Astronomy & Celestial Mechanics
the sky, computed
Generator

The patched-conics generator

How small a sphere of influence is
How small a sphere of influence is. Each planet's sphere of influence as a fraction of its own orbital radius, against that radius, both logarithmic. The largest belongs to Jupiter at 6.19% and the smallest to Mercury at 0.194%. The patched-conic method treats a trajectory as heliocentric everywhere outside these, and the figure is the argument for why that costs so little: they are thousandths of the journey.

Each planet's sphere of influence as a fraction of its own orbital radius, against that radius, both logarithmic. The largest belongs to Jupiter at 6.19% and the smallest to Mercury at 0.194%. The patched-conic method treats a trajectory as heliocentric everywhere outside these, and the figure is the argument for why that costs so little: they are thousandths of the journey.

5 essays call patched-conics. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

How small a sphere of influence is. Each planet's sphere of influence as a fraction of its own orbital radius, against that radius, both logarithmic. The largest belongs to Jupiter at 6.19% and the smallest to Mercury at 0.194%. The patched-conic method treats a trajectory as heliocentric everywhere outside these, and the figure is the argument for why that costs so little: they are thousandths of the journey. Spaceflight

One trajectory, stitched from three two-body problems

An interplanetary flight is a problem with no closed solution. It is flown by cutting it into pieces that each have one, and the seams are places where the model is knowingly false.

 A patch that throws away 2.5 to 436 metres a second. The velocity error a patched-conic approximation makes at each planet's sphere of influence, on a logarithmic scale, for an approach at 3 kilometres a second. The sphere of influence is where the two ways of writing the problem — planet-centred with the Sun as a perturber, or Sun-centred with the planet as one — become equally bad, and at that radius the neglected solar tide is exactly two times the fifth root of the planet's mass ratio times the planet's own pull. That is drawn beside each planet, it runs from 0.09 to 0.50, and it is not the same for all of them — a factor of 5.6 that the definition does not remove. Because both neglected terms are largest exactly at the surface where the switch is made, the trajectory has a discontinuity in its acceleration and an accumulated velocity error of metres a second. That is negligible for a mission design and enormous for a navigation solution, which is why patched conics are used to find a trajectory and never to fly one: the real trajectory is obtained by numerically integrating the full n-body problem, differentially corrected onto the patched-conic solution as a starting guess. Spaceflight

The discontinuity a patched conic hides

An interplanetary trajectory is designed as two exact solutions glued along a surface where neither is valid. At that surface both neglected forces are at their largest, so the stitched path has a kink no real trajectory has — and the size of the kink is metres a second, which is a rounding error for a mission design and a catastrophe for a navigation solution.

 How far a small error at departure moves the arrival at Mars. The distance by which a craft on the Hohmann transfer from the Earth's orbit to Mars's misses its aim point at the nominal arrival time, against an error in its departure speed, on logarithmic axes, integrated numerically for the 259-day flight. An error along the direction of motion (solid) moves the arrival by 76 km for every millimetre a second; one across it (dashed) by 23 km. The relation is linear over the whole range drawn: a metre a second at departure — the precision of a good launch — is 76 thousand kilometres at Mars, several times the planet's own radius and a thousand times what a landing needs. The ratio is the corresponding element of the state transition matrix, and it is the number every navigation plan is built around. Spaceflight

A millimetre a second at launch, seventy-six kilometres at Mars

A craft on its way to Mars arrives seventy-six kilometres from where it was aimed for every millimetre a second its departure speed was wrong. No launch is that good, so no trajectory is flown as designed; it is flown as corrected, by Newton's method run on the matrix that says how the arrival depends on the departure — which doubles its correct digits at every step and makes a two-body sketch into a real trajectory in two iterations.

 How fast an error in the departure grows, near a neck and on a conic. How much a small error in a spacecraft's velocity is magnified in its later position — the largest singular value of the block of the state-transition matrix that maps velocity errors to position errors, in the Earth–Moon system's units — against time in days, integrated from the variational equations of the restricted three-body problem. A trajectory held at the L1 point between the two bodies (blue) amplifies errors exponentially, by a factor of e every 1.48 days — the rate the linearised dynamics there predict, 2.932 in the system's units, which the integration reproduces to within two per cent. A two-body circular orbit at half the separation (green), integrated the same way with the second body's mass switched off, amplifies them only secularly — an error in speed is an error in period, and that grows along the track in proportion to time. Over 14 days the neck has multiplied an error by 2,359; the conic by 9. Spaceflight

A trajectory with no good first guess

A trajectory designed from conics is corrected onto the real one by Newton's method, and the method works because the arrival depends on the departure only polynomially. Near a Lagrange point the dependence is exponential: every day and a half near the Earth–Moon neck multiplies an error by e. The first guess has to be exponentially good, the digits of the departure run out in weeks, and the only cure is to stop treating the trajectory as one piece.

 An outward path and a return path that cross, seen from a frame turning with the Moon. A trajectory in the Earth–Moon restricted three-body problem, in the frame rotating with the Moon. It leaves a 185 km parking orbit at 10.95 km/s, passes 716 km behind the Moon 3.0 days later, and returns to a perigee of -9 km without an engine firing, 6.0 days after departure. The crossing of the two legs is why it is called a figure of eight. The Moon's pull turns the direction of travel, and in a stationary frame the Earth is a much weaker influence on the far side; neither fact alone makes the geometry close. The Earth and Moon are at true scale. Spaceflight

A way home that needs no engine

A spacecraft can leave the Earth so that the Moon's gravity turns it round and sends it home with no further burn. Drawn in the frame that turns with the Moon the path is a figure of eight. It was flown to the Moon by every early Apollo mission for one reason: if the engine fails, the crew comes back. That safety is exquisitely narrow. A tenth of a metre a second at departure moves the perilune by tens of kilometres and the return by thousands, and the perilune it allows is not one that most landing sites can use.

The whole library · All essays