NotesVersion 2 · Calculator, drive and sky

The math behind the mission.

The flight-plan calculator, the pair-beam drive, the relativistic sky and the worked numbers below are computed live in your browser from these equations. The other figures on the site come from the sources listed at the end. The model is deliberately ideal: it shows what physics allows, so the engineering gap is easy to see.

01Photon rocket

Mass ratio for an exhaust that leaves at c

In the ship's instantaneous rest frame, a sliver of rest mass dM is annihilated and leaves as light with energy c²|dM|. If the drive points that light with directional efficiency η (the exhaust's momentum times c, divided by its energy), the light carries momentum η c|dM| and the ship gains the same amount:

M d v^{'} = - η c d M \Rightarrow d φ = - η \frac{d M}{M}

φ = atanh β is the rapidity, which adds linearly between frames

Integrating from launch mass M₀ to dry mass M₁ gives the relativistic rocket equation for a photon drive:

β = \tanh (η \ln R) \Leftrightarrow R = \frac{M_{0}}{M_{1}} = \exp (\frac{atanh β}{η})

η = 1 recovers Ackeret's result R = √((1 + β)/(1 − β))

For a perfect drive, reaching 0.2c takes R = 1.2247: about 18 percent of the launch mass is propellant. Braking to a stop at the destination with the same drive multiplies the two burns' ratios, so a rendezvous needs R².

02Burns and coast

Constant proper acceleration, then coast

Each burn holds the acceleration felt on board, a, constant. Over a burn that reaches rapidity φ:

τ = \frac{c}{a} φ t = \frac{c}{a} \sinh φ x = \frac{c^{2}}{a} (\cosh φ - 1)

τ: probe clock · t: Earth clock · x: distance covered, Earth frame

The probe then coasts at β = tanh φ, its clock running slow by γ = cosh φ. If the burns cannot reach the chosen speed before running out of distance, the calculator burns all the way (flyby) or to the midpoint (rendezvous) and reports the peak actually reached. Mass, energy and power always use that peak.

03Energy and power

Where the numbers in the calculator come from

m_{prop} = m_{dry} (R - 1) m_{anti} = \frac{m_{prop}}{2} E = m_{prop} c^{2} P = \frac{M_{0} a c}{η}

Half the propellant is antimatter · P is the beam power at launch

A perfectly collimated beam of light with power P pushes with a force of P/c: about 3.3 newtons per gigawatt. That is why the calculator's beam power at launch is large even for small probes, and why the drive must be extraordinarily good at turning annihilation energy into directed light.

"Share of a year of world energy" divides the energy released by humanity's annual primary energy use, about 600 EJ in 2025.

04Pointing the light

Directional efficiency η

η is the exhaust's momentum, times c, per unit of fuel energy used. For light it is the fraction of the light's momentum that points straight back. It measures how well the drive aims its exhaust, the core research problem.

η = 1: every photon leaves exactly aft. The theoretical limit.
η = 2/3: a flat, glowing surface radiating thermally (a Lambertian emitter), the absorb-and-re-radiate baseline.
η = 1/2: light sprayed evenly over the rear hemisphere.
η = β: electron-positron pairs annihilating while co-moving aft at speed β, the SoLeV drive. Their light is beamed along their motion. Counting the energy spent accelerating them, the whole drive's effective exhaust speed is 0.42 to 0.84 of light speed for 30 to 50 percent power conversion, depending on the radiators: section 05.

Because η divides the exponent, a drive at η = 0.5 needs the square of a perfect drive's mass ratio. Collimation is worth more than almost anything else in the design.

05The pair beam

Aim the pairs, and the light follows

No mirror can turn a gamma ray around, so the SoLeV drive never tries. It accelerates electrons and positrons, merges them into one beam pointed aft and lets them annihilate behind the probe. This section works out the geometry and speed that send the light out the back, how magnets do the aiming, and what it all costs.

The 2017 paper flagged this question and left it open: it proposed studying how annihilation changes when the pair has "net linear momentum greater than zero", and noted that the result depends on the particles' energies and the angle between their paths. What follows is that study.

The closest published designs cross the two beams head-on at the focus of a gamma-ray mirror (Smith and Webb 2001; Semyonov 2014), where the light leaves in random directions and the mirror does not exist. We have not found prior work that merges the beams so they travel together instead.

