Is the moth chasing the light, so put out the fire?

"Moth to the fire" is a natural phenomenon, but its reason has always been misunderstood as phototaxis, but it is not. Especially moths, there is a question on the Zhihu called "Why do insects rush to the sun because of their different phototaxis?" ",this question is very good, because just see the title, the in the mind have begun to doubt" phototaxis ".

In fact, moths don't put out fires because they like bright light. They mainly use bright light to navigate for themselves. For example, at night, moths navigate by moonlight. In fact, they tried to fly in a straight line without accident. After all, the straight line is the shortest.

Since there are moths in this world, it seems that all this seems natural. However, later, humans learned to use "fire". A significant difference between this artificial light source and moonlight formed by sunlight or reflected sunlight at the boundary is that artificial light sources are mostly point light sources, such as candlelight, and sunlight or moonlight can be regarded as parallel light, but the history of human learning to use fire is obviously short enough compared with the history of moths, and human influence on nature is limited after all. Natural selection is far from teaching all moths in the world to distinguish between natural light and artificial light, so the moth is still the moth that will put out the fire forever, but human beings gradually see the truth of its "fire extinguishing".

Because moths are used to parallel light in nature, they think that as long as they keep a certain angle with light, their flight trajectory is straight. This is like, "whether a cat walks in a straight line depends on a mouse;" Whether moths fly in a straight line depends on the light. "In the dim moonlight night, moths fly very well, but the result is close to a much brighter candlelight, and the candlelight light is radial. Assuming that the moth's flight trajectory and the center point of candlelight are in the same plane, and the moth's flight direction and the light direction always keep an angle of 45 degrees (of course, it can also be other angles, here for convenience of calculation), then obviously its flight trajectory will be an "equiangular spiral"!

If the above picture is not convincing, then there are time-lapse photos to prove it:

The moth just circled around the light source until it met a fire, and then it was gone.

But today, I suddenly felt that I might as well calculate this trajectory and see what the hell an equiangular spiral is. Yes, it is full. It may be that after reading the differential equation, it is specially called for all speeds and dip angles!

Assuming that the candlelight is at the origin of coordinates and the position of the moth is (x, y), first put the position of the moth in the first quadrant, which is convenient for calculation, and then let the angle between the moth and the candlelight and the positive direction of the X axis be α.

Obviously, there is tan α = y/X. Let the flying track of moths be y = y (x), then there are:

Therefore, the homogeneous differential equation is obtained:

Using the inverse method, let z = y/x, then y = zxy' = z'x+z

So there are:

Two-sided integrated:

This general solution looks ugly, let's express it in polar coordinates!

Perfect! Thanks to Euler formula ~

What you get is a perfect exponential spiral. In nature, this equiangular motion around a point forms an exponential spiral curve with a natural bottom. Such as conch, which may be one of the reasons why natural bases are called "natural".

For the origin of natural base, please use WeChat official account "Research Dog" and WeChat official account ID: Research Dog will push "How can natural base E be" natural "in the next issue? 》。