Touching the Sky

 Touching the Sky

Produced by Aubrey Lieberman with ChatGPT 5.0 turbo — September 2025

The speed of an orbiting satellite is given by the formula:

v = √(GM / r).

Here G is the gravitational constant, M the mass of Earth, and r the orbital radius. For the GPS satellites high above us, this works out to about 3.9 kilometers per second. Each carries an atomic clock, ticking with exquisite regularity, adjusted for relativity so that our maps and phones don’t drift into nonsense. These clock towers built with physics and mathematics are indeed as high as the sky.

For a circular GPS orbit the speed is v = √(GM/r) and the period is T = 2π√(r³/GM), with GM = 3.986×10¹⁴ m³/s² and orbital radius r = R_E + h ≈ (6,371 + 20,200) km = 2.6571×10⁷ m. Plugging in: v ≈ √((3.986×10¹⁴)/(2.6571×10⁷)) ≈ 3.87×10³ m/s = 3.87 km/s. The period is T ≈ 2π√((2.6571×10⁷)³/(3.986×10¹⁴)) ≈ 4.31×10⁴ s ≈ 11.97 h, almost exactly half a sidereal day (~23.93 h), so a GPS satellite orbits about twice per sidereal day.

But you don’t need to wrestle with the square roots and constants. Take my hand and look at the sky around us, with swarms of machines, invisible during the day, sometimes apparent to the naked eye at night, amongst the moon and the stars.

The satellites are like the sparks in a web encircling the earth, linked to it by communicating receivers and transmitters all over the planet, a marvel of gravitation, electromagnetism, and human imagination.

Imagine a far more expansive web around the sun, capturing its energy and fueling our needs, the physicists’ Dyson sphere, also becoming the communication hub for the solar system and the space beyond.

These are the days of awe.

Aubrey Lieberman,

September 2025