The Motion of Satellites in Circular Orbits

Introduction

Nicolaus Copernicus postulated that Earth and the other planets have a circular orbit around the Sun. He also discovered that as the distance from the Sun increased, so did the orbital periods. These orbits were later discovered to be ellipses by Kepler, but the orbits of most planets in the solar system are roughly circular. The Earth’s orbital distance from the Sun fluctuates by only 2%. The eccentric orbit of Mercury, whose orbital distance changes by roughly 40%, is an exception.

Since estimating the orbital speed of the satellite and period is considerably easier in circular orbits, we will assume circular orbits for the sake of analysis. We focus on objects orbiting the Earth, but our findings can be applied to other situations.

A Satellite is a Projectile

The key thing to grasp when it comes to satellites is that they are projectiles. A satellite, in other words, is an object whose only force is gravity. The sole factor that governs a satellite’s motion once it is sent into orbit is gravity. Newton was the first to propose that a projectile shot at a high enough speed might circularly orbit the planet. Consider launching a missile horizontally from the famed Newton’s Mountain, far above the impact of air drag. The force of gravity would draw the projectile downward as it moved horizontally in a tangent to the Earth’s surface.

So, what is the minimum launch speed for a satellite to orbit the Earth? The solution comes from a fundamental fact regarding the Earth’s curvature. The Earth’s surface slopes downward by approximately 5 metres for every 8000 metres measured along the horizon. So, if you look out horizontally over the Earth’s horizon for 8000 metres, you’ll notice that the Earth curves downwards for a distance of 5 metres below this straight-line course. A missile must travel 8000 metres horizontally for every 5 metres of vertical fall to orbit the planet. It just so happens that the vertical distance travelled by a horizontally launched projectile in its first second is the same as the vertical distance travelled by a vertically launched projectile in its first second.

Vectors of Velocity, Acceleration, and Force

An orbiting motion of a planet can be characterised using the same motion characteristics as any other circular object. At every point along its course, the satellite’s velocity would be tangent to the circle. The satellite’s acceleration would be directed toward the circle’s centre or the central body around which it is circling. And this acceleration is the result of a net force acting in the same direction as the acceleration.

Elliptical Orbits of Satellites

Sometimes, satellites will orbit in elliptical routes. The centre body is positioned at one of the ellipse’s foci in such circumstances. Satellites travelling on elliptical routes have similar motion characteristics. The satellite’s motion is perpendicular to the ellipse. The satellite accelerates in the direction of the ellipse’s focal point. The net force acting on the satellite, according to Newton’s second law of motion, is directed in the same direction as the acceleration – towards the ellipse’s focus.

The gravitational attraction between the core body and the circling satellite provides this net force once more. There is a component of force in the same direction (or opposite direction) as the motion of the planet in the case of elliptical routes. The elliptical motion of satellites, unlike uniform circular motion, is not defined by a constant speed.

Conclusion

In a nutshell, satellites are projectiles that orbit around a big central body rather than colliding with it. Because they are projectiles, they are affected by gravity, which is a universal force that also works over huge distances between two masses. The motion of satellites is regulated by Newton’s laws of motion, just like any other projectile. As a result, the mathematics of these satellites is derived from a mathematical application of Newton’s universal law of gravitation to circular motion.