One equation for the speed anywhere, and the eccentricity is not in it

Ask how fast a body is moving at some point of its orbit and the answer needs, it seems, quite a lot of information: how big the orbit is, how elongated, where in it the body has got to, and which direction it is heading.

It needs two numbers. The distance from the primary at that instant, and the semi-major axis of the orbit. Nothing else — not the eccentricity, not the position angle, not the orientation, not the mass of the orbiting body.

$$v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right).$$

That is the vis-viva relation, and the striking thing about it is what has gone missing. Two bodies at the same distance from the Sun, one on a nearly circular orbit and one on a comet’s path, move at completely different speeds — but only because their semi-major axes differ, never because their shapes do. A circle and a wildly eccentric ellipse of the same $a$ cross at two points, and at those crossings the two bodies are moving at identical speeds.

It is conservation of energy, wearing a different hat #

The relation is not an extra fact about orbits. It is the statement that energy is conserved, solved for the speed.

The energy per unit mass of an orbiting body is kinetic plus potential,

$$\varepsilon = \frac{v^2}{2} - \frac{GM}{r},$$

and for an inverse-square force this is constant along the orbit. Evaluating it at periapsis and at apoapsis, where the motion is purely transverse and the angular momentum makes the algebra collapse, gives

$$\varepsilon = -\frac{GM}{2a}.$$

That is the whole content. The total energy depends on the semi-major axis and on nothing else — not on the eccentricity, not on the angular momentum. Substituting it back into the first expression and solving for $v$ produces the vis-viva relation directly.

Which means the surprise has a one-line explanation. Speed is fixed by energy and distance; energy is fixed by $a$; therefore speed is fixed by $a$ and $r$. The eccentricity is absent from the speed law because it is absent from the energy.

Angular momentum, by contrast, depends on both: $h = \sqrt{GMa(1-e^2)}$. So the two conserved quantities of a two-body orbit divide the labour cleanly. Energy fixes the size, angular momentum fixes the shape, and the speed at a given distance is a question about energy alone.

Three cases in one expression #

The relation covers the whole conic family, and it does so by letting $a$ take values that sound illegal.

For a circle, $r = a$ everywhere, and the bracket collapses to $1/r$: $v = \sqrt{GM/r}$, the circular speed.

For an ellipse, $a$ is positive and $2/r > 1/a$ throughout, so $v$ is always real and always less than the escape speed.

For a parabola, $a$ is infinite and $1/a$ vanishes, leaving $v^2 = 2GM/r$ — escape speed exactly, at every distance.

For a hyperbola, $a$ is negative. The term $-1/a$ becomes an addition, and $v^2$ approaches $-GM/a$ rather than zero as $r$ runs to infinity. That residual is $v_\infty^2$, and it is the quantity interplanetary mission design actually budgets in, usually squared and called $C_3$.

A negative semi-major axis is a strange object, and it is the right bookkeeping. A hyperbola has no far end, so “half the long axis” measures nothing; what the algebra needs is a quantity that carries the energy, and $-GM/2a$ carries it whatever the sign.

What a burn does to it #

The relation is not merely descriptive. It is the working tool of orbital manoeuvring, and the reason is that a burn changes exactly one of its two inputs.

A rocket firing changes the velocity at the point where it fires and does not change the position — the fact that makes two burns the minimum for a transfer. So $r$ is the same immediately before and after, $v$ has changed by the amount burned, and the equation returns the new $a$ in one step. A manoeuvre is a jump from one curve to another at fixed $r$, and the whole of trajectory design is the arithmetic of such jumps.

The direction of the burn does not enter the energy bookkeeping, which is a fact worth stating carefully because it is half true. The magnitude of the new velocity is what fixes $a$, so a burn that changes direction without changing speed leaves the orbit’s size and period untouched while altering its shape and orientation completely. That is what a pure plane change is, and it is why plane changes are so expensive: they cost a great deal of $\Delta v$ and buy no energy at all.

Reading the relation the other way gives the fact that makes deep-space missions possible. For a fixed $\Delta v$, the change in $v^2$ — and therefore in energy — is $2v\,\Delta v + \Delta v^2$, which grows with the speed the vehicle already has. Burning where the vehicle is fastest, deep in the gravity well, buys the most energy. That is the Oberth effect, and it falls straight out of the relation being quadratic in speed.

The map the whole solar system is planned on #

Because every manoeuvre is a difference of two vis-viva speeds, the cost of going anywhere can be tabulated once and consulted forever. The result is the delta-v map, and it is the single most consulted object in mission design.

Its entries are all the same calculation. Low Earth orbit to geostationary transfer: 2.44 km/s, then 1.47 to circularise. Low Earth orbit to escape: 3.22. Escape to a Mars transfer: 0.60. Mars capture into low orbit: 2.09, or nearly nothing if the atmosphere is used. Low lunar orbit to the surface: 1.87. Every one of them is a difference of two speeds at one radius. Each number is two evaluations of one equation and a subtraction, and each is accurate to the width of the line it is printed on.

