Titius–Bode law

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 a function of planetary sequence. The formula suggests that, extending outward, each planet would be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit and was eventually superseded as a theory of Solar System formation. It is named for Johann Daniel Titius and Johann Elert Bode.'... 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 five.'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.This latter point seems in particular to follow from the astonishing relation which the known six planets observe in their distances from the Sun. Let the distance from the Sun to Saturn be taken as 100, then Mercury is separated by 4 such parts from the Sun. Venus is 4+3=7. The Earth 4+6=10. Mars 4+12=16. Now comes a gap in this so orderly progression. After Mars there follows a space of 4+24=28 parts, in which no planet has yet been seen. Can one believe that the Founder of the universe had left this space empty? Certainly not. From here we come to the distance of Jupiter by 4+48=52 parts, and finally to that of Saturn by 4+96=100 parts. 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 a function of planetary sequence. The formula suggests that, extending outward, each planet would be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit and was eventually superseded as a theory of Solar System formation. It is named for Johann Daniel Titius and Johann Elert Bode. The law relates the semi-major axis a {displaystyle a} of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10: where x = 0 , 3 , 6 , 12 , 24 , 48 , … {displaystyle x=0,3,6,12,24,48,ldots } with the exception of the first step, each value is twice the previous value.There is another representation of the formula: a = 2 n × 3 + 4 {displaystyle a=2^{n} imes 3+4} where n = − ∞ , 0 , 1 , 2 , … {displaystyle n=-infty ,0,1,2,ldots } .The resulting values can be divided by 10 to convert them into astronomical units (AU), resulting in the expression for m = − ∞ , 0 , 1 , 2 , … {displaystyle 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. 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 A similar sentence, likely paraphrased from Gregory, 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 ; but we are not sure that there are no more.' To this, in his 1766 translation of Bonnet's work, Johann Daniel Titius added two of his own paragraphs, at the bottom of page 7 and at the beginning of page 8. The new interpolated paragraph is not found in Bonnet's original text, nor in translations of the work into Italian and English.