Vol. Science · Astronomy · The burn
The same impulsive Δv does not buy the same orbital energy everywhere. Specific energy is ε = v²/2 − μ/r. On a bound Keplerian ellipse that is also −μ/(2a). A burn along the velocity vector changes the speed from v to v + Δv, so the energy changes by Δε = v·Δv + ½(Δv)². The cross term is the Oberth effect: a larger v, deep in the well, multiplies the same Δv into a larger energy gain. The ½(Δv)² term does not care how fast the craft was already going. Start on a circle, vc = √(μ/r). A prograde burn leaves periapsis at the burn point. If the new energy is still negative, a' = −μ/(2ε') and the apoapsis is 2a' − r. If ε' ≥ 0 the craft is unbound. Escape from the circle takes Δvesc = vc(√2 − 1), the gap between the two speeds on the Escape desk. 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. A classroom circle 400 km up has r = 6,771 km, circular speed 7.673 km/s, local escape 10.851 km/s, and Δv to escape 3.178 km/s. The 400 km altitude is an example. The fact sheet does not print it. It is the low orbit on the Hohmann and vis-viva desks. The comparison circle at the geosynchronous radius is 42,164.190 km, from r³ = μ T² / (4π²) with the sheet’s sidereal rotation period, 23.9345 hours. That is 35,793.190 km above the volumetric radius. It is a classroom circle at that r, not a mission. A 1 km/s burn gains 8.173 km²/s² at 400 km and 3.575 km²/s² on that higher circle. Speed on the ellipse after the burn is vis-viva. The half-ellipse between two circles is Hohmann. The year is Kepler.
Classroom schematic. Point-mass, two-body, one impulsive burn along the velocity. No atmosphere, no drag, no oblateness, and no third body. Not an ephemeris and not a mission planner. Top-down, from the north, so a prograde burn points counterclockwise. The two circles are to scale with each other. When the low circle hugs the body, the body is drawn smaller so the gap stays visible, and the hint says so. The 400 km and 10,000 km altitudes are classroom examples. The geosynchronous radius is computed from the fact sheet. It is not an altitude the sheet prints. The solar preset uses Earth’s semimajor axis and Mars’s semimajor axis as the radii of two circles. Those fact-sheet orbits are not circles.
From the north · Earth at the focus · same Δv on both circles
Preset
Altitudes above the volumetric mean radius, in km
Enter a Δv of zero or greater, and a higher circle outside the low circle.
Earth · two circles
Δv = 1.000 km/s
Low · v before
7.673 km/s
r = 6,771 km
Higher · v before
3.075 km/s
r = 42,164.190 km
Low · Δε
8.173 km²/s²
v·Δv + ½(Δv)²
Higher · Δε
3.575 km²/s²
Same Δv, smaller v
The low burn gains more energy because circular speed is higher. Both 1 km/s burns stay bound.
The burn arrows are the same Δv. The bright path leaves the low circle.
What the formula is
ε = v²/2 − μ/r. μ is GM of the central body. r is the distance from that body. v is the speed. On a circle, vc = √(μ/r), so ε = −μ/(2r). On a bound ellipse the same energy is −μ/(2a). The Escape desk is the two speeds at one r: circular, and √2 times circular. The vis-viva desk is the speed the ellipse already has, v = √[μ (2/r − 1/a)].
Δε = v·Δv + ½(Δv)² for an impulse along the velocity. The new speed is v' = v + Δv. Expanding v'²/2 − v²/2 gives the cross term and the square. The cross term grows with v. The square does not. That is why the same burn, deep in the well, changes the energy more than it does higher up, where the circular speed is lower.
From a circle, a prograde burn puts periapsis at the burn point. If ε' is negative, a' = −μ/(2ε') and ra = 2a' − r. If ε' is zero, the path is a parabola and the excess speed at infinity is zero. If ε' is positive, the path is a hyperbola and v∞ = √(2ε'). Escape from the circle still costs Δvesc = vc(√2 − 1). That increment is larger on the lower circle, because vc is larger. The higher circle was already closer to leaving.
A burn that escapes the low circle can leave more residual speed than the same Δv higher up. It does, once Δv is past the average of the two circular speeds. Below that, the higher burn can finish with a larger v∞, because it started higher in the well, even though it gained less energy. The readout prints both. A 6 km/s classroom burn from 400 km leaves 8.319 km/s at infinity. The same 6 km/s on the geosynchronous circle leaves 7.965 km/s. The Δv that only just escapes the 400 km circle, 3.178 km/s, leaves no excess there and 4.493 km/s from the higher circle.
