Radiation Pressure Calculator: Solar Sails, P = I/c and 2I/c
Radiation Pressure Calculator: the push you get from light
Light carries momentum, so it pushes. Absorb it and the pressure is I/c; reflect it and the pressure is 2I/c, because the light leaves again going the other way. Sunlight at the Earth gives nine millionths of a pascal on a mirror. This page works out the pressure, the force on a sail and the acceleration it produces.
Radiation pressure, sail force and the acceleration it gives
A mirror sail of 32 m² is held square-on to sunlight at the Earth, where the irradiance is 1361 W/m². The whole craft, sail included, has a mass of 5 kg. Find the radiation pressure, the force, the acceleration, and how long it would take to pick up 1 km/s.
One line, and the factor of two that decides the answer
- P
- radiation pressure, in pascals. A pressure, so it does not depend on how big the surface is — only on the irradiance and on what the surface does with the light.
- I
- irradiance, in watts per square metre: the power the light delivers to each square metre of surface. 1361 W m−2 at the Earth, by the IAU 2015 nominal value.
- c
- speed of light, exactly 299 792 458 m s−1. Dividing by it is why the answer is always so small: a kilowatt per square metre buys you three millionths of a pascal.
- the factor 2
- momentum reversed rather than absorbed. An absorbed photon hands over p; a reflected one arrives with p and leaves with −p, so the surface takes 2p. Same light, twice the push — which is the whole reason sails are mirrors.
- (1 + R)
- the real case. R = 0 gives I/c and R = 1 gives 2I/c, and an aluminised film at R = 0.88 gives 1.88 I/c. The missing (1 − R) is absorbed and has to leave as heat.
- F = PA, a = F/m
- the only two steps that bring in the spacecraft. The pressure is a property of the light; the force needs the area and the acceleration needs the mass, which is why sail design is a fight over grams per square metre.
Worked example
A mirror sail of 32 m² is held square-on to sunlight at the Earth, where the irradiance is 1361 W/m². The whole craft, sail included, has a mass of 5 kg. Find the radiation pressure, the force, the acceleration, and how long it would take to pick up 1 km/s.
Start with the absorbing case, because it is the one you can check. P = I/c = 1361 / 2.99792458×108 = 4.5398×10−6 Pa. That is 4.54 micropascals. The units work: W m−2 divided by m s−1 is J m−3, and an energy per volume is a pressure.
Double it, because this is a mirror. P = 2I/c = 9.0796×10−6 Pa = 9.0796 µPa, which is the headline. Reflecting the light rather than absorbing it costs nothing and doubles the thrust, and that single fact decides what a sail is made of.
Now the force on 32 m². F = PA = 9.0796×10−6 × 32 = 2.905×10−4 N, about 0.29 millinewtons. For scale, that is roughly the weight of a grain of rice on Earth, spread over a sail the size of a small bedroom.
Divide by the mass for the acceleration. a = F/m = 2.905×10−4 / 5 = 5.81×10−5 m s−2, about six millionths of g. Nothing you could feel, and nothing that would get you off the ground.
Then remember that it never switches off. In one day the speed gained is a × 86 400 = 5.02 m/s; in a year, 1.83 km/s. Reaching 1 km/s takes about 199 days of sailing. A chemical rocket would do that in a few minutes and then be out of fuel for ever; the sail keeps going for as long as the Sun shines on it.
One check worth doing. The craft is 5 kg over 32 m², which is 156 g per square metre. A mirror sail stops falling towards the Sun altogether at about 1.53 g per square metre — a hundred times lighter — so this craft is nowhere near escaping the Sun on light alone. That ratio is the lightness number, and the calculator gives it as 0.00980.
