# 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 $L = 2\pi r/\sin(\alpha)$, where $\alpha$ is the pitch angle of the field line, namely $\tan(\alpha) = B_\theta/B_\phi$. Assuming $B_\phi \gg B_\theta$, one has $\sin(\alpha) \simeq \tan(\alpha)$, so that [1]

$L = 2 \pi r \frac{B_\phi}{B_\theta} = 2 \pi R q$

where q is the safety factor, approximated by $q = r B_\phi / R B_\theta$.

## 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