FGT Sagita Size

Updated on Fri, 2012-03-16 12:24. Originally created by sgliske on 2012-03-14 15:30.

Some notes about the size of the sagita in the FGT for various pT tracks.

Radius of Curvature

Following the change of units, one has the formula $p_T = 0.15 r$ with $p_T$ in GeV/c and $r$ in meters. Thus, one can form the table

p_T [GeV/c]	r [m]
1	3.3
3.75	25
87.5	550

Delta Phi

The z position of the helix parameterization can be written as $z = z_0 \pm (r \sinh\eta ) \phi$ . Thus

$\delta\phi = \frac{\delta z}{ r \sinh\eta }$

The maximum change in z-position is roughly 0.5 m, and thus the change in angle is always quite small, generally below 0.05 rad.

Track Length

One can form the triangle in the xy plane of the first and last track position, and see the relation that the distance $\ell$ between the first and last position is related to the radius and change in angle as

$\frac{\ell}{2} = r\sin\frac{\delta\phi}{2} \approx r\frac{\delta\phi}{2}$

$\ell \approx \frac{\delta z}{\sinh\eta}$

For all revelant angles, the approximation is good to a part in 1000 or better.

The sixth FGT disk is at 1.19 m, and the radius is approximately 0.10 to 0.40 m. This corresponds to an eta range of about 1.8 to 3.2, or $\sinh\eta$ between 3.0 and 11.8. This means that the track length $\ell$ is in the range 0.04 to 0.17 m, independent of the pT of the track.