Fig. 1 · Pair-beam drive, schematic and not to scale. Electrons (solid) and positrons (dashed) arrive tilted by α. The dipole's force (amber) bends both onto the axis, and the merged beam leaves aft. In a real beam, most pairs would annihilate far behind the probe or drift apart first (part 5).

1 · Momentum decides where the light goes

Take one electron and one positron, each with Lorentz factor γ, crossing the thrust axis at half-angle α. Their sideways momenta cancel, so the pair carries

E = 2 γ m c^{2} p_{∥} = 2 γ β m c \cos α \Rightarrow η = \frac{p_{∥} c}{E} = β \cos α

Energy and momentum are conserved, so this holds for two photons, three, or anything else the pair becomes

The ideal angle is zero. Beams that travel together, parallel and aft, give η = β. Beams that collide head-on (α = 90°) give η = 0: light in every direction, which is ordinary annihilation at rest. The cost of a nonzero angle is small at first, since cos α is flat near zero: 10° loses 1.5 percent. Crossing also gives the pair internal energy,

\sqrt{s} = 2 m c^{2} γ \sqrt{1 - β^{2} \cos^{2} α}

Pair-frame energy · at α = 0 it is exactly 2mc² = 1.022 MeV: the particles are at rest relative to each other

At α = 0 the pair annihilates as gently as it would at rest, only moving. That is the low-energy annihilation the 2017 paper wanted, without the need to make any new particles.

Half-angle α	η at γ = 2	η at γ = 5	Loss vs parallel	Pair-frame energy, γ = 2

2 · Speed decides how tight the cone is

In the pair's own frame the light leaves evenly in all directions. Seen from the probe, the pair is moving, so each photon's direction is aberrated toward the beam:

\cos θ = \frac{\cos θ^{*} + β}{1 + β \cos θ^{*}} \frac{d N}{d Ω} = \frac{1}{4 π γ^{2} {(1 - β \cos θ)}^{2}}

θ* in the pair frame, θ from straight aft in the probe frame · here β is the pair's speed, β cos α in general

Every photon emitted into the pair's aft hemisphere lands inside a cone of half-angle

θ_{1 / 2} = \arccos β = \arcsin \frac{1}{γ} \approx \frac{1}{γ}

Half the photons, and more than half the energy, because forward photons are also blueshifted

So "near straight out the back" is a speed requirement. Pairs at γ = 2 (0.87c) put half their photons within 30°; at γ = 10, within 5.7°, carrying 94 percent of the light's energy within 10°. Spread does not waste momentum by itself: the sideways components cancel in pairs, and the net is always η = β. A few photons do head forward, toward the probe: 6.7 percent at γ = 2, carrying 1.3 percent of the energy.

γ	Speed	η = β	Half the photons within	Photons within 10°	Energy within 10°	Photons heading forward	Rest mass share of exhaust energy

3 · Magnets do the aiming

A magnetic field pushes a moving charge sideways, with F = q v × B, so electrons and positrons bend in opposite directions on circles of radius

r = \frac{γ β m c}{e B} = 1.7045 mm \times \frac{γ β}{B [T]} \int B d s = \frac{γ β m c}{e} α

Left: bending radius · right: the field integral along the beam's path that turns it through angle α (radians)

That opposite bending is what makes a single dipole a merger. With the field into the page and the beams heading aft, the force pushes positrons down and electrons up, so positrons enter from below and electrons from above, each tilted by α, and both leave on the axis. Run backward, the same magnet is a spectrometer that splits pairs by charge, the separator of the 2017 paper. The positrons come from the stored antimatter and the electrons are ordinary matter, so nothing has to be made on board. A solenoid after the merger keeps the two beams overlapped, and once merged the beam is electrically neutral, so the probe does not charge up.

The fields are modest. A γ = 2 beam bends on a 3 mm radius in a 1 tesla field, and turning it through 10° takes about 5 millitesla over 10 cm.

