In Newton's formulation, Kepler's laws sound like this::
First law: under the influence of gravity, one celestial body can move in relation to another in a circle, ellipse, parabola and hyperbola. It must be said that it is valid for all bodies between which mutual attraction acts.
- the formulation of Kepler's second law is not given, since this was not necessary.
- Kepler's third law was formulated by Newton as follows: the squares of the sidereal periods of the planets, multiplied by the sum of the masses of the Sun and the planet, are related as the cubes of the semi-major axes of the planets' orbits.
Kepler's First Law (Lawellipses)
Kepler's first law.
Every planetsolar system contacts byellipse, in one of the focuses of which isSun.
The shape of the ellipse and the degree of its similarity to a circle is characterized by the relation , where c- distance from the center of the ellipse to its focus (half the interfocal distance), a- major semi-axis. Magnitude e is called the eccentricity of the ellipse. At c= 0 and e= 0 the ellipse turns into a circle.
Proof of Kepler's first law
Law universal gravity Newton states that "every object in the universe attracts every other object along a line connecting the centers of mass of the objects, in proportion to the mass of each object, and inversely proportional to the square of the distance between the objects." This assumes that the acceleration a has the shape
Let us remember that in polar coordinates
In coordinate form we write
Substituting and into the second equation, we get
which is simplified
After integration we write the expression
for some constant, which is the specific angular momentum (). Let
The equation of motion in the direction becomes equal
Newton's law of universal gravitation relates force per unit mass to distance as
Where G- universal gravitational constant and M- mass of the star.
As a result
This differential equation It has common decision:
for arbitrary integration constants e and θ 0 .
Replacing u by 1/ r and putting θ 0 = 0, we get:
We got the equation conical section with eccentricity e and the origin of the coordinate system at one of the foci. Thus, Kepler's first law follows directly from Newton's law of universal gravitation and Newton's second law.
Kepler's Second Law (Law of Areas)
Kepler's second law.
Each planet moves in a plane passing through the center of the Sun, and in equal times the radius vector connecting the Sun and the planet sweeps out sectors of equal area.
In relation to our Solar system, two concepts are associated with this law: perihelion- the point of the orbit closest to the Sun, and aphelion- the most distant point of the orbit. Thus, from Keppler’s second law it follows that the planet moves unevenly around the Sun, having a larger linear speed than at aphelion.
Every year at the beginning of January, the Earth moves faster when passing through perihelion, so the apparent movement of the Sun along the ecliptic to the east also occurs faster than the average for the year. At the beginning of July, the Earth, passing aphelion, moves more slowly, and therefore the movement of the Sun along the ecliptic slows down. The law of areas indicates that the force controlling orbital motion planets, directed towards the Sun.
Proof of Kepler's second law
By definition, the angular momentum of a point particle with mass m and speed is written as:
.where is the radius vector of the particle and is the momentum of the particle.
A-priory
.As a result we have
. Let's differentiate both sides of the equation with respect to time
because the vector product parallel vectors is equal to zero. notice, that F always parallel r, since the force is radial, and p always parallel v a-priory. Thus we can say that is a constant.
Kepler's Third Law (Harmonic Law)
The squares of the periods of revolution of the planets around the Sun are related as the cubes of the semimajor axes of the planets' orbits.
Where T 1 and T 2 are the periods of revolution of two planets around the Sun, and a 1 and a 2 - the lengths of the semimajor axes of their orbits.
Newton found that gravitational attraction The formation of a planet of a certain mass depends only on the distance to it, and not on other properties, such as composition or temperature. He also showed that Kepler's third law is not entirely accurate - in fact, it includes the mass of the planet: 
, Where M is the mass of the Sun, and m 1 and m 2 – planetary masses.
Since motion and mass are found to be related, this combination of Kepler's harmonic law and Newton's law of gravity is used to determine the mass of planets and satellites if their orbits and orbital periods are known.
Proof of Kepler's third law
Kepler's second law states that the radius vector of a revolving body sweeps out equal areas for equal periods of time. If we now take very small periods of time at the moment when the planet is at points A And B(perihelion and aphelion), then we can approximate the area with triangles with heights, equal to the distance from the planet to the Sun, and the foundation, equal to the product planet speed over time.
