Titius-Bode law, referred to as "Bode's law", is a simple geometric rule about the orbits of planets in the solar system. It was discovered in 1766 by Johann Daniel Titius (1729~1796), a German middle school teacher. It was later summarized into an empirical formula by Johann Elert Bode, director of the Berlin Observatory.

Meaning

Empirical law of the average distance of planets from the sun. In 1766, the German Titius proposed to take a sequence of 0, 3, 6, 12, 24, 48, 96, 192..., then add 4 to each number, and then divide by 10, you can approximately get The average distance of each planet from the Sun in astronomical units. In 1772, the German astronomer Bode further studied this problem and published this rule, so it was named Titius-Bode's rule, sometimes referred to as Titius' rule or Bode's rule. This rule can be expressed as: Calculated from nearest to far from the sun, corresponding to the nth planet (for Mercury, n is not taken as 1, but -∞), its distance from the sun a =0.4 0.3× 2n-2) (astronomical unit)

After the Titius-Bode rule was proposed, two discoveries gave it strong support. First, in 1781 F.W. Herschel discovered Uranus, which was almost exactly in the orbit predicted by the rules. Second, Tydeus predicted at that time that there should be a celestial body between Mars and Jupiter at a distance of 2.8 astronomical units from the sun. In 1801, Italian astronomer Piazzi discovered Ceres at this distance. Since then, astronomers have discovered many asteroids near this distance. However, this rule also has some shortcomings. For example, the calculated values ??for Neptune and Pluto are inconsistent with the observed values, and for Mercury, n is not set to 1 but to -∞, which is also difficult to understand. In addition, the average distance between some satellites and the planet they belong to also has a law similar to the Titius-Bode rule. Regarding the origin of the Titius-Bode rule, although some people have proposed some explanations, there is no conclusion yet.

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2011-7-22 10:12:00

In 1772, the German astronomer Bode wrote in his "Guide to Starry Sky Research" This book summarizes and publishes a rule about planetary distances proposed by Titius, a German physics professor 6 years ago. The main content of the rule is: take a sequence of 0, 3, 6, 12, 24, 48, 96... add 4 to each number and divide by 10 to get the actual distance of each planet from the sun. approximation.

For example, the average distance from Mercury to the sun is (0 4)/10 = 0.4 (astronomical units); the average distance from Venus to the sun is (3 4)/10 = 0.7; the average distance from the earth to the sun is ( 6 4)/10=1.0　The average distance from Mars to the Sun is (12 4)/10=1.6　If this continues, the distance of the next planet should be: (24 4)/10=2.8　But there is no planet at this distance, so There is no other celestial body. Bode believed that the "Creator" would not intentionally leave a blank in this place; Tidus believed that perhaps an undiscovered satellite of Mars was at this location, but in any case Tideus - Bode was Then there is a discontinuity at "2.8 (astronomical units)".

The two farthest planets known at that time were Jupiter and Saturn. If we continue to extrapolate according to the rules, the situation is encouraging. The comparison between the data given by the rules and the actual situation is as follows:

The data given by the rule are the actual distances of each planet from the sun (astronomical units)

Mercury 0.4　　0.387

Venus 0.7　　0.723

Earth 1.0 1.000

Mars 1.6 1.524 2.8

Jupiter 5.2 5.203

Saturn 10.0 The distances were very similar, so everyone began to believe that there should be a large planet to fill in the place of "2.8". Bode appealed to other astronomers, hoping that colleagues could organize to search for this "lost" planet.

Some enthusiastic astronomers began to search for the "lost" planets. Several years passed without any results. Just when everyone was a little discouraged and ready to give up this endless search, in 1781 the British astronomer Herschel accidentally discovered the seventh largest planet in the solar system, Uranus. Surprisingly, the average distance between Uranus and the sun is 19.2 astronomical units, which is in good agreement with the result calculated by the Titius-Bode rule (192 4)/10=19.6. At this time, the status of the rule suddenly rose. Almost everyone believed in it without a doubt, and completely believed that there must be a large planet in the vacant position of "2.8". However, the method was not appropriate, so it has never been found. .

But more than ten years have passed quickly, and there is still no news about this "lost" planet.

Until the beginning of 1801, an astonishing news came out from Sicily, Italy. Piazzi, the director of a remote observatory there, discovered a new celestial body during a routine observation. After calculating its The distance is 2.77 astronomical units, which is very similar to "2.8". The new celestial body was therefore considered to be the large planet that many people were desperately searching for but had never found, and was named "Ceres".

