Bullen parameter

Remarks

By the Adams–Williamson equation, $\eta = 1$ in any shell of uniform composition and adiabatic temperature. With $K = \rho\Phi$, the bulk modulus along the profile, and $dP/dr = -\rho g$, one finds the equivalent form $\eta = dK/dP + g^{-1} d\Phi/dr$. The check: $dK/dP = (\Phi\, d\rho/dr + \rho\, d\Phi/dr)/(-\rho g) = \eta - g^{-1} d\Phi/dr$. Example: if $\rho$ satisfies the Adams–Williamson equation, then $d\rho/dr = -\rho g/\Phi$, and the formula gives $\eta = 1$. Non-example: a shell in which density does not change with depth while $\rho$, $g$, $\Phi$ are positive has $\eta = 0 \ne 1$; it cannot be adiabatic and uniform. A superadiabatic temperature gradient gives $\eta < 1$, and a gradual change of composition or phase toward denser material with depth gives $\eta > 1$.