# Connection length

The connection length is understood to be the distance between two points, measured along a magnetic field line passing through these points.

In the fusion context, a distinction is made between closed field lines (inside the Last Closed Flux Surface) and open field lines (outside).

## Closed field lines

In a toroidally confined plasma, inside the Last Closed Flux Surface, the connection length is commonly defined as the length, measured along the magnetic field, to complete a poloidal turn.

In a circular tokamak, the poloidal circumference is 2πr. The connection length is ${\displaystyle L=2\pi r/\sin(\alpha )}$, where ${\displaystyle \alpha }$ is the pitch angle of the field line, namely ${\displaystyle \tan(\alpha )=B_{\theta }/B_{\phi }}$. Assuming ${\displaystyle B_{\phi }\gg B_{\theta }}$, one has ${\displaystyle \sin(\alpha )\simeq \tan(\alpha )}$, so that [1]

${\displaystyle L=2\pi r{\frac {B_{\phi }}{B_{\theta }}}=2\pi Rq}$

where q is the safety factor, approximated by ${\displaystyle q=rB_{\phi }/RB_{\theta }}$.

## Open field lines

Outside Last Closed Flux Surface, the connection length associated with a given point is defined as the shortest distance from that point to any material surface measured along the field line through that point.

## References

1. K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, Springer-Verlag (2005) ISBN 3540242171