The same sail at different distances from the Sun
| Where | Distance (AU) | Irradiance I (W/m²) | Pressure on a mirror (µPa) | Force on 32 m² (mN) |
|---|---|---|---|---|
| Mercury, at perihelion | 0.307 | 14 400 | 96.1 | 3.07 |
| Mercury, mean | 0.387 | 9090 | 60.6 | 1.94 |
| Venus | 0.723 | 2600 | 17.4 | 0.556 |
| Earth | 1.000 | 1361 | 9.08 | 0.291 |
| Mars | 1.524 | 586 | 3.91 | 0.125 |
| Jupiter | 5.20 | 50.3 | 0.336 | 0.0107 |
| Saturn | 9.58 | 14.8 | 0.0989 | 0.00316 |
| Neptune | 30.1 | 1.50 | 0.0100 | 0.000320 |
Why you have never felt it
| Situation | Irradiance | Pressure | Compare with |
|---|---|---|---|
| A 60 W bulb 1 m away | ≈5 W/m² | 1.7×10−8 Pa | nothing you own can measure it |
| Full sunlight on your hand | 1000 W/m² | 3.3×10−6 Pa | 3×10−11 of atmospheric pressure |
| Sunlight on a mirror in space | 1361 W/m² | 9.1×10−6 Pa | the weight of a sand grain per m² |
| A gentle breeze on your hand | — | ≈1 Pa | about 100 000 times the sunlight figure |
| Standard atmospheric pressure | — | 101 325 Pa | 1010 times the sunlight figure |
Why light pushes, and why it matters only where nothing else does
A photon has no mass but it does have momentum, p = E/c. When light lands on a surface that momentum has to go somewhere, and it goes into the surface. That is radiation pressure. The whole of this page is one line of Newton’s second law applied to a stream of photons, which is why the impulse and momentum page is the right place to go if the mechanism rather than the number is what you want.
The factor of two is the examinable part. An absorbed photon delivers p. A reflected photon arrives with p and leaves with −p, so the change in momentum is 2p and the surface feels twice the push. That is why every solar sail ever flown is a mirror and none of them is black. If a question gives you a shiny surface and you use I/c, you lose the mark; if it gives you a matt black one and you use 2I/c, you lose it the other way. Read the surface before you write anything down.
Why the number is so small. You are dividing by the speed of light. A kilowatt per square metre — about full sunlight — gives three millionths of a pascal, which is 3×10−11 of the atmospheric pressure around you. Light does push on you when you walk outside: the force on a square metre of you is about 3 micronewtons, the weight of a third of a milligram. The air you are standing in is pushing on the same square metre with a hundred thousand newtons.
And why it is not negligible in space. Nothing else is pushing. A force that never stops, applied to something light enough, wins against any amount of arithmetic about how small it is: at the Earth a mirror sail gains about 5 m/s of speed per day for every 156 g per square metre of craft, and it keeps doing that for years. Solar sailing is not a way of getting a large acceleration; it is a way of getting a small one for free, indefinitely, with no propellant to run out. Compare that with the rocket equation, where every bit of speed has to be bought with mass you brought along.
Where it genuinely dominates: small things. Radiation pressure scales with area, and gravity with volume, so as an object gets smaller the ratio of light-push to weight rises as 1/size. For a dust grain about a micrometre across the two are comparable, and grains roughly between a tenth of a micrometre and a micrometre get pushed onto escaping paths and leave the Solar System instead of orbiting it. That size range is approximate, because how hard the light pushes depends on how well a grain that small scatters it. This is not a curiosity: it is why comet tails point away from the Sun rather than trailing behind the comet, why the inner Solar System is swept clear of the finest dust, and why a grain’s fate depends on its size rather than on what it is made of. The same competition, run inside a star, is what sets the maximum mass a star can have: above roughly a hundred solar masses the radiation pressure of the star’s own light exceeds the gravity holding its outer layers on.
Where this goes wrong in homework. Three places, in order of frequency. The factor of two, as above. Then forgetting that P is a pressure: the question usually wants a force, and a force needs the area, so P alone earns nothing. Then the angle: this page and almost every textbook formula assume the light arrives square-on, and a sail tilted at θ both intercepts less light and throws the reflected light off sideways, so the useful thrust falls roughly as cos²θ. A sail at 45° to the Sun keeps about half its thrust along the beam, not seven tenths.
Frequently asked questions
Does light really push on me when I stand in the Sun?
Yes, and the number is easy. Full sunlight is about 1000 W/m², so the pressure on your skin is 3.3×10−6 Pa. Over about a square metre of you that is roughly 3 micronewtons, the weight of a third of a microgram. The air in the room is pushing on the same area with 101 325 newtons, which is why you have never noticed. Nothing about the effect is subtle or disputed; it is simply ten orders of magnitude too small to feel.
Why is a mirror pushed twice as hard as a black surface?
