What Are Kepler's Laws?

Kepler's law is the three major laws on planetary motion proposed by German astronomer Kepler. The first and second laws, published in 1609, were summarized by Kepler from astronomer Tycho's observations of Mars' position; the third law was published in 1619. These three laws are also called ellipse law, area law and harmony law.

Kepler's first law

Kepler's second law

Kepler's third law

Chinese name: Kepler's law
Foreign name: Kepler's law
nickname: Kepler's law

Presenter: Kepler
Presentation time: 1618
Applied discipline: astronomy
Scope of application: aerospace

Kepler's law

Law of Ellipse All planets orbit around the sun are ellipses, and the sun is at a focal point of the ellipses.

Law of Area The line connecting the planet and the sun sweeps the same area in the same time interval.

Stellar time of all planets around the sun

The square of) is proportional to the cube of their major axis (a _i ), ie

Since then, scholars have modified the first law so that the orbits of all planets (and comets) belong to a conic curve, and the sun is at one of their focal points. The second law is accurate only if the mass of the planet is much smaller than that of the sun. If you consider that planets also attract the sun, this is a two-body problem. The exact formula of the modified third law is:

(Where m ₁ and m ₂ are the masses of the two planets; m _a is the mass of the sun).

Mathematical Derivation of Kepler's Law

Kepler's law is about the movement of planets around the sun, while Newton's law is more broadly about the movement of several particles due to the attraction of each other by gravity. There are only two particles, one of which is super lighter than the other. Under these special conditions, light particles will move around heavy particles, just like a planet moves around the sun according to Kepler's law. However, Newton's law allows other solutions. Planetary orbits can be parabolic or hyperbolic. This is unpredictable by Kepler's law. Under the condition that one particle is not super lighter than another particle, according to the solution of the generalized two-body problem, each particle moves around their common center of mass. This is also unpredictable by Kepler's law.

Kepler's law uses either geometric language or equations to link the coordinates and time of the planets with the orbital parameters. Newton's second law is a differential equation. Kepler's law is guided by the art of solving differential equations. We will guide Kepler's second law first, because Kepler's first law must be guided by Kepler's second law. ^[1]

Mathematical proof of Kepler's law

Proof of the first law ^[2]

set up

Thus, the angular velocity is

Differential of time and differential of angle have the following relationship:

According to the above relationship, the derivative of radial distance with respect to time is:

Find the derivative again:

Substituting into the equation of radial motion

Divide this equation by

, You can get a simple constant coefficient non-homogeneous linear total differential equation to describe the planetary orbit:

To understand this differential equation, first list a special solution

Then solve the remaining homogeneous linear total differential equations with constant coefficients,

Its solution is

Here,

versus

Is constant. Combining the special solution and the homogeneous equation solution, we can get the general solution

Select the axis and let

. Back

among them,

Is the eccentricity.

This is the polar coordinate equation of the conic curve, and the origin of the coordinate system is one of the focal points of the conic curve. if

,then

Described are elliptical orbits. This proves Kepler's first law. ^[3]

Proof of the second law ^[2]

Kepler's second law says this: in equal time, planets and stars

Kepler's law The lines swept across the same area. O is the star, and the straight line AC is the trajectory of the planet when it is not attracted to gravity. Let the time interval t between the planets from A to B and from B to C be equal, the time at A is t1, B is t2, and C is t3. Assuming that the planet is not affected by the gravitational force of O, the areas SABO and SBCO swept at this time are equal (equal base and same height). The planet is subject to gravity, because the direction of gravity points to the star at all times, so the period from t1 to t3

In time, the total effect of the direction of gravity of the planet should be in the direction of BO (this requires a bit of vector knowledge). Therefore, the position C 'of the planet at time t3 should be obtained by adding two vectors: vector AC + vector CC' (as CC 'is parallel to BO, so a vector along the BO direction is equivalent to CC'). In this way, SBCO = SBC'O (same height as the bottom). Therefore, SBC'O = SABO. Because t is arbitrary, the area swept by the line between the planet and the star is the same in equal time. ^[4]

Proof of the third law ^[2]

