Sections · Solar sail

Vol. Science · Astronomy · The push

Solar sail
and the light

Sunlight carries momentum. At 1 au the nominal irradiance is 1,361 W/m², the IAU 2015 Resolution B3 conversion constant, not a claim that the Sun is always that bright. It thins as 1/r². Divided by c = 299,792,458 m/s, a perfect absorber, face-on, feels 4.540 μN/m². A perfect reflector sends the light back, so the momentum change doubles: 9.080 μN/m², about 9.1 micronewtons on a square metre. On a flat sail, fraction ρ of the light is reflected and the rest is absorbed. Tilt θ is the angle of the sail normal from the outward Sun line. The force is (S A cos θ / c) · [(1−ρ) along the Sun line + 2 ρ cos θ along the normal]. Characteristic acceleration is that sail face-on at 1 au. The lightness number is the radiation acceleration divided by solar gravity, with the same resolution’s GM☉ = 1.3271244×10²⁰ m³ s⁻². Kopp & Lean 2011 measured 1360.8 ± 0.5 W/m² at the 2008 minimum. That measurement is not the number in the formula. A burn’s price is the rocket desk. The wait for the light itself is the light-time desk.

Educational calculator. Ideal flat sail. No thermal re-radiation, no wrinkles, no planet’s shadow, and no atmosphere. The Δv line holds this acceleration fixed: Δv = a × t. A real sail moves, the irradiance changes, and the orbit curves. Not a trajectory and not for mission design.

Interactive · A flat sail

Sun at the left · outward is to the right · prograde is up

Light and force
Sail
Prograde

Distance from the Sun

Area

Mass

Reflect

Tilt

Area and mass from a cited spacecraft, then a perfect reflector at 1 AU

Time, if this acceleration stayed constant

Days

Opens on 1 m² and 1 kg, a perfect reflector, face-on, at 1 AU. The arrow length is the force beside that perfect reflector at this distance. The direction is the force. Prograde is a circular orbit’s direction, so a positive tilt points toward it.

1 m² · 1 kg · perfect reflector · face-on

9.080 μm/s²

9.080 μN/m² · 1 AU · 1 day

Irradiance1,361.0 W/m²
Pressure9.080 μN/m²
Force9.080 μN
Acceleration9.080 μm/s²
Characteristic9.080 μm/s²
Lightness1.531×10⁻³
Solar gravity5.930 mm/s²
Δv, constant a0.784 m/s

Irradiance at 1 AU — the distance where the nominal irradiance is defined — is 1,361.0 W/m², the nominal 1,361 W/m² divided by the square of the distance in AU. A perfect absorber, face-on, would feel 4.540 μN/m². A perfect reflector, face-on, would feel 9.080 μN/m², twice that, because the light leaves with the opposite momentum. This sail is 1 · perfect reflector, tilt 0°, area 1 m², mass 1 kg. The pressure on it is 9.080 μN/m² and the force is 9.080 μN. The force is radial, straight out from the Sun. Face-on, on a circular orbit, the velocity is perpendicular to this push, so at that instant the force does no work. The acceleration is 9.080 μm/s². Characteristic acceleration, the same area, mass, and reflectivity held face-on at 1 AU, is 9.080 μm/s². Solar gravity here is 5.930 mm/s², so the lightness number is 1.531×10⁻³. Both accelerations fall as 1/r², so that ratio does not change with distance. If this acceleration stayed constant for 1 day (86,400 s), the speed would change by 0.784 m/s. That line holds the acceleration fixed. A real sail moves, the irradiance changes, and the orbit curves. It is not a trajectory.

