Kepler's Laws of Planetary Motion: A Detailed Guide On It!!

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INTRODUCTION TO MOTION OF PLANETS

Since the time of the astronomer Copernicus, it has been known that several planets orbit the Sun. However, in order to calculate how the planets spin and their motions, it was necessary to know their exact positions in the sky (at different times). Astronomer Tycho Brahe studied the planets’ positions without a telescope for many years (Galileo invented the telescope after Brahe’s death) and recorded lots of data. The data was then analyzed by scientist Johannes Kepler, who discovered three laws governing planet motion. These are referred to as Kepler’s laws of planetary motion.

KEPLER'S LAWS

Kepler’s laws are discussed below.

First Law: Each planet revolves around the Sun in an elliptical orbit and the Sun is at the focus of that ellipse.
Second Law: A straight line joining the Sun and a planet traverses equal areas in equal periods of time.
Third Law: The square of the period of revolution of a planet around the Sun is proportional to the cube of the semi-parameter of its orbit.

In Kepler’s laws, the motion and orbit of a planet are referred to as Keplerian motion and Keplerian orbit respectively.

Note that the motion of a natural or man-made object (such as a satellite) orbiting an object (e.g. a planet) is similar to the motion of the planets orbiting the Sun. That is, the motion of the satellites also obeys Kepler’s laws. However, it should be remembered that in cases where Kepler’s laws apply, the mass of orbiting objects is negligible compared to the object at focus, as in our solar system. Although the orbits of the planets of the solar system are elliptical, they are almost circular, that is, the eccentricity of the orbits is very low. Since the circle is a special region of the ellipse (eccentricity of the circle, e = 0), Kepler’s laws are equally applicable to circular orbits.

All of Kepler’s laws apply to circular and elliptical orbits (i.e. orbits where 0 ≤ e ≤ 1). Since hyperbolas and parabolas are open conic sections, Kepler’s first and third laws do not apply to such orbits. For example, a comet that makes a single trip past the Sun follows an elliptical or parabolic path.

PROOF OF KEPLER'S SECOND LAW

Let us assume that a planet moves from point P to point Q in very small time ∆t and the distance of that planet from the sun is r. Note that the eccentricity of the planets’ orbits is so low that when drawn accordingly the orbits appear circular; for this reason, the orbit is shown to be elliptical with the centre raised in the figure.

Let the straight line joining the Sun and the planet pass through the area ∆A at time ∆t. Now, the area of a triangle with height r and area r · ∆θ is approximately equal to ∆A. that is,

∆A ≈ 1/2 · r · r · ∆θ = 1/2 r²∆θ ……..(1)

As the value of ∆t (and hence the value of ∆θ) approaches zero, the value of ∆A equals 1/2 r²∆θ. That is, from (1) we can write,

lim ∆→0 (∆A/∆t) = ½ r²[lim ∆→0 (∆θ/∆t)]

or, dA/dt = ½ r²(∆θ/∆t) = ½ r²ω ……..(2)

Where, ω = dθ/dt is the angular velocity of the straight line joining the Sun and the planet.

The figure shows the linear momentum of the planet →P and the components P_r and P_n along its radius and perpendicular to the radius respectively. Let the mass of the planet be m. Now, the angular momentum of the planet relative to the Sun is the value of →L,

L = r P_n = r · mωr

or, L = mωr² ……..(3)

From (2) and (3) it follows,

dA/dt = L/2m …….(4)

L is a constant if no external force acts on the system consisting of the sun and the planet. Hence from (4) we can say,

dA/dt = constant

ORBITAL VELOCITY AND PERIOD OF REVOLUTION OF PLANETS

ORBITAL VELOCITY

Let us assume that a planet of mass m is orbiting the Sun of mass M₀ in an orbit of radius r.

The speed at which a planet travels in its orbit is called its orbital velocity (v). The value of this orbital velocity is a constant for circular orbits. Centripetal force required to move the planet in circular orbit = mv²/r. The mutual gravitational force between the Sun and the planets provides this centripetal force. This gravitational force is GM₀m/r².

