Calculating the light that can reach a tree in a forest?

Question

I was wondering if there was some way to calculate the portion of light that could reach a tree in a forest only from the horizontal direction, in which case the factor to figure out would be the combined amount of light blocked by the trees as it's getting to a given tree. I realize that the problem doesn't work with a forest of trees where they're placed completely randomly, but what if they were planted partly randomly so on average, they were placed in a square grid pattern, so it could be approximated that way. Also that they all have the same trunk diameter. Any suggestions as to how to approach this problem? I imagine that you could approximate it as a square grid and determine the relative light blockage of light surrounding the forest for each square ring of trees surrounding given center trees. Then you'd be able to add these up for the total amount of light that all the tree trunks can block. I guess it could also be thought of as the opposite, where setup is the same but you'd be finding the amount of light that could escape from a lantern on a tree in the forest. Or just general calculations with a grid and light blockage/absorption. I'd appreciate any thoughts on this. I have some ideas but I was wondering if there was some mathematics that deals with this kind of thing.

Answer 1

You should think about the distribution of trees in rings around the origin. If trees have (an average) radius $r$ a tree at distance $R$ from the origin will intercept (on average and once $R \gg r$) $\frac {2r}{2\pi R}$ of the light. If there are $n(R)dR$ trees at a distance between $R$ and $R+dR$ from the origin, that annulus will block $\frac {2rn(R)dR}{2\pi R}$ of the light. As with Olbers's paradox which is the 3D version of this, if the area density of trees is a constant all the light will eventually be absorbed. The absorbtion of trees goes down as $\frac 1R$ because they are farther away, but the number in an annulus goes up as $R$ so the fraction absorbed is constant in each annulus.

For a more general $n(R)$ distribution it is easier to work with the logarithm of the light remaining. The annulus from $R$ to $R+dR$ reduces the light by a factor $1-\frac {2rn(R)dR}{2\pi R}$ or subtracts $\frac {2rn(R)dR}{2\pi R}$ from the log. The log of the light remaining at radius $R$ is then decreased by $\int_0^R \frac {2rn(\rho)d\rho}{2\pi \rho}$ The approximation $R \ll r$ came from the fact that a large tree will block a little more light than this would suggest. You might just decree that there are no trees closer than, say $5r$ and raise the lower limit on the integral to that.