# Law of Areas

Kepler`s Second Law: The imaginary line connecting a planet and the sun crosses equal areas of space at the same time intervals as the orbiting planet. Basically, planets do not move along their orbits at a constant speed. On the contrary, their speed varies, so that the line connecting the centers of the sun and planet sweeps equal parts of an area at the same time. The closest approach point to the Sun to the planet is called perihelion. The point of greatest separation is aphelion, so according to Kepler`s second law, a planet moves faster when it is at perihelion and slower at aphelion. A line connecting a planet and the sun sweeps equal areas at equal time intervals.  The eccentricity of the Earth`s orbit makes the time between the March equinox and the September equinox about 186 days, unequal to the time between the September equinox and the March equinox, about 179 days. A diameter would cut the orbit in equal parts, but the plane through the sun parallel to the Earth`s equator intersects the orbit in two parts with areas in a ratio of 186 to 179, so the eccentricity of the Earth`s orbit is about Q2. Kepler`s second law states that the radius vector of a planet scans equal regions of the sun at equal time intervals. This law is a consequence of the preservation of: This law is known as the law of the territories. The line connecting a planet to the sun sweeps equal areas at equal time intervals. The rate of change in the area over time will be constant. We can see in the figure above that the sun is at the point and the planets revolve around the sun.

The sectoral areas are characterized by| z s p | = b a ⋅ | z s x |. {displaystyle |zsp|={frac {b}{a}}cdot |zsx|.} Kepler`s second law – sometimes called the law of equal surfaces – describes the speed at which a particular planet moves in orbit around the sun. The speed at which a planet moves through space is constantly changing. A planet moves faster when it is closest to the sun and slower when it is farthest from the sun. However, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would cross the same area in equal periods of time. For example, if an imaginary line were drawn from Earth to the Sun, then the swept area of the line in each 31-day month would be the same. This is illustrated in the diagram below. As can be seen from the diagram, the areas formed when the Earth is closest to the Sun can be approached as a wide but short triangle; while areas formed when the Earth is furthest from the Sun can be approached as a narrow but long triangle.

These areas are the same size. Since the base of these triangles is shortest when the Earth is farthest from the Sun, the Earth would have to move more slowly so that this imaginary region is the same size when the Earth is closest to the Sun. Replacing the expression for v2 in the above equation gives: The story of our best understanding of planetary motion could not be told without the work of a German mathematician named Johannes Kepler. Kepler lived in Graz, Austria, in the early 17th century. Due to religious and political difficulties, which were common at that time, Kepler was banished from Graz on 2 August 1600. The orbital radius and angular velocity of the planet in elliptical orbit vary. This is shown in the animation: the planet moves faster when it approaches the sun, and then slower when it is farther from the sun. Kepler`s second law states that the blue sector has a constant area.

where θ^ {displaystyle {hat {boldsymbol {theta }}}} is the unit vector whose direction is 90 degrees counterclockwise of r^ {displaystyle {hat {mathbf {r} }}}, and θ {displaystyle theta } is the polar angle, and where a point above the variable is the differentiation with respect to time. 2. Galileo is often credited with the early discovery of four of Jupiter`s many moons. Moons orbiting Jupiter follow the same laws of motion as planets orbiting the sun. One of the moons is called Io – its distance from Jupiter`s center is 4.2 units and orbits Jupiter in 1.8 Earth days. Another moon is called Ganymede; it is 10.7 units from the center of Jupiter. Make a prediction of the Ganymede period using Kepler`s law of harmonies. For eccentricity 0 ≤ e <1, E < 0 implies that the body has limited movement. A circular orbit has an eccentricity e = 0 and an elliptical orbit has an eccentricity e < 1.

Kepler`s problem assumes an elliptical orbit and the four points: how long it takes a planet to orbit the sun (its period, P) is related to the planet`s average distance from the sun (d). That is, the square of the period, P * P, divided by the cube of the mean distance, d * d * d, is equal to a constant. For each planet, regardless of its period or distance, P*P/(d*d*d) is the same number. The problem is to calculate the polar coordinates (r,θ) of the planet from perihelion t. The right side of the equation above is the same value for each planet, regardless of the mass of the planet. Subsequently, it is reasonable for the T2/R3 ratio to be the same value for all planets if the force that keeps the planets in their orbits is gravity. Newton`s universal law of gravity predicts results consistent with known planetary data and provides a theoretical explanation of Kepler`s law of harmonies. Moreover, the current use of Kepler`s second law is somewhat misleading. Kepler had two versions that were related in a qualitative sense: the “Distance Selling Act” and the “Territorial Law”. The “territorial law” is what has become the second law of the group of three; but Kepler himself did not favor it in this way.

 Kepler`s first law – sometimes called the law of ellipses – explains that planets orbit the sun in an orbit described as an ellipse. An ellipse can be easily constructed with a pencil, two pens, a string, a sheet of paper and a piece of cardboard. Stick the sheet of paper with both pens on the cardboard. Next, tie the cord into a loop and wrap the loop around the two pins. Take your pencil and pull the string until the pencil and two pens form a triangle (see diagram on the right). Then begin to trace a path with the pencil, holding the string tightly wrapped around the pens. The resulting shape is an ellipse. An ellipse is a special curve in which the sum of the distances from each point in the curve to two other points is a constant. The other two points (represented here by the staples) are called the focal points of the ellipse.

The closer these points are to each other, the more the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are in the same place. Kepler`s first law is quite simple: all planets orbit the sun in an ellipse-like orbit, with the sun at one of the focal points of this ellipse. This means that the time ` T ` is directly proportional to the cube of the semi-major axis, i.e. `a`. Let`s derive the equation from Kepler`s 3rd law. Suppose this equation gives M as a function of E. The determination of E for a given M is the inverse problem. Iterative numerical algorithms are often used. The transverse acceleration a θ {displaystyle a_{theta }} is zero: The method for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of time t from perihelion consists of the following five steps: But note: The Cartesian position coordinates with respect to the center of the ellipse are (a cos E, b sin E) This is the characteristic of an ellipse, that the sum of the distances of a planet from two foci is constant.

The elliptical orbit of a planet is responsible for the appearance of the seasons. Mathematically, an ellipse can be represented by the formula: The sun plays an unjustified asymmetric role. This is what he assumed in Newton`s law of universal gravity: Kepler`s third law – sometimes called the law of harmonies – compares the orbital period and radius of a planet`s orbit with those of other planets. Unlike Kepler`s first and second laws, which describe the motion properties of a single planet, the third law compares the motion properties of different planets. The comparison is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for each of the planets. To illustrate, consider the orbital period and the average distance from the Sun (orbital radius) for Earth and Mars, as shown in the table below. If the two bodies of the solar system are the Moon and the Earth, the acceleration of the Moon Since the velocity of an object in a near-circular orbit can be approximated as v = (2 * ft * R)/T, After calculating the eccentric anomaly E, the next step is to calculate the actual anomaly θ.

Опубликовано