06 · Expensive
Escape
Escape is √2 times circular at the same r. From LEO the extra is ~3.2 km/s — less than the climb from the pad. Orbit is halfway to anywhere.
In English
To never come back from this altitude, go √2 times circular speed. From LEO that extra is about 3.2 km/s — smaller than the launch that got you there. Hence the slogan: orbit is halfway to anywhere.
- Try this
- Start at 400 km. Note the extra. Drag toward GEO: circular is slower, extra is smaller still.
- Keep this
- Escape = √2 × circular. LEO extra ~3.2 km/s.
Lab · √2 times circular
Escape is √(2μ/r) = √2 times circular at the same r. From LEO that extra is only ~3.2 km/s — less than the climb from the pad to LEO. Leaving Earth is two acts: get into orbit, then leave the well.
From 400 km circular
- Circular
- 7.67 km/s
- Escape
- 10.9 km/s
- Extra
- 3.18 km/s
vesc = √(2μ/r) = √2 vcirc. From the surface, 11.2 km/s. Nobody goes that way through the air. Oberth: burn deep in the well, where the same Δv buys more energy.
C3 is leftover energy after escape. TMI and TLI are specified as C3, not as “escape plus a bit.” The lab stops at the √2 so the slogan is earned.