Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

A drive with no rocket equation

Radiation pressure and solar gravity both fall as the inverse square, so their ratio is a constant of the vehicle. A sun-facing sail therefore only rescales the central mass — it has to be tilted to do anything, and the best tilt throws away sixty-two per cent of the thrust.

Assumes Low-thrust transfer and Rocket equation.

18 min read 5 figures Everything is a conicLight is the only messenger

Every vehicle in this field pays an exponential. The propellant fraction is 1−e−Δv/c1 - e^{-\Delta v/c}, and the entire art of the subject consists of making Δv\Delta v smaller or cc larger, because the exponential punishes any failure to do either.

A sail does not pay it. It carries no propellant, so there is no mass ratio, no cc and no exponential — and a velocity budget, which is the currency every comparison so far has been denominated in, ceases to be the right quantity.

What replaces it is one dimensionless number and one angle.

A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: IKAROS, 2010 at 1607 g/m² gives β = 9.5e-4; LightSail 2, 2019 at 156 g/m² gives β = 9.8e-3; a 5 µm film with no structure at 7 g/m² gives β = 0.219, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat.
Fig. 1 The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail’s normal and the sunline. The force is along the normal and goes as cos⁡2α\cos^2\alpha — one cosine for the area presented to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos⁡3α\cos^3\alpha and the transverse one as cos⁡2αsin⁡α\cos^2\alpha\sin\alpha. A sun-facing sail has no transverse push at all, and the transverse peak is at 35.26°, where it is 0.385 of the face-on force.

The number that replaces the mass ratio

Solar radiation pressure at a distance rr is P=L⊙/(4πr2c)P = L_\odot/(4\pi r^2 c), and solar gravity is GM⊙/r2GM_\odot/r^2. Both fall as the inverse square, so their ratio depends on nothing but the sail’s areal density.

Define the lightness number β\beta as that ratio. For a perfectly reflecting sail it is

β=σ∗σ,σ∗=L⊙2πGM⊙c=1.53 g/m2,\beta = \frac{\sigma^*}{\sigma}, \qquad \sigma^* = \frac{L_\odot}{2\pi G M_\odot c} = 1.53\ \mathrm{g/m^2},

where σ\sigma is the sail’s mass per unit area. At σ=σ∗\sigma = \sigma^* the Sun pushes exactly as hard as it pulls.

One and a half grams per square metre is the whole specification, and it is a demanding one: a kilometre-square sail at that density weighs 1.53 tonnes, which is the whole vehicle including its structure, its payload and every mechanism that deploys it. Nothing built has come close.

A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: IKAROS, 2010 at 1607 g/m² gives β = 9.5e-4; LightSail 2, 2019 at 156 g/m² gives β = 9.8e-3; NEA Scout, 2022 at 45 g/m² gives β = 0.034; a bare 2.5 µm film at 3.5 g/m² gives β = 0.437, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat.
Fig. 2 The same thrust curve with four sails’ lightness numbers. IKAROS, the first to fly and demonstrate photon propulsion, had β=9.5×10−4\beta = 9.5\times10^{-4}; a small square sail of the current generation reaches a few times 10−210^{-2}; a bare film of 2.5 µm aluminised plastic with no structure at all would reach 0.44. The last of those is not a spacecraft — it is the material by itself, with nothing attached — which is the honest statement of how far the achievable β\beta is from one.

The consequence of the two inverse squares

Because the two forces have the same distance dependence, a sail held perpendicular to the sunline does something remarkable: nothing interesting.

The net radial force is (1−β)GM⊙/r2(1-\beta)GM_\odot/r^2, which is the gravity of a Sun of mass (1−β)M⊙(1-\beta)M_\odot. The orbit is still a conic; Kepler’s laws still hold; the period is still given by the third law with μ\mu replaced by μ(1−β)\mu(1-\beta). The vehicle does not spiral outward and it does not gain energy.

That is the single most counterintuitive statement about sails and it follows in one line. A continuous outward force does not raise an orbit if it points at the central body, because a purely radial force does no work on a circular orbit and exerts no torque.