In the figure, A and B are the perihelion and distant points of the planet's motion, respectively.

with

Respectively the speed of the planet at this point, as the speed is along the tangent of the orbit, it can be seen

with

The directions of both are perpendicular to the long axis of this ellipse, and the corresponding area velocities of the planets at these two points are

{1}

According to Kepler's second law, there should be

, So get

{2}

The total mechanical energy E of the planetary motion is equal to the sum of its kinetic energy and potential energy. When he passes the perihelion and distant points, his mechanical energy should be

{3}

According to the conservation of mechanical energy, there should be

, Therefore

{4}

Solve from {2} {4}

{5}

From {5} and {1}, the area velocity is

The area of the ellipse is

, Then this planetary motion period is

{6}

Squaring both sides of the {6} formula, we have

Note:

Is the semi-major axis,

Is the semi-minor axis,

Is the half focal length

Scope of Kepler's Law

Kepler's second law

Kepler's law applies to all celestial bodies in the universe. It has universal significance in the field of macro low-speed celestial body motion. For high-speed celestial motion, Kepler's law provides an equation for its return to low-speed states.

In other words, Kepler's second law and its inferences apply not only to all planets orbiting the sun, but also to satellites centered on planets, but also to the case where a single planet or satellite moves along an elliptical orbit.

Only suitable for celestial bodies moving at low macro speed. When it was proposed, no strict proof was given, but it laid the foundation for the later proofs of many laws. ^[4]

Kepler's third law

Kepler's law is a universal law applicable to all two-body problems. Kepler's law is not only applicable to the solar system. He holds true for both gravitational systems with central celestial bodies (such as planet-satellite systems) and binary star systems. For several celestial bodies moving around the same central celestial body, the ratio (R ^ 3 / T ^ 2) of the cube of their orbital radius to the square of the period is equal to (GM / 4 ^ 2), which is the central celestial body mass. This ratio is a constant that has nothing to do with planets, and only depends on the mass of the central body, so the same ratio is the same for M. ^[5]

A brief history of Kepler's law

Kepler's law discovery background

Kepler's law is the law of planetary motion discovered by Kepler. He published two laws on planetary motion in his published "New Astronomy" in 1609, and in 1618 he discovered the third law. Kepler was fortunate enough to obtain very accurate astronomical data observed and collected by the famous Danish astronomer Tycho Brahe for more than 20 years. About 1605, according to Brah's planetary position data, following Copernicus's theory of uniform circular motion, through 4 years of calculations, the data observed in Tycho was found to have an error of 8 '. The Kepler believed that The data is correct, so he questioned the "perfect" divine movement (circular motion at constant speed). After a lot of calculations for nearly 6 years, Kepler came to the first and second laws, and after another 10 years, After a lot of calculations, the third law was obtained. Kepler's law challenged the Aristotles and Ptolemies to astronomy and physics. He argued that the earth is constantly moving; the planetary orbits are not epicycles, but rather elliptical; the speed of the planets' revolutions is not constant. These arguments greatly shaken the astronomy and physics of the time. After nearly a century of studying the stars and the moon, abandoning sleep and forgetting to eat, physicists were finally able to explain the truth in physical theory. Newton used his second law and the law of universal gravitation to mathematically prove Kepler's law and let people understand its physical meaning.

Kepler's law planet orbit

The sun is the center of the universe. The earth revolves around the sun like other planets. In the 16th century, astronomer Copernicus proposed a new era-leading theory of the solar system with his bold insight, which brought about a technological revolution. But it wasn't until half a century later that German mathematician Kepler used observational data provided by Danish astronomer Budi Valley Brahe to draw the first accurate map of the solar system. Kepler's toil reinforced Copernicus's theory. He fought alone, and finally used the data of Tycho Brahe to accurately explain the movement of the planet. During his lifetime, his achievements were not acknowledged, but his insight remains the basis of modern universe theory.

Kepler's Law Kepler

Kepler portrait Kepler (Johannes Kepler, 1571-1630), a German astronomer. Kepler was born in a small German family on December 27, 1571. As soon as he came to earth, he suffered many misfortunes. Smallpox made him pockmarked, and scarlet fever broke his eyes.

