Gravity and Orbits
Orbits and GPS
Falling so fast you keep missing the ground
Right now, dozens of satellites are falling toward the Earth as fast as they can, and none of them ever gets any closer. That single strange sentence is what an orbit is. Understand it and you understand why the space station does not drop out of the sky, why a geostationary dish can stare at one satellite forever, and how the phone in your pocket knows where it is to within a few meters.
The last lesson left us with a puzzle. Astronauts on the station are firmly held by gravity and are in constant free fall, yet the station never crashes. How does something fall forever without landing? Newton answered this with a thought experiment about a cannon, and we can now play with his cannon directly.
Predict first: fire the cannon harder
Play with it: Newton's cannonball
The widget below is a tiny gravity simulator. It launches a satellite sideways from a fixed height and lets real gravity bend its path, step by step. Try the preset buttons first, then drag the launch-speed slider slowly upward and watch the shape of the path change.
In orbit: it keeps falling toward the planet and keeps missing.
Fire too slowly and the path curves down into the planet: a crash, exactly like a thrown stone. Speed up and the arc stretches out. At one special speed the ball falls toward the planet at precisely the rate the planet's surface curves away beneath it, so it keeps the same height all the way around: a circle. That is an orbit. The satellite is not fighting gravity or beyond it, it is falling the whole time, and continually missing. Faster still and it swings out into a long ellipse, and beyond one more threshold it leaves for good.
The circular-orbit speed, derived
We can pin down that special speed with two ideas we already have. Anything moving in a circle is constantly accelerating toward the center, and that has size . For an orbit, the thing providing that inward pull is gravity, whose strength is the inverse-square law from last lesson. Set the gravity the planet supplies equal to the centripetal pull the circle needs:
One factor of cancels, and solving for the speed gives a clean, derived result:
Read the surprise in that formula: a bigger orbit needs a slower speed. Low satellites race around in about ninety minutes; the Moon, far out, takes a month. Push the radius to just the right value and the orbit takes exactly one day, so the satellite keeps pace with the ground turning beneath it and appears to hang motionless overhead. That is a orbit, the trick behind fixed satellite dishes. It is not hovering, which would take a constant rocket burn, it is orbiting at just the speed that matches Earth's spin.
The CS bridge: how GPS finds you
Those orbits are useful because we can time them precisely, and that is exactly what GPS runs on. Each satellite carries an atomic clock and endlessly broadcasts the same message: this is who I am, and this is the time I sent this. Your phone hears it, subtracts the send time from the time it arrived, and gets the signal's travel time. Multiply by the speed of light and that travel time becomes a distance.
Knowing your distance from one satellite places you somewhere on a huge sphere around it. A second satellite's distance narrows you to where two spheres cross. A third pins it down. This is , and the widget's second tab shows it in two dimensions: drag the receiver and watch three range circles, one per satellite, always meet exactly where you are.
In practice GPS uses a fourth satellite too, and the reason is a beautiful catch. Your phone's clock is a cheap quartz crystal, nowhere near atomic accuracy, so its idea of "now" is off by some unknown amount, and that same error corrupts every distance identically. A fourth measurement gives the math enough to solve for that clock error as an extra unknown and cancel it, which quietly turns every phone into the proud owner of an atomic-grade clock correction.[2]
Why nanoseconds decide meters
C = 299_792_458 # speed of light, meters per second
def distance_to(sat):
dt = time_received(sat) - sat.time_sent # signal travel time
return C * dt # distance = c * travel time
# one distance -> a sphere you are somewhere on
# three distances -> the spheres intersect at your position
# a fourth -> solves for your cheap clock's unknown offset too
fix = trilaterate([distance_to(s) for s in visible_satellites])Lock it in
- An orbit is continuous free fall: the satellite falls toward the planet as fast as the planet curves away, so it never lands.
- Circular orbital speed is v = sqrt(G M over r), so higher orbits move slower.
- A geostationary orbit has a one-day period, so it stays over one spot; it is orbiting, not hovering.
- GPS turns each satellite's signal travel time into a distance, and several distances trilaterate your position.
- Timing is everything: a nanosecond of clock error is about 30 cm of position error, which is why the satellites carry atomic clocks.
Check yourself
Compared with a low satellite, a satellite in a much higher circular orbit moves:
GPS works out how far you are from each satellite by measuring the:
Why doesn't the space station simply fall down out of orbit?
Orbit as sideways motion plus falling. Try to state it, then check.
Match each launch speed to the path it produces (from the widget).
the arc dips into the planet: a crash
a circle at constant height: a stable orbit
a stretched ellipse swinging out and back
escape: it leaves and never returns
Primary source
Richard Pogge, Real-World Relativity: The GPS Navigation SystemA clear, quantitative walk through how GPS turns satellite timing into position, and why the timing has to be good to nanoseconds. The natural bridge into the relativity lesson.
Sources