What it does do is change the relation between speed and radius. A sail released at the Earth’s orbital speed with β=0.5\beta = 0.5 finds itself moving at the circular speed for a central mass twice what it now feels, so its orbit becomes an ellipse with perihelion where it started — and it drifts outward and back, without ever gaining energy.

So a sail that only points at the Sun is a vehicle that has changed the solar system’s gravitational constant for itself and gone nowhere. To do anything it has to tilt.

The angle that costs two thirds

An ideal flat sail reflects light specularly, so the momentum imparted is along the sail’s normal regardless of where the sail is pointing. The magnitude carries two cosines of the cone angle α\alpha between that normal and the sunline: one because the sail intercepts cos⁡α\cos\alpha of the light that would fall on its full area, and one because only cos⁡α\cos\alpha of each photon’s returned momentum lies along the normal.

Resolving into the orbit frame gives a radial component ∝cos⁡3α\propto\cos^3\alpha and a transverse one ∝cos⁡2αsin⁡α\propto\cos^2\alpha\sin\alpha, and the transverse component is what raises or lowers an orbit.

Differentiating cos⁡2αsin⁡α\cos^2\alpha\sin\alpha gives tan⁡α=1/2\tan\alpha = 1/\sqrt2, so

αopt=arctan⁡12=35.264°,transverse thrust=233=0.385.\alpha_{\text{opt}} = \arctan\frac{1}{\sqrt2} = 35.264°, \qquad \text{transverse thrust} = \frac{2}{3\sqrt3} = 0.385 .

Sixty-two per cent of the available thrust is the price of pointing any of it somewhere useful. That is a large tax, and it is the closest thing a sail has to the rocket equation’s exponential — an irreducible loss built into the geometry rather than into the propellant.

It has the same character as the factor of π/2 a continuous plane change pays: both are the cost of a thrust direction being constrained by something other than what the trajectory wants, and both are pure numbers with no engineering in them. A sail cannot be improved past 0.385 any more than a continuous turn can be improved past π/2, because neither loss is a loss to anything that could be made better.

A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: a bare 2.5 µm film at 3.5 g/m² gives β = 0.437; the same with minimal booms at 10 g/m² gives β = 0.153; a realistic small sail at 45 g/m² gives β = 0.034, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat.
Fig. 3 The same curves against three sails an order of magnitude apart in areal density. The thrust curve does not move at all — the cone-angle geometry is independent of β\beta entirely — and only the legend changes, from β=0.437\beta = 0.437 for a bare film down to 0.0340.034 once a real structure is attached. That separation is the useful part of the problem: β\beta sets how strong the force is, α\alpha sets where it points, and the two are chosen independently. A rocket has no such separation, since its thrust direction and its magnitude both come from the same engine.

An orbit that is still a conic, with a different μ

The rescaling of μ\mu is worth following further, because it turns every result about two-body orbits into a result about sails with no work at all.

Every orbit one force allows is a conic, and the shape depends on the speed relative to the local circular speed. With a sun-facing sail the local circular speed becomes μ(1−β)/r\sqrt{\mu(1-\beta)/r}, which is smaller — so a vehicle at the Earth’s orbital speed is now moving faster than circular and is at the perihelion of an ellipse.

At β=0.5\beta = 0.5 the effective μ\mu is halved, and a body at circular speed for the full μ\mu has exactly the escape speed for the halved one, since escape speed is 2\sqrt2 times circular. A sail with β=0.5\beta = 0.5 released at any circular orbit is on a parabolic escape trajectory, and one with β>0.5\beta > 0.5 is hyperbolic — with no manoeuvre, no steering and no propellant, simply by being unfolded.

Kepler’s third law rewrites the same way: a sail at β=0.19\beta = 0.19 on a circular orbit at one astronomical unit has a period of 1/0.81=1.111/\sqrt{0.81} = 1.11 years, so it drifts backwards relative to the Earth by forty days a year while remaining at the same distance. That is the mechanism behind every proposal to station a spacecraft somewhere the two-body problem forbids.

