A primary rainbow occurs when sunlight enters raindrops, refracts, reflects once off the back of the droplet, and then refracts again as it exits toward the observer. The common perception is that seeing a second, outer arc—a double rainbow—is a rare event. This secondary bow is much harder to observe than its brighter inner counterpart. Understanding the physics explains why this phenomenon is not necessarily rare in occurrence, but is certainly infrequent in visibility.
The Science Behind Double Rainbow Formation
The mechanism that creates a primary rainbow is a single internal reflection of sunlight within a water droplet. When light enters the droplet, it splits into its spectral colors, reflects off the interior back surface, and then exits, forming the familiar arc. The formation of the secondary bow requires an additional step inside the raindrop. This outer arc is generated by light that undergoes two internal reflections before exiting the droplet toward the observer’s eye.
The extra reflection causes two major differences: the reversal of color and the reduction in brightness. Each time light reflects off the inner surface of the water droplet, some energy is lost, making the secondary bow significantly dimmer. The light travels a longer path inside the droplet, and more light escapes at the water-air boundary. This double reflection process drastically reduces the light intensity, causing the outer arc to be fainter than the inner one.
The secondary rainbow forms at a larger angle relative to the anti-solar point—the spot directly opposite the sun—than the primary bow. The primary bow forms at roughly 42 degrees, while the secondary bow forms at about 51 degrees. This difference means the secondary arc is always positioned outside and above the primary arc. The requirement for two reflections means that every primary rainbow is accompanied by a secondary one, though it is often too faint to be seen.
Visual Characteristics of the Secondary Bow
The most immediate visual difference is the reversal of the secondary arc’s color order compared to the primary one. In the primary rainbow, colors range from red on the outside of the arc to violet on the inside. Because the light has undergone a second internal reflection, the secondary bow’s color order is flipped. Violet appears on the outer edge and red on the inner edge of the bow.
The secondary bow also appears noticeably wider and more diffuse than the primary bow. This results from the light being spread out over a greater angle due to the second reflection. The light intensity is estimated to be about 43 percent less than the primary bow, contributing to its washed-out, pastel appearance. The combination of reduced intensity and increased width makes the secondary arc a subtle feature of the sky.
A distinct area of dark sky, known as Alexander’s Dark Band, separates the two rainbows. This occurs because the physical angles of light deviation prevent sunlight from being reflected toward the observer in the region between the two arcs. The primary rainbow focuses light inside its arc, and the secondary rainbow spreads light outside its arc. This leaves the space between the 42-degree and 51-degree angles relatively unlit. This dark band enhances contrast, making the secondary rainbow easier to perceive when conditions are favorable.
Frequency Versus Visibility
The core question of how rare a double rainbow is centers on the distinction between its occurrence and its visibility. The conditions that produce a primary rainbow—sunlight hitting raindrops at the correct angle—simultaneously create the opportunity for a secondary rainbow. Therefore, a secondary bow is technically always present whenever a primary bow is visible. The phenomenon is not intrinsically rare, but is an almost constant companion to the primary arc.
The rarity lies solely in the ability of an observer to detect the secondary bow. Because the outer arc is fainter, it requires specific atmospheric conditions to be seen. Favorable conditions include a low sun angle, which maximizes the arc’s height, and a dark background sky, such as a receding storm cloud. This dark background provides the necessary contrast to make the secondary bow’s dim, pastel colors distinguishable.
If the sky behind the primary rainbow is bright, the faint light of the secondary rainbow is overwhelmed and washed out. Thus, the double rainbow is an infrequent sighting, not an infrequent occurrence. Witnessing a double rainbow is a sign that the atmospheric conditions are near-perfect for maximum light intensity and contrast.