A secondary bow is dimmer than a primary rainbow due to additional reflective losses and greater absorption of light within the raindrops.
Work Step by Step
A secondary rainbow, with the colors reversed, is created from the double reflection of light within each raindrop. This additional reflection means that the light spends more time in the droplet, so the water absorbs (and scatters) more light. Also, some of the additional reflections result in losses as light exits the droplet. This is discussed on page 530.