γ	Momentum	Rigidity Bρ	Radius at 1 T	Field in a 10 cm magnet to turn 10°

4 · The energy bill sets the best speed

Faster pairs make tighter light, but accelerating them takes energy, and the only source on board is the antimatter itself. Suppose a share f of the fuel annihilates at rest in the power core, a converter turns a fraction η_c of that into beam energy, and the waste heat leaves through radiators with directional efficiency η_th. Balancing the energy, per unit of fuel:

η_{c} f = (γ - 1) (1 - f) \Rightarrow f = \frac{γ - 1}{γ - 1 + η_{c}}

The beam's kinetic energy comes from the share burned in the core

The whole drive's effective exhaust speed, the momentum it puts out per unit of fuel consumed, is then

\frac{v_{eff}}{c} = γ β \cos α (1 - f) + η_{th} (1 - η_{c}) f

This is the η of the rocket equation in section 01 and of the flight-plan calculator

With the radiators facing sideways (η_th = 0) and α = 0, setting the derivative to zero gives a closed form:

γ^{*} = \frac{1}{1 - η_{c}} \frac{v_{eff}^{*}}{c} = \sqrt{\frac{η_{c}}{2 - η_{c}}} f^{*} = \frac{1}{2 - η_{c}}

At 50 percent conversion: γ* = 2 (0.87c), v_eff = 0.577c, and two thirds of the fuel is burned for power

With the radiators facing sideways the best speed is modest, a Lorentz factor of 1.4 to 2 for 30 to 50 percent conversion, and the light's half-cone is 44° to 30°. Pushing the pairs faster for a tighter cone would make the drive worse, because thrust depends on γ only through v_eff.

Radiators that face aft change that. Waste heat then earns thrust too, so the best speed rises: at 50 percent conversion and η_th = 2/3, the best Lorentz factor is about 5.8, the drive reaches 0.84c, and half the photons leave within 10°. In that design the efficient speed and the straight-back geometry agree.

One comparison keeps this honest. A white-hot absorber that simply radiates all of the annihilation heat aft reaches η = 2/3 with no accelerators at all. With sideways radiators, the pair beam beats it only above 8/13, or 62 percent, conversion. With radiators facing aft, the pair beam is that absorber plus a beam, so it always comes out ahead. Turning 511 keV gamma rays into beam power at tens of percent has never been demonstrated, which makes the converter the crux of the design. The explorer below computes all of this live.

Conversion η_c	Radiators sideways: best γ	Half-cone	v_eff	Radiators aft: best γ	Half-cone	v_eff	Antimatter per kg, 0.2c mission

Pair Lorentz factor γ2.00

Crossing half-angle α0°

Power conversion ηc50%

Radiators

Pair speed: 0.866 c
Light's aft momentum share η: 0.866
Half the photons within: 30.0°
Photon energy within 10°: 17%
Fuel burned for power: 67%
Effective exhaust speed: 0.577 c
Antimatter per kg, 0.2c mission: 0.51 kg

5 · Where and how fast they annihilate

Co-moving pairs barely move relative to each other, which is the slow limit of Dirac's annihilation cross-section. There the rate per positron does not depend on the relative speed:

Γ^{*} = n^{*} π r_{e}^{2} c τ = \frac{1}{Γ^{*}} L = γ β c τ

n*: density of each species in the pair frame · r_e = 2.818 fm · τ: time for half the pairs to annihilate, since both species deplete · L: distance covered meanwhile, probe frame

That is slow at drive-like densities. A one-newton beam at γ = 2 squeezed through a square centimeter holds only about 2.0 × 10¹⁰ pairs per cubic centimeter in its own frame, and half of them would annihilate only after about 23 astronomical units, even if the beam never spread. Real beams spread, so most pairs would drift apart before annihilating. The exhaust is mostly a neutral stream of electrons and positrons, some of them perhaps bound into long-lived positronium atoms.

That is fine for thrust. The push happened when the accelerators pushed the pairs, and annihilation only decides what the exhaust ends up as. It also answers the usual objection to pair-annihilation rockets, that annihilation is too slow to finish near the ship (Semyonov 2014, 2020): in this design nothing depends on it.

Two effects change the rate but not the direction. Coulomb attraction between slow pairs raises it (the Sommerfeld enhancement), and in a cold beam pairs can bind into positronium faster than they annihilate directly, a rate we have not yet calculated. Positronium in its ground state decays into two photons in 125 picoseconds or three in 142 nanoseconds, measured in its own frame. Either way energy and momentum are conserved, so the light's aft share stays η = β.