The map’s most useful property is that it is a graph with edges that add. The cost of a route is the sum of its steps, so the cheapest path between two points can be found by inspection, and the surprising routes stand out immediately. Reaching the surface of Mercury from low Earth orbit costs more than reaching Pluto. Reaching the Sun costs more than leaving the solar system. Both are consequences of the same relation, and neither is obvious before it is tabulated.

There is a caution attached to reading it, and it matters. The map’s numbers are impulsive — they assume each burn is instantaneous. A real burn takes minutes, during which the vehicle moves and gravity keeps pulling, so some of the thrust is spent going the wrong way. The resulting gravity loss adds a few percent to a departure from low orbit and adds nearly two kilometres per second to a launch from the ground. The map is exact about the physics and optimistic about the engineering, and every mission carries a margin for the difference — which is what the rocket equation then converts into propellant.

What was actually measured #

The relation is a theorem, not an observation, so the honest question is what it has been tested against — and the answer is that it is checked continuously, to extraordinary precision, by tracking.

A deep-space spacecraft’s radio signal is returned coherently, and the Doppler shift on the return gives its line-of-sight velocity. Integrated over a pass, the measurement is good to something like $0.05$ millimetres per second. The predicted velocity comes from a trajectory model in which the vis-viva relation, applied through the patched-conic decomposition and refined by numerical integration, is the underlying dynamics. Agreement at that level, over years and billions of kilometres, is the test.

It is worth being precise about which quantity in the relation is well known. $GM$ for the Sun is known to about nine significant figures because it is exactly what planetary orbits measure; $G$ alone is known to five. So the relation is used in a form where $GM$ appears as a single symbol and is never split. A trajectory model that needed the Sun’s mass in kilograms would immediately be four digits worse.

The one number in the equation that carries real observational uncertainty is $r$ — where the spacecraft actually is — and it is obtained from the same tracking, by ranging: a coded signal is returned and its round-trip time gives the distance to a few metres at a range of $10^9$ kilometres. The relation is then a consistency check between two independent radio measurements, one of range and one of range rate, and the residuals between them are where the small unmodelled forces show up.

Those residuals are not always zero, and the most famous case is instructive about the limits. The two Pioneer spacecraft showed an unexplained sunward acceleration of about $8.7 \times 10^{-10}$ m/s², visible only because the tracking was good enough to see a discrepancy of that size accumulate over decades. It was eventually attributed to the anisotropic emission of waste heat from the spacecraft’s own radioisotope generators, pushing gently in one direction. Nothing was wrong with the dynamics. The spacecraft was glowing more brightly on one side.

The energy that is not in the orbit #

Because the relation ties speed to $a$ alone, it produces a way of talking about a mission’s cost that is independent of any particular trajectory, and this is the generalisation worth carrying away.

Define the characteristic energy $C_3 = -GM/a = v_\infty^2$, the square of the speed a spacecraft retains after climbing out of a body’s well. Every departure from the Earth has a $C_3$, quoted in km²/s², and it is the single number a launch vehicle’s performance is advertised against. A trans-lunar injection is about $-2$ km²/s² — negative, so still bound. A minimum-energy transfer to Mars is about $9$. New Horizons left at $158$, the highest ever, which is why it crossed the Moon’s orbit in nine hours.

The utility of the quantity is that it is additive across gravity wells in a way speeds are not. A spacecraft’s heliocentric orbit is fixed by its energy relative to the Sun; the Earth departure fixes its energy relative to the Earth; and the two are stitched together at the edge of the Earth’s sphere of influence by adding velocity vectors, not speeds. That stitching is the patched-conic method, and $C_3$ is the currency it trades in.

The same quantity is what makes a gravity assist legible. In the planet’s frame the flyby cannot change $v_\infty$, so the spacecraft’s $C_3$ with respect to the planet is untouched; what changes is the direction, and adding the planet’s own motion back produces a different $C_3$ with respect to the Sun. The manoeuvre is a conversion between two energies measured against two different bodies, and neither of them is violated.

The two special values the relation contains #

Two particular speeds fall out of the relation as limiting cases, and both are used so often that their derivation from it is worth making explicit.

Setting $r = a$ gives $v = \sqrt{GM/r}$: the circular speed, the speed at which an orbit neither rises nor falls. It is the value the relation returns whenever the body is at a distance equal to its own semi-major axis, which for a circular orbit is everywhere and for an eccentric one is at exactly two points per revolution — the two places where a body on an ellipse is moving at the circular speed for its current altitude while not being on a circular orbit at all.

Letting $a \to \infty$ removes the second term entirely and gives $v = \sqrt{2GM/r}$, which is the escape speed. The famous factor of $\sqrt2$ between them is nothing more than the ratio of $2/r$ to $1/r$ inside the bracket, appearing at every distance because both terms scale the same way.