Presets at 1 AU · perfect reflector · face-on · Δv over 1 day if the acceleration stayed constant
Sail	Area	Mass	Force	Acceleration	Lightness	Δv, 1 day
1 m² · 1 kg	1 m²	1 kg	9.080 μN	9.080 μm/s²	1.531×10⁻³	0.784 m/s
IKAROS	196 m²	307 kg	1.780 mN	5.797 μm/s²	9.775×10⁻⁴	0.501 m/s
LightSail 2	32 m²	4.93 kg	290.548 μN	0.0589 mm/s²	9.938×10⁻³	5.092 m/s
ACS3	80 m²	16 kg	726.369 μN	0.0454 mm/s²	7.656×10⁻³	3.922 m/s

The buttons need JavaScript. The arithmetic is still S = 1361 / (distance in AU)², pressure S/c for a perfect absorber and 2S/c for a perfect reflector, and force (S A cos θ / c) · [(1−ρ) r̂ + 2 ρ cos θ n̂]. At 1 AU, face-on, a perfect reflector feels 9.080 μN/m². The 1 m², 1 kg reference accelerates at 9.080 μm/s². One day of that, held constant, is 0.784 m/s. IKAROS, 196 m² and 307 kg, is 5.797 μm/s². LightSail 2, 32 m² and 4.93 kg, is 0.0589 mm/s². ACS3, 80 m² and 16 kg, is 0.0454 mm/s². Not a trajectory.

What the push is

Tiny, and it does not switch off. At 1 AU a perfect reflector, face-on, feels 9.080 μN/m², about 9.1 micronewtons on a square metre. On the 1 m², 1 kg reference that acceleration is 9.080 μm/s². A day of it, held constant, is 0.784 m/s. A Julian year of 365.25 days, still held constant, is 286.53 m/s. The push is tiny beside a rocket burn, and the sunlight does not switch off. Lightness 1, the same perfect reflector held face-on, would take 1.531 g/m². The reference sail is 1 kg/m².

Reflection is a second push. An absorbed photon hands the sail its momentum, E/c. Face-on, the pressure is the irradiance divided by c, 4.540 μN/m² at 1 AU. A reflected photon leaves carrying that momentum the other way, so the sail is pushed twice: once by the arrival and once by the departure. The face-on reflector is 9.080 μN/m². The slider is the fraction between those two. Thermal re-radiation of the absorbed energy is left out.

Tilt is how the orbit spirals. On a circular orbit the velocity is sideways, perpendicular to the Sun line. Face-on, the force is radial, so at that instant it does no work. Tilt the sail and a reflected photon leaves at an angle. For a perfect reflector the force follows the sail normal. Tip that normal forward, along the motion, and the sail gains orbital energy and spirals outward. Tip it backward and the sail loses energy and spirals inward. A perfect absorber has no transverse force in this model. The photons it catches were traveling radially, and absorbing them does not turn that momentum sideways.

What the mission buttons are

1 m² and 1 kg is a ruler. It is not a spacecraft. Face-on, the acceleration in micrometres per second squared equals the pressure in micronewtons per square metre. The other buttons replace the area and the mass, then set a perfect reflector, face-on, at 1 AU, so the comparison is this page’s ideal. None of those spacecraft was that reflector, and none of them is given an orbit here.

IKAROS is a side and a wet mass. The JAXA project page prints a square of side 14 m, a cross section of 20 m after deployment, and a liftoff mass of about 310 kg. Tsuda et al. print sail size 14×14 m and a wet mass of 307 kg, including fuel 20 kg and the sail 16 kg. The button is 14×14, which is 196 m², and 307 kg. The public about 310 kg is not substituted. A 20 m diagonal is the same square’s corner-to-corner length, not a second area.

LightSail 2 is 32 m² and 4.93 kg. Mansell et al. 2023, Table 2, prints mass 4.93 kg and sail area 32 m². Bruce Betts of The Planetary Society is a coauthor. The mission’s SciTech table prints the same pair. That paper’s introduction says 5 kg. The FCC orbital debris assessment prints 4.85 kg and 32 m². The button uses 4.93 kg and 32 m². The 5 kg and the 4.85 kg are not substituted. LightSail 2 flew an on/off cycle in Earth orbit. This button does not.