Pair-frame density	Half annihilated after, pair frame	Distance behind probe, γ = 2	Distance behind probe, γ = 6

6 · What is still hard

The converter. Two thirds or more of the antimatter would be burned to make electricity: about a third of a gigawatt of core heat for every newton of thrust, at γ = 2 and 50 percent conversion. No one has turned annihilation gamma rays into beam power at that efficiency, and the converter and radiators, not the magnets, would dominate the drive's mass.
The merger. A single dipole bends particles of different momentum by different angles, so the two beams must be matched in momentum, and the field must end sharply at the merge point or it splits them again. The magnet's angled entrance edge also focuses the beam, which the optics must account for. Laser-made pairs have broad energy spreads, so they would need momentum selection or cooling first.
Current. A newton of thrust from the beam alone needs about 170 amperes of each species at γ = 2, or 50 amperes at γ = 6. Holding unmerged beams that intense together against their own space charge is a real accelerator problem, and the two currents must match to about a part per million, or the probe charges up within a second and starts pulling one species back.
Beam stability. Dense pair beams can break up through plasma instabilities. Small angular spread is cheap: an rms spread σ (radians) lowers η by a factor of about 1 − σ²/2, so 5° costs 0.4 percent.
Shielding. The power core still emits 511 keV gamma rays in every direction, and the forward half has to be stopped before it reaches the probe.
Antimatter. Everything above assumes the fuel exists. Making and storing kilograms of positrons remains the largest open problem on the path to launch.

06The relativistic sky

Why the stars crowd forward and turn blue

The hero animation and the speed demo each place 14,000 to 16,000 stars on the celestial sphere and transform each one every frame. The bright stars around Alpha Centauri (Hadar and the Southern Cross), the dark Coalsack nebula and the tilt of the Milky Way come from J2000 coordinates; the faint background population is synthetic. A star at angle θ from the direction of travel appears at θ′, and its light is Doppler shifted by D:

\cos θ^{'} = \frac{\cos θ + β}{1 + β \cos θ} D = \frac{1}{γ (1 - β \cos θ^{'})}

Color: blackbody at temperature D·T · Point stars' total flux grows as D² · The Milky Way's surface brightness as D⁴

At 0.2c, 60 percent of the sky appears in the forward hemisphere and light from dead ahead arrives 22 percent bluer, so a star there delivers 1.5 times as much energy, summed over all wavelengths (for a Sun-like star such as Alpha Centauri A, that is also about 1.5 times as much visible light). The Milky Way glow is mapped through the same transform, pixel by pixel. Alpha Centauri sits less than a degree from the galactic plane, which is why it appears inside the band. There is no twinkle: that is an effect of Earth's atmosphere.

07Worked numbers

Antimatter needed per kilogram of probe

Perfect drive (η = 1). Multiply by the probe's dry mass.

Cruise speed	Flyby: mass ratio	Flyby: antimatter / kg	Rendezvous: mass ratio	Rendezvous: antimatter / kg	4.34 ly at that speed

08What this ignores

The model is ideal on purpose

All propellant annihilates and every joule leaves as light. Real conversion and containment losses add mass.
Dry mass is fixed. In practice radiators, shielding and the power system grow with beam power.
No drag from interstellar gas, no dust erosion, no gravity from the Sun or the destination stars.
Antimatter production and storage are treated as solved. They are not: see the hard problems on the main page.
The distance is 4.344 light-years to Alpha Centauri A and B, from the 2021 orbital parallax of 750.81 milliarcseconds. The often-quoted 4.37 comes from an older measurement.

09The 2017 paper

What we've learned since 2017

The undergraduate paper that started SoLeV is shared as written. Its core idea, aiming annihilation light out the back, still stands. Several of its numbers have been revised:

Thrust. At 10²⁴ pairs per second, annihilation at rest releases about 164 GW, so even a perfectly aimed beam gives at most about 546 N. The paper's 37,455 N case implies about 35 MeV per photon rather than 0.511 MeV, and the speeds in its Figs. 6 and 7, including the 1.99 percent of light speed case, inherit that.
Scaling. Thrust grows linearly with the pair rate, as the paper's own Fig. 8 shows, not exponentially.
Pair rate. 10²⁴ positrons per second is the instantaneous rate inside a picosecond laser pulse. Measured yields are about 10¹⁰ to 10¹² positrons per shot.
Energy. Pairs made on board return only the energy the power plant put in, so an interstellar mission has to carry antimatter produced in advance.
Positronium. About 125 ps (1.244 × 10⁻¹⁰ s) is the lifetime of para-positronium. Ortho-positronium lives about 142 ns in vacuum.
Pair production. In millimeter-thick high-Z targets, the Bethe-Heitler process dominates over the trident process.

10Sources

References

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