Kepler’s 3rd Law.

The square of a planet’s orbital period is proportional to the cube of its average distance from the sun. This can be mathematically expressed as; P1^2/P2^2 = a1^3/a2^3

Newton’s Universal Law of Gravitation.

A fundamental principle that describes how every object in the universe attracts every other object. This can be mathematically expressed as; Fg=GM1M2/r^2, where G is the constant 6.67x10^-11 Nm^2/kg^2

Centripetal Acceleration.

This is the acceleration directed toward the center of a circular path. In terms of the solar system, this is the inward acceleration that keeps them orbiting the Sun.

Kepler’s discoveries were revolutionary in explaining the movement of celestial bodies around a star. However, they were established long before Sir Isaac Newton discovered gravity. Consequently, Kepler’s laws did not account for gravitational forces among celestial bodies. Thus, when Newton made his discovery, it further elucidated the unknown forces underlying Kepler’s laws. In present physics, both Kepler's and Newton’s discoveries offer a deeper understanding of the dynamics within the solar system and explanations of planetary orbits. Newton’s law of gravitation establishes that every particle or body attracts another particle or body with a force proportional to their masses and inversely proportional to the distance between their centers. This principle can be applied to Kepler’s notion of planetary motion, wherein his discoveries describe the movements of celestial bodies. One way to view it is through Kepler’s laws providing a contextual map of how planets move, while Newton’s law serves as the underlying mechanism explaining why these celestial bodies move in the first place.

Planets are big celestial bodies that have huge masses. Hence, they have enough gravitational force to attract smaller celestial bodies (e.g. moons) into their orbit. The same goes for the solar system. The Sun attracts all of the planets because its mass is greater than those around it; therefore, having a greater gravitational force around its orbit. So, we can infer that the greater the mass of a body the greater its gravitational attraction; as Sir Newton’s law establishes.

Additionally, the gravitational force that pulls the two masses provides the centripetal force necessary to keep the objects in orbit. This occurs because gravity acts as the mediator of the centripetal force. We know that planets move in an elliptical orbit, meaning they constantly change direction as they orbit the Sun. The reason for this is the gravitational force of the Sun, which continuously attracts planets inward. As the Sun pulls planets inward, their velocity is perpendicular to the gravitational attraction, allowing them to sustain their orbit around the Sun. So, in a way, the centripetal force serves as the bridge between the motion of planets and the underlying force acting upon them.

Application

So, where can planetary motion/Kepler’s law be useful and how can professionals use such information? Planetary motion is used in many fields regarding space. For example, In astronomy, where Labster (n.d) stated that this field studies the motion of planets, asteroids, and other space objects that orbit the sun with the use of satellites since this visualizes how they move with the help of calculations using such laws.

This is why it is also related to the engineering field since they create and launch satellites within the orbit of the Earth with the information they gathered about the motions of space objects. According to Unacademy (n.d.), satellites are launched into elliptical orbits or almost circular orbits, meaning they talk about the motion of the central body and the forces that propel it into Earth’s orbit.

Since the moon and other space objects also orbit the Earth and are considered natural satellites, this information accompanied by how the Earth orbits the sun, can be used to gather information and predict the time at which a significant event may occur in space such as solar, lunar eclipses, comet shows, asteroid impacts, etc.

Kepler’s laws may be used as a tool for discovering new planets outside the solar system which are called exoplanets. The transit method is a tool that is based on Kepler's law which detects planets orbiting from distant stars. As planets orbit around the star, its starlight dims lighty which indicates the presence of the planet.

With the use of Kepler’s laws and understanding how planets move, scientists may take advantage of designing fuel-efficient routes for spacecraft to travel far distances consuming little fuel. Making spacecraft for further exploration of the universe. These laws have broad applications, including in astronomy for understanding planetary motion, in astrodynamics for spacecraft trajectory calculations, and in engineering for optimizing satellite orbits and propulsion systems. One example is the calculation of an artificial satellite orbiting at an average altitude above the Earth's surface. This example directly relates to astrodynamics and spacecraft engineering, where understanding Kepler's third law helps determine the orbital characteristics and timing of artificial satellites, crucial for designing and operating satellite systems efficiently.