Then the diameter of Ceres was measured, and it was more than 700 kilometers. This confused everyone. Why is it not a big planet but a small planet? But the shocking things were yet to come. In the next year, in March 1802, the German doctor Obers discovered another planet - Pallas, between the orbits of Mars and Jupiter. In addition to being slightly smaller, Pallas The star was almost the same as Ceres, and the distance was basically the same. Then a third star - Juno and a fourth star - Vesta were discovered. In the end, the total number of asteroids discovered was as many as 500,000. They were all concentrated in a specific area between Mars and Jupiter, the so-called "asteroid belt", and its center position coincided with Tydeus. ——Data given by Bode's rule.

Why did the big planet turn into 500,000 asteroids? At that time, some people speculated whether the originally existing large planet exploded for some reason that was temporarily unknown to people?

In 1846 and 1930, Neptune and Pluto were discovered successively. These two discoveries were setbacks for the Tidius-Bode rule. Compare their rule values ??and actual distances as follows :

The actual distance between the regular value and the sun

Neptune (384 4) / 10 = 38.830.2

Pluto (768 4) / 10 = 77.239 .6

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Gaussian and small Discovery of Planets

Name: Karl Friedrich Gauss

Introduction: JOHANN KARL FRIEDRICH GAUSS: 1777-1855

German mathematician, born in Braunschweig, studied and taught at the University of G?ttingen. His achievements in life were extremely rich, involving almost all fields of mathematics, and he enjoyed the reputation of "Prince of Mathematics". His image was also printed on the German 10-mark banknote.

Quotation: DIE MATHEMATIK IST DIE K?NIGIN DER WISSENSCHAFTEN Mathematics is the queen of science.

In 1766, a German middle school teacher discovered that the distances between the planets in the solar system and the sun conformed to certain rules. Later, the director of the Berlin Observatory in Germany, Chang Bode, attributed this to the following empirical formula: The so-called Titius Bode's rule:

Where D is the distance from the planet to the sun, expressed in astronomical units. After substituting different N values ??into the solution, the following interesting results can be obtained:

Obviously, the known planets at that time all conformed to Tidius Bode's rule, but when N=3 But there is an empty space, and there is no celestial body corresponding to it. So many astronomers speculated that there should be an unknown planet there, so many astronomers began their own searches.

In 1781, British astronomer William Herschel accidentally discovered a new planet, Uranus, during a sky survey. After observations, people found that Uranus was not the planet they were looking for, but The discovery of Uranus strengthened people's trust in Tidius Bode's rule. Because the actual distance between Uranus and the Sun (19.2184 astronomical units) is very close to the calculated value of 19.6 when N=6. This gives people a huge boost of confidence.

On January 1, 1801, the good news came with the New Year's bell. Italian astronomer Piazzi discovered a new celestial body at the Palermo Observatory in Sicily. Of course, Piazzi did not confirm that it was a new planet at first. He once thought it was a comet. But later Piazzi had to suspend the observation because of illness. Later, Ceres was submerged in the glow of the sun, making observation difficult, so Piazzi was unable to continue the follow-up observation. This celestial body was "lost" in this way.

Piazzi made his observation results public and asked scholars from various countries to help find the missing celestial body. After hearing the news, the great German mathematician Gauss immediately started calculations. Gauss invented a new method to solve the approximate orbit of the planet based on the data of three observations. According to this new method, the problem was quickly solved, and the calculation was completed in a few days. Later, astronomers predicted it as predicted by Gauss. The planet was re-observed near its location. Gauss's method has been widely used in astronomy since then.

This newly discovered celestial body was later confirmed to be a planet between Jupiter and Mars and named Ceres. However, people soon discovered that there were many similar celestial bodies between Mars and Jupiter. Since they were not very large, they were collectively called asteroids. However, Tidius Bode's rule, which once played an important role in the search for asteroids, is rarely believed by anyone anymore. This is because Neptune, discovered in 1846, no longer complies with Tidius Bode's rule (calculated value 38.8, actual distance 30.1104 astronomical units). Some people believe that Titius Bode's rule may reflect some characteristics of solar system dynamics, but how to explain it and how to treat anomalies are difficult to solve. Therefore, after making a major contribution to the discovery of asteroids, Titius Spode's rule gradually faded out of the stage of history.

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About the planets The law of distance. Also known as Bode's rule. The empirical relationship was first proposed by J.D. Titius of Germany in 1766. In 1772, J.E. Bode of Germany published the summarized formula: an = 0.4 + 0.3 × 2n-2, where an is the nth number expressed in astronomical units. The average distance of planets from the sun, n is the order from nearest to far from the sun (but Mercury n=-∞ is an exception). The discovery of Uranus in 1781 coincided with the position of n=8, which prompted people to search for n=5 celestial bodies. In 1801, an asteroid was discovered (consistent with a5=2.8). However, the physical meaning of Bode's formula is unclear, and Neptune discovered in 1846 and Pluto discovered in 1930 deviate greatly from this formula. Therefore, many people still hold a negative attitude and believe that it is at best an empirical formula to help memory. With the deepening of research, many planetary distance formulas have been proposed, and the more commonly used form is an 1: an = β (β is a constant related to the mass of the planet). Moreover, in some satellite systems, regular satellites also have similar relationships. The physical meaning of this rule remains to be further explored.

