Thursday, July 17, 2008

A Light Burden

Did you know that radiation carries with it a kind of pressure known as radiation pressure? That means that when I shine a torch on you, it's supposed to be pushing you as it gets absorbed or scattered off of you. Let's try and determine the force exerted on the Earth by the light rays from the Sun, and you'll see what a staggering figure this is!

Assuming that the Sun is a perfect sphere, of radius R = 695 500 km and that it is a perfect blackbody with emissivity e = 1.00, and a roughly constant surface temperature of T = 5 778 K, we can use the Stefan-Boltzmann Law to determine the power of radiation emitted by the sun:

Where σ is the Stefan Boltzmann constant and A is the surface area of the Sun. And yes I've conveniently assumed the Earth is of so a low temperature with respect to the Sun such that it hardly radiates back any radiation back to the Sun.

Let us denote the distance of the Earth from the sun to be d (which is an enormous d = 149 600 000 km) and its radius as r = 6371 km. With these figures, we can then determine the intensity of solar light received at the position of our Earth, which is the initial power emitted by the Sun divided by the surface area of a sphere of radius d since the light from the Sun is isotropic in all directions:

Since all light rays coming from such a far off source will be parallel to one another, we can take the cross sectional area of the Earth to be the area that intercepts the light rays, which can be taken to be:

The power intercepted by the Earth is therefore:

Now, we know that for electromagnetic radiation, pc = E, where p is the equivalent momentum of a quantum of energy, c is the speed of light in a vacuum, and E is the energy of radiation considered. This simply means that the power is related to the momentum as such:

Don't forget, we know that the force exerted by the radiation on Earth is equivalent to the rate of change of momentum of the radiation photons (of course, we're assuming all radiation is absorbed, which is an approximation!), and thus the force exerted on Earth by the Sun's light is:

And this force is about 10^8 N! Which turns out to be an astronomical number! :) So why do you think the Earth is not being pushed away by the Sun? Easy, there's still the force of gravity, which when worked out by Newton's Law of Gravitation:

And this amounts to about 10^22 N! Do you see the difference in the two forces? The gravitational force far outweighs the force exerted by light, and thus continues to aid us in our journey through intergalactic space! :)

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