Created by Dominik Czernia, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Jun 05, 2023

This radiation pressure calculator will help you estimate the radiation pressure inside and outside of stars is. We can describe this type of pressure in two ways:

• As a force exerted on the surface by light particles – photons; and
• As a pressure in the medium in which the electromagnetic radiation propagates.

In the text below, we have explained what solar pressure is and how you can estimate it using the radiation pressure equation.

The radiation pressure that reaches the Earth is almost negligible compared to other types of pressure. However, it gains significance at high temperatures, for example, in the interior of stars.

If you are interested in more mundane matters, you can also check out our general pressure calculator, where you can find the definition of pressure.

Have you ever heard about the Cosmos 1 project to test a solar sail (lightsail) in space? In this mission, the spacecraft named Cosmos 1 was supposed to increase its velocity using radiation pressure. Read on if you want to know how solar sails work and what is the physics behind them.

## What is the radiation pressure?

Because electromagnetic waves carry energy, we can also expect them to carry momentum. This electromagnetic radiation can be viewed in terms of particles which are known as photons (read our De Broglie wavelength calculator for more). When photons hit a surface, they exert a force on that surface. The surface will absorb the momentum of photons (if the material is opaque) or reflect the photons (well, partially reflect).

Is radiation pressure a significant phenomenon? Let's consider the hot interior of a star. All matter with a temperature higher than absolute zero emits electromagnetic radiation described by Planck's law. On the other hand, you probably know that stars consist of particles held together by the gravitational force. Now, if there isn't any radiation pressure, the star would collapse! We know that stars can exist for billions of years, so we should consider radiation pressure too.

Our radiation pressure calculator can handle two situations. The first is to calculate the solar pressure outside the star. We have used the below radiation pressure equation:

$p_{\rm out} = \frac{xL\cos^2(\alpha)}{4\pi R^2c},$

where:

• $p_{\rm out}$ – Radiation pressure;
• $x$ – Determines the type of surface: $\small x = 1$ – opaque surface, $\small x = 2$ – reflective surface;
• $L$Luminosity of star;
• $\alpha$ – Angle between the light beam and the surface of absorbing/reflecting surface;
• $R$ – Distance from the star; and
• $c$ – Speed of light $\small c \approx 2.99792458 \times 10^8\ \rm m/s$.

In practice, materials are neither completely reflecting nor absorbing, so the $x$ will be between 1 and 2. In the simple mode of our radiation pressure calculator, we assumed that the light falls perpendicular to the surface $\small (\alpha = 0 \degree)$. If you want to change it, just go to the advanced mode of the calculator.

With the second solar radiation equation, you can estimate pressure radiation inside a star:

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

where:

• $p_{\rm int}$ – Internal radiation pressure;
• $T$ – Temperature; and
• $\sigma$ – Stefan-Boltzmann constant $\small \sigma = 5.670367 \times 10^{-8}\ \rm W/(m^2 \cdot K^4)$.

By comparison, the pressure of ordinary molecules in gases increases linearly with temperature $\small \propto T$ (see ideal gas law calculator). Let's use our radiation pressure calculator to compare what is the solar pressure on Earth and inside the Sun.

If an absorbing surface is opaque, $\small R = 1\ \rm au$ (one astronomical unit $\small 1\ \rm au$ is an average distance between Earth and Sun) and $\small L = 1\ L_{\bigodot}$ (one solar luminosity equals $\small 1\ L_{\bigodot} = 3.828 \times 10^{26}\ \rm W$) we obtain $\small p_{\rm int} = 4.54\ \text{μPa}$.

On the other hand, if we want to compute what is solar pressure inside the corona of the Sun $(\small T = \text{5,000,000 K})$, we will get $\small p = 157.6\ \rm GPa$, which is about $\small 10^{17}$ times bigger than that on Earth!

You might want to express these pressures, for example, in atmospheres (atm). Check our pressure converter to learn how to do this.

## How do solar sails work?

Solar sail (also called lightsail or photon sail) is a theoretical method of traveling in space using radiation pressure exerted by sunlight on large mirrors. It is just like a sailing boat when the wind blows a sail, but with light.

People wanted to test the solar sail concept, and in 2005, they launched for the first time an orbital spacecraft named Cosmos 1 which had eight sail blades (15 m long with a total surface area of 600 square meters). Unfortunately, in the 83rd second of the flight, the engine broke down, and the whole vehicle fell back into the sea. However, since then, several other satellites with the lightsail equipment have been successfully launched into Earth orbit.

Dominik Czernia, PhD
Pressure outside the Sun
Type of surface
Opaque
Luminosity
L☉
Distance
au
Pressure
bar
Pressure inside the Sun
Temperature
K
Pressure
GPa
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