The Big Dipper is an asterism, not a constellation, and its stars are not all part of the same star system. So there isn't a single, definitive brightness ranking for all the stars. However, we can look at the brightest stars within the asterism itself.

Here are the stars of the Big Dipper ordered from brightest to dimmest, along with their Bayer designations:

1. Alioth (ε Ursae Majoris): The brightest star in the Big Dipper.

2. Dubhe (α Ursae Majoris): Second brightest.

3. Merak (β Ursae Majoris):

4. Phecda (γ Ursae Majoris):

5. Megrez (δ Ursae Majoris):

6. Mizar (ζ Ursae Majoris): Although visually appearing as a single star, Mizar is a binary star system with a faint companion known as Alcor.

7. Alkaid (η Ursae Majoris): The star at the end of the Big Dipper's handle.

This list is based on the apparent magnitude of the stars, which is how bright they appear from Earth. The actual luminosity of the stars can vary greatly, but this is the order we see them from our perspective.