He had extraordinary mathematical abilities. At the beginning of the 17th century, as a result of many years of observations of the motion of the planets, as well as on the basis of an analysis of the astronomical observations of Tycho Brahe, Kepler discovered three laws, which were later named after him.
Kepler's first law(law of ellipses). Each planet moves in an ellipse with the Sun at one of its foci.
Kepler's second law(law of equal areas). Each planet moves in a plane passing through the center of the Sun, and in equal time intervals, the radius vector connecting the Sun and the planet sweeps equal areas.
Kepler's third law(harmonic law). The squares of the orbital periods of the planets around the Sun are proportional to the cubes of the semi-major axes of their elliptical orbits.
Let's take a closer look at each of the laws.
Kepler's first law (the law of ellipses)
Every planet solar system revolves around an ellipse with the Sun at one of its foci.
The first law describes the geometry of the trajectories of planetary orbits. Imagine a section of the lateral surface of a cone by a plane at an angle to its base, not passing through the base. The resulting shape will be an ellipse. The shape of the ellipse and the degree of its similarity to a circle is characterized by the ratio e = c / a, where c is the distance from the center of the ellipse to its focus (focal distance), a is the semi-major axis. The value of e is called the eccentricity of the ellipse. For c = 0, and hence e = 0, the ellipse turns into a circle.
The point P of the trajectory closest to the Sun is called the perihelion. Point A, the furthest from the Sun, is aphelion. The distance between aphelion and perihelion is the major axis of the elliptical orbit. The distance between aphelion A and perihelion P is the major axis of the elliptical orbit. Half the length of the major axis, semi-axis 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.
Kepler's second law (law of areas)
Each planet moves in a plane passing through the center of the Sun, and for equal periods of time, the radius vector connecting the Sun and the planet occupies equal areas.
The second law describes the change in the speed of the planets around the Sun. 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. The planet moves around the Sun unevenly, having a greater linear velocity at perihelion than at aphelion. In the figure, the areas of the sectors highlighted in blue are equal and, accordingly, the time it takes for the planet to pass each sector is also equal. The Earth goes through perihelion in early January and aphelion in early July. Kepler's second law, the law of areas, indicates that the force that governs the orbital motion of the planets is directed towards the Sun.
Kepler's third law (harmonic law)
The squares of the orbital periods of the planets around the Sun are proportional to the cubes of the semi-major axes of their elliptical orbits. It is true not only for the planets, but also for their satellites.
Kepler's third law allows you to compare the orbits of the planets with each other. The farther the planet is from the Sun, the longer the perimeter of its orbit, and when moving along the orbit, its complete revolution takes longer. Also, with increasing distance from the Sun, the linear velocity of the planet decreases.
where T 1 , T 2 are the periods of revolution of the planet 1 and 2 around the Sun; a 1 > a 2 are the lengths of the semi-major axes of the orbits of planets 1 and 2. The semi-axis is the average distance from the planet to the Sun.
Later, Newton found that Kepler's third law is not entirely accurate - in fact, it also includes the mass of the planet:
where M is the mass of the Sun, and m 1 and m 2 are the mass of the planet 1 and 2.
Since motion and mass are related, this combination of Kepler's harmonic law and Newton's law of gravity is used to determine the masses of planets and satellites if their orbits and orbital periods are known. Just knowing the distance of the planet to the Sun, you can calculate the length of the year (the time of a complete revolution around the Sun). Conversely, knowing the length of the year, you can calculate the distance of the planet from the Sun.
Three laws of planetary motion discovered by Kepler gave an accurate explanation of the uneven motion of the planets. The first law describes the geometry of the trajectories of planetary orbits. The second law describes the change in the speed of the planets around the Sun. Kepler's third law allows you to compare the orbits of the planets with each other. The laws discovered by Kepler later served Newton as the basis for creating the theory of gravitation. Newton proved mathematically that all Kepler's laws are consequences of the law of gravity.
As soon as "whistleblowers" appeared on the site, claiming that mathematics is heresy, and gravitational attraction between planets does not exist at all, let's see how the law gravity allows us to describe phenomena established empirically. Below is the mathematical justification for Kepler's first law.
1. Historical digression
First, let's remember how this law came into being in general. In 1589, a certain Johannes Kepler (1571 - 1630) - a native of a poor German family - finishes school and enters the University of Tübingen. There he studied mathematics and astronomy. Moreover, his teacher Professor Mestlin, being a secret admirer of the ideas of Copernicus (the heliocentric system of the world), teaches at the university the "correct" theory - the Ptolemy's system of the world (ie geocentric). That, however, does not prevent him from introducing his student to the ideas of Copernicus, and soon he himself becomes a staunch supporter of this theory.
In 1596, Kepler published his Cosmographic Mystery. Although the work is of dubious scientific value even for those times, nevertheless it does not go unnoticed by the Danish astronomer Tycho Brahe, who has been making astronomical observations and calculations for a quarter of a century. He notices the independent thinking of the young scientist and his knowledge of astronomy.
Since 1600, Johann has been working as an assistant to Brahe. After his death in 1601, Kepler began to study the results of the work of Tycho Brahe - the data of many years of astronomical observations. The fact is that by the end of the 16th century, Prussian tables (tables of the movement of celestial bodies, calculated on the basis of the teachings of Copernicus) began to give significant discrepancies with the observed data: the error in the position of the planets reached 4-5 0 .
To solve the problem, Kepler was forced to complicate the Copernican theory. He abandons the idea that the planets move in circular orbits, which ultimately allows him to solve the problem of the discrepancy between theory and observed data. According to his findings, the planets move in elliptical orbits, with the Sun at one of its foci. So the distance between the planet and the Sun changes periodically. This conclusion is known as Kepler's first law.
