Titius-Bode                                             Modified 12-14-08
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 which is twice as far away as Saturn or Pluto which is four times away at 128. There are new planets or planetoids farther out to be added to the list. Do they fit Titius-Bode? 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. An "au" is the distance from the sun to the earth. The graph and table show it is not perfect but close enough to be very 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 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 Earth 2 1.0 1.000 5.98 1.000 149.6 25.73 1.013 31.559 17.267 Mars 4 1.6 1.524 0.642 1.874 227.9 26.15 1.029 59.374 17.899 Asteroids 8 2.8 2.7 4.436 403.9 26.72 1.022 140.01 18.757 Jupiter 16 5.2 5.203 1900 11.86 778.3 27.38 1.025 374.55 19.741 Saturn 32 10 9.539 570 29.46 1427 27.99 1.022 929.77 20.650 Uranus 64 19.6 19.18 87 84.07 2871 28.69 1.025 2650.9 21.698 Neptune 30.06 102 164.8 4497 29.13 5201.2 22.372 Pluto 128 38.8 39.53 0.7 247.7 5913 29.41 1.025 7843.6 22.783

The rule: y = 0.3*x +.4, where y is the distance to the sun, in astronomical units, and
x = 0,1,2,4,8,16,32,64,128. The most interesting feature is the "x" values, doubling as planets are added, to the right. 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:
We have left out Neptune because it kills the pattern. 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. A big event for Mars watchers.
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.

Periods: The period of a circular orbit is
p = 2*pi*r/vt, r is radius, the distance to the sun and vt is the velocity, tangent to the orbit. If the radius doubles then the period doubles, in an octave like 2:1 orbital resonance, if vt stays the same, as might be seen in a Hubble expansion.

Kepler and Titius-Bode:
m*vt2/r = G*m*M/r2, The centrifugal force equals the gravitational force. This is just another way of writing Kepler's third law.
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?
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.

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.

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. In each example, the radius has doubled, the period has doubled and the multiplier times G*M/(4*pi2) has doubled.

Plutor/Mercuryr = 5913E9_m/57.9E9_m = 102.1: 102.13/2 = 1032
Plutop/Mercuryp = 7843E6_s/7.598E6_s = 1032: 10322/3 = 102.1

More mass matters?
The density, at a certain radius is,
{m/r}/(4*pi*r2),
where {m/r} is the mass/radius ratio. The gas cloud that formed our solar system, may have followed the mass/radius ratio system, that we observed for our galaxy.

Spherical layers: The volume contained in spherical layers that double in radius is,
4/3*pi*r3- 4/3*pi*(r/2)3
4/3*pi*7/8*r3
7/6*pi*r3

The mass = density * volume, contained in each onion layer shell of the 8*r3 = 2*4*p2*G*M/(4*pi2),
(2*r)3 = (2*p)2*2*G*M/(4*pi2),
sphere is,
{m/r}/(4*pi*r2) * 7/6*pi*r3 = {m/r}*7/24*r = {m/r}*.29*r, but the mass of the planets seems unrelated to their radius.

Toroids: The volume contained in toroids in the spherical layers that double in radius is,
pi*r2*2*pi*R
pi*(1/2*r)2*pi*2*(3/4*r)
3/32*pi2*r3

The mass = density * volume, contained in each toroid, in each onion layer shell of the sphere is,
{m/r}/(4*pi*r2) * 3/32*pi2*r3 = {m/r}*pi/24*r = {m/r}*.13*r, but the mass of the planets again, seems unrelated to their radius.

The masses are trivial in relationship to the forces which give us Titius-Bode. This suggest that the source of the spacing of the planets is orbital resonance or electromagnetic.

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