So, mv²/r = GM₀m/r²

or, v² = GM₀/r

or, v = √GM₀/r ………(1)

Note that orbital velocity (v) is not constant in case of elliptical orbit. In this case, v ∝ 1/r if the distance r from the Sun to the planet at any instant.

ORBITAL ANGULAR VELOCITY

The angular velocity at which a planet travels in its orbit is called its angular velocity (ω). Its relation to rotational speed is ω = v/r. While ω is variable in elliptical orbits, it is constant in circular orbits. (1) from equation no.

ω = v/r = √GM₀/r³ ………(2)

ORBITAL PERIOD

The time it takes for a planet to complete one revolution around the Sun is called its orbital period (T). The planet completes a complete circular path in time T, i.e. travels a distance of 2πr. So, the orbital speed,

v = 2πr/T

or, T = 2πr/v = 2πr √r/GM₀ [from equation (1)]

or, T = 2π√r³/GM₀ ………(3)

So, T² = (4π²/GM₀) · r³

or, T² = constant X r³ ………(4)

That is T² ∝ r³. This is Kepler’s third law. From this formula, it can be said that the farther a planet is from the Sun, the longer its orbital period. Mercury, the closest planet to the Sun, has an orbital period of only 88 days, while Neptune, the farthest planet, has an orbital period of about 165 years. Incidentally, in 2006, the International Astronomical Union classified Pluto as a dwarf planet; that is, Pluto is no longer recognized as a planet.

Note in equations (1), (2) and (3) that none of the orbital speed, orbital angular velocity or orbital period of a planet depends on the mass m of the planet, although each depends on the mass of the Sun M₀. As a result, if the distance (r) of a planet from the sun and the orbital period (T) of that planet are known, it is possible to determine the mass of the sun from equation (3), but it is not possible to determine the mass of the planet. Obviously, the above discussion applies not only to the motion of the planets around the Sun, but also to the motion of any object orbiting the Sun or its satellites around a planet.

CONCLUSION

Kepler’s laws of planetary motion offer a fundamental knowledge of how planets orbit the Sun. The first law, the Law of Ellipses, demonstrates that planetary orbits are elliptical, with the Sun at one focus, calling into question the traditional assumption of completely circular orbits. The second law, the Law of Equal Areas, states that a planet moves faster closer to the Sun and slower farther away, while maintaining angular momentum. Finally, the third law, the Law of Harmonies, provides a precise link between a planet’s orbital period and its distance from the Sun, emphasising that farther away planets take longer to complete an orbit. Together, these equations provide fundamental insights into celestial mechanics, illustrating the predictability of planetary motion and laying the path for Newton’s theory of gravitation. Kepler’s principles are still used today in astronomy and space research to help us understand other planets.

REMINDER

The force by which any two particles of matter in this universe attract each other is called gravitation.
Any two particles in the universe attract each other along their connecting straight lines. The value of this attractive force is proportional to the product of the masses of the two particles and inversely proportional to the square of the distance.
The force with which two particles of unit mass attract each other at a distance of unit distance is called the gravitational constant.
The gravitational force acting on an object of unit mass at any point in a gravitational field is called the gravitational field strength at that point.
The amount of work done by an external agent to bring an object of unit mass from an infinite distance to a point in the gravitational field is called the gravitational field at that point.
The force with which the earth attracts any object on or near the surface is called gravity. The acceleration due to the force of gravity on a freely falling object is called gravitational acceleration.
Kepler’s Laws of Motion of Planets and Satellites:

First Law: Each planet revolves around the Sun in an elliptical orbit and the Sun is at one focus of that ellipse.
Second Law: A straight line joining the Sun and a planet traverses equal areas in equal periods of time.
Third Law: The square of the orbital period of a planet around the Sun is proportional to the cube of the semi-radius of the orbit.

The minimum velocity at which an object can be thrown from the surface of the Earth or any other planet or satellite beyond its gravitational attraction is called the release velocity.
If the relative angular velocity of an artificial satellite with respect to the Earth’s angular momentum is zero and the satellite is always at the equator, it appears from the surface that the satellite is fixed in the same place in the sky. Such satellites are called geosynchronous satellites.

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Kepler’s Laws of Planetary Motion: A Detailed Guide About It!!

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