Log 000 · Why this manual exists
You are holding a manual, but not the reading kind. Every figure below is a working instrument: the Earth in Figure 1 genuinely pulls, the trajectories are computed by a real two-body integrator, and the velocity arrows obey your hand. We built it this way because orbits refused to enter our heads any other way. A paragraph can tell you that an orbit is a fall that keeps missing; only your own hand on the cannon makes you believe it.
So the Society asks one thing of you: predict first, then drag. The difference between what you expected and what the orbit does is the entire lesson. Scrolling will move you down the page. It will never move the sky. Only your hands do that.
Log 001 · A tall mountain and no wind
Falling Forever
In 1687 Newton proposed a thought experiment: carry a cannon up an impossibly tall mountain and fire it sideways, harder and harder. Below is that experiment with the thinking made touchable. The mountain stands 300 kilometers tall (real ones are shorter; the exaggeration buys us a vacuum), and gravity in this figure is the real law, an inverse square field with μ = GM = 398,600 km³/s².
Slow shots arc over and strike the ground. Faster ones strike farther, because the Earth's surface curves away beneath them. At 7.73 kilometers per second the ground curves away exactly as fast as the ball falls toward it, and the shot comes home from behind: a circular orbit. The ball is still falling. It never stops falling. It simply keeps missing. And past 10.93, it outruns gravity's interest entirely.
Exercise 001. Drag the arrow. Find the speed where falling becomes missing. Then find the one where the ball leaves and never writes home.
- Muzzle speed
- 7.73 km/s
- Result
- circular orbit
- Period
- 1h 30m
- Shots fired
- 3
What your hand just learned is the whole secret: orbit is not about going up. It is about going sideways fast enough that the ground never arrives.
Log 002 · One position, one velocity, one future
The Shape of Speed
An orbit is not a track laid in the sky. It is a consequence. The spacecraft below has its entire future written by two arrows: where it is, and how fast it is going. Drag the ship along its path and the velocity adjusts to match that spot. Drag the tip of the velocity arrow and you are choosing a different future, instantly, everywhere: the ellipse redraws, and the two labeled points slide to their new homes. Apoapsis is the high point, periapsis the low. The Society is named after the view from the top.
Try the trade we prize most: speed spent here becomes altitude there, half an orbit away. Pull the arrow longer at the low point and watch the far side leap. Aim it off the tangent and the whole ellipse rotates, because whatever else changes, the new orbit must still pass through the one point where you fired.
- Semi-major axis
- 12,811 km
- Eccentricity
- 0.47
- Apoapsis alt
- 12,500 km
- Periapsis alt
- 380 km
- Speed at ship
- 4.03 km/s
- Period
- 4h 00m
Log 003 · The quartermaster prices every burn
The Transfer
In 1925 Walter Hohmann, a city architect who did celestial mechanics after hours (our kind of person), published the cheapest known road between two circular orbits: half an ellipse that kisses both. It costs two burns. The first, prograde, stretches the far side of your orbit up to the target. The second, fired at apoapsis, lifts the low side to circularize. Fuel is delta-v, and delta-v is everything: your tanks hold 5.00 km/s and the optimum for this pair of orbits is 3.43. Spend little, and the Society takes notice.
Exercise 003. Slide the diamond node around your orbit, pull its handle prograde to plan a burn, then arm it and run the clock. Reach the target ring at 42,164 km (where a satellite hovers over one meridian) and circularize within 3 percent.
- Δv spent
- 0.00 km/s
- Hohmann optimum
- 3.43 km/s
- Planned burn
- +0.00 km/s
- Status
- holding, awaiting your plan
Notice where the cheap road runs: you never point at the target. You raise your far side until it touches the ring, coast the long half ellipse, and buy the circle at the top.
Log 004 · The calendar is a navigation instrument
The Window
Mars is not where you aim. Your ship coasts the long half ellipse for 259 days, so you aim at where Mars will be, a point the instrument draws for you as a ghost. The geometry only works when Mars leads Earth by about 44 degrees at departure. Miss the moment and physics does not punish you; it just hands you the synodic wait, about 26 months, until the angle comes around again.
Scrub the calendar with your hand. Watch the phase angle fall toward 44 degrees and the ghost slide toward the arrival mark. Launch when you believe. The sky keeps its own calendar.
- Calendar
- Sol 0
- Phase angle, Mars ahead of Earth
- 182.8°
- Angle that works
- 44.3°
- Next window
- in 300 days
- Verdict
- no probe aloft
Log 005 · Final exercise, probation ends
The Rendezvous
The final exam is a handshake 400 kilometers up. Your station runs a circular orbit, one lap every 92 minutes, and you trail it by about 470 kilometers. Here is the sentence every probationary member gets wrong on the first try: to catch up, brake. Braking drops you to a lower orbit, and lower orbits are faster, so the gap closes. Then, and this is the half everyone forgets, undo your cleverness: burn prograde as you arrive, raise back up, and match speeds. Skip that step and you will sail past the station, waving.
If you docked, you now hold the two ideas this manual exists to hand you: an orbit is a velocity, and patience is propellant.
Log 006 · Adjourned
That is the whole manual. No formula was required to begin, and yet you have now used vis-viva, conic sections, symplectic integration, phase angles, and the synodic period, all through your fingertips. The equations are waiting in the appendix whenever you want to shake their hands properly.