The general statement is that a sun-facing sail is not a propulsion system at all; it is a change of the constant in the force law. Everything that follows from the inverse-square law still follows, with one number altered.

What a sail is actually good at

Given the tax, it is worth asking what sails are for, and the answer is not “transfers”.

Station-keeping at a point that is not an equilibrium. A sail can hold a spacecraft at a position where gravity alone could not, by supplying a continuous force that nothing else could supply for years. A solar observatory placed sunward of the Earth–Sun L1L_1 point — closer to the Sun than the Lagrange point, and therefore giving earlier warning of solar storms — is the standard example, and it is possible only with a continuous radial thrust that costs no propellant. An electric engine could supply the same force and would exhaust its propellant in a fraction of the mission.

Very high Δv\Delta v over very long times. A sail’s acceleration is small and unending. Integrated over a decade, a β\beta of 0.05 at the Earth’s distance gives about 0.3 mm/s², which is 100 km/s over ten years — a velocity budget no chemical or electric vehicle can approach, delivered at the cost of a decade.

Missions with no schedule pressure at all, which is the same argument the specific-impulse trade ended on, taken to its limit: a vehicle with an unbounded velocity budget and an unbounded flight time has no trade left to make, and the orbit that has to be paid for every year is exactly the kind of obligation it dissolves.

And going inward, where the acceleration grows. A sail’s thrust rises as 1/r21/r^2, so a mission to the inner solar system accelerates as it goes. That is the reverse of a solar-electric vehicle, whose power collapses with distance, and it makes sails the preferred concept for missions that have to reach the solar poles or approach the Sun closely.

A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: at the Earth's distance at 100 g/m² gives β = 0.015; the same sail at 0.3 AU at 9 g/m² gives β = 0.170; the same sail at 5 AU at 2500 g/m² gives β = 6.1e-4, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat.
Fig. 4 One sail of areal density 100 g/m², with its effective lightness number expressed as the sail it is equivalent to at the Earth’s distance. Nothing about the sail has changed; the labels record that its thrust is eleven times larger at 0.3 AU and twenty-five times smaller at Jupiter’s distance. Strictly, β\beta is a constant of the vehicle and does not vary with distance — it is the acceleration that does, as 1/r21/r^2, exactly as gravity does — so the drawn equivalence is a way of reading the acceleration and not a change in the sail.

The displaced equilibrium

The first of those applications deserves the arithmetic, because it is the clearest case of a sail doing something no other vehicle can.

A spacecraft between the Earth and the Sun feels solar gravity inward, terrestrial gravity outward and needs a centripetal acceleration to keep pace with the Earth. Those three balance at exactly one place, which is L1L_1, 1.5 million kilometres sunward. Five places keep station in the two-body problem and no more.

Add a continuous radial force and the balance can be struck elsewhere. A sail held sun-facing supplies exactly that, so the equilibrium moves sunward by an amount depending on β\beta — and a spacecraft parked there sees the solar wind earlier than one at L1L_1 does, which is the entire operational value of the concept.

The numbers are unforgiving. Doubling the warning time requires moving the station to about twice the distance from the Earth, which needs a β\beta of order 0.05 — an areal density near 30 g/m² for the whole spacecraft including instruments, which is an order of magnitude beyond anything flown.

So the application that is most clearly worth having is the one just out of reach, which has been the state of the field since the concept was published in the 1990s.

Where the idealisation fails

The flat specular sail is an idealisation and the deviations are all in the same direction.

Reflection is not perfect. A real aluminised film reflects about 88 per cent of the incident light and absorbs the rest, which it re-emits thermally. Absorbed light gives momentum along the incoming direction rather than along the normal, so the net force tilts away from the normal towards the sunline — which reduces the transverse component and moves the optimum cone angle a few degrees.

Reflection is not entirely specular. Some fraction is diffuse, which imparts momentum along the normal in a different proportion, and a sail that has wrinkled or degraded scatters more.

And a sail is not flat. A real sail billows under the very pressure it is collecting, so its normal varies across its surface and the net thrust direction is the average over a curved membrane. The billow is small and its effect is a percent-level correction to the thrust and a larger one to the torque, which is the part that matters.

