Kepler’s law | Law of Planetary Motion | First, Second, & Third Laws

In finite motion, the particle has total negative Energy (E < 0) and two or more points. Total energy always equals the particle’s potential Energy, and the particle’s kinetic Energy becomes ‘0’.Eccentricity 0 ≤ e < 1, E < 0 indicates that the body has finite motion. A circular orbit has eccentricity e = 0, and an elliptical orbit has eccentricity e < 1.

Motions are classified into two types:

• Bounded Motion
• Unbounded Motion

Kepler’s law

In unrestricted motion, the particle has total positive Energy (E > 0) and has an extreme point where the total Energy is equal to the particle’s potential Energy, i.e., the kinetic Energy becomes zero. Eccentricity e ≥ 1, E > 0 means the body is in uncontrolled motion. A parabolic orbit has eccentricity e = 1, and a hyperbolic path has eccentricity e > 1.

Kepler’s First Law

Kepler’s first law, also called the Law of Ellipticity, states that the planets are revolved around the Sun in an Elliptical pattern. The Law of Orbits is the common name for Kepler’s first law. As a rule, the orbit of any planet around the Sun is an ellipse, with the Sun at 1 of the two focal points of the ellipse. We already know that the Planets revolve around the Sun in a circular orbit. According to Kepler, the Planets revolve around the Sun, though not in circular orbits. However, it is centered on the ellipse.

An ellipse has two centers (S & S’). Sun (S) is one of the focal points of the ellipse. Perihelion (PS) represents the point at. The planet is closest to the Sun, while aphelion (AS) represents the point at which the planet is farthest from the Sun. Central axis (2a) and minor axis (2b). The sum of the distances of any planet from the two centers is constant. The elliptical orbit of the planet causes the seasons.

Kepler’s Second Law

Kepler’s second law often called the law of regions, determines how fast each planet rotates Orbits the Sun. The rate at which each planet rotates through space is constantly changing. When planets are closer to the Sun, it rotates faster, and when it is farther from the Sun, it rotates slower. “According to Kepler’s second law, it accelerates when an orbiting satellite approaches focus. As an orbiting object gets closer the object, pushing it into its orbit,

 its gravitational effect increases.”

According to the mathematical formulae of the law, the area covered by a planet or rotating object is equal in giving time regardless of the distance to the object in view. Since the areas are equal, the arc at a greater distance is shorter, resulting in lower velocity. This applies to all objects in orbit. The developed areas can be calculated as broad but small triangle Regions formed. On the other hand, Earth is farthest from the Sun and can be approximated in a narrow but long triangle.

These areas are roughly the same size. The Earth can be moved away from the Sun, and the base of these triangles becomes smaller. For this imaginary region to be the same size as when the Earth is close to the Sun, the Earth must move very slowly when it is far from the Sun.

Law of Conservation of momentum:

This law can be verified as the conservation of angular momentum law, which states that a planet’s motion orbiting a star in a fixed orbit is always constant in its orbit. Angular momentum of the planet at position B, LB = m rB vB.

Angular momentum of the planet at position D, LD = m rD vD.

Now, since L is constant at any time, LB = LD

⇒ rB vB = rD vD

The above expression shows that the distance and velocity of a planet in orbit are inversely related.

Kepler’s Third Law

Kepler’s third law often called the law of periods, establishes a comparison between the orbital period of one planet and the radius of the orbits of other planets. The 3rd law compares the motion of different planets in contrast to Kepler’s first and second laws, which describe the motion of a single planet. Kepler’s third law defines the relationship between a planet-star system’s mass, the planet’s distance from the star, and its orbital period.

 It states that the square of a planet’s period is directly proportional to the cube for a semimajor

the axis of the planet’s orbit and inversely proportional to the sum of masses of the star and the planet. The principle of Kepler’s third law is stated as follows:

T2 = (4 π2 a3) ⁄ (G (M + m))

Where T is the period, M is the mass of the Sun, m is the planet’s mass, R is the length of the semimajor axis, and G is the gravitational constant.

Conclusion:

Kepler’s laws are still valid and have an important place in the history of science, astronomy and cosmology. They were the critical step in the revolution, which moved from the Earth-centered model to the heliocentric model and led to the discovery of Newton’s laws.