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Astronomical unit:

———————— Astronomical Unit (English: Astronomical Unit, abbreviated as AU) is a unit of length, approximately equal to the average distance between the Earth and the Sun. One of the astronomical constants. The basic unit for measuring distances in astronomy, especially the distance between celestial bodies in the solar system. The average distance from the Earth to the Sun is one astronomical unit. One astronomical unit is approximately equal to 149.6 million kilometers. In 1976, the International Astronomical Union defined an astronomical unit as the distance between a particle with negligible mass, an undisturbed orbit and an orbital period of 365.2568983 days (one Gaussian year) and an object with a mass equal to about one sun. The currently accepted astronomical unit is 149,597,870,691 ±30 meters (approximately 150 million kilometers or 93 million miles).

When the astronomical unit was first used, its actual size was not very clear, but the distances of the planets could be calculated in astronomical units by using heliocentric geometry and the laws of planetary motion. Later, the actual size of the astronomical unit was finally determined accurately through parallax and, more recently, radar. However, due to the uncertainty of the gravitational constant (only five or six significant bits), the mass of the sun cannot be very accurate. If SI units are used when calculating planetary positions, their accuracy will inevitably be reduced during the unit conversion process. So these calculations are usually in units of solar mass and astronomical units, rather than kilograms and kilometers.

The distance of one astronomical unit.

Equivalent to the average distance from the earth to the sun, about 1.496×10^8km

In life, "astronomical units" (astronomical numbers) are often used to describe a very large number...

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Historical derivation

　Astronomical constants one. The basic unit for measuring distance in astronomy, especially the distance between celestial bodies in the solar system, represented by A.

Before 1938, the astronomical unit refers to the average distance from the center of mass of the Earth-Moon system to the Sun without the perturbation of a large planet (see perturbation theory), or in other words, it is the center of mass of the Earth-Moon system The semi-major diameter of a perturbation-free elliptical orbit around the sun. According to Kepler's law, there is the following relationship between Gauss's gravitational constant k, the mass of the sun S, the mass of the Earth-Moon system m, the average distance A between the Earth-Moon system and the Sun, and the Earth's revolution period around the sun T: [1]

When taking the mass of the sun as the astronomical mass unit (that is, taking S=1) and the average distance from the Earth-Moon system to the sun as the astronomical distance unit (that is, taking A=1), Gauss based on the inaccurate T at that time and the m/S value, k=0.01720209895. In 1938, the Sixth International Astronomical Union decided to fix the k value and not change it. According to this k value, when S=1, A=1 and m=0, the T value can be calculated to be 365.2568983263 calendar days.

Therefore, the definition of the astronomical unit can be changed to: when the revolution period is 365.2568983263 epochal days, the semi-major diameter of the elliptical orbit of a hypothetical non-perturbed planet with zero mass is equal to one Astronomical unit. Based on the accurate T value and m/S value, it can be calculated that the semi-major diameter of the daily orbit of the Earth-Moon system is 1.00000003 astronomical units. Since the movement of the Earth is affected by the perturbations of other celestial bodies, the average distance between the Sun and the Earth is actually 1.0000000236 astronomical units.

Before the 1960s, the astronomical unit was derived based on the measurement of solar parallax π⊙. In Newcomb's astronomical constant system, the solar parallax π⊙ = 8 minus 80, and the corresponding astronomical unit length is equal to 149,500,000 kilometers. Since the 1960s, radar astronomy has achieved precise results. Therefore, the astronomical unit is derived based on the speed of light c and the light travel time per unit distance τA. In 1964, the International Astronomical Union's astronomical constant system took A as 149,600×10 meters and used it as the basic constant.

This value started in 1968 and was used until 1983. In 1976, the International Astronomical Union's astronomical constant system took A as 1.49597870×10 meters and changed it to a derived constant. This value will be uniformly adopted from 1984.

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Calculation

In the 1976 Astronomical Constant System, it is defined as the distance that light travels in a vacuum in 499.004782 seconds. Its value is 1.49597870×10^11 meters (^ represents power). 1 astronomical unit is equal to 1.58129×10^(-5) light years, or 4.84813×10^(-6) parsecs.

Light year (ly) Light year is a unit of length, not a unit of time. A light year is the distance traveled by light in a vacuum in one year.

The speed of light in vacuum is constant (the speed is about 300,000 kilometers/second).