That torque is the real engineering problem. The centre of pressure of a billowed, imperfectly reflecting sail does not coincide with its centre of mass, so the vehicle experiences a continuous disturbing torque, and attitude control for a structure of that size and flimsiness is much harder than the propulsion is.

A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: IKAROS, 2010 at 1607 g/m² gives β = 9.5e-4; a solar-polar concept at 20 g/m² gives β = 0.076; an interstellar precursor at 5 g/m² gives β = 0.306, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat.
Fig. 5 The flown sail set against two concepts that have been designed and not built. A solar-polar orbiter needs β\beta near 0.08 to reach a high heliographic inclination in a few years, and an interstellar precursor aiming to reach 200 astronomical units within a working lifetime needs 0.3. The gap between the first entry and the other two is three orders of magnitude in β\beta and is entirely structural, since the reflective film for all three is the same material at a similar thickness.

There is one route past the structural problem that changes the arithmetic rather than improving it, and it is the reason the subject has revived. A sail that is very small — a few square metres, carrying a gram-scale payload — has a structure that scales differently, because the membrane stresses fall with size and the booms can be omitted entirely. The lightness numbers reachable that way are much higher, and the payload is correspondingly absurd.

Whether a gram-scale spacecraft can do anything is a separate question from whether it can be accelerated, and the second has a clean answer while the first does not.

A critical density from defined constants, and one flown spacecraft

The critical areal density is a combination of measured constants — the solar luminosity, the gravitational parameter of the Sun, and the speed of light — and it is exact to the precision those are known.

The cone-angle relation is geometry and a reflection law, both exact within the idealisation.

The flown sails’ areal densities are as-built figures. IKAROS was 315 kg over 196 m², which is a whole spacecraft including a bus that happened to be attached to a sail; LightSail 2 was 5 kg over 32 m². Neither was built to maximise β\beta, and quoting their lightness numbers as a state of the art understates what a purpose-built sail would reach.

The most important measurement is the one nobody comments on: IKAROS demonstrated in 2010 that a sail accelerates by the predicted amount, from Doppler tracking of the spacecraft against a model of every other force acting on it. The predicted acceleration was of order 10−410^{-4} m/s² and the measurement confirmed it to a few per cent, which is a hard measurement and is the entire experimental foundation of the field. Separating a photon push of that size from everything else acting on a spacecraft is the same problem that an anomalous acceleration turning out to be a spacecraft’s own heat took thirty years to resolve, and it is worth remembering that the resolution there was a thermal recoil of about the same magnitude.

No curve here is a trajectory

No curve here is a trajectory. The thrust components against cone angle say what a sail can do at an instant and nothing about what sequence of angles gets it anywhere, and the sequence is the whole of trajectory design for sails.

The sail’s own temperature is absent. An absorbing sail close to the Sun heats until it radiates what it absorbs, and the limiting perihelion for a given film is set by its melting point rather than by anything dynamical.

And there is no time axis, which for a vehicle whose entire advantage is accumulated over years is the most conspicuous omission. A sail’s competitiveness is a statement about decades, and a plot of instantaneous thrust cannot contain one.

Why the tilt is not steady either

The 35.26° optimum is the angle that maximises the transverse thrust at an instant. A sail raising its orbit does not hold it.

The reason is the same one Edelbaum’s steering law had: the useful direction of the thrust rotates as the vehicle goes round its orbit, so a sail spiralling outward has to rotate its normal once per revolution to keep the transverse component pointing forward. And it has to reverse the sign of the tilt every half-revolution if it is changing the plane rather than the size.

That is a large attitude manoeuvre for a structure that is kilometres across and grams per square metre thick, repeated every orbit for years. It is the operation that dominates a sail mission’s difficulty, and it is why the flown demonstrations have used spin stabilisation — a spinning sail is stiffened by its own rotation and its attitude is changed by precessing the spin axis, which is slow and reliable and cannot be done quickly.

So a sail’s steering bandwidth is low in a way no other vehicle’s is, and trajectories are designed around a cone angle that changes over days rather than minutes.