ACS3 is 80 m² and 16 kg, as a design table. Wilkie, Small Satellite Conference 2021, SSC21-II-10, prints a total deployed sail area of 80 m². The appendix table prints a total spacecraft mass, including the sail, of 16 kg, and each quadrant at 20 m². NASA’s mission page says about 80 square meters. The button uses 80 m² and 16 kg. The same table prints an effective characteristic acceleration of 0.045 mm/s² and an effective lightness number of 0.0077, with specular reflectance 0.74 on the aluminum face. This page’s perfect reflector computes 0.0454 mm/s² and lightness 7.656×10⁻³. Those effective figures are not copied. Wilkie also prints the order of a perfect reflector at 1 AU as 9×10⁻⁶ Pa. This page’s 9.080 μN/m² is the nominal 1,361 W/m² divided into c, twice.

Literacy · the words on the readout

Irradiance: Power per area at the sail’s distance. S = 1,361 W/m² divided by the square of the distance in AU. One astronomical unit is 149,597,870,700 m exactly, IAU 2012 Resolution B2. The planet buttons are each fact sheet’s J2000 semimajor axis in AU. Earth’s is 1.00000011, which is not the 1 AU button. At the tenth of a watt this page prints, both read 1,361.0 W/m².
Radiation pressure: Force per area. Face-on, an absorber feels S/c and a reflector feels 2S/c. With a tilt, the card’s pressure is the force magnitude divided by the sail area, so the projected area and the reflectivity are already in it. c is 299,792,458 m/s exactly.
Reflectivity: ρ from 0 to 1. Zero is a perfect absorber. One is a perfect specular reflector. The fraction in between is reflected; the rest is absorbed and given no thermal push. A real film sits between the ends and is not a single number on this slider.
Tilt: The angle of the anti-sunward normal from the outward Sun line. Zero is face-on. Positive tips that normal toward prograde, the direction drawn upward. Negative tips it the other way. At ±90° the sail is edge-on and the force is zero. The back of the sail is not lit.
Characteristic acceleration: The acceleration this area, mass, and reflectivity would have at 1 AU, face-on. It does not use the current tilt or the current distance. It is the sail’s yardstick. The card’s acceleration is the one at the distance and tilt you set.
Lightness number: Radiation acceleration divided by solar gravitational acceleration at the same distance. GM☉ is 1.3271244×10²⁰ m³ s⁻². Both accelerations fall as 1/r², so the ratio stays put when you move the distance and changes when you change the sail, the reflectivity, or the tilt. Lightness 1, face-on and perfectly reflecting, is 1.531 g/m². A sail heavier than that still feels the push. It does not cancel the Sun’s gravity.

