Momentum compaction

The momentum compaction or momentum compaction factor is a measure for the 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 accelerators (like synchrotrons), and for astronomical objects that are bound by gravitation.

For a perturbed orbit, the momentum compaction factor is defined as the derivative of normalized path length difference to normalized momentum{{cite book

| last1 = Conte | first1 = Mario

| last2 = McKay | first2 = William W.

| title = An Introduction to the Physics of Particle Accelerators

| publisher = World Scientific

| edition = 2nd

| date=Apr 2008

| isbn= 978-981-277-961-8

| url= http://www.worldscibooks.com/physics/6683.html

}}{{cite book|last1=Minty|first1=Michiko G.

|last2=Zimmermann|first2=Frank

|title=Measurement and Control of Charged Particle Beams|year=2003|publisher=Springer-Verlag|location=Berlin, Heidelberg, New York|isbn=978-3-540-44187-8|page=159}}

$\backslash alpha\_p\; =\; \backslash frac\{\backslash mathrm\{d\}L\; /\; L\}\; \{\backslash mathrm\{d\}p\; /\; p\}\; =\; \backslash frac\{p\}\{L\}\; \backslash frac\{\backslash mathrm\{d\}L\}\{\backslash mathrm\{d\}p\}\; =\backslash frac\{1\}\{L\}\; \backslash oint\; \backslash frac\{D\_\{x\}(s)\}\{\backslash rho(s)\}\backslash mathrm\{d\}s$.

Furthermore, the momentum compaction is closely connected to the so-called slip-factor{{cite book

| last1 = Steinhagen | first1 = R. J.

| title = CERN Accelerator School Beam Diagnostics / Tune and chromaticity diagnostics

| publisher = CERN

| date=August 2009

| page=343

| editor = Daniel Brandt

}} $\backslash eta$ with the horizontal dispersion $D\_x$ and the gyroradius $\backslash rho$

$\backslash alpha\_p\; =\; \backslash frac\{1\}\{\backslash gamma^\{2\}\}+\backslash eta$

wherein $\backslash gamma$ is the Lorentz factor.