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

momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

dependence of the recirculation path length for an object that is bound in cyclic motion (closed orbit). It is used in the calculation of particle paths in circular

particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental ...

s (like synchrotrons), and for astronomical objects that are bound by

gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

. For a perturbed

orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...

, the momentum compaction factor is defined as the derivative of normalized path length difference to normalized momentum

\alpha_p = \frac  = \frac \frac =\frac \oint \frac\mathrms

. Furthermore, the momentum compaction is closely connected to the so-called slip-factor

\eta

with the horizontal dispersion

D_x

and the gyroradius

\rho

\alpha_p = \frac+\eta

wherein

\gamma

is the

Lorentz factor The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...

References

{{Accelerator-stub Accelerator physics Equations of astronomy Momentum Orbits