Because the momentum is reversed rather than just stopped. Think of throwing a ball at a wall. A ball of putty sticks: the wall takes the ball’s momentum, p. A rubber ball bounces straight back: the wall takes p to stop it and another p to send it back, so 2p. Photons behave the same way. The consequence is that absorbing light is worse for thrust than reflecting it, even though absorbing is what heats the surface up — you get less push and more heat, which is the wrong end of both trades.
Could a solar sail lift off from the ground?
No, not by any margin. The pressure on a mirror in full sunlight is about 9×10−6 Pa, so to hold up even 1 kg (about 9.8 N) you would need roughly a million square metres of sail — a square kilometre — and the sail itself would weigh far more than that. Sails work in space because there is no weight to beat, only inertia, and because they have years in which to act. Set the lightness number row to 1 or more on this page and you can see what it would take: about 1.5 grams per square metre for the entire craft.
Is radiation pressure the same as the solar wind?
No, and confusing them is common. Radiation pressure is photons: it is what this page computes, and at the Earth it is about 9 µPa on a mirror. The solar wind is a stream of actual particles — mostly protons and electrons — and its dynamic pressure at the Earth is of order a nanopascal, roughly thousands of times below the photon pressure. Sails are pushed overwhelmingly by light, not by wind, despite the name. A magnetic sail is a different idea that does use the particles.
Why do comet tails point away from the Sun?
Two separate mechanisms, and radiation pressure is one of them. The dust tail is dust grains pushed outwards by sunlight: the push goes as the grain’s cross-section while its weight goes as its volume, so for small enough grains the light wins and they are blown outwards, curving behind as they go. The ion tail is ionised gas dragged by the solar wind and the magnetic field it carries, and it points almost exactly away from the Sun. Both tails lead the comet when it is heading away from the Sun, which looks wrong until you realise neither has anything to do with the direction of travel.
What does the lightness number mean?
It is the radiation push on a craft divided by the Sun’s gravitational pull on the same craft. Both fall off as the inverse square of distance from the Sun, so the ratio is the same everywhere — it is a property of the spacecraft, not of where it is. At a lightness number of 1 the two cancel and the craft neither falls towards the Sun nor is pushed away; above 1 it can leave the Solar System on sunlight alone. For a mirror sail that needs the whole craft to weigh about 1.53 g per square metre of sail. No sail flown so far has been anywhere near that light, and the sail is only part of the mass — the spacecraft has to come too.
Related calculators
References
- NIST, Fundamental Physical Constants (SP 961, 2022 wallet card), and E. Tiesinga et al., CODATA Recommended Values of the Fundamental Physical Constants: 2022. The constants on this page come from here: c = 299 792 458 m s−1, h = 6.626 070 15×10−34 J s and k = 1.380 649×10−23 J K−1 are exact by the 2019 SI definitions; G = 6.674 30(15)×10−11 m3 kg−1 s−2 is not, and is the worst-known constant used anywhere in this category.
- International Astronomical Union, Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties (XXIX General Assembly, 2015), together with the IAU definitions of the astronomical unit (exactly 149 597 870 700 m) and the parsec (648 000/π au). Resolution B3 fixes the nominal total solar irradiance at 1361 W m−2, the nominal solar luminosity at 3.828×1026 W and the nominal solar radius at 6.957×108 m. The resolution is explicit that these are agreed conversion factors chosen close to the best estimates, not measurements.
- The Planetary Society, The Planetary Society’s LightSail 2 mission reenters atmosphere, completes mission (press release, November 2022). Read for this page: it gives the sail as four Mylar sections totalling 32 m², a launch in June 2019 on a Falcon Heavy and re-entry on 17 November 2022. It does not state the spacecraft’s mass, so the 5 kg default on this page is a round figure for a shoebox-sized craft and is not LightSail 2’s own.
- D. J. Griffiths, Introduction to Electrodynamics, 4th edition, Cambridge University Press (2017), §8.2. The momentum density of the electromagnetic field, from which P = I/c for absorption and 2I/c for normal reflection follow directly. Named as the level to read next for the derivation; cited for provenance, not re-read for this page.
- E. F. Nichols and G. F. Hull, A preliminary communication on the pressure of heat and light radiation, Physical Review 13, 307–320 (1901), and the independent measurement by P. Lebedev in the same year. The first laboratory confirmations that light exerts the pressure Maxwell’s theory predicted. Cited for provenance; the original papers were not read for this page.