Hail Mary Science
Spaceflight and relativity

Photon propulsion calculator

Light has no mass, but it does carry momentum. If an object emits light one way, it gets pushed the other way. Set the mass and the emitted power and the calculator gives you the resulting force and acceleration.

Key formula

p = E / c · F = P / c · a = F / m

Try it

How to use the simulator

Light has no mass, but it does carry momentum: a photon with energy E carries p = E/c. If an organism emits light in one direction, it gets pushed the opposite way, like a rocket that throws out photons instead of gas. The simulator works out the force produced by the light emitted each second and the acceleration that force gives a tiny cell.

Controls

  • Organism mass: 1 to 100 picograms (a typical astrophage is around 20 pg).
  • Emitted energy: on a log scale, from a billionth to a thousandth of a joule per second.

Things to try

  1. At the starting values the force is only a few femtonewtons, yet the acceleration comes out at tenths of a m/s², because the mass is so tiny.
  2. Double the mass and check that the acceleration halves (Newton's second law, a = F/m).
  3. Raise the energy tenfold and the force rises tenfold too: with light, pushing harder costs proportionally more energy.

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The momentum of light

A photon with energy E carries momentum p = E/c. When an object emits light, total momentum has to be conserved, so the object recoils the other way. If it emits a power P continuously in one direction, the resulting force is F = P/c.

Because c is huge, that force is tiny: one watt of light produces about 3.3 nanonewtons. A 10 W LED bulb shining all its light one way would push with about the force of a speck of dust resting on your hand.

Where it does matter

The force is small, but acceleration depends on mass (a = F/m). For a microscopic object the same force can produce noticeable acceleration. And for a large craft, the push works if it is kept up for a long time without spending mass.

Solar sails use the effect the other way round: they reflect sunlight and receive twice the momentum. Japan's IKAROS probe demonstrated it in 2010 with a sail 14 m on a side. The Breakthrough Starshot project proposes pushing gram-scale sails with lasers from Earth.

Worked example

Example: a cell that emits light

  1. Take a 20-picogram cell (2·10⁻¹⁴ kg) emitting one microwatt (10⁻⁶ W) of light in a single direction.
  2. The force is F = 10⁻⁶ / 3·10⁸ ≈ 3.3·10⁻¹⁵ N, about 3 femtonewtons.
  3. The acceleration is a = 3.3·10⁻¹⁵ / 2·10⁻¹⁴ ≈ 0.17 m/s². For its size, that is a strong push.
  4. Double the mass in the calculator and the acceleration halves. Raise the power tenfold and the force rises tenfold too.

In Project Hail Mary

The novel's fictional microorganisms move by emitting infrared light, and the ship uses the same principle at large scale as its engine. This calculator shows how much force emitted light really provides.

Frequently asked questions

Does a flashlight push backwards?

Yes, but very little: a 1 W flashlight pushes with about 3 nanonewtons, tens of thousands of times less than the weight of a feather.

Why does a solar sail get twice the push?

Because reflecting light reverses its momentum: the sail absorbs it on arrival and gives it back on the bounce. A surface that absorbed the light without reflecting it would get half.

Could a photon rocket be built?

In theory it is the most efficient rocket possible per kilogram of fuel. In practice it would need to turn mass into light with an efficiency that today only matter-antimatter annihilation approaches.

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