# Momentum compaction

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The '''momentum compaction''' or '''momentum compaction factor''' is a measure for the [momentum](/source/momentum) 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](/source/particle_accelerator)s (like [synchrotron](/source/synchrotron)s), and for astronomical objects that are bound by [gravitation](/source/gravitation).

For a [perturbed](/source/perturbation_theory) [orbit](/source/orbit), the momentum compaction factor is defined as the derivative of normalized path length difference to normalized momentum<ref>{{cite book
 | last1 = Conte | first1 = Mario
 | last2 = McKay | first2 = William W.
 | title = An Introduction to the Physics of Particle Accelerators
 | publisher = [World Scientific](/source/World_Scientific)
 | edition = 2nd
 | date=Apr 2008
 | isbn= 978-981-277-961-8
 | url= http://www.worldscibooks.com/physics/6683.html
}}</ref><ref>{{cite book|last1=Minty|first1=Michiko G.
|last2=Zimmermann|first2=Frank
|title=Measurement and Control of Charged Particle Beams|year=2003|publisher=[Springer-Verlag](/source/Springer-Verlag)|location=Berlin, Heidelberg, New York|isbn=978-3-540-44187-8|page=159}}</ref>

<math>\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</math>.

Furthermore, the momentum compaction is closely connected to the so-called slip-factor<ref>{{cite book
 | last1 = Steinhagen | first1 = R. J.
 | title = CERN Accelerator School Beam Diagnostics / Tune and chromaticity diagnostics
 | publisher = [CERN](/source/CERN)
 | date=August 2009
 | page=343
 | editor = Daniel Brandt
}}</ref> <math>\eta</math> with the horizontal dispersion <math>D_x</math> and the [gyroradius](/source/gyroradius) <math>\rho</math>

<math>\alpha_p = \frac{1}{\gamma^{2}}+\eta</math>

wherein <math>\gamma</math> is the [Lorentz factor](/source/Lorentz_factor).

==References==
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Category:Accelerator physics
Category:Equations of astronomy
Category:Momentum
Category:Orbits

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