# Solar Sail Calculator - Radiation Pressure, Lightness Number, Spiral Orbits

> Solar sail simulator: radiation pressure I/c vs 2I/c, oblique force, lightness number, spiral orbits at the 35° optimum and the Starshot laser sail.

- URL: https://qilingtools.com/en/physics/solar-sail
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Starting from photon momentum p = h/λ, this tool works out the push of sunlight on a sail: I/c when absorbed, 2I/c when reflected, and a force along the sail normal falling as cos²θ at oblique incidence. Sail loading then gives the characteristic acceleration and lightness number β, an RK4 integration follows the sail spiralling through the solar system, and real missions such as IKAROS and LightSail 2 are compared. The last tabs estimate a Breakthrough Starshot laser sail's trip to Alpha Centauri and clear up common myths.

## How to use

1. On the pressure tab set the distance from the Sun, incidence angle, reflectivity and area, and see the light rays and force components
2. On the loading tab enter mass per square metre to get the characteristic acceleration and β, and compare with past missions
3. On the orbit tab choose β and the cone angle α, play the spiral and read when it reaches the orbit of Mars or Venus
4. Use the laser and photon tabs to estimate interstellar travel time, photon counts and forces, and read the misconceptions

## Features

- Absorbed vs reflected pressure, normal and in-plane force components and force–angle curve
- Characteristic acceleration a_c = 2ηS₀/(cσ), lightness number β and the 1.53 g/m² critical loading
- RK4 heliocentric orbit: outward or inward spiral animation, r–t graph and the 35.26° optimum
- Approximate figures for IKAROS, LightSail 2, NEA Scout and ACS3
- Laser sail with relativistic Doppler thrust, diffraction-limited range and travel time to Alpha Centauri
- Photon momentum calculator plus notes on comet tails and the Crookes radiometer

## FAQ

### Why does a reflecting sail feel twice the pressure of an absorbing one?

A photon carries momentum p = E/c. When absorbed, it hands over p; when reflected, its momentum changes from +p to −p, so the sail receives 2p. Light of irradiance I at normal incidence therefore presses I/c on an absorber and 2I/c on a mirror. Near Earth sunlight is about 1361 W/m², so the mirror pressure is only about 9 μPa.

### What happens when light hits the sail at an angle?

At incidence θ the sail intercepts power reduced by cosθ, and on a perfect mirror each photon transfers only its normal component 2p cosθ, so the force is 2I cos²θ/c, always along the sail normal. On an absorber the force points along the light with size I cosθ/c. Because a mirror's force follows its normal, tilting the sail steers the thrust sideways, which is how a sail changes its orbit.

### How can a solar sail fly towards the Sun, and what is the best angle?

Light can only push the sail away from the Sun, but tilting the normal against the direction of motion gives the thrust a component opposite the velocity. That brakes the sail, lowers its orbital energy and makes it spiral inward. The along-track component is proportional to cos²α·sinα, which peaks at α = arctan(1/√2) ≈ 35.26°, so about +35° spirals out fastest and about −35° spirals in fastest.

### What is the lightness number β, and why doesn't it depend on distance?

β is the ratio of the sail's radiation-pressure acceleration to the Sun's gravitational acceleration. Both fall as 1/r², so the ratio depends only on the sail loading σ (mass per square metre): β = σ*/σ with σ* ≈ 1.53 g/m² for a perfect mirror. Flown sails have β of only 0.001–0.01; reaching β near 1 would need the sail and payload together to weigh just a gram or two per square metre.

### Is a Crookes radiometer turned by light pressure?

No. Light pressure would push the shiny faces harder, so they would retreat, but in practice the black faces retreat. The vanes turn because of thermal effects in the thin gas (thermal transpiration and edge effects) caused by the temperature difference between the faces, which are far stronger than radiation pressure; in a hard vacuum the radiometer barely moves. A comet's dust tail is the genuine example of light pressure, while the ion tail is blown mainly by the solar wind.

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