Everything between those two values is a bound orbit and everything above is not. The relation contains the whole classification, and the two named speeds are simply the values of one parameter at its extremes.

Drawn as speed against distance rather than as energy against distance, the same two limits become the ends of a single curve.

That picture is worth holding onto, because it makes the relation’s strangest property obvious. Two spacecraft on completely different orbits, passing the same point at the same distance from the Sun, are moving at the same speed if and only if their semi-major axes are equal — and their semi-major axes are equal if and only if their periods are. So a period, which is a statement about time, fixes a speed at every distance, which is a statement about the present instant. Nothing about the shape of the path enters, and nothing about where the body is going. The curve above is the whole of the two-body problem’s speed content, and every conic is a segment of it.

The one place the starting speed is not zero #

Every manoeuvre above begins from an orbit, and there is one that begins from the ground. Applying the relation there exposes a term that orbital mechanics usually has no reason to mention.

A launch site is not stationary. It is carried east by the Earth’s rotation at 465 metres per second at the equator, falling as the cosine of the latitude — 410 at Cape Canaveral’s 28.5 degrees, 325 at Baikonur’s 45.6, and nothing at all at a pole.

That speed is free. A vehicle launched eastward starts with it, so the velocity it has to supply to reach orbital speed is reduced by exactly that amount, and the saving is a few per cent of a launch budget of about nine kilometres per second. It is small in fractional terms and large in propellant, because the propellant cost is exponential in the velocity change rather than proportional to it.

The consequence is that launch sites are chosen for latitude, and that a site near the equator with an ocean to the east is worth a great deal. It is also why a launch to a retrograde or polar orbit is more expensive: the rotation is no longer helping, and for a retrograde orbit it is opposing.

There is a second term at launch that the relation cannot supply and that dominates the difference between theory and practice. Reaching orbital speed from the ground requires climbing out of the atmosphere as well as accelerating, and during the climb the vehicle is fighting gravity with a thrust that is not horizontal. The velocity lost that way — the gravity loss — is well over a kilometre per second, and the drag loss adds a hundred metres or so more.

So a launch costs about nine and a half kilometres per second to reach an orbit whose speed is seven and eight-tenths, and the difference is entirely made of terms that a two-body relation has no place for. The relation is exact about the destination and silent about the journey.

There is a reason the relation is stated with $a$ rather than with the energy it stands for, and it is practical rather than aesthetic. A semi-major axis is measurable from a period by Kepler’s third law, and a period is the easiest orbital quantity to measure — it needs only two observations far enough apart and no absolute calibration at all. The energy is not directly measurable by anything. So the relation is written in terms of the observable that fixes it, which is why it is used as an engineering formula rather than quoted as the conservation law it is. Every Δv budget in this subject is a difference of two square roots taken with two values of $a$, and the energy never appears. That is a choice about notation rather than about physics, and it is worth noticing that a subject whose central quantity is conserved almost never writes the conserved quantity down.

Where the model stops #

Two bodies. The energy is conserved for the pair. A third body exchanges energy with them continuously, and the semi-major axis becomes a slowly varying quantity rather than a constant. In the restricted three-body problem the vis-viva relation is replaced by the Jacobi integral, which is a different conserved quantity and is not an energy.

A point-mass primary. An oblate body’s potential is not $-GM/r$, so the relation acquires correction terms that are small and are routinely modelled.

No thrust, no drag. A continuously thrusting or decelerating vehicle has a semi-major axis that changes every second, and the relation holds only instantaneously.

Newtonian gravity. Close to a compact object the potential is not the Newtonian one and the orbit does not close, so there is no fixed $a$ for the relation to refer to.

There is also a limitation shared by every figure on this page, and it is worth naming because the relation invites the error. All of these plots put speed against distance, which suppresses time entirely. A body on a highly eccentric orbit spends almost all of its period in the slow, distant part of the curve and flashes through the fast part in a few percent of it — so the left-hand end of every speed curve here, where the interesting numbers are, is where the body almost never is. The relation says what the speed is at each distance. It says nothing whatever about how long the body spends there, and supplying that requires solving Kepler’s equation.

The ladder from here #

Later rungs on this anchor: the derivation from the energy integral in full. The Jacobi integral, and what replaces vis-viva when a third body is present. $C_3$, $v_\infty$, and the launch-vehicle performance curves quoted against them. The Oberth effect worked through numerically. Plane changes, and why they buy no energy. Low-thrust spirals, where the relation holds instant by instant and the integrated cost is completely different. Patched conics and the sphere of influence. The relativistic correction to the energy, measurable at the galactic centre. And the specific angular momentum as the other conserved quantity, which fixes everything about the orbit that vis-viva does not.

The name is Latin — vis viva, living force — and belongs to a seventeenth-century argument between Leibniz and the Newtonians about whether the quantity conserved in a collision went as $v$ or as $v^2$. Leibniz was right, the quantity is twice the kinetic energy, and the name survives in exactly one equation.