The check
Earth’s GM is 0.39860 × 106 km3/s2. The volumetric mean radius is 6,371.000 km. Escape speed at that radius is 11.186 km/s, the figure the Earth Fact Sheet lists, and the figure the Escape desk prints at altitude 0. Add a classroom 400 km and r = 6,771 km. Circular speed is 7.673 km/s. Local escape is 10.851 km/s. The difference is 3.178 km/s, and that difference is vc(√2 − 1). The vis-viva desk prints the same 7.673 km/s and 10.851 km/s on its 400 km circle.
The geosynchronous radius uses that same GM and the sheet’s sidereal rotation period, 23.9345 hours: r³ = μ T² / (4π²). The radius is 42,164.190 km. The altitude above the volumetric radius is 35,793.190 km. A 1 km/s prograde burn gains 8.173 km²/s² at 400 km and 3.575 km²/s² on that circle. The cross terms are 7.673 km²/s² and 3.075 km²/s². The ½(Δv)² term is 0.500 km²/s² either way. A classroom circle 10,000 km up, r = 16,371 km, gains 5.434 km²/s² from the same 1 km/s. These altitudes are examples. The fact sheet does not print them.
Words
Classroom picture of one burn at two radii. From 400 km, with Earth’s GM, Δv to escape is 3.178 km/s. A 1 km/s burn gains 8.173 km²/s² there and 3.575 km²/s² on the computed geosynchronous circle. Point-mass, two-body, impulsive. Not an ephemeris and not a mission planner. The 400 km and 10,000 km altitudes are examples, not fact-sheet lines. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The speed
Circular orbit and escape at one altitude. √2 apart. The Δv on this page is the gap between those two speeds.
The coast
Two burns and half an ellipse between two circles. The first burn is an Oberth burn at the inner orbit.
The ellipse
Speed anywhere on the ellipse this burn just made. v = √[μ (2/r − 1/a)].
The year
P² = a³ in years and AU. The new semimajor axis on this page is what sets the new period.
Two circular orbits, one impulsive Δv along the velocity at each. Δε = v·Δv + ½(Δv)². The picture is a classroom schematic. It is not a trajectory to fly.
| 400 km and geosynchronous | Low circle: Earth Fact Sheet volumetric radius 6,371.000 km plus 400 km. Higher circle: the radius whose period equals the sheet’s 23.9345-hour sidereal day. μ is Earth’s GM. The 400 km altitude is a classroom example. The geosynchronous altitude is computed. Neither is a line the fact sheet prints as an orbit. |
|---|---|
| 400 km and 10,000 km | Both altitudes are classroom examples above the same volumetric radius. Same Earth GM. The fact sheet does not print them. |
| 1 AU and Mars’s a | Two classroom circles around the Sun. The inner radius is Earth’s semimajor axis, 149.598 × 106 km. The outer radius is Mars’s semimajor axis, 227.956 × 106 km. μ is the Sun’s GM. These are not the eccentric orbits on the sheets. |
| Arrow keys | With a preset button focused, Left and Right move through the presets. They do nothing while a field is focused, and nothing while this dialog is open. |
| Δv | Kilometers per second, prograde, the same number on both circles. Zero leaves both circles as they are. The field accepts a typed value. |
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| 1 km/s | Puts the burn back to one kilometer per second. A modest burn. Around Earth both results stay bound at the presets. |
| Δv to escape the low circle | Sets vc(√2 − 1) at the low radius. The low path opens into a parabola. The higher circle, which needs less Δv to escape, is already unbound. |
| Low excess wins | On the Earth presets this sets 6 km/s. That is past the average of the two circular speeds, so both paths escape and the low burn’s hyperbolic excess is the larger one: 8.319 km/s from 400 km, 7.965 km/s from the geosynchronous circle. On the solar preset it sets Δv just above that average. |
| Low and higher | Around Earth, altitudes above the volumetric mean radius. Around the Sun, radii in millions of kilometers. The higher circle stays outside the low one. Moving either one leaves the preset. |
| v before | √(μ/r) on each circle. |
|---|---|
| Δε | v·Δv + ½(Δv)². Larger on the low circle whenever Δv is greater than zero. |
| Energy | ε before the burn and ε' after it. Negative is bound. Zero or positive is escape. |
| Result | Semimajor axis a' = −μ/(2ε') when bound. Escape, parabolic or with v∞, when it is not. |
| Apoapsis or excess | ra = 2a' − r when the new orbit is an ellipse. v∞ = √(2ε') when it is not. |
| Picture | The burn sits on the right, and the arrow points up the page: counterclockwise, from the north. A path that runs off the frame still has its apoapsis or excess in the readout. |
| ? | Opens this dialog. Escape closes it. Tab stays inside until it closes. |
|---|---|
| Escape | Closes the dialog and returns focus to the ? button. |