How Bright is the Sun?

The brightness of the Sun is inversely proportional to our distance from it (the further way from it we are, the fainter it appears to be). It varies with the square of the distance. Thus we say that its brightness follows an inverse square law.

Light Dims with Distance
Two 1000 watt light-bulbs are shown. When one is shifted to lie two times as far away as the other, its output drops by a factor of two squared (four) to 250 watts. When is it shifted to lie ten times as far away as the other, its output drops by a factor of ten squared (one hundred) to 10 watts.
(click on picture for animation)

Consider rays of light emitted from the Sun. You can see that as the rays move away from the Sun, they spread farther and farther apart from each other.

The Sun is shown as a circle, with straight rays of light being emitted in all directions. As the rays extend out further and further from the surface of the Sun, the distance between neighboring rays increases.

Let us examine the light coming from a small piece of the Sun, as shown below. We have marked along the horizontal axis points at 1, 2, and 3 units. These could represent the position of the Earth relative to the Sun, and then positions that are twice and three times as far away. The red lines show the length needed to span the separation between the end sunbeams at each radius.

The Sun is shown as a circle, with straight rays of light being emitted in all directions; we focus upon the rays being emitted from a small portion of the Sun's surface. As these rays extend further from the Sun, distance between neighboring rays increases. At a distance of one Earth orbital radii, the length of a line segment subtending the complete set of rays is drawn. At a distance of two Earth orbital radii, it takes a line segment twice as long to subtend the same rays, and at three radii it takes a line segment three times as long.

Now we explore the relationship between the radius, the distance we lie away from the Sun, and the length needed to span the separation between end sunbeams. This is a linear relation. This means that if we double the radius, the length increases by a factor of two. If we triple the radius, the length increases by a factor of three. This shows us that as we move from a radius of 1 to a radius of 3, it will take a length three times longer to encompass a single set of sunbeams.

The previous figure is shown as an XY plot, where X is the orbital radius and Y is the length of the line segment spanning the bundle of light rays. A line is drawn on the plot running through the points (0,0), (1,1), (2,2), and (3,3).

This means that the spacing between neighboring sunbeams will increase by a factor of three. The figure below shows a set of sunbeams hitting a surface at a radius of 1, 2 and 3 from the Sun. The side view shows the light coming from the left and hitting the surface at the right, while the front view shows the light shining out of the page directly towards us. We can see that a unit length of ground will receive 6 sunbeams at a radius of 1 (left), 3 sunbeams at a radius of 2 (middle) and 2 sunbeams at a radius of 3 (right). Another way to say this is that the intensity of the light will fall off inversely with radius.

The light rays are shown from the side (where they appear as parallel arrows) and head-on (where they appear as dots) As we shift to larger distances from the Sun, the distance between neighboring rays increases in both views.

The final figure is the same, only it shows the sunbeams falling on a surface with width and length rather than just length. This is equivalent to a patch of land on the surface of the Earth. The side view shows the light coming from the left and hitting the surface at the right, while the front view shows the light shining out of the page directly towards us. We can see that our unit patch of ground will receive 6 × 6 = 36 sunbeams at a radius of 1 (left), 3 × 3 = 9 sunbeams at a radius of 2 (middle) and 2 × 2 = 4 sunbeams at a radius of 3 (right). Another way to say this is that the intensity of the light will fall off inversely with the square of the radius.

Because the surface of the Sun is a two-dimensional structure, we must add a second dimension to our visualizations. Rather than examining a line of rays, we add a width to the length of the region and shift from counting the number of rays per unit length to rays per unit area. The number of rays shifts from changing with distance to changing with the square of distance. This is shown again from the side and head-on.