What is left of the rocket equation

It is worth being precise about what a sail escapes, because “no rocket equation” is a slogan and the underlying statement is narrower.

The rocket equation is a consequence of momentum conservation for a vehicle that carries its reaction mass. A sail carries none; its reaction mass is the Sun’s photons, supplied externally and free. So the mass ratio disappears, and with it the exponential.

What does not disappear is the need for a force. The sail still has to produce one, and the force it produces is proportional to its area and inversely proportional to its mass — so a sail’s performance is limited by a structural problem, which is how to hold a very large very thin membrane flat, with a mass that scales badly.

The exponential has therefore been traded for a scaling law: not mp/m0=1−e−Δv/cm_p/m_0 = 1 - e^{-\Delta v/c} but a∝A/ma \propto A/m, with the awkwardness that mm grows with AA as soon as the structure has to be real. A sail is not free of a hard limit; it has a different one, and the different one has not yet been engineered past.

Still open: whether a useful β has been reached

Nothing here is in doubt physically. The question is entirely one of materials.

The largest sail flown is 32 m² of aluminised polyester at a whole-spacecraft β\beta near 10−210^{-2}. Concepts for missions that would be genuinely enabled by sails — a solar polar orbiter, a sunward-displaced space weather monitor, an interstellar precursor — need β\beta between 0.05 and 0.5, which is one to two orders of magnitude away.

The films exist. A 2.5 µm aluminised plastic already reaches 3.5 g/m² by itself, which is β=0.44\beta = 0.44 for the material alone. Everything between that number and the flown ones is structure, and whether the structure can be made light enough is the open question and has been for fifty years.

From here: the optimisation that has been assumed and never derived

Each argument so far has computed a cost and compared it against another. What none has done is ask how a low-thrust trajectory is found rather than priced — the steering law has been taken as optimal, and its optimality has been stated rather than derived.

That derivation is Pontryagin’s principle applied to a spacecraft, in which the thrust direction is a free function of time and the answer comes from a costate vector that has no physical meaning and points where the thrust should go. It is the one piece of the theory every comparison here has leaned on and never opened.

Where this goes next

There is a best exhaust speed, and the calendar picks it. Payload fraction against exhaust speed for a 11 km/s mission, at three thrusting durations, with a power plant of 0.025 kilograms per watt of jet power. The vehicle is payload plus power plant plus propellant and the arithmetic is one line: λ = e^(−Δv/c) − (αc²/2t)(1 − e^(−Δv/c)), the first term the rocket equation and the second the mass of the machine that makes the jet. They pull opposite ways. A slow exhaust burns propellant; a fast one needs power, and the power per newton rises in proportion to c, so the plant's mass rises as c². Neither end is where anybody builds, and the optimum is 3,211 s over 200 days, 6,523 s over 700 days, 11,424 s over 2000 days — the same mission, the same Δv, and the best engine for it changes by a factor of 3.6 depending only on how long there is to do it. A gridded ion engine at 3,100 seconds sits at 30.4 km/s, which suits the shortest of these and is slow for the longest. What the curve cannot show is that α is not a constant either: a solar array's mass per watt falls as the fourth power of the distance from the Sun, so the same vehicle is a different point on this plot at Mars and at Jupiter. The engine is chosen by the calendar A chemical stage's exhaust speed is fixed by chemistry. An electric one's is a dial, and turning it up costs power — so there is a best setting, and it is decided by how long the mission has rather than by how far it is going. How much of a rocket has to be propellant. Propellant fraction against the velocity change bought, for three real propellant combinations. The curves approach 1 and cannot reach it, so every extra kilometre per second costs a larger share of what remains — which is why staging exists. The exponential that decides what can be flown A rocket carries its own reaction mass, so every kilogram of propellant has to be accelerated by the propellant beneath it. The result is exponential, and it is the reason spaceflight is hard. Every orbit one force allows. Circle, 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. 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.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Conic sectionΔvGravitational parameterLightness numberLow-thrust transferOrbital transferRadiation pressureRocket equationSolar sailSpecific impulse