Satellite Orbits

The cannonball that never lands

Newton had a genuinely unhinged idea, and it turned out to be right. Imagine a cannon on a mountaintop so tall it pokes above the atmosphere. Fire the cannonball gently and it arcs down and thumps into the ground a mile away. Fire it harder and it lands farther off. Fire it hard enough and something delightful happens: the ground curves away beneath it just as fast as the ball falls toward the ground. It keeps falling — and keeps missing.

That's an orbit. A GNSS satellite 20,000 km up is not hovering, and it has emphatically not "escaped" gravity. It is falling, constantly, with the enthusiasm of a dropped anvil. It simply has enough sideways speed that it never arrives. Douglas Adams described flying as "throwing yourself at the ground and missing." He was joking. He was also describing orbital mechanics with unreasonable accuracy.

The idea, before any mathematics

Strip away the drama and an orbit is a shape plus a schedule. Kepler worked out three rules that, four centuries later, still run every GNSS constellation:

• First law — it's an ellipse. The satellite traces an ellipse with Earth at one focus (not the center). A circle is just the boring special case where the two foci coincide.
• Second law — it trades altitude for speed. The line from Earth to the satellite sweeps out equal areas in equal times. Near Earth (perigee) that line is short, so the satellite must race to sweep the same area; far away (apogee) it can afford to loiter.
• Third law — bigger orbits are slower. The period depends only on the orbit's size (its semi-major axis), and it grows faster than the size does. Double nothing else — just climb higher — and your lap time stretches.

To pin an orbit down completely, engineers use six orbital elements. Two set the ellipse itself — semi-major axis $a$ (size) and eccentricity $e$ (how squashed). Three orient that ellipse in space — inclination (tilt of the plane vs. the equator), RAAN (which way the tilted plane is turned), and argument of perigee (which way the ellipse points within its plane). The last one, true anomaly, says where along the path the satellite is right now. Six numbers, and you know where a satellite is and where it's going. That's the whole magic trick.

Seeing it

The two laws that feel most abstract — "equal areas" and "size sets the clock" — become obvious the moment you watch them. Keep your eye on two things: how the satellite's speed changes as it rounds the ellipse, and how the whole lap time changes when you raise the orbit. One is Kepler's second law; the other is his third. Both are below, live.

Try it yourself

Earth sits at the focus of the orbit (the small circle). The satellite carries a velocity arrow whose length tracks its actual speed. The perigee (closest) and apogee (farthest) points are marked.

Two knobs: altitude grows the orbit (semi-major axis), and eccentricity squashes it. The presets jump you to real regimes — hug-the-planet LEO, the GPS shell, distant GEO, and a lopsided Molniya-like ellipse.

LEOGPS (MEO)GEOMolniya-like

Altitude (semi-major axis)20,180 kmEccentricity0.00Pause

Orbital period

11 h 58 min

Perigee alt.

20,180 km

Apogee alt.

20,180 km

Current speed

3.87 km/s

The velocity arrow stretches at perigee and shrinks at apogee — equal areas swept in equal times. One lap on screen ≈ 11 h 58 min in reality (about 3,075× time-lapse). Raise the altitude and watch the real period balloon: geometry sets the clock.

Three things worth provoking on purpose:

1. Crank eccentricity up and watch the velocity arrow pump — long at perigee, stubby at apogee. That pulsing is Kepler's second law you can see.
2. Drag altitude from LEO to GEO and watch the orbital period read-out balloon from ~90 minutes to 24 hours. Nothing but size changed.
3. Push eccentricity high on a low orbit until the warning fires — perigee has dropped below the surface, and your "orbit" is now a very expensive lawn dart.

Now the mathematics

Kepler's third law is the workhorse. With $GM$ the Earth's gravitational parameter and $a$ the semi-major axis, the period is

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

Notice what's absent: eccentricity. Two orbits with the same $a$ share a period even if one is a circle and the other a cigar. Notice also the $a^3$: altitude punishes period brutally, which is why geostationary orbit sits way out at $a \approx 42{,}164$ km to earn its 24-hour lap.

Speed comes from the vis-viva equation, energy conservation in disguise. At radius $r$,

$$v = \sqrt{GM\left(\frac{2}{r} - \frac{1}{a}\right)}$$

Small $r$ (perigee) gives large $v$; large $r$ (apogee) gives small $v$ — Kepler's second law falling out of the algebra. The radius itself, as a function of the true anomaly $\nu$, is

$$r(\nu) = \frac{a\,(1 - e^2)}{1 + e\cos\nu}$$

which is just the equation of an ellipse with the focus at the origin.

There's one honest complication. To find where the satellite is at a given time, you'd like to advance a clock-like angle uniformly. That angle is the mean anomaly $M$, and it grows steadily at the mean motion $n = \sqrt{GM/a^3}$. But $M$ isn't the real geometric angle. They're linked by Kepler's equation:

$$M = E - e\sin E$$

where $E$ is the eccentric anomaly. The catch: you can't algebraically solve this for $E$ — it's transcendental. So every propagator on Earth does the same thing our code does: Newton–Raphson iteration, converging in a handful of steps (we use a threshold of $10^{-12}$ rad and cap at 50 iterations; near-circular GNSS orbits converge in three or four). Then a stable conversion turns $E$ into the true anomaly $\nu$. That little loop is the quiet engine behind the animation above.

How this appears in engineering

Here's the payoff, and the reason this lesson lives on a positioning platform. A GNSS receiver computes your position from how long signals took to arrive — but travel time is meaningless unless you know where each satellite was when it transmitted. That "where" is an orbit evaluation.

Every GPS satellite therefore broadcasts a set of ephemeris parameters: essentially Keplerian elements plus small correction terms for the perturbations the tidy two-body model ignores (Earth's equatorial bulge, the Moon, the Sun, solar radiation pressure). Your receiver runs almost exactly the pipeline above — solve Kepler's equation, find the true anomaly, place the satellite in its orbital plane, then rotate through inclination, RAAN, and argument of perigee into ECEF. The result is a satellite position in meters, ready to feed the pseudorange solver. Get the orbit wrong by a few meters and your fix is wrong by a few meters — orbit error is a direct error source, not an academic footnote.

The orbit geometry matters too: whether the visible satellites are spread wide across the sky or bunched together is set by the constellation's orbits, and that spread is exactly what dilution of precision measures.

Check your understanding

• A satellite is "falling" the entire time it's in orbit. Why does it never get any closer to the ground?
• Two satellites share the same semi-major axis, but one orbit is nearly circular and the other strongly elliptical. Which has the longer period — and why is that a trick question?
• On an eccentric orbit, where is the satellite moving fastest, and what conserved quantity forces that?
• GEO sits far higher than the GPS shell. Using only Kepler's third law, argue why its period is longer without computing anything.
• Why is knowing a satellite's orbit a prerequisite for a receiver computing your position at all?

Where to go next

You can now produce a satellite's position from six numbers and a clock — which is precisely the input the rest of GNSS has been quietly assuming. Carry it into pseudorange, where those positions anchor the distance measurements, and into dilution of precision, where the arrangement of those orbits decides whether your fix is crisp or mushy. The coordinate frames you met earlier are the stage; orbits are what finally walk out onto it.