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Geostationary Satellites

The following text is not mine; it is a translation of a fragment extracted from the book "Eso no estaba en mi libro de la exploración espacial" by Pedro León Guadalmazán, cited in the Bibliography. I highly recommend reading it. The translation is by no means professional but sufficiently illustrates the concept. The p5.js construction above aims to illustrate what the book details in these paragraphs.

Let's Fire Cannons for Science

Let's start by thinking of our planet Earth as a sphere with a curved surface that is not immediately apparent but becomes evident over long distances. Knowing the radius of our planet and using simple mathematical calculations, it can be demonstrated that for every 8 kilometers we travel, the surface "drops" by 5 meters. In other words, if we had a perfectly straight 8-kilometer-long iron beam resting on the ground at one end, the other end would be 5 meters above the surface due to Earth's curvature.

If you've ever watched a ship sail away from the shore while lying on the beach, you may have noticed that the hull disappears first, followed gradually by the sails or chimney, making it look as though the ship is slowly sinking. This is the same reason why lighthouses are built so tall—so that ships far out at sea can still see them and avoid getting lost beyond the horizon.

Now that we understand this concept, let's revisit the reasoning Newton presented in his book Principia (1687), where he details the concept of orbit based on the earlier ideas of Galileo and Kepler to better understand gravity. Newton imagined that if we climbed to the top of a mountain with a powerful cannon and fired a projectile toward the horizon, at first, it would travel in a straight line but gradually succumb to gravity and atmospheric resistance, eventually hitting the ground a few kilometers away. If we kept firing with increasing power, the projectile would travel farther and farther each time, covering more of the planet.

At a certain theoretical point, if we fired the cannonball with enough power, it would travel all the way around the Earth and come back to hit us in the back of the head.

Newton Principia

Source: Public domain image from Newton's Principia

Speed Limits in Orbits

Earlier, we noted that the Moon is 380,000 km away from Earth and is pulled by our planet's gravity with a small acceleration. However, it does not fall onto Earth because it is orbiting, constantly trying to escape, with gravity keeping it in balance. To maintain this equilibrium, the Moon moves around our planet at a speed of 1 km/s (or 3600 km/h), which allows it to remain in orbit at that distance—just like any other object we place in orbit at that height.

As we get closer to Earth, gravity strengthens, so we need to travel faster to maintain orbital equilibrium. At 36,000 km, where geostationary communication satellites are positioned, the required speed is 3 km/s (10,800 km/h). At around 400 km, where the International Space Station orbits, the necessary speed is 7.7 km/s (27,700 km/h). In low Earth orbit, at 200-250 km, an object must travel at 8 km/s (28,800 km/h) to stay in orbit.

Remember Newton's cannonball on the mountain? We've come full circle—our satellites (cannonballs) need to travel at 8 km/s to match Earth's curvature and remain in orbit. Long live Newton!

In summary, we now understand that at greater distances from Earth, gravity is weaker, and thus a satellite farther from our planet can travel at a lower speed to stay in orbit. In fact, an orbit is simply a balance between gravity and velocity. Once in orbit, no further propulsion is needed to maintain it (except for small adjustments), which is why our Moon doesn’t need engines to keep orbiting us.

Object Altitude above Earth Speed (km/s)
Satellite in low orbit 200 km 8
International Space Station 420 km 7.7
Geostationary satellite 35,800 km 3
The Moon 380,000 km 1

The "Magic" of Geostationary Orbit

You may already be using this type of orbit without realizing it—just check if you have a satellite dish at home. If you do, your dish is pointing directly at a satellite (Astra, Hispasat, Intelsat...) in a geostationary orbit.

If all satellites constantly orbit Earth at high speeds, why does your satellite dish always point at the same spot? Shouldn’t it have to move to track the satellite across the sky? In theory, yes—but by employing a special trick, we can keep the satellite "fixed" in the sky so that your dish doesn’t have to move, ensuring uninterrupted TV shows and movies.

A satellite orbiting at 300 km takes about 90 minutes to complete one full revolution around Earth. As we increase the orbit’s altitude, the circumference grows, and the satellite moves more slowly, taking longer to complete an orbit. At 1000 km, the orbital period increases to 105 minutes; at 5000 km, it extends to 3 hours and 20 minutes. If we keep increasing the altitude, we eventually reach 35,786 km, where the orbital period matches Earth's rotation: 23 hours and 56 minutes.

This means the satellite orbits Earth at the same speed as Earth's rotation, appearing stationary in the sky. Science is better than magic!

Thanks to this, we can use geostationary orbits for weather and telecommunications satellites to transmit TV, radio, internet, and data without appearing to move.

Although many attribute this idea to science fiction writer Arthur C. Clarke, the concept was first proposed in 1928 by Slovenian engineer Herman Potočnik in his book The Problem of Space Travel, where he suggested building a wheel-shaped space station to simulate Earth's gravity and remain fixed in space at nearly 36,000 km. Clarke later popularized the concept in 1945 in Wireless World under the title "Extra-Terrestrial Relays."

If you search for operational geostationary satellites, you'll see names like Hispasat 30W, Astra 19.2E, or Intelsat 35E. These indicate their longitudinal position over Earth's equator: W for west and E for east. Several satellites from the same company may occupy the same position, requiring frequent repositioning to maintain spacing and prevent drifting.

GeoGebra construction by Rafael Losada:
GeoGebra - Rafael Losada

GeoGebra - Rafael Losada

Bibliografía