Momentum compaction

The momentum compaction or momentum compaction factor is a measure for the

For a perturbed orbit, the momentum compaction factor is defined as the derivative of normalized path length difference to normalized momentum^[1][2]

${\displaystyle \alpha _{p}={\frac {\mathrm {d} L/L}{\mathrm {d} p/p}}={\frac {p}{L}}{\frac {\mathrm {d} L}{\mathrm {d} p}}={\frac {1}{L}}\oint {\frac {D_{x}(s)}{\rho (s)}}\mathrm {d} s}$.

Furthermore, the momentum compaction is closely connected to the so-called slip-factor^[3] ${\displaystyle \eta }$ with the horizontal dispersion ${\displaystyle D_{x}}$ and the gyroradius ${\displaystyle \rho }$

${\displaystyle \alpha _{p}={\frac {1}{\gamma ^{2}}}+\eta }$

wherein ${\displaystyle \gamma }$ is the Lorentz factor.

References

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