Using the law of conservation of energy to total energy planets at points A And B, let's write
Now that we have found V B, we can find the sectorial velocity. Since it is constant, we can choose any point of the ellipse: for example, for the point B we get
However total area ellipse is equal 
(which is equal to π ab, because the 
). Time full turn, thus equals
The assumption of uniform circular motion of the planets of the solar system was not consistent with the heliocentric system of the world of N. Copernicus, since the discrepancies between the calculated and real positions of the planets in certain periods of time were significant. This contradiction was resolved by the outstanding German astronomer AND. Kepler . Based on many years of observations of the movements of the planets and the study of the works of his predecessors, Kepler discovered three laws, later named after him.
Kepler's first law, also called law of ellipses, was formulated by a scientist in 1609.
Kepler's first law: All planets in the solar system move in elliptical orbits, with the sun at one focus.
Point closest to the Sun P trajectory is called perihelion, point A, farthest from the Sun, is aphelion. The distance between aphelion and perihelion is the major axis of the elliptical orbit. Half the length of the major axis, semi-axle a, is the average distance from the planet to the Sun.
The average distance from the Earth to the Sun is called an astronomical unit (AU) and is equal to 150 million km.
The shape of the ellipse and the degree of its difference from a circle are determined by the ratio c/a, Where c— distance from the center of the ellipse to the focus, a- semimajor axis of the ellipse.
The greater this ratio, the more elongated the orbit of the planet’s motion (Fig. 37), the foci are farther apart from each other. If this ratio is zero, then the ellipse turns into a circle, the foci merge into one point - the center of the circle.
The orbits of the Earth and Venus are almost circular, for the Earth the ratio is c/a is 0.0167, for Venus - 0.0068. The orbits of other planets are more flattened. The most elongated orbit of Pluto, for which c/a = 0.2488. Not only planets around the Sun move in elliptical orbits, but also satellites (natural and artificial) around the planets. The point of motion of the satellite closest to the Earth is called perigee, the most distant is called apogee.
Kepler's second law (area law): the radius vector of the planet describes equal areas in equal periods of time.
Figure 38 illustrates Kepler's second law. It is clear from the figure that the radius vector is a segment connecting the focus of the orbit (essentially, the center of the Sun) and the center of the planet at any point in its motion along the orbit. In accordance with Kepler's second law, the areas of the sectors highlighted in color are equal to each other. Then it turns out that during the same period of time the planet travels different distances in its orbit, i.e. the speed of movement is not constant: v 2 >v 1 . How closer planet to perihelion, the faster its movement, as if it were trying to quickly get away from the scorching sun rays.Material from the site
Kepler's third law (harmonic): the squares of the periods of revolution of two planets around the Sun are related to each other, like the cubes of the semi-major axes of their orbits.
Remembering that the length of the semi-major axis of the orbit is considered the average distance from the planet to the Sun, we write mathematical expression Kepler's third law:
T 2 1 /T 2 2 =a 3 1 /a 3 2 ,
Where T1,T 2— periods of revolution of planets 1 and 2; a 1 >a 2— the average distance from planets 1 and 2 to the Sun.
Kepler's third law is true for both planets and satellites, with an error of no more than 1%.
Based on this law, it is possible to calculate the length of the year (the time of a complete revolution around the Sun) of any planet if its distance to the Sun is known. And vice versa - using the same law, you can calculate the orbit, knowing the period of revolution.