Sources · the constants and the sails

Cited, not invented

NIST — speed of light in vacuum and BIPM — the metre — c = 299,792,458 m/s exactly. Pressure is irradiance divided by that c, and twice that for a face-on reflector.
IAU 2012 Resolution B2 — the astronomical unit is 149,597,870,700 m exactly. Distance in metres is the AU value times that length. Solar gravity uses the same metre.
IAU 2015 Resolution B3 (Prša et al. 2016) — nominal total solar irradiance 1 S^N_☉ = 1361 W m⁻², and nominal solar mass parameter 1 (GM)^N_☉ = 1.3271244 × 10²⁰ m³ s⁻². Both are conversion constants, chosen near the 2015 estimates, not new measurements. The resolution text is also at the IAU archive. The notes to Resolution B2 in that document give a cycle-23 average of 1361 (±1) W m⁻² and cite Kopp & Lean 2011 for the TIM scale. This page divides by 1361, not by 1360.8.
Kopp & Lean 2011, Geophysical Research Letters 38, L01706 — the most accurate value they give for total solar irradiance during the 2008 solar minimum is 1360.8 ± 0.5 W m⁻², from TIM on SORCE. Cited here. Not substituted for the nominal 1361.
NASA NSSDCA — Mercury Fact Sheet — Semimajor axis (AU) 0.38709893. The same line the Kepler desk uses. Last updated 11 January 2024.
NASA NSSDCA — Venus Fact Sheet — Semimajor axis (AU) 0.72333199. Last updated 11 January 2024.
NASA NSSDCA — Earth Fact Sheet — Semimajor axis (AU) 1.00000011. Not exactly 1 au. Last updated 15 November 2024.
NASA NSSDCA — Mars Fact Sheet — Semimajor axis (AU) 1.52366231. Last updated 19 May 2025.
NASA NSSDCA — Jupiter Fact Sheet — Semimajor axis (AU) 5.20336301. Last updated 2 October 2024.
JAXA — IKAROS — membrane a square of side 14 m, cross section 20 m after deployment, mass at liftoff about 310 kg. The 20 m is the diagonal of that square. The about 310 kg is not the button.
Tsuda et al., Journal of Space Technology and Science — “Challenges and Results on Attitude Control Operation of World’s First Solar Power Sail IKAROS.” Table of basic specifications: wet mass 307 kg, including fuel 20 kg and sail 16 kg, sail size 14×14 m. The button is 196 m² and 307 kg.
Mansell, Bellardo, Betts, Plante, and Spencer 2023, Aerospace 10, 579 — LightSail 2, Table 2: mass 4.93 kg, sail area 32 m². The button uses that pair.
Mansell, Spencer, et al., including Betts and Nye — LightSail 2 SciTech paper — the characteristics table prints 32 m² and 4.93 kg. The introduction says each LightSail has a mass of 5 kg and a 32 m² sail. The 5 kg is not substituted.
FCC — LightSail 2 orbital debris assessment — mass 4.85 kg and a 32 m² sail. Not substituted. The area agrees. The mass does not.
Wilkie — ACS3, Small Satellite Conference 2021, SSC21-II-10 — total deployed sail area 80 m². Appendix table: total spacecraft mass including sail 16 kg, sail quadrant area 20 m², effective characteristic acceleration 0.045 mm/s², effective lightness 0.0077, specular reflectance of the aluminum face 0.74. The button uses 80 m² and 16 kg only. The same paper’s perfect-reflector order of magnitude is 9×10⁻⁶ Pa at 1 AU.
NASA — Advanced Composite Solar Sail System — the deployed sail is about 860 square feet, about 80 square meters, and about 9 meters on a side. The about 80 square meters is the public rounding of the 80 m² in the design paper.

Classroom picture of radiation pressure on a flat sail. The irradiance is the IAU 2015 nominal 1361 W/m², not the Kopp & Lean 1360.8. c and the astronomical unit are exact. GM☉ is the IAU 2015 nominal 1.3271244×10²⁰ m³ s⁻². A perfect reflector, face-on, at 1 AU feels 9.080 μN/m². Mission buttons are cited area and mass, then this ideal. Not a trajectory and not for mission design. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The burn

Irradiance	1,361 W/m² over the square of the distance in AU.
Pressure and force	The force from the flat-sail formula, and that force divided by the area.
Acceleration	Force divided by mass, at this distance and tilt.
Characteristic	Face-on acceleration at 1 AU for this area, mass, and reflectivity.
Lightness	This acceleration divided by solar gravity at this distance. It does not change when only the distance changes.
Δv	Acceleration times the time you set. Labeled as a constant-acceleration simplification.

Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Solar sail
and the light

Cited, not invented

Rocket

Light time

Hohmann

Kepler

Solar map

Solar-sail desk

Controls

Readouts

Keyboard and touch

Distance	0.2 to 40 AU, on a logarithmic slider. 1 AU is the nominal irradiance. Mercury, Venus, Earth, Mars, and Jupiter are the fact-sheet semimajor axes. Earth is 1.00000011 AU.
Area and mass	Area from 1 to 10,000 m². Mass from 0.5 to 2,000 kg. Both sliders are logarithmic.
Reflectivity	0 is a perfect absorber. 1 is a perfect reflector. The number accepts the same range.
Tilt	−90° to 90°. Zero is face-on. Positive tips the push toward prograde. The diagram’s Left and Right keys step it by one degree.
Presets	1 m² · 1 kg, IKAROS, LightSail 2, and ACS3 set the area and the mass, then a perfect reflector, face-on, at 1 AU. They do not store a flight reflectivity.
Time	0 to 3,652.5 days. The buttons are 1 day, 30 days, and 365.25 days. Δv = a × t, with the acceleration held fixed. A day is 86,400 seconds.

Solar sailand the light

Cited, not invented

Rocket

Light time

Hohmann

Kepler

Solar map

Solar sail
and the light