In his book The God Delusion, Richard Dawkins briefly mentions the notion that moths are drawn towards candles, and hence commit suicide on a pretty regular basis. This would seem pretty antithetical to evolution. One possible reason for this (Robert Persig mentions the same idea) is that moths use natural light sources, i.e. the Moon and stars, as navigational references. Since celestial objects are, for all practical purposes, an infinite distance away, their light rays are more or less parallel, and so they make excellent beacons.
The problem with artificial light sources, such as candles, is that they are much closer. Maintain an angle of, say, 30° to a nearby object, and the eventual result is a spiral in to said object, thus resulting in toasted moth, and proving that creationists were right all along.
Hold up a second, there, Dick. A spiral? And a logarithmic one at that? Why not a circle? Intuition would suggest that an object that maintained a constant angle to another object would travel in an infinite circle. One needs to look at our own solar system to see proof of that. (Yeah, I know. Circle. Ellipse. Whatever.)
Mario Livio suggests that hawks do not actually circle their prey, they (logarithmically) spiral in on it in much the same way.
So what separates out winged friends from our heavenly bodies? Is there some way to mathematically prove this spiral effect?
An article in the Indianapolis Star the other day claimed that some birds navigate the same way, keeping the moon to one side. Many of them are lured by the flashing beacon atop antenna towers. They fly in circles around the tower. Many eventually die, either from exhaustion, or from colliding with guy wires or the tower itself.
The proliferation of cell phone towers has greatly increased the number of birds that are confused to death, the article claimed.
How do you measure the “angle of attack”? Is it the angle between the instantaneous velocity vector and the vector connecting moth to flame (planet to sun)? If so, then it needs to be 90° for the trajectory to be a circle; otherwise the tangent vector of the flight path (planetary orbit) has a radial component pointing inward, and the distance between moth and flame (planet and sun) will decrease over time, resulting in the spiral that Dawkins mentions.
I thought so. Dawkins would have done well to expand on this a little.
In fact, one can visualize the problem easily if one imagines the tightest angle possible, 0°, or the opposite, 180°, in which case the moth would fly directly towards or away from the candle.
I’m just wondering what the math behind this might be.
Slight hijack but how is this “antithetical to evolution”? If anything it looks to me like it’s antithetical to creationism because an “intelligent designer” would build in a way for the moth to distinguish candles from natural light sources while evolution would only kick in once some random change points in that direction.
Excellent point. But from a shallow and unthinking perspective, it seems counter-intuitive that deliberate immolation is a survival skill likely to be passed on to offspring.
One must keep in mind that insect navigation predates the invention of the candle by millions(?) of years.
Restricting our attention to planar trajectories, we could start by denoting the desired trajectory as [x(t), y(t)]. With the intention of eventually recovering the polar equation for a logarithmic spiral, let’s require that the components have the particular form [r(t) cos(t), r(t) sin(t)]. The number of dependent variables is thus reduced to 1, and to justify this simplification it must be argued that only the shape of trajectory is relevant (not the speed at which the moth flies it), so by reparametrization the curve may be recast in a form more amenable to polar plotting.
Now differentiate the trajectory components with respect to t and obtain the instantaneous tangent vector (velocity vector) [r’(t) cos(t) - r(t) sin(t), r’(t) sin(t) + r(t) cos(t)]. Take the inner product of the velocity and position vectors, yielding r(t)r’(t) after simplification. On the other hand, the geometric formulation of inner product (as the product of vector magnitudes and the cosine of the angle between them) forces us to equate r(t)r’(t) with r(t) sqrt{r’(t)^2 + r(t)^2} cos(A), where A is the (constant) angle of attack held by the flying moth. If A is constant, then so also is the ratio r’(t)/r(t), as can be seen by manipulating the preceding equation so that r and its derivative are on one side. Thus r’(t)=kr(t), and the constant k is related to A by k=1/tan(A)^2. If A is obtuse, then k is positive, while if A is acute, then k is negative. The differential equation for r(t) therefore has exponentially growing (decaying) solution r(t)=r(0)e^{kt} if the angle of attack is obtuse (acute). As mentioned in my last post, if A is a right angle then the trajectory is simply a circle.
Finally, plotting the polar equation r=e^{k*theta} for different values of k gives an idea of what this so-called logarithmic spiral looks like.