The motion of planets in circular orbits

Introduction

A body moving in a uniform circular motion experiences a constant centripetal acceleration, which is given by an equation. A force is necessary to create this acceleration, according to Newton’s second law. The force at work in the case of an orbiting planet is gravity. The Sun’s gravitational attraction acts on Earth as an inward (centripetal) force. The centripetal acceleration of the orbital motion is caused by this force.

Before these principles can be described numerically, it’s helpful to understand why a force is required to keep a body in a constant-speed orbit. The reason for this is that the planet’s velocity is tangent to the orbit at all times. In the absence of gravity, the planet would obey Newton’s first law of inertia and fly off in a straight line at constant speed in the direction of velocity. The planet’s inertial tendency is overcome by the force of gravity, which keeps it in orbit.

A body in uniform circular motion experiences a centripetal acceleration given by the equation at all times (40). According to Newton’s second law, a force is necessary to create this acceleration. The sun’s gravitational attraction acts on Earth as an inward (centripetal) force. This force causes the centripetal acceleration of the orbital motion.

Johannes Kepler thoroughly evaluated the positions in the sky of all the known planets and the Moon, calculating their positions at regular intervals, using the accurate data obtained by Tycho Brahe. He created three rules based on this study, which we will discuss in this part.s).

Kepler’s Laws

The rules of motion of planets established by Johannes Kepler are as follows:

1. The orbits of all planets are ellipses.
2. In the same time interval, a line between the Sun and the planet sweeps out the same amount of area.
3. The cube of a planet’s semi-major axis is directly proportional to the square of its orbital period.

The most common application of Kepler’s first law is to exactly characterise the geometry of an innermost stable circular orbit: an ellipse unless other objects upset it. If a comet or other object is detected to have a hyperbolic path, Kepler’s first law states that it will only visit the sun once until it collides with a planet.

Kepler’s rules are applicable to the motions of natural and man-made satellites, as well as star systems and extrasolar planets. Of course, the laws do not take into account the gravitational interactions (as perturbing effects) of the many planets on each other as formulated by Kepler. The basic challenge of reliably forecasting the motions of more than two bodies under mutual attraction is exceedingly difficult; analytical solutions to the three-body problem are only possible in a few exceptional instances. It should be noted that Kepler’s laws apply not only to gravitational forces, but also to all other inverse-square-law forces, as well as the electromagnetic forces within the atom, if relativistic and quantum effects are taken into account..

Kepler’s third evaluates the correlation between the masses of two mutually rotating objects and the calculation of orbital parameters. Study the possibility of a tiny star orbiting a larger one. Both stars spin around a shared mass centre, known as the barycenter. This holds regardless of the size or mass of the things involved. One way to discover planetary systems linked with distant stars is to measure a star’s motion around its barycenter with a huge planet.

These principles apply to a two-dimensional representation of the motion of planets, which is all that is required to describe orbits. The passage of the sun across space would be included in a three-dimensional representation of motion.

Kepler’s First Law Describes the Shape of an Orbit

A planet’s orbit around the sun (or a satellite’s orbit around a planet) isn’t a complete circle. It’s an ellipse, or a circle that has been “flattened.” One ellipse’s foci are the sun (or the planet’s centre). One of the two interior locations that determine the geometry of an ellipse is the focus. The distance between one focus and any point on the ellipse and the distance between one focus and the second focus is always the same.

The Second Law of Kepler describes how the speed of an item varies along its orbit.

The orbital speed varies based on its distance from the sun. The stronger the sun’s gravitational influence on a planet and the faster it moves, the closer it is to the sun. The weaker the sun’s gravitational attraction becomes further from the sun, the slower it moves in its orbit.

The third law of Kepler compares how an object’s speed varies along its orbit.

Because the Sun’s gravitational force on it is smaller, a planet farther from the Sun not only has a longer route than a planet closer to it, but it also moves slower. As a result, the greater a planet’s orbit is, the longer it takes to complete.

Conclusion

Kepler discovered that the orbits of the planets followed three laws based on astronomical observations and records of  , a wealthy scientist who believed in an Earth-centred model of the cosmos. To summarise, Kepler’s laws are nevertheless relevant today and play a vital role in science, astronomy, and cosmology history. They were a pivotal step in the transition from the Earth-centred to the heliocentric paradigm, and they were instrumental in the discovery of Newton’s laws.