What Is Keplers Second Law

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the sun.

Kepler first law – The law of orbits
Kepler'southward second police – The police force of equal areas
Kepler's third law – The law of periods

Tabular array of Content:

Introduction
First police force
Second Law
3rd Law

Introduction to Kepler's Laws

Motility is e'er relative. Based on the energy of the particle nether motion, the motions are classified into two types:

Bounded Motility
Unbounded Move

In bounded motility, the particle has negative full energy (East < 0) and has two or more farthermost points where the total energy is always equal to the potential free energy of the particle, i.e., the kinetic energy of the particle becomes zero.

For eccentricity 0 ≤ eastward <1, E < 0 implies the body has bounded motion. A circular orbit has eccentricity e = 0, and an elliptical orbit has eccentricity e < i.

In unbounded motion, the particle has positive total energy (E > 0) and has a unmarried farthermost point where the total free energy is always equal to the potential energy of the particle, i.eastward., the kinetic energy of the particle becomes zero.

For eccentricity e ≥ 1, E > 0 implies the torso has unbounded motion. Parabolic orbit has eccentricity east = 1, and Hyperbolic path has eccentricity e > ane.

Also Read:

Gravitational Potential Free energy
Gravitational Field Intensity

Kepler's laws of planetary motion can be stated every bit follows:

Kepler First law – The Law of Orbits

According to Kepler'south commencement police force," All the planets circumduct around the sun in elliptical orbits having the dominicus at i of the foci". The point at which the planet is close to the dominicus is known as perihelion, and the signal at which the planet is further from the sunday is known as aphelion.

It is the characteristic of an ellipse that the sum of the distances of any planet from 2 foci is constant. The elliptical orbit of a planet is responsible for the occurrence of seasons.

Kepler's Laws of Planetary Motion

Kepler First Police – The Law of Orbits

Kepler's 2nd Law – The Law of Equal Areas

Kepler'due south 2nd law states, " The radius vector fatigued from the sun to the planet sweeps out equal areas in equal intervals of time"

Every bit the orbit is non circular, the planet's kinetic energy is not abiding in its path. It has more kinetic energy virtually the perihelion, and less kinetic energy nigh the aphelion implies more speed at the perihelion and less speed (v_min) at the aphelion. If r is the distance of planet from sunday, at perihelion (r_min) and at aphelion (r_max), and so,

r_min+ r_max = 2a × (length of major axis of an ellipse) . . . . . . . (1)

Keplers Second Law

Kepler'south Second Law – The constabulary of Equal Areas

Using the law of conservation of angular momentum, the law tin can be verified. At any point of time, the angular momentum can be given as, L = mrⁱⁱω.

Now consider a small area ΔA described in a pocket-sized fourth dimension interval Δt and the covered angle is Δθ. Let the radius of curvature of the path exist r, so the length of the arc covered = r Δθ.

ΔA = 1/2[r.(r.Δθ)]= 1/2r²Δθ

Therefore, ΔA/Δt = [ 1/2rⁱⁱ]Δθ/dt

Taking limits on both sides as, Δt→0, we get;

\(\begin{array}{fifty}\lim_{\Delta t\rightarrow 0}\frac{\Delta A}{\Delta t}=\lim_{\Delta t\rightarrow 0}\frac{1}{ii}r^{2}\frac{\Delta \theta }{\Delta t}\finish{array} \)

\(\begin{array}{l}\frac{dA}{dt}=\frac{1}{2}r^{ii}\omega\end{array} \)

\(\begin{array}{l}\frac{dA}{dt}=\frac{Fifty}{2m}\stop{array} \)

At present, past conservation of angular momentum, L is a constant

Thus, dA/dt = constant

The area swept in equal intervals of fourth dimension is a constant.

Kepler's second police can too be stated as "The areal velocity of a planet revolving around the sun in elliptical orbit remains constant, which implies the angular momentum of a planet remains constant". As the angular momentum is constant, all planetary motions are planar motions, which is a direct outcome of central force.

Check: Acceleration due to Gravity

Kepler's Third Constabulary – The Police of Periods

According to Kepler'southward law of periods," The square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major centrality".

T^two ∝ a^three

Shorter the orbit of the planet around the lord's day, the shorter the time taken to complete one revolution. Using the equations of Newton'due south law of gravitation and laws of motility, Kepler's 3rd law takes a more full general form:

P^two= 4π² /[K(M_ane+ 1000₂)] × a³

where M_i and M₂ are the masses of the two orbiting objects in solar masses.

Frequently Asked Questions on Kepler's police force

What does Kepler's first law explicate?

According to Kepler'due south showtime law, all the planets circumduct around the Sun in elliptical orbits with the Sun every bit 1 of the foci.

What does Kepler's 2d explicate?

According to Kepler'southward second constabulary, the speed at which the planets move in space continuously changes. The second law helps to explain that when the planets are closer to the Sunday, they will travel faster.

What is Kepler's 3rd law?

Kepler'southward third law, also called the law of periods, states that the square of the orbital period is proportional to the cube of its mean distance R.

Why are the orbits of the planets non circular?

For the orbits to exist round, it requires the planets to travel with a certain velocity, which is extremely unlikely. If in that location is whatever modify in the velocity of the planet, the orbit will be elliptical.