2. Mathematical justification
Let us now see how Kepler's first law agrees with the law of universal gravitation. To do this, we derive the law of motion of a body in a gravitational field with spherical symmetry. In this case, the law of conservation of angular momentum of the body $\vec(L)=[\vec(r),\vec(p)]$ is satisfied. This means that the body will move in a plane perpendicular to the vector $\vec(L)$, and the orientation of this plane in space is unchanged. In this case, it is convenient to use the polar coordinate system $(r, \phi)$ with origin at the source of the gravitational field (ie, the vector $\vec(r)$ is perpendicular to the vector $\vec(L)$). Those. we place one of the bodies (the Sun) at the origin of coordinates, and below we derive the law of motion of the second body (the planet) in this case.
The normal and tangential components of the velocity vector of the second body in the selected coordinate system are expressed by the following relations (hereinafter, the dot means the time derivative):
$$ V_(r)=\dot(r); V_(n)=r\dot(\phi) $$
The law of conservation of energy and angular momentum in this case has the following form:
$$E = \frac(m\dot(r)^2)(2)+\frac(m(r\dot(\phi))^2)(2)-\frac(GMm)(r)=const \hspace(3cm)(2.1)$$ $$L = mr^2\dot(\phi)=const \hspace(3cm)(2.2)$$
Here $G$ is the gravitational constant, $M$ is the mass of the central body, $m$ is the mass of the "satellite", $E$ is the total mechanical energy of the "satellite", $L$ is the magnitude of its angular momentum.
Expressing $\dot(\phi)$ from (2.2) and substituting it into (2.1), we get:
$$ E = \frac(m\dot(r)^2)(2)+\frac(L^2)(2mr^2)-\frac(GMm)(r) \hspace(3cm)(2.3) $ $
We rewrite the resulting ratio as follows:
$$ dt=\frac(dr)(\sqrt(\frac(2)(m)(E-\frac(L^2)(2mr^2)+\frac(GMm)(r)))) \hspace (3cm)(2.4)$$
From relation (2.2) it follows:
$$ d\phi=\frac(L)(mr^2)dt $$
Substituting expression (2.4) instead of $dt$, we get:
$$ d\phi=\frac(L)(r^2)\frac(dr)(\sqrt(2m(E-\frac(L^2)(2mr^2)+\frac(GMm)(r) ))) \hspace(3cm)(2.5) $$
To integrate the resulting expression, we rewrite the expression under the root in brackets in the following form:
$$ E-((\frac(GMm^(3/2))(\sqrt(2)L))^2 - \frac(GMm)(r) + \frac(L^2)(2mr^2) ) + (\frac(GMm^(3/2))(\sqrt(2)L))^2=$$ $$ =E-(\frac(GMm^(3/2))(\sqrt(2 )L)-\frac(L)(r\sqrt(2mr)))^2 + (\frac(GMm^(3/2))(\sqrt(2)L))^2=$$ $$ = \frac(L^2)(2m)(\frac(2mE)(L^2)+(\frac(GMm^2)(L^2))^2-(\frac(GMm^2)(L^ 2)-\frac(1)(r))^2) $$
Let us introduce the following notation:
$$ \frac(GMm^2)(L^2)\equiv\frac(1)(p) $$
Continuing the transformations, we get:
$$ \frac(L^2)(2m)(\frac(2mE)(L^2)+(\frac(GMm^2)(L^2))^2-(\frac(GMm^2)( L^2)-\frac(1)(r))^2)=$$ $$\frac(L^2)(2m)(\frac(2mE)(L^2) + \frac(1)( p^2)-(\frac(1)(p)-\frac(1)(r))^2)=$$ $$\frac(L^2)(2m)(\frac(1)(p ^2)(1+\frac(2EL^2)((GM)^2m^3))-(\frac(1)(p)-\frac(1)(r))^2) $$
Let's introduce the notation:
$$ 1+\frac(2EL^2)((GM)^2m^3) \equiv e^2 $$
In this case, the converted expression takes the following form:
$$ \frac(L^2e^2)(2mp^2)(1-(\frac(p)(e) (\frac(1)(p)-\frac(1)(r)))^2 ) $$
For convenience, we introduce the following variable:
$$ z=\frac(p)(e) (\frac(1)(p)-\frac(1)(r)) $$
Now equation (2.5) takes the form:
$$ d\phi=\frac(p)(er^2)\frac(dr)(\sqrt(1-z^2))=\frac(dz)(\sqrt(1-z^2))\ hspace(3cm)(2.6) $$
We integrate the resulting expression:
$$ \phi(r)=\int\frac(dz)(\sqrt(1-z^2))=\arcsin(z)-\phi_0 $$
Here $\phi_0$ is the integration constant.
Finally, we get the law of motion:
$$ r(\phi)=\frac(p)(1-e\sin((\phi+\phi_0))) $$
Setting the integration constant $\phi_0=\frac(3\pi)(2)$ (this value corresponds to the extremum of the function $r(\phi)$), we finally obtain:
| $$r(\phi)=\frac(p)(1+e\cos(\phi)) \hspace(3cm)(2.7)$$ $$p=\frac(L^2)(GMm^2) $$ $$e=\sqrt(1+\frac(2EL^2)((GM)^2m^3))$$ |
It is known from the course of analytic geometry that the expression obtained for the function $r(\phi)$ describes curves of the second order: an ellipse, a parabola and a hyperbola. The parameters $p$ and $e$ are called the focal parameter and the eccentricity of the curve, respectively. The focal parameter can take any positive value, and the eccentricity value determines the type of trajectory: if $e\in)
