years.
Resonances move planets. As planets approach, they accelerate, energy changes, orbits change, periods change. As planets retreat, they decelerate, energy changes, orbits change, periods change.All this nudges all the planets. Eventually all the planets reach some sort of partial equilibrium. Titius-Bode may represent this equilibrium state. We can see some of these dynamic interactions in orbital simulators.

The expanding solar system?
If the solar system expands in the same way the universe expands, according to Hubble's constant. The middle slope line shows the size of the solar system 4.5 billion years ago. The x values remain the same. Titius-Bode still works. Only the slope of the line changes, as the solar system expands, and with it the radial distance from the sun to the planets. This graph is a very convenient way to show a Hubble expansion.

Kepler and Titius-Bode:
m*vt2/r = G*m*M/r2, The centrifugal force, measured from the barycenter, equals the gravitational force. This is just another way of writing Kepler's third law.
p = 2*pi*r/vt, The period of an orbit is p. r is the radius, the distance to the Sun and vt is the velocity, tangent to the orbit, G is the gravitational constant, M is the mass of the Sun.
p = 2*pi*r/vt, therefore vt2 = {4*pi2*r2/p2}
m*{4*pi2*r2/p2}/r = G*m*M/r2,
substituted for vt2
r3 = p2*G*M/(4*pi2), collected terms. G*M/(4*pi2) = 3.3615E18_m3/s2. m the mass of the planet has canceled. M is the mass of the Sun. This is a r3 and p2 equation. The cube of the radius is proportional to the square of the period if G*M/(4*pi2) is constant. This is Kepler's third law but how can cubes and squares relate to the seeming linear form of Titius-Bode?

Examples:
r3 = p2*G*M/(4*pi2), Kepler.
(1)*r3 = (1)*p2* (1)*G*M/(4*pi2), multiplied both sides by (1).
(1*r)3 = (1*p)2* (1)*G*M/(4*pi2), collect terms for Venus.

r3 = p2*G*M/(4*pi2), Kepler.
(8)*r3 = (4)*p2* (2)*G*M/(4*pi2), multiplied both sides by (8).
(2*r)3 = (2*p)2* (2)*G*M/(4*pi2), collect terms for Earth.

r3 = p2*G*M/(4*pi2), Kepler.
(64)*r3 = (16)*p2* (4)*G*M/(4*pi2), multiplied both sides by (64).
(4*r)3 = (4*p)2* (4)*G*M/(4*pi2), collect terms for Mars.

r3 = p2*G*M/(4*pi2), Kepler.
(512)*r3 = (64)*p2* (8)*G*M/(4*pi2), multiplied both sides by (512).
(8*r)3 = (8*p)2* (8)*G*M/(4*pi2), collect terms for the asteroids.

r3 = p2*G*M/(4*pi2), Kepler.
(4096)*r3 = (256)*p2* (16)*G*M/(4*pi2), multiplied both sides by (4096).
(16*r)3 = (16*p)2* (16)*G*M/(4*pi2), collect terms. for Jupiter

r3 = p2*G*M/(4*pi2), Kepler.
(32768)*r3 = (1024)*p2* (32)*G*M/(4*pi2), multiplied both sides by (32768).
(32*r)3 = (32*p)2* (32)*G*M/(4*pi2), collect terms for Saturn.

r3 = p2*G*M/(4*pi2), Kepler.
(262144)*r3 = (4096)*p2* (64)*G*M/(4*pi2), multiplied both sides by (262144).
(64*r)3 = (64*p)2* (64)*G*M/(4*pi2), collect terms for Uranus.

r3 = p2*G*M/(4*pi2), Kepler.
(1000000)*r3 = (10000)*p2* (100)*G*M/(4*pi2), multiplied both sides by (1000000).
(100*r)3 = (100*p)2* (100)*G*M/(4*pi2), collect terms for Neptune.

r3 = p2*G*M/(4*pi2), Kepler.
(2097152)*r3 = (16384)*p2* (128)*G*M/(4*pi2), multiplied both sides by (2097152).
(128*r)3 = (128*p)2* (128)*G*M/(4*pi2), collect terms for Pluto.

In each example, except Neptune, the x value has doubled. The radius and period has increased. There are minimum harmonics, which might perturb the orbits and remove planets, in this expotential series. These are safe orbits for the planets to endure for a long time.

Orbital radius increase examples using ln/ln values starting with Mercury:
57.9E9_m^1.025 = 107.58E9_m, for Venus
107.58E9_m^1.013 = 149.678E9_m, for Earth
149.678E9_m^1.016 = 225.923E9_m, for Mars
225.923E9_m^1.022 = 401.557E9_m, for the Asteroids
401.557E9_m^1.025 = 783.143E9_m, for Jupiter
783.143E9_m^1.022 = 1430.56E9_m, for Saturn
1430.56E9_m^1.025 = 2880.01E9_m, for Uranus
2880.01E9_m^1.016 = 4557.68E9_m, for Neptune
4557.68E9_m^1.009 = 5924.78E9_m, for Pluto

Orbital period increase examples using ln/ln values starting with Mercury:
7.598E6_s^1.059 = 19.349E6_s, for Venus
19.349E6_s^1.029 = 31.475E6_s, for Earth
31.475E6_s^1.0137 = 59.621E6_s, for Mars
59.621E6_s^1.048 = 140.804E6_s, for the Asteroids
140.804E6_s^1.052 = 373.547E6_s, for Jupiter
373.547E6_s^1.046 = 926.134E6_s, for Saturn
926.134E6_s^1.051 = 2654.45E6_s, for Uranus
2654.45E6_s^1.031 = 5201.37E6_s, for Neptune
5201.37E6_S^1.018 = 7780.50E6_m, for Pluto

The masses seem unrelated to the forces which give us Titius-Bode. This suggest that the source of the spacing of the planets is orbital clearing or electromagnetic.

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