I took the above panoramic view, spanning something like 120 degrees, in a local park towards the end of last year. The sun was almost on the horizon to the southwest, at right of frame. The moon was well risen in the southeast, framed by the little red box in the image above. After taking the panorama, I zoomed in for the enlarged view of the moon shown in the inset, to demonstrate the apparent problem. The moon is higher in the sky than the sun, but its illuminated side is pointing slightly upwards, rather than being orientated, as one might expect, with a slight downward tilt to face the low sun.

This appearance is quite common, whenever the moon is in gibbous phase (between the half and the full), and therefore separated by more than 90 degrees from the sun in the sky. Every now and then someone notices the effect, and decides that they have to overthrow the whole of physics to explain it. I could offer you a link to a relevant page, but I won’t—firstly, I don’t like to send traffic to these sites; secondly, you might be driven mad by the experience and I’d feel responsible.

Actually, the illuminated part of the moon is pointing directly towards the sun; it just doesn’t look as if it is. So (as with my previous post “Why Do Mirrors Reverse Left And Right But Not Up And Down?”) the title of this post is an ill-posed question—it assumes something that isn’t actually so.

Here’s a diagram showing the arrangement of Earth, moon and sun in the situation photographed above:

The Earth-bound observer is looking towards the setting sun. Behind and above him is the moon, its Earth-facing side more than half-illuminated. The sun is so far away that its rays are very nearly parallel across the width of the moon’s orbit. In particular the light rays bringing the image of the setting sun to the observer’s eyes are effectively parallel to those shining on the moon—the divergence is only about a sixth of a degree.

But we know that parallel lines are affected by perspective. They appear to converge at a vanishing point. The most familiar example is that of railway lines, like these:

But there’s a problem with this sort of perspective. To illustrate it, I took some photographs of the top of the very low wall that surrounds the park featured in my first photograph:

The views look north and south towards two opposite vanishing points. The surface of the wall is marked with the remains of the old park railings, which were sawn off and removed during the Second World War. These provide a couple of reference points, which I’ve marked with numbers. The parallel sides of the wall appear to diverge as they approach the camera towards Point 1; and they appear to converge as they recede from the camera beyond Point 2. But what happens between 1 and 2?

I used my phone camera again to produce this rather scrappy and unconventional panorama, looking down on the top of the wall and spanning about ninety degrees:

The diverging perspective at Point 1 curves around to join the converging perspective at Point 2. It’s mathematically inevitable that this should happen—what’s surprising is that we’re generally unaware of it. In part, that’s because our normal vision spans a smaller angle than we can produce in a panoramic photograph; but it’s also because our brains are very good at interpreting the raw data from our eyes so that we see what we need to see. In this case, as we scan our eyes along the length of this wall, we have the strong impression that its sides are always parallel, despite the fact that its projection on our retinas is more like a tapered sausage with a bulge in the middle.

So: our brains are good at suppressing this “curve in the middle” feature of parallel lines in perspective, at least for simple local examples like railway lines and walls.

Now let’s go back to those parallel light rays coming from the sun and illuminating the moon. Like railway tracks, they’re affected by perspective. In the photograph below, the setting sun is projecting rays from behind a low cloud:

Although the rays are in fact parallel, perspective makes them seem to radiate outwards in a fan centred on the sun. I’ve written about these crepuscular rays in a previous post, and at that time suggested that whenever you see them you should turn around and look for anticrepuscular rays, too:

These converge towards the antisolar point—the point in the sky directly opposite the sun—and they’re produced by exactly the same perspective effect. Which means solar rays have to do the same “diverge, curve, converge” trick as the sides of my park wall. Unfortunately, crepuscular rays tend to fade into invisibility a relatively short distance from the sun, and to reappear as anticrepuscular rays only a relatively short distance from the antisolar point. So we can’t visually track their grand curves across the sky.

But we can see the effect of that perspective curvature when the low sun illuminates a gibbous moon. Here’s a diagram of a sheaf of parallel solar rays, as they would appear when projected on to the dome of the sky:

Perspective makes the sun’s rays diverge when the observer looks towards the sun, but converge when the observer turns and looks at the antisolar point. Because the sun is sitting on the horizon, all the rays in my diagram above are not only parallel to each other, but also to the horizon. And because the gibbous moon is more than ninety degrees away from the sun, it’s illuminated by rays that are apparently converging towards the antisolar point on the horizon, rather than spreading outwards from the sun.

So the impression that that the moon’s illuminated portion doesn’t point towards the sun is a very strong one. This is because the scale of the moon-sun perspective is very much larger than the examples for which our brains have learned to compensate. The moon is the only illuminated object we see which is further away than a few kilometres, and our brains otherwise never have to deal with grand, horizon-spanning perspectives in illumination. So our intuitions tell us that the light rays illuminating the moon in the diagram above can’t possibly have come from the sun, since they’re apparently descending towards the antisolar point.

Standing in the open, observing the illusion, I find it impossible to mentally sketch the curve from sun to moon and see that it’s a straight line. Nothing that rises from one horizon and descends to the other horizon can possibly be a straight line, my brain insists, despite its cheerful acceptance that the straight, parallel sides of my park wall can appear to diverge and then converge in exactly the same way.

In the old days the approved way of demonstrating that there really was a straight line connecting the sun to the centre of the illuminated portion of the moon was with a long bit of string held taut between two hands at arm’s length. Placing one end of the string over the sun, and then fiddling with the other end until it intersected the moon, one could eventually produce a momentary impression that the straight line of the taut string really did align with the illuminated side of the moon. But it was all a bit unsatisfactory.

But now we have panorama apps on our phones. The one I use stitches together multiple images, and provides an on-screen guide to ensure that each successive image aligns with its vertical edge parallel to the image before—it forces the user to stay aligned in a single plane as they shift the viewing direction between successive frames. Usually, the object of the exercise is to scan along the horizon to obtain a wide-angle view of the scenery. But (as my odd little downward-looking panorama of the park wall demonstrated) it isn’t necessary to start the panorama with a vertically orientated camera aimed at the horizon.

So, back in the park and shortly after I took the image at the head of this post, I aimed my phone camera at the moon, and tilted it sideways so that it aligned with the tilted orientation of the moon’s illuminated portion. Then I triggered my panorama exposures and followed the on-screen guides—which led me across the sky in a rising and falling arc until I arrived at the setting sun!

Here’s the result:

So now perspective makes the horizon appear to curve implausibly, while the illuminated portion of the moon quite obviously faces directly towards the sun.