On this page there is material on the following topics:
Kepler's second law report
Anatomy of Kepler's law
Kepler's harmonic law
Kepler's laws astronomy message
Questions about this material:
An important role in the formation of ideas about the structure of the solar system was also played by the laws of planetary motion, which were discovered by Johannes Kepler (1571-1630) and became the first natural science laws in their modern understanding. Kepler's work created the opportunity to generalize the knowledge of mechanics of that era in the form of the laws of dynamics and the law of universal gravitation, later formulated by Isaac Newton. Many scientists up to early XVII V. believed that the movement celestial bodies should be uniform and occur along the “most perfect” curve-circle. Only Kepler managed to overcome this prejudice and establish the actual shape of planetary orbits, as well as the pattern of changes in the speed of movement of planets as they revolve around the Sun. In his searches, Kepler proceeded from the conviction that “number rules the world,” expressed by Pythagoras. He looked for relationships between various quantities characterizing the motion of planets - the size of orbits, the period of revolution, speed. Kepler acted virtually blindly, purely empirically. He tried to compare the characteristics of the movement of the planets with the patterns of the musical scale, the length of the sides of the polygons described and inscribed in the orbits of the planets, etc. Kepler needed to construct the orbits of the planets, move from the equatorial coordinate system indicating the position of the planet on celestial sphere, to a coordinate system indicating its position in the orbital plane. He used his own observations of the planet Mars, as well as many years of determinations of the coordinates and configurations of this planet carried out by his teacher Tycho Brahe. Kepler considered the Earth's orbit (to a first approximation) to be a circle, which did not contradict observations. In order to construct the orbit of Mars, he used the method shown in the figure below.
Let us know angular distance Mars from the vernal equinox point during one of the planet's oppositions is its right ascension "15 which is expressed by the angle g(gamma)Т1М1, where T1 is the position of the Earth in orbit at this moment, and M1 is the position of Mars. Obviously, after 687 days (this is the sidereal period of Mars’ orbit), the planet will arrive at the same point in its orbit.
If we determine the right ascension of Mars on this date, then, as can be seen from the figure, we can indicate the position of the planet in space, more precisely, in the plane of its orbit. The Earth at this moment is at point T2, and, therefore, the angle gT2M1 is nothing more than the right ascension of Mars - a2. Having repeated similar operations for several other oppositions of Mars, Kepler obtained a whole series of points and, drawing a smooth curve along them, constructed the orbit of this planet. Having studied the location of the obtained points, he discovered that the speed of the planet’s orbit changes, but at the same time the radius vector of the planet describes equal areas in equal periods of time. Subsequently, this pattern was called Kepler's second law.
The radius vector is called in this case a segment of variable size connecting the Sun and the point in the orbit in which the planet is located. AA1, BB1 and CC1 are the arcs that the planet traverses in equal periods of time. The areas of the shaded figures are equal to each other. According to the law of conservation of energy, the total mechanical energy of a closed system of bodies between which gravitational forces act remains unchanged during any movements of the bodies of this system. Therefore, the sum of kinetic and potential energy of a planet that moves around the Sun is constant at all points in its orbit and is equal to its total energy. As the planet approaches the Sun, its speed increases and its kinetic energy increases, but as the distance to the Sun decreases, its potential energy decreases. Having established the pattern of changes in the speed of motion of the planets, Kepler set out to determine the curve along which they revolve around the Sun. He was faced with the need to choose one of two possible solutions: 1) assume that the orbit of Mars is a circle, and assume that in some parts of the orbit the calculated coordinates of the planet diverge from observations (due to observation errors) by 8"; 2) assume that the observations do not contain such errors, and the orbit is not a circle. Confident in the accuracy of Tycho Brahe's observations, Kepler chose the second solution and found that the best way The position of Mars in its orbit coincides with a curve called an ellipse, while the Sun is not located at the center of the ellipse. As a result, a law was formulated, which is called Kepler's first law. Each planet revolves around the Sun in an ellipse, with the Sun at one focus.
As is known, an ellipse is a curve in which the sum of the distances from any point P to its foci is a constant value. The figure shows: O - center of the ellipse; S and S1 are the foci of the ellipse; AB is its major axis. Half of this value (a), which is usually called the semimajor axis, characterizes the size of the planet’s orbit. Point A closest to the Sun is called perihelion, and point B farthest from it is called aphelion. The difference between an ellipse and a circle is characterized by the magnitude of its eccentricity: e = OS/OA. In the case when the eccentricity is equal to O, the foci and the center merge into one point - the ellipse turns into a circle.
