Titius–Bode law

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The Titius–Bode law (sometimes termed just Bode's law) is a hypothesis that the bodies in some orbital systems, including the Sun's, orbit at semi-major axes in an exponential function of planetary sequence. The hypothesis correctly predicted the orbits of Ceres and Uranus, but failed as a predictor of Neptune's orbit.

Contents

Formulation

The law relates the semi-major axis a of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10:

where \scriptstyle n = 0, 3, 6, 12, 24, 48 \ldots, each value of \scriptstyle n > 3 twice the previous value. The resulting values can be divided by 10 to convert them into astronomical units (AU), which would result in the expression

for \scriptstyle m = -\infty, 0, 1, 2 \ldots

For the outer planets, each planet is predicted to be roughly twice as far from the Sun as the previous object.

History

The first mention of a series approximating Bode's Law is found in David Gregory's The Elements of Astronomy, published in 1715. In it, he says, "...supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety six."[1] A similar sentence, likely paraphrased from Gregory,[1] appears in a work published by Christian Wolff in 1724.

In 1764, Charles Bonnet said in his Contemplation de la Nature that, "We know seventeen planets that enter into the composition of our solar system [that is, major planets and their satellites]; but we are not sure that there are no more."[1] To this, in his 1766 translation of Bonnet's work, Johann Daniel Titius added the following unattributed addition, removed to a footnote in later editions:

Take notice of the distances of the planets from one another, and recognize that almost all are separated from one another in a proportion which matches their bodily magnitudes. Divide the distance from the Sun to Saturn into 100 parts; then Mercury is separated by four such parts from the Sun, Venus by 4+3=7 such parts, the Earth by 4+6=10, Mars by 4+12=16. But notice that from Mars to Jupiter there comes a deviation from this so exact progression. From Mars there follows a space of 4+24=28 such parts, but so far no planet was sighted there. But should the Lord Architect have left that space empty? Not at all. Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope. Next to this for us still unexplored space there rises Jupiter's sphere of influence at 4+48=52 parts; and that of Saturn at 4+96=100 parts.

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