Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

The speed that does not come back, and the √2 that separates it

Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving.
11 min read 6 figures Everything is a conicScales that defeat intuition

Gravity has infinite range. It never switches off, and there is no distance at which a body has finished escaping. So the phrase “escape velocity” ought to be meaningless — whatever speed something leaves at, gravity is still pulling on it.

It is not meaningless, because the total pull over an infinite distance is finite. The potential well has a definite depth, and a body arriving with more kinetic energy than that depth will still be moving when the pull has run out.

Circular and escape speedOrbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.2468101200.511.5distance, in body radiiescape speedcircular speedthe surfacea high orbitgeostationarythe gap is always a factor of √2
Fig. 1 Circular and escape speed against distance, in units of the circular speed at the body’s surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.

The energy statement

Set the total energy to zero — the boundary between bound and unbound:

12v2−GMr=0⟹vesc=2GMr.\frac{1}{2}v^2 - \frac{GM}{r} = 0 \quad\Longrightarrow\quad v_{\text{esc}} = \sqrt{\frac{2GM}{r}}.

The circular speed at the same radius comes from balancing gravity against the centripetal requirement, vcirc=GM/rv_{\text{circ}} = \sqrt{GM/r}. Divide one by the other and everything cancels:

vescvcirc=2.\frac{v_{\text{esc}}}{v_{\text{circ}}} = \sqrt{2}.

No mass, no radius, no constants. The ratio is 2\sqrt{2} at every distance from every body in the universe, and the gap visible in the figure is the same gap everywhere.

The consequence for spaceflight is encouraging. Reaching low Earth orbit takes 7.8 km/s; leaving Earth entirely takes 11.2 km/s. Getting into orbit is 71% of the speed needed to leave the planet, which is the origin of the observation that low orbit is halfway to anywhere. In energy terms it is exactly half, since energy goes as the square.

The direction that does not matter

A curious feature of the escape condition: it says nothing about direction.

Energy is a scalar. A body launched at 11.2 km/s straight up escapes; so does one launched at 11.2 km/s at 45°, or horizontally, or at any angle that does not run into the ground. The trajectories are wildly different — one radial, one a wide parabola — and all of them are unbound.

That is a real consequence of energy conservation and it is worth checking against intuition, which insists that going up must be more effective. It is not. What direction changes is the shape of the escape trajectory and the time taken, not whether escape happens.

There is a strong practical caveat. Launching horizontally at ground level means traversing the atmosphere at 11.2 km/s, which is not survivable. Real launches go up first to clear the air and then turn, and the turn is a compromise between drag and gravity losses rather than anything to do with escape.

Every orbit one force allowsCircle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.common periapsiscircleellipse, e = 0.5ellipse, e = 0.9parabola, e = 1hyperbola, e = 1.5e < 1 returnse ≥ 1 never does
Fig. 2 The four conics, sharing a focus. The parabola is the escape trajectory exactly: zero total energy, the boundary between the ellipses that return and the hyperbolas that leave with speed to spare.

Escape speed produces a parabola, which as a member of the conic family is the case of exactly zero energy. A body on it arrives at infinity with exactly zero speed — it escapes, and only just. Any excess puts it on a hyperbola, arriving with a residual speed called v∞v_\infty, and that residual is the quantity interplanetary mission design actually cares about.

The numbers, and what they decide

Body Escape speed
Ceres 0.51 km/s
Moon 2.4 km/s
Mars 5.0 km/s
Earth 11.2 km/s
Jupiter 59.5 km/s
Sun (from surface) 618 km/s
Sun (from Earth’s orbit) 42.1 km/s

The last two rows matter more than they look. Leaving the solar system from Earth’s orbit requires 42.1 km/s relative to the Sun, and Earth already provides 29.8 of it. A spacecraft that escapes Earth in the right direction inherits that, and needs only the difference — which is why every outer-planet mission launches along Earth’s orbital motion, and why a gravity assist is worth so much.

Escape speed also decides which atmospheres survive. Gas molecules have a distribution of speeds, and the fastest tail of that distribution can exceed escape speed and leak away. Since lighter molecules move faster at a given temperature, they leak first. The Moon has 2.4 km/s and no atmosphere. Earth has 11.2 and has kept nitrogen and oxygen while losing essentially all its hydrogen and helium. Titan is smaller than Mars with an escape speed of 2.6 km/s and has a thick atmosphere, because it is cold enough that the tail never reaches. The composition of a planet’s air is a competition between escape speed and temperature.

The deeper the well, the more a burn is worth

Escape speed rises steeply as a body is approached, and that steepness makes the timing of a burn matter enormously.

A burn adds a fixed amount of velocity. The energy it adds is not fixed — the change in kinetic energy is Δ(v2/2)\Delta(v^2/2), which for a given Δv\Delta v grows with the speed the spacecraft already has. Burning deep in a gravity well, where the spacecraft is moving fastest, converts propellant into orbital energy far more efficiently than burning far out.

This is the Oberth effect, and it is the reason a departure burn is made at the lowest point of the orbit rather than at leisure. Parker Solar Probe uses it in reverse: each Venus flyby lowers its perihelion, so each subsequent burn happens deeper in the Sun’s well and does more.

The effect has an unintuitive corollary. To reach the Sun is harder than to leave the solar system. Earth’s 29.8 km/s of orbital motion has to be cancelled to fall inward, against 12.3 km/s to escape outward from Earth’s orbit. Getting to the Sun is more than twice as expensive as getting to interstellar space, which is why the Parker probe needed seven Venus flybys and why nothing has ever simply flown at the Sun.

