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Orbits: Falling Around the World

Turn the falling-around idea into real numbers: derive orbital speed and period, see why higher orbits are slower, understand why astronauts float, and compute the special altitude of a geostationary satellite.

What actually keeps a satellite up?

A satellite is not held up by rockets or by being 'beyond gravity.' It is simply in continuous free fall — like the falling elevator, but moving sideways so fast that the Earth curves away beneath it exactly as fast as it drops. It falls forever and never lands. Everything about orbits follows from taking that one sentence seriously.

One picture, many outcomes. As launch speed increases, the same gravity produces a crash, then a circle skimming the planet, then a widening ellipse, and finally an unbound escape path. Speed alone decides the shape.

Drive it yourself: set the tangential speed and watch a circular orbit form when gravity exactly matches the turn. A little slower and the satellite dips toward the planet; a little faster and the orbit stretches into an ellipse.

The circular-orbit condition

For a circular orbit of radius r, gravity must supply exactly the centripetal force mv^{2}/r needed to keep bending the path into a circle. Set the gravitational pull equal to that requirement:

\frac{GMm}{r^{2}} = \frac{mv^{2}}{r}

Gravity (left) provides exactly the centripetal force (right). The satellite's mass m appears on both sides and cancels.

Orbital speed and period

Cancel the satellite's mass and one factor of r, and solve for the speed. This is the orbital velocity — the one speed at which a circular orbit at radius r is possible.

v = \sqrt{\frac{GM}{r}}

Orbital speed for a circular orbit. Strikingly, it does not depend on the satellite's own mass — a bolt and a space station at the same altitude orbit at the same speed.

The time for one lap is the circumference divided by the speed, T = 2\pi r / v. Substituting v gives the orbital period:

T = 2\pi\sqrt{\frac{r^{3}}{GM}}

The orbital period grows with r^{3/2}. A low Earth orbit (r ≈ R⊕, v ≈ 7.9 km/s) laps the planet in about 90 minutes.

Why astronauts float

Worked example: the geostationary orbit

Communications and weather satellites often sit in a geostationary orbit: they circle once per day, in step with Earth's rotation, so they hover over the same spot on the equator. Let us find the radius that makes T equal one day. Rearrange the period formula:

r = \left(\frac{GM\,T^{2}}{4\pi^{2}}\right)^{1/3}

Solve the period relation for r. For a geostationary orbit use the sidereal day, T ≈ 86 164 s, and GM⊕ ≈ 3.99×10¹⁴ m³/s².

Plugging in gives r \approx 4.22\times 10^{7}\ \mathrm{m} — about 42,200 km from Earth's center. Subtract Earth's radius (6,370 km) and the satellite hovers at an altitude of roughly 35,800 km above the equator. Every geostationary satellite in the world — thousands of them — must live on this one thin ring, which is why orbital 'slots' there are a scarce, regulated resource.