Orbital Mechanics

Have you ever thought about why the Moon doesn't come crashing down onto Earth? Fortunately for us, Earth's gravity has trapped the Moon, as well as many other space objects, in its orbit. But how exactly do orbital forces work?

Let’s start with a thought experiment called Newton’s Cannonball. When a cannonball is fired, gravity and air resistance bend its path and bring it back to Earth. For this experiment, let’s focus on gravity.

Imagine you are firing cannonballs off of an extremely high mountain. The Earth’s gravity would bend the path of the cannonball, causing it to fall back to Earth. The faster the cannonball is fired, the farther it would go before falling back to Earth. But if the cannonball was going fast enough, it would not fall back to Earth’s surface at all. That’s because the Earth would be curving away from the cannonball at the same rate that the cannonball is falling. The cannonball would be in orbit - just like the Moon is.

What are Kepler’s Laws?

Around 1605, the German astronomer and mathematician Johannes Kepler presented his three laws of planetary motion. These laws form the basis of our understanding of satellite and planetary orbits. Kepler's theories were based on our solar system.That’s why they often refer to our Sun. But astronomers have discovered other planetary systems since his time. Kepler’s Laws can be applied to these planetary systems, too.

What is Kepler’s First Law?

Kepler’s First Law states that the orbit of each planet is an ellipse with the Sun at its focus. The point at which a planet is at its closest to the Sun is called the perihelion. The point at which a planet is at its furthest from the Sun is called the aphelion.

What is Kepler’s Second Law?

Kepler’s Second Law states that a line joining a planet to the Sun sweeps out equal areas in equal times. As you just learned, planets orbit in ellipses. This means that planets move faster when they are close to the Sun, and slower when they are further away.

This applies to satellites and other spacecraft, too. But instead of the Sun, satellites orbit the Earth. We could restate Kepler’s Second Law to make it relevant for satellites and spacecraft: “A line joining a spacecraft to the centre of the Earth sweeps out equal areas in equal times.” In other words, a satellite moves faster when it is close to the Earth and more slowly when it is further away.

What is Kepler’s Third Law?

Kepler’s Third Law states that the square of the period of a planet’s orbit (P) is proportional to the cube of its mean distance from the Sun (a). In other words, the further a planet is from the Sun, the longer it takes to complete an orbit.

Again, we can apply this to satellites. The higher a satellite is from the surface of the Earth, the longer it takes to complete an orbit.

What is the physics behind Kepler’s Laws?

Kepler’s Laws describe orbital motion. But it was the English mathematician and physicist Isaac Newton who would later discover the underlying physics. Newton published his Three Laws of Motion in his most famous book Philosophiae Naturalis Principia Mathematica in 1687. These laws are the basis of what scientists call “Newtonian” or “classical” mechanics.

Newton’s Laws of Motion are as follows:

• Newton’s First Law – An object at rest will tend to stay at rest unless an outside force acts upon it. Similarly, an object in motion will tend to keep moving in a straight line unless an outside force acts upon it.
• Newton’s Second Law – The acceleration of an object is proportional to the force applied on it, and is in the same direction as that force.
• Newton’s Third Law – For every action there is an equal and opposite reaction.

Newton’s Third Law is probably the best known of the three. This law is really what makes spaceflight possible. Rocket engines work by propelling exhaust out the back of the spacecraft. Since every action has an equal and opposite reaction, the spacecraft is propelled forward.

Newton also had a Law of Universal Gravitation. It states that the force of gravitational attraction between any two objects is directly proportional to the product of their masses. It is also inversely proportional to the square of the distance between them. In other words, the bigger the objects and the closer they are together, the greater the force of gravitational attraction between them. The reverse is true, too. 

As far as we know, the same principle applies everywhere in the universe. That’s why we call it a “universal” law.

The motion that results from this gravitational attraction is called two-body motion.

What are the different types of orbits?

There are different types of orbits. The type that a satellite is launched into depends on its mission. Low Earth Orbit (LEO) is the simplest orbit to achieve. The majority of artificial objects orbiting the Earth are in the LEO “corridor.” That is an area between 160 km and 1 000 km above the Earth’s surface. It is bounded on the low end by atmospheric drag factors and at the high end by the lower Van Allen radiation belt.

A polar orbit is an orbit that passes over the Earth’s poles, traveling from north to south. Sun-synchronous orbit (SSO) is a type of polar LEO. In SSO, the Earth’s uneven gravitational field “twists” the orbit of a satellite at a rate of one revolution per year. The result is that the orbital plane of the satellite will always maintain the same angle with respect to the Sun. Also, the satellite’s orbit will cross the equator at the same local time every orbit. This means that the satellite will experience the same lighting conditions every time it passes over a particular point on the Earth’s surface.
This is why remote sensing, weather, and reconnaissance (military observation) missions use Sun-synchronous orbits. For example, these systems can detect motion by detecting a change in shadows.

Geostationary orbit (GEO) occurs at an altitude of about 35 700 km. In GEO, a satellite travels around Earth’s equator at the same rate that the Earth is rotating. This means that from Earth, the satellite appears stationary.  British science fiction author Arthur C. Clarke first proposed that this orbit could be used by communications satellites in a Wireless World article published in 1945. This is why GEO is sometimes called the “Clarke orbit.” The majority of communications satellites and many weather satellites are in GEO orbits. 

Scientists and engineers have launched approximately 8 000 human-made objects into space since the launch of Sputnik 1 in 1957. Fortunately for us, there’s a lot of physics in motion up there. Thanks to physics, these objects stay in space instead of crashing down towards us on Earth!