At the age of 17, Kepler entered the University of Tübingen to study theology, and in 1591 he obtained a master's degree in theology. But because his father was heavily in debt, he had to drop out of school. Due to his infirmity and illness, his parents thought he was only suitable as a priest because the profession was easier. However, Kepler was very talented in mathematics. When he learned some theories about natural science, he abandoned the idea of being a priest and finally taught natural science in a university in Austria.

In 1600, Kepler, 30, wrote a letter to the unknown Danish astronomer Tycho. He told Tycho the results and ideas of his astronomy. After Tycho saw it, he was amazed by Kepler's talent, and immediately wrote to invite him to be his assistant. But just 10 months after Kepler came to Tycho, the old man died. Kepler inherited the valuable information left by the old man, including the old man's observations of Mars movement.

Kepler Kepler used the observation data accumulated in Tycho Valley for many years, carefully analyzed and found that the planets were moving in elliptical orbits, and proposed the three laws of planetary motion (ie Kepler's law), which laid the foundation for Newton's discovery of the law of universal gravitation.

Based on Tycho's work, Kepler compiled a "Rudolph Star Table" after a lot of calculations, which lists the positions of 1005 stars. This catalogue is much more accurate than other catalogues, so until the middle of the eighteenth century, the Rudolph Catalogue was still regarded as a treasure by astronomers and navigators, and its form has remained almost unchanged to this day.

Kepler's main works include "The Mystery of the Universe", "Optics", "The Theory of Cosmic Harmony", "Summary of Copernicus Astronomy", "The Theory of Comet" and "The Strange 1631 Astrology" and so on. Among them, Kepler found the simplest world system in "Cosmic Harmony", and only 7 ellipses can describe the system of celestial movement; in "Cometism", he pointed out that the tail of a comet always carries a back The sun is caused by the repulsion of the material from the comet, which is a correct prediction of the existence of radiation pressure half a century ago; in addition, Kepler also found an approximate law of atmospheric refraction. In honor of Kepler's achievements, the International Astronomical Union decided to name the asteroid 1134 as Kepler asteroid. ^[1]

Kepler's law discovery process

In 1601, Tycho died. John Kepler took over Tycho's job and began compiling Rudolph's catalog. But Kepler's interest and attention are more focused on improving and perfecting Copernicus' heliocentric theory, and studying the nature of planetary orbits. He found that Tycho's observations were not consistent with the Copernican and Ptolemy systems. He was determined to find the cause of this inconsistency and the true orbit of the planet.

Kepler The initial research began with Mars, where the differences between observations and theories are prominent. He used traditional uniform circular motion plus eccentric circles to calculate, but all failed. After nearly 70 calculations of various planetary orbital shapes for 4 years, Kepler realized that the uniform velocity circular motion and eccentric circular orbital patterns of the Copernican system did not match the actual orbits of Mars. So he boldly abandoned the "uniform circular motion" prejudice that has ruled human thought for 2000 years, and tried to use other geometric curves to represent the shape of Mars' orbit. He believes that the focus of the planetary orbit should be on the sun that produces the center of gravity, and further concluded that the linear velocity of Mars' motion is not uniform, faster near the sun and slower at the far sun, and concluded that the diameter of the sun from Mars The area swept in one day is equal. Kepler generalized this conclusion to other planets, and the results were also consistent with the observed data. In this way, he first got the law of equal area of the planets. He later discovered that Mars's orbit was not a perfect circle, but an ellipse with the focus on the sun. He also applied this conclusion to other planets. So he got the law of elliptical orbit of the planet. These two laws were published in his book "New Astronomy" published in 1609. But he was not satisfied with what he had achieved. He was eager to find a general model that fits all planets, linking the planets together. He firmly believed that there was a simple rule that linked all the planets together.

