Vol. Science · Astronomy · The departure

Characteristic
energy

Characteristic energy is how a hyperbolic departure is counted in the planet’s frame. C3 = v_∞². Specific orbital energy is ε = v²/2 − μ/r, and on a hyperbola C3 = 2ε. At a periapsis radius r, escape speed is √(2μ/r) and circular speed is √(μ/r). A craft already on a circular parking orbit at that r reaches the hyperbola with one impulsive prograde burn, Δv = √(C3 + 2μ/r) − √(μ/r). C3 = 0 is barely escape. v_∞ is zero, and the path is a parabola. A positive C3 is a hyperbolic excess. Around Earth, μ is the Earth Fact Sheet GM, 0.39860 × 10⁶ km³/s², and the volumetric mean radius is 6,371.000 km. At altitude 0 those two figures give 11.186 km/s, the escape velocity the sheet lists. The Escape desk is that pair of speeds. A classroom circle 400 km up has r = 6,771 km. The fact sheet does not print that altitude. Circular speed there is 7.673 km/s and local escape is 10.851 km/s, so C3 = 0 costs 3.178 km/s. That is the Δv the Oberth desk prints, and the figure the rocket desk prices. At that desk’s classroom 450 s, with g₀ = 9.80665 m/s², m_f/m₀ = 0.4867. At 300 s it is 0.3395. A classroom excess of v_∞ = 3 km/s is C3 = 9 km²/s². Periapsis speed is 11.258 km/s and the burn is 3.585 km/s. The same C3 on the geosynchronous classroom circle, 42,164.190 km from the center, costs 2.208 km/s. The lower circle costs more, because it starts deeper in the well. Oberth holds Δv fixed and watches the energy change. This desk holds C3 fixed. Speed on a bound ellipse is vis-viva. A heliocentric transfer, such as the Hohmann desk’s Earth-to-Mars burn, is not an Earth-relative C3. A flyby that turns a v_∞ the craft already has is the gravity assist desk. The sphere where the planet’s frame is the useful one is Hill.

Classroom schematic. Impulsive burn, point-mass Earth, no atmosphere. The parking orbit is a circle. The departure is a hyperbola, or a parabola when C3 is zero. No drag, no gravity losses, no oblateness, and no third body. Not a launch window, not Horizons, and not mission planning. The 400 km altitude is a classroom example. The geosynchronous radius is computed from the fact sheet’s sidereal rotation period, 23.9345 hours. The sheet does not print either altitude. The 300 s and 450 s figures are the rocket desk’s classroom examples. They are not engines. The remaining mass fraction is a comparison, not a propellant load.

Classroom comparison

Three presets from the formulas on this page. The 400 km row is the Oberth desk’s low circle. The higher circle is that desk’s geosynchronous radius, 42,164.190 km, from the fact sheet’s 23.9345 hour sidereal day. The remaining fractions use g₀ = 9.80665 m/s² and the unrounded Δv. A highlighted row is where the controls sit. Not a propellant load.

What the formula is

Words

C3

Characteristic energy, v_∞², in km²/s². It is twice the specific energy of the departure hyperbola. Zero is barely escape. Positive is a hyperbolic excess. Negative would still be bound. This desk starts at zero.

v∞

The speed relative to the planet, far away, after the engines are already off. On this page it is √C3. The gravity-assist desk turns a v_∞ that arrives from somewhere else. This desk is the burn that produces one.

Parking circle

A circular orbit at radius r = R + altitude. R is Earth’s volumetric mean radius, 6,371.000 km. Speed √(μ/r). The 400 km altitude is a classroom example shared with the Oberth and gravity-assist desks. The fact sheet does not print it.

Periapsis speed

v_p = √(v_∞² + 2μ/r). The fast point on the hyperbola, at the parking radius, just after the impulsive burn. When C3 is zero this equals local escape speed.

The two circles

The readout keeps Δv for this C3 at 400 km and on the higher circle, whatever the slider is doing. The higher circle is r = 42,164.190 km, 35,793.190 km above the volumetric radius, from the sidereal period 23.9345 hours. It is the Oberth desk’s comparison circle. It is not a mission orbit.

What the picture leaves out

Air, a spinning Earth, an equatorial bulge, a finite burn, gravity losses, and every third body. The disk is a volumetric sphere. When the 400 km gap would be a hairline, the disk is drawn smaller so the parking orbit can be seen, and the hint says so. The speeds still use the true radius. Not a launch window, not Horizons, and not mission planning. No named flight, and no mission C3.

Classroom picture of one impulsive departure from a circular parking orbit. At 400 km, C3 = 0 is a 3.178 km/s burn and C3 = 9 km²/s² is a 3.585 km/s burn. The same C3 = 9 on the geosynchronous classroom circle is 2.208 km/s. Fact-sheet GM and volumetric radius. The 400 km altitude and the 3 km/s excess are classroom examples. Not a launch window, not Horizons, and not a mission. Earthrise preview: NASA / Bill Anders (public domain).

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