1ly=9.46x10^12km

Parsec (pc) 1pc refers to the angle of 1AU when viewed from a certain celestial body and the solar system is orthogonal to the line of sight in one year. 〃.

1pc=2.06×105AU=3.26ly

1PC is approximately equal to 30835997962819660.8 meters

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Calculation method

Astronomers use triangular parallax method, spectroscopic parallax method, star cluster parallax method, and statistical parallax method. Method, Cepheid parallax method and mechanical parallax method, etc., to determine the distance between stars and us. The determination of the distance between stars is of great significance for studying the spatial position of stars and determining the luminosity and motion speed of stars.

There are more than 50 stars within 16 light-years of the sun. The nearest star is Proxima Centauri, which is about 4.2 light-years away from the sun, or about 40 trillion kilometers. Trigonometric Parallax Method Measuring the distance between celestial objects is not an easy task. Astronomers divide the celestial objects that need to be measured into several levels according to their distance.

The celestial bodies that are relatively close to us are no more than 100 light years away from us (1 light year = 9.46?1012 kilometers). Astronomers use the trigonometric parallax method to measure their distances. The trigonometric parallax method places the celestial body being measured at the vertex of a very large triangle. The two ends of the diameter of the Earth's orbit around the sun are the other two vertices of the triangle. By measuring the angle of view from the Earth to that celestial body, and then using the Knowing the diameter of the earth's orbit around the sun, we can calculate the distance from that celestial body to us by relying on trigonometric formulas.

We cannot use the triangular parallax method to measure the distance between slightly more distant celestial bodies and the earth, because their parallax can no longer be accurately measured on the earth.

Moving star cluster method

At this time we need to use the kinematic method to measure the distance. The kinematic method is also called the moving star cluster method in astronomy, and is determined based on their moving speed. distance. However, when using the kinematic method, it must also be assumed that all stars in the moving cluster move in the Milky Way at equal and parallel speeds. For celestial bodies outside the Milky Way, kinematic methods cannot determine the distance between them and the Earth. Cepheid parallax method Cepheid parallax method is also called the standard candlelight method.

There is a formula in physics about the relationship between luminosity, brightness and distance. S∝L0/r2

Measure the luminosity L0 and brightness S of the celestial body, and then use this formula to know the distance r of the celestial body. Luminosity and brightness have different meanings. Brightness refers to how bright the luminous object we see is, which we can directly measure on the earth. Luminosity refers to the luminous ability of a luminous object itself. The key is to know how to get the distance. Astronomer LeWitt discovered "Cepheid variable stars" and there is a definite relationship between their light variation period and luminosity.

So the luminosity can be determined by measuring its light change period, and then the distance can be found. If there is a Cepheid variable star in a galaxy outside the Milky Way, then we can know the distance between this galaxy and us. For those more distant galaxies that cannot even be observed whether they contain Cepheids, of course we have to find another way.

The triangular parallax method and the Cepheid parallax method are the two most commonly used distance measurement methods. The scale of the former is hundreds of light years, and the scale of the latter is millions of light years. In the middle ground statistical and indirect methods are used. The largest celestial ruler is Hubble's law method, with a scale of the order of 10 billion light-years.

Hubble's Law Method

In 1929, Edwin Hubble studied the relationship between the radial velocity and distance of extragalactic galaxies. At that time, the radial velocities of only 46 extragalactic galaxies were available, and only 24 of them had calculated distances. Hubble derived a roughly linear proportional relationship between radial velocity and distance.

Modern precise observations have confirmed this linear proportional relationship

V = H0×d

Where v is the recession speed, d is the galaxy distance, H0=100h0km ． s-1Mpc (the value of h0 is 0. Using Hubble's law, you can first measure the red shift Δν/ν, find V through the Doppler effect Δν/ν=V/C, and then find d.

Hubble's law reveals that the universe is constantly expanding. This expansion is a uniform expansion of the entire space. Therefore, an observer at any point will see exactly the same expansion. From the perspective of any galaxy, all galaxies will see the same expansion. Taking it as the center and spreading out in all directions, the farther away the galaxies are from each other, the greater the speed.

Example: Pluto is 39.5 astronomical units from the sun.

Jupiter is 5.2 AU from the Sun.

Betelgeuse has an average diameter of 2.57 AU.

The moon is 0.0026 AU from the Earth.

Approximate conversion value

1 Astronomical Distance Unit (AU) = 1.49597870 × 10^11 meters = 149,600,000 kilometers = 92,960,000 miles = 490,800,000,000 feet

1 light year (ly)=9.4605536×10^15 meters=63239.8 astronomical distance units

1 parsec (PC)=3.085678×10^16 meters=206264.8 astronomical distance units=3.261631 Light years

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