Free Radiation Pressure Calculator

Understanding Radiation Pressure

Radiation pressure is the force per unit area exerted by electromagnetic radiation on a surface. This phenomenon arises because photons, the quanta of light, carry momentum; when they collide with an object, they transfer that momentum. This radiation pressure calculator enables you to compute two key scenarios: the pressure experienced by a surface outside a star (e.g., on a solar sail) and the pressure that builds inside a star due to its own radiation.

Although the radiation pressure we encounter on Earth is minuscule (on the order of micropascals at Earth’s distance from the Sun), it becomes enormous at the high temperatures found in stellar interiors. In fact, internal radiation pressure provides a crucial force that counterbalances gravity, preventing stars from collapsing under their own weight. This balance is the reason stars can shine stably for billions of years.

The Radiation Pressure Equations

Pressure from Starlight (External)

When a surface is placed at a distance $R$ from a star of luminosity $L$, the external radiation pressure is given by:

$$p_{\text{ext}} = \frac{x\, L\, \cos\alpha}{4\pi R^2 c}$$

where:

• $x$ describes the surface's reflectivity: $x = 1$ for a perfectly absorbing (opaque) surface and $x = 2$ for a perfectly reflecting surface. Real materials have $x$ between 1 and 2.
• $\alpha$ is the angle between the incoming light beam and the surface normal ($\alpha = 0^\circ$ for perpendicular incidence, giving $\cos\alpha = 1$).
• $c \approx 2.99792458 \times 10^8\ \text{m/s}$ is the speed of light.

In the default mode of this calculator, the light is assumed to strike the surface perpendicularly. You can, however, adjust the angle to fit your exact scenario. This equation is the basis for all lightsail pressure calculations.

Pressure in Stellar Interiors (Internal)

Inside a star, the radiation field is so intense that it contributes a substantial pressure. This internal radiation pressure depends only on the temperature $T$:

$$p_{\text{int}} = \frac{4\sigma}{3c}\, T^4$$

Here, $\sigma = 5.670367 \times 10^{-8}\ \text{W/(m}^2\!\cdot\!\text{K}^4)$ is the Stefan‑Boltzmann constant, and the factor $4/(3c)$ converts the radiative energy density into pressure. Unlike ideal gas pressure (which scales linearly with $T$), radiation pressure grows with the fourth power of temperature, making it dominant in the hot cores of massive stars.

Practical Examples

Let’s apply the solar pressure calculator to two extreme scenarios.

Example 1: Earth orbit (external)
Take a perfectly absorbing surface at Earth’s distance from the Sun ($R = 1\ \text{au} = 1.496 \times 10^{11}\ \text{m}$) and assume perpendicular incidence. The Sun’s luminosity is $L = 3.828 \times 10^{26}\ \text{W}$.

$$p_{\text{ext}} = \frac{1 \times 3.828 \times 10^{26}}{4\pi \times (1.496 \times 10^{11})^{2} \times 2.998 \times 10^{8}} = 4.54 \times 10^{-6}\ \text{Pa} \quad (4.54\ \mu\text{Pa}).$$

Example 2: Solar corona (internal)
Inside the Sun’s corona, the temperature reaches $T = 5\,000\,000\ \text{K}$. Using the internal pressure formula:

$$p_{\text{int}} = \frac{4 \times 5.670367 \times 10^{-8}}{3 \times 2.99792458 \times 10^{8}} \times (5 \times 10^{6})^{4} = 1.576 \times 10^{11}\ \text{Pa} = 157.6\ \text{GPa}.$$

The internal pressure is thus about $10^{17}$ times larger than the external pressure at Earth – a vivid illustration of how radiation pressure dominates inside stars. (For convenience, pressures can be converted to other units, such as atmospheres, using standard conversion factors.)

Solar Sails and Photon Propulsion

A solar sail (or lightsail) is a spacecraft propulsion concept that directly exploits the external radiation pressure described above. By deploying a large, reflective membrane, the sail catches photons from the Sun, gradually accelerating the vehicle without consuming propellant. The maximum thrust is obtained when the sail is highly reflective ($x$ close to 2) and oriented face‑on to the Sun ($\alpha = 0^\circ$).

The historic Cosmos 1 mission (2005) was one of the first attempts to test solar sail technology in orbit. It carried eight sail blades, each 15 m long, giving a total area of $600\ \text{m}^2$. Although the launch failed 83 seconds after liftoff, subsequent missions have successfully demonstrated lightsail deployment and acceleration. This photon pressure calculator allows you to explore the numbers behind these concepts: input your own star’s luminosity, distance, sail reflectivity, and tilt angle, and instantly see the resulting radiation pressure.