<Dd> β max = cos − 1 ⁡ (2 − 1 + n 2 3 n) ≈ 40.2 ∘ (\ displaystyle \ beta _ (\ text (max)) = \ cos ^ (- 1) \ left ((\ frac (2 (\ sqrt (- 1 + n ^ (2)))) ((\ sqrt (3)) n)) \ right) \ approx 40.2 ^ (\ circ)). </Dd> <P> Substituting back into the earlier equation for φ yields 2φ ≈ 42 ° as the radius angle of the rainbow . </P> <P> Secondary rainbows are caused by a double reflection of sunlight inside the raindrops, and are centred on the sun itself . They are about 127 ° (violet) to 130 ° (red) wide . Since this is more than 90 °, they are seen on the same side of the sky as the primary rainbow, about 10 ° above it at apparent angles of 50--53 ° . As a result of the "inside" of the secondary bow being "up" to the observer, the colours appear reversed compared to the primary bow . The secondary rainbow is fainter than the primary because more light escapes from two reflections compared to one and because the rainbow itself is spread over a greater area of the sky . Each rainbow reflects white light inside its coloured bands, but that is "down" for the primary and "up" for the secondary . The dark area of unlit sky lying between the primary and secondary bows is called Alexander's band, after Alexander of Aphrodisias who first described it . </P> <P> Unlike a double rainbow that consists of two separate and concentric rainbow arcs, the very rare twinned rainbow appears as two rainbow arcs that split from a single base . The colours in the second bow, rather than reversing as in a secondary rainbow, appear in the same order as the primary rainbow . A "normal" secondary rainbow may be present as well . Twinned rainbows can look similar to, but should not be confused with supernumerary bands . The two phenomena may be told apart by their difference in colour profile: supernumerary bands consist of subdued pastel hues (mainly pink, purple and green), while the twinned rainbow shows the same spectrum as a regular rainbow . The cause of a twinned rainbow is the combination of different sizes of water drops falling from the sky . Due to air resistance, raindrops flatten as they fall, and flattening is more prominent in larger water drops . When two rain showers with different - sized raindrops combine, they each produce slightly different rainbows which may combine and form a twinned rainbow . A numerical ray tracing study showed that a twinned rainbow on a photo could be explained by a mixture of 0.40 and 0.45 mm droplets . That small difference in droplet size resulted in a small difference in flattening of the droplet shape, and a large difference in flattening of the rainbow top . </P>

When do the colors of the rainbow appear in reverse order
find me the text answering this question