The red arc of a primary rainbow appears at an angular radius of 42° centred on the anti-solar point (which corresponds to the shadow of your head or the shadow of your camera). The appearance of a rainbow depends on the altitude α of the sun. For example, when the sun is setting (α = 0°), the arc of the rainbow is almost vertical where it meets the horizon – as shown in Fig. 1 above. In such cases, the rainbow can appear as a gigantic semi-circular arch apparently stretching from one side of the sky to the other. Note that rainbows are so large that it is impossible to take a picture of the entire rainbow – unless your camera has an ultra-wide angle lens (e.g. a fish-eye lens). If you have taken a picture of a small circular rainbow, it is more likely to be a glory – as described on the next page.
The top of the rainbow's arc appears at an elevation of (42° - α) where α is the altitude of the sun. For example, when α = 0°, the top of the red arc of the rainbow is at an elevation of 42° above the horizontal. On the other hand, when the sun's altitude α = 39°(as in Fig. 2 below), the top of the rainbow appears just 3° above the horizontal.
Note that when α > 42°, the top of the rainbow will appear below the horizontal. This explains why rainbows are not usually seen in the middle of the day in midsummer. Rainbows are seen much more frequently later in the day because of this geometrical constraint and, also, because rain is more frequent in the late afternoon/evening!
How can we prove that rainbows are circular? We rarely see the full circle of the rainbow because the raindrops causing the rainbow tend to be up in the sky. However, if you use a garden hose, you can illuminate the raindrops below you (near your feet) – and, if you are lucky, you will see the full circle of your own personal rainbow. Note that the bottom of the rainbow occurs at an elevation of -(42° + α). For example, at sunset when α = 0°, the bottom of the rainbow is at -42° (i.e. 42° below the horizontal) and the top of the rainbow is at +42° (i.e. 42° above the horizontal). Similarly, when α = 48°, the bottom of the rainbow is at -90° (directly underneath you) and the top of the rainbow is at -6°. To see a rainbow, you normally have to have your back towards the sun, but if α > 48° you may need to turn around to face the sun to see the bottom of the rainbow which will appear in front of your feet.
If the sun is vertically overhead (α = 90°) and your feet are getting wet, you might see a horizontal rainbow surrounding the lower part of your body. Of course, to have the sun vertically overhead, it needs to be midday and you need to be in the tropics – AND it needs to be raining! However, it is possible to achieve this unusual effect by simulating the overhead sun using a bright light pointing downwards and using garden sprinklers to spray water around your knees – as shown in the diagram in Fig. 3.
Simple trigonometry based on Fig. 3 shows that, when H = 5 metres and W = 2 metres, the camera needs to be about 0.75 metres above the water drops and tilted at β = 37° below the horizontal to record an apparently horizontal circular rainbow with an angular radius δ = 53° rather than the "normal" radius of 42°. This disparity in the size of the rainbow is due to the fact that the light source in Fig. 3 is diverging (i.e. the rays are not parallel, unlike the rays from the sun).
Waterfalls are great places to see rainbows because they can generate large amounts of spray above the horizon (as in Fig. 2 above) and below the horizon. If you are really lucky, you might see a large part of the circular arc of a rainbow – as shown in Fig. 4 below.
Fig. 4 Composite image of a rainbow seen at Victoria Falls in Zambia when the sun's altitude α ≈ 5°
By being in exactly the right place at the right time, you might even see a full circle rainbow - as recorded in Fig. 5 below.
Page updated on 4 August 2012
|Previous page: Rainbows: use of Lee diagrams||
Next page: Glories and the Brocken spectre