A Hohmann transfer, 4 to 1 in radiusTwo circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation.burn 1: +0.265burn 2: +0.184total 0.449 — and no way to spend lesscoast: half an ellipse, 1.98 of an inner-orbit yearspeeds in units of the inner circular speed
Fig. 3 A transfer to four times the radius. The first burn happens at the low point, where the spacecraft is moving fastest — which is where a given change in speed buys the most energy.

Where it stops being a speed

Push the formula to its limit and it produces the strangest object in astrophysics.

Escape speed grows as the square root of M/rM/r, so a body compact enough will have an escape speed exceeding the speed of light. Setting vesc=cv_{\text{esc}} = c gives

rs=2GMc2,r_s = \frac{2GM}{c^2},

which for the Sun is 3 km and for the Earth about 9 mm.

Michell in 1783 and Laplace in 1796 both did this calculation and both concluded that sufficiently massive stars would be invisible. Their reasoning was Newtonian and their formula is right, which is a coincidence: the correct general-relativistic derivation gives the identical expression for entirely different reasons. The Newtonian picture has light slowing down and falling back like a thrown ball, which is wrong; the relativistic one has the geometry of spacetime closing off every outward path, which is not a statement about speed at all.

At that point “escape velocity” stops being a useful concept. Nothing escapes, not because it would need to move faster than light, but because there is no outward direction to move in.

Just below the line

The interesting region is not above the escape curve but just under it, where an orbit is bound and only barely.

An orbit at eccentricity 0.97An orbit of eccentricity 0.97. The primary sits at a focus, offset from the centre by 0.97 of the semi-major axis, and the closest and furthest points differ by a factor of 65.67.empty focusrperiapsisapoapsis
Fig. 4 An orbit at eccentricity 0.97. The body is on a bound path by a very small margin, spends nearly all its time far from the primary, and passes through the whole of the interesting part of its orbit in a matter of weeks.

Long-period comets live here, and the margin is tiny. A comet with an aphelion at 50,000 AU is bound by an energy difference of about one part in 10510^5 of its perihelion kinetic energy — small enough that a passing star, or a nudge from Jupiter, can push it either way. The boundary between returning and leaving is crossed routinely by objects that were never far from it.

Speed against distance, for orbits of the same periodOrbital speed against distance from the primary. Every orbit with the same semi-major axis follows the same curve; the eccentricity decides only which stretch of it the body uses.00.511.50123distance from the primary (a = 1)e = 0e = 0.4e = 0.8circular speed at r = a
Fig. 5 Speed against distance. Near the primary the curves for very different orbits converge, which is why a comet’s perihelion speed is almost exactly the escape speed at that distance and why it is so hard to tell the two apart from a short arc of observations.

That convergence is a practical difficulty as well as a curiosity. Determining whether a newly discovered object is bound requires measuring an energy that is nearly zero, from an arc of orbit that is nearly a parabola whatever the answer, which is why the interstellar visitors needed weeks of tracking before anybody would say so.

Where the model stops

Point masses. The formula uses the distance from the centre, which is only right outside a spherical body. Inside one, the escape speed is smaller and the calculation needs the mass enclosed.

Circular and escape speedOrbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.10203040506000.511.5distance, in body radiiescape speedcircular speedthe surfacegeostationarythe Moonthe gap is always a factor of √2
Fig. 6 The same relation out to the Moon’s distance. The curves fall steeply and then flatten, which is why almost all of the cost of leaving is paid in the first few body radii — and why a burn deep in the well is worth so much more.

One body. Escaping the Earth is not escaping the Sun. Every escape is from a specific well, and the solar system’s structure is a set of nested wells with boundaries where one dominates. A spacecraft on a hyperbola with respect to Earth is usually still on a perfectly ordinary ellipse with respect to the Sun.

No atmosphere. Drag makes the ballistic calculation irrelevant for any launch from a body with air.

Impulsive departure. A continuously thrusting spacecraft never needs escape speed at any moment — it can spiral out arbitrarily slowly. Escape speed is the requirement for an unpowered coast, not for leaving.

Newtonian gravity. As above, at the extreme.

The figure has a limitation shared by every plot of this kind: it shows speeds in units of the surface circular speed, which makes all bodies look the same. That is the point — the 2\sqrt{2} is universal — and it also hides that the actual numbers span three orders of magnitude between a small asteroid and the Sun. Universality on one axis is bought by suppressing everything on the other.

The ladder from here

Later rungs: gravitational potential and potential energy, and the sign conventions that make them confusing. Escape from a rotating body, and why equatorial launch sites help. Atmospheric escape mechanisms — Jeans, hydrodynamic, and non-thermal — and which planets lost what. The Oberth effect worked through. v∞v_\infty and the characteristic energy C3C_3. Spheres of influence and patched conics. The rocket equation, which converts these speeds into propellant mass and delivers the bad news. And the Schwarzschild radius done properly, where the Newtonian coincidence stops.

Michell’s 1783 paper describes objects whose light cannot escape and suggests they might be detected by their effect on companion stars. That is exactly how the first stellar-mass black hole was found, 188 years later.

Where this goes next

common periapsiscircleellipse, e = 0.5ellipse, e = 0.9parabola, e = 1hyperbola, e = 1.5e < 1 returnse ≥ 1 never does Every orbit one force allows, and the number that picks between them Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back. burn 1: +0.202burn 2: +0.158total 0.360 — and no way to spend lesscoast: half an ellipse, 1.21 of an inner-orbit yearspeeds in units of the inner circular speed The cheapest way between two orbits, and why it is so slow Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait. the planet's velocityinout70°speed before: 0.505speed after: 0.661gained 0.157, and burned nothingthe circle: constant speed in the planet's frame Stealing speed from a planet, which does not notice A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.