In calculating Kepler’s first law an example is when we are calculating the distance of a moon from a planet when it is a certain distance from one of the foci of its elliptical orbit. This problem involves understanding the geometric properties of ellipses, such as the constant perimeter of the triangle formed by the foci and the moon's position. By applying the formula in axes, we can solve problems related to the dimensions and characteristics of elliptical orbits, including the positions of objects within those orbits.

Calculate quantities related to planetary or satellite motion

Kepler's third law of planetary motion: The squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. Mathematically, it is expressed as:
P² = a³
P = orbital radius
a = semi-major axes

It can also be expressed as the ratio between the two quantities:

P1^2/P2^2 = a1^3/a2^3

This formula is used to calculate either the orbital period or distance of a planet to a certain object.

1. Mercury orbits the sun at a distance of 0.4 AU (1 AU = 1.5 x 10^8 km). What is mercury’s orbital period in Earth years?

P² = a³
P² = 0.4au³
P² = 0.064au
P = 0.25 years

2. Use the values from before to calculate and prove the mass of the sun.

A^3/P^2 = GMs / 4π^2

Using such P = 0.25 yrs and corresponding a = 0.4 au values from the last question:
(6x10^10)³ / (7884000 sec)2 = (6.67×10−11m^3)(Ms) / 4π2
(0.4 AU)(4π2) = (0.25 years) (6.67×10^−11 N m^3/kg)(Ms)
15.79 = (1.6675x10^-11) (Ms)
Ms = 2.06 x 10^30 Kg (roughly the same mass)

3. Calculate the force of gravity that exists between a 60kg person and an 80kg person, who is 0.5m away from one another. (G = 6.67×10^−11 N m^3/kg)

F = GM1M2 / r2
F = (6.67×10^−11 N m^3/kg) (60 kg)(80 kg) / (0.5m)2
F = 1.28 x 10^-6 N

Summary

Kepler’s Law was proposed by Johannes Kepler who introduced 3 laws relating to their movement in space which are the Law of Ellipsis (1st Law), the Law of Equal Areas (2nd Law), and the Law of Harmonies (3rd Law). In classical and astronomical physics, Kepler’s Planetary motion tackles celestial bodies’ motion in the Solar System where the Sun is considered the foci and how planets orbit around it. It also stated that gravitational force is one of the central factors that maintains and launches the orbits since gravity must be balanced to the orbital velocity of a smaller body which is the planet. Concerning Gravity, it is also the fundamental force that explains why an object falls at the same rate, and objects such as planets in space are naturally attracted to one another. It is also fundamentally derived from Newton’s Universal Law of Gravitation, a fundamental concept with the formula Fg=GM1M2/r2, which states that all objects in the universe are attracted to one another. G is equal to 6.67x10-11 Nm2/kg2.

To expound more on Kepler’s law, the Law of Ellipsis states “All planets move around the Sun in an elliptical orbit, with the Sun being one of its foci.” this means that the orbits of the planets are not circular but rather elliptical, and ellipses in geometry have two foci which states that the sun is not the center of our solar system. The Law of Equal Areas states “The radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time.” the imaginary lines that connect the planets and the sun will sweep equal areas at equal times as each planet revolves around the sun which implies that velocity of planets’ orbit and the sun is not constant. Lastly, The law of Harmonies states “The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distances from the Sun.” This implies an orbital period based on the distance of planets to the sun such as compared to planets closer to the Sun with lower orbital periods, the planet furthest from the Sun will have a longer orbital period. The mentioned laws can be used to determine how planets and satellites work which astronomers and engineers can use to monitor and evaluate planets whether it is where we can live or not.

The square of a planet’s orbital period is proportional to the cube of its average distance from the sun.

This can be mathematically expressed as; P1^2/P2^2 = a1^3/a2^3

Newton’s Universal Law of Gravitation.

A fundamental principle that describes how every object in the universe attracts every other object. This can be mathematically expressed as; Fg=GM1M2/r^2, where G is the constant 6.67x10^-11 Nm^2/kg^2

Centripetal Acceleration.

This is the acceleration directed toward the center of a circular path. In terms of the solar system, this is the inward acceleration that keeps them orbiting the Sun.