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 with in GeV/c and in meters.  Thus, one can form the table

Delta Phi

The z position of the helix parameterization can be written as .  Thus

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 between the first and last position is related to the radius and change in angle as

or

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 between 3.0 and 11.8.  This means that the track length is in the range 0.04 to 0.17 m, independent of the pT of the track.

 Sagita Size

The formula relating the sagita with the radius and track length is

.

Some sagita values at various eta, pT values are given in the following table.  Note, the sagita is in mm, while the other distances are in m.

Conclusions

Even for low pT (0.5 GeV) tracks, the sagita size is quite small.