Kepler contributed greatly to astronomy Encouraged by this belief, Kepler endured the great misfortunes of individuals in the family. Under the difficult conditions that few people knew and supported, after nine years of repeated calculations and assumptions, he finally found a large amount of observation data in 1618. The harmony of the hidden numbers behind: the square of the planet's revolution period is proportional to the cube of their average distance from the sun. This is the law of cycles. In 1619, he introduced the third law in the book "Harmony of the Universe", and he couldn't help writing: "Recognizing this truth, this is beyond my best expectations. The big picture is set, this book It was written, it may be read by some people today, or it may be read by future generations. It will probably wait for a century before there are worshippers, and I do nt care about that. "

Kepler's Three Laws is another revolution in astronomy. It completely destroys Ptolemy's complicated current round universe system, and perfects and simplifies Copernicus' heliocentric universe system. Kepler's greatest contribution to astronomy was his attempt to establish celestial dynamics and explain the dynamics of the structure of the solar system on a physical basis. Although he puts forward that the magnetic force from the sun drives the planets to orbit. But it has great inspiration for future generations to find out the mystery of the structure of the solar system. It also gives important hints for the establishment of classical mechanics and the discovery of Newton's law of universal gravitation.

Known as the "King of the Stars", Tycho Brahe has achieved many achievements in astronomical observations, leaving over 20 years of observation data and a precise star catalog after his death. His assistant Kepler used these observations and star tables to compile new star tables. However, at the beginning of the work, I encountered difficulties. It is not feasible to compile Mars's running table according to the orbit of the circle. The "sly guy" on Mars always does not follow the command and always loves to deviate. After analysis and calculation, Kepler found that if Mars's orbit is not a perfect circle, but an ellipse, then the contradiction will disappear. After a long period of detailed and complicated calculations, he finally found that the planets orbit in an ellipse in the plane passing through the sun, and the sun is at a focal point of the ellipse. This is the first law of planetary motion, also known as "the law of orbits". ^[6]

As Kepler continued his research, Mars, the "stupid", lied to him again. It turned out that Kepler and his predecessors studied planetary motion as constant velocity. He worked hard for one year according to this method, but still could not get results. Later, he discovered that the speed of the planets running in an elliptical orbit was not constant, but that the area swept by the line between the planet and the sun was the same in equal time. This is the second law of planetary motion, also known as the "area law".

After 9 years of hard work, Kepler has found the third law of planetary motion: the ratio of the square of the revolution period of all planets in the solar system to the cube of the semi-major diameter of the planetary orbit is a constant.

Influence of Kepler's Law

First of all, Kepler's law shows an extremely brave creative spirit in scientific thinking. Long before Copernicus created the heliocentric universe system, many scholars put forward different opinions on the concept of heaven and earth and movement. But the idea that celestial bodies follow perfect uniform circular motion has never been doubted. Kepler resolutely denied it. This is a very bold idea. Copernicus knew that several circles could be combined to produce an ellipse, but he had never used an ellipse to describe the orbit of a celestial body. As Kepler said, "Copernicus was unaware of the wealth he had at his fingertips."

Secondly, Kepler's law completely destroyed Ptolemy's current system, liberating the Copernican system from his Majesty, and brought it full integrity and rigor. Copernicus abandoned the preconceived opinion of the ancient Greeks, that is, the essential difference between heaven and earth, and obtained a much simpler system. But it still has to use more than thirty circles to explain the apparent movement of celestial bodies. Kepler found the simplest world system and solved it with only seven ellipse theory. Since then, the motion of the planet can be easily and accurately estimated without the need for any wheel and eccentric circle.

Third, Kepler's law gives people a clear idea of planetary motion. It proves that the planetary world is a shapely (or "harmonious" as Kepler calls it) system. The central celestial body of this system is the sun, which is dominated by some unified force from the sun. The sun is in one of the focal points of each planet's orbit. The revolution period of a planet depends on the distance between each planet and the sun, and has nothing to do with mass. In the Copernican system, although the sun is located at the "center" of the universe, it does not play this role, because no planet's orbital center coincides with the sun.

There are many scientific discoveries made in the history of science due to the use of scientific experiments and recorded data made by predecessors. But like the discovery of the laws of planetary motion, from the intensive observation of Tycho's more than 20 years to Kepler's long and careful calculation, it is rare that the road is so difficult and the results are so brilliant. It's all obtained without a telescope! ^[7]