Multi-layer core golf ball

1. A golf ball comprising a core and a cover, wherein the core comprises:

a solid inner core layer formed from an unfoamed first polybutadiene composition and having a diameter of 1.10 inch or less and a center Shore C hardness (H center ) of 40 or less,

one or more optional intermediate core layers, and

an outer core layer formed from a second polybutadiene composition and having a thickness of 0.150 inches or greater and an outer surface Shore C hardness (H outer surface ) of 80 or greater,

wherein H outer surface >H center , and H outer surface −H center ≧40, and

wherein the core has a gradient quotient, Q, of from 7 to 12, as defined by the equation:

G

+

T

10

×

COR

=

Q

where G is the hardness gradient of the core in Shore C as defined by H outer surface minus H center , T is the percent of trans-polybutadiene isomer at the surface of the outer core layer, and COR is the coefficient of restitution of the core measured at an incoming velocity of 125 ft/s; and

wherein the hardness gradient, G, is from 60 to 90; the percent of trans-polybutadiene isomer at the surface of the outer core layer, T, is from 1 to 5; and the COR of the core is from 0.790 to 0.840.

2. The golf ball of claim 1 , wherein H outer surface −H center ≧45.

3. The golf ball of claim 1 , wherein H outer surface −H center ≧55.

4. The golf ball of claim 1 , wherein H outer surface −H center ≧60.

5. The golf ball of claim 1 , wherein the core has an overall compression of ≧70.

6. The golf ball of claim 1 , wherein the inner core layer composition is a thermoset composition consisting essentially of:

polybutadiene;

from 0.1 phr to 2.0 phr of a peroxide;

optionally 5 phr or less of a metal oxide;

optionally 5 phr or less of a metal coagent;

optionally 5 phr or less of a metal carbonate; and

optionally one or more additional components selected from the group consisting of fillers, colorants, and antioxidants.