It is noteworthy that the book in which Kepler published the first two laws he discovered in 1609 was called “ New astronomy, or the Physics of the Heavens, set forth in studies of the movement of the planet Mars...". Both of these laws, published in 1609, reveal the nature of the motion of each planet separately, which did not satisfy Kepler. He continued his search for “harmony” in the movement of all planets, and 10 years later he managed to formulate Kepler’s third law:
T1^2 / T2^2 = a1^3 / a2^3
The squares of the sidereal periods of revolution of the planets are related to each other, like the cubes of the semimajor axes of their orbits. This is what Kepler wrote after the discovery of this law: “What 16 years ago I decided to look for,<... >finally found, and this discovery exceeded all my wildest expectations..." Indeed, the third law deserves the most highly appreciated. After all, it allows you to calculate the relative distances of the planets from the Sun, using already known periods their revolutions around the Sun. There is no need to determine the distance from the Sun for each of them; it is enough to measure the distance from the Sun of at least one planet. Size of semi-major axis earth's orbit - astronomical unit(a.e.) - became the basis for calculating all other distances in the Solar System. Soon the law of universal gravitation was discovered. All bodies in the Universe are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
F = G m1m2/r2
Where m1 and m2 are the masses of bodies; r is the distance between them; G - gravitational constant
The discovery of the law of universal gravitation was greatly facilitated by the laws of planetary motion formulated by Kepler and other achievements of astronomy in the 17th century. Thus, knowledge of the distance to the Moon allowed Isaac Newton (1643 - 1727) to prove the identity of the force that holds the Moon as it moves around the Earth and the force that causes bodies to fall to the Earth. After all, if the force of gravity varies in inverse proportion to the square of the distance, as follows from the law of universal gravitation, then the Moon, located from the Earth at a distance of approximately 60 of its radii, should experience an acceleration 3600 times less than the acceleration of gravity on the Earth's surface, equal to 9. 8 m/s. Therefore, the acceleration of the Moon should be 0.0027 m/s2.
The force that keeps the Moon in orbit is the force gravity, weakened by 3600 times compared to that acting on the Earth’s surface. You can also be convinced that when the planets move, in accordance with Kepler’s third law, their acceleration and the gravitational force of the Sun acting on them are inversely proportional to the square of the distance, as follows from the law of universal gravitation. Indeed, according to Kepler’s third law, the ratio of the cubes of the semi-major axes of the orbits d and the squares of the orbital periods T is a constant value: The acceleration of the planet is equal to:
A= u2/d =(2pid/T)2/d=4pi2d/T2
From Kepler's third law it follows:
Therefore, the acceleration of the planet is equal to:
A = 4pi2 const/d2
So, the force of interaction between the planets and the Sun satisfies the law of universal gravitation and there are disturbances in the movement of the bodies of the Solar System. Kepler's laws are strictly satisfied if the motion of two isolated bodies (the Sun and the planet) under the influence of their mutual attraction is considered. However, there are many planets in the Solar System; they all interact not only with the Sun, but also with each other. Therefore, the motion of planets and other bodies does not exactly obey Kepler's laws. Deviations of bodies from moving along ellipses are called perturbations. These disturbances are small, since the mass of the Sun is much more mass not only of an individual planet, but of all planets as a whole. The greatest disturbances in the movement of bodies in the solar system are caused by Jupiter, whose mass is 300 times greater than the mass of the Earth.
The deviations of asteroids and comets are especially noticeable when they pass near Jupiter. Currently, disturbances are taken into account when calculating the positions of planets, their satellites and other bodies of the Solar System, as well as trajectories spacecraft, launched for their research. But back in the 19th century. calculation of disturbances made it possible to make one of the most famous discoveries in science “at the tip of a pen” - the discovery of the planet Neptune. Conducting another survey of the sky in search of unknown objects, William Herschel in 1781 discovered a planet, later named Uranus. After about half a century, it became obvious that the observed motion of Uranus does not agree with the calculated one, even when taking into account disturbances from all known planets. Based on the assumption of the presence of another “subauranian” planet, calculations were made of its orbit and position in the sky. This problem was solved independently by John Adams in England and Urbain Le Verrier in France. Based on Le Verrier's calculations, German astronomer Johann Halle discovered on September 23, 1846, a previously unknown planet - Neptune - in the constellation Aquarius. This discovery was a triumph heliocentric system, the most important confirmation of the validity of the law of universal gravitation. Subsequently, disturbances were noticed in the movement of Uranus and Neptune, which became the basis for the assumption of the existence of another planet in the solar system. Her search was crowned with success only in 1930, when, after viewing large quantity photographs of the starry sky, Pluto was discovered.