TPC and Magnet survey - 09/15-16/03

1. Survey frame definition

The following is the header of the survey file (.xls) defining the reference frame for the survey file:

"This is the survey data taken September 15 & 16, 2003. The Magnet is in the assembly building.
The Star Magnet is held in its design location, based on the average least squares value of targets located on the face of the magnet. The TPC and SVT-SSD are located and related to the magnet.
0,0,0 is the design center of the magnet, positive "X" is away from the center of RHIC, positive "Y" is west, and positive "Z" is above the beam height, units are meters. "

Translated into STAR's standard refence frame we have the following relationship:
Xsurvey = Xstar
Ysurvey = Zstar
Zsurvey = Ystar

Note: In the following plots, the titles of the histograms refer to the survey coordinate system.

2. TPC in survey coordinates

This analysis was done under the following assumptions:

The TPC is a perfect cilynder
Each end of the TPC is formed by 12 identical sectors
The survey targets are embedded in each sector at the same position

The absolute position of the targets in TPC coordinates is unknown (although it cool be know by accesing TPC darwings).

2.1 TPC center

To determine the TPC center in survey coordinates we use the following strategy:

Find two symmetric targets with respect to the center of the TPC (i.e. target "a" in sector W12 and target "a" in sector E18)
Calulate the middle of the line joining the two targets
Histogram the x, y, and z coordinates of that point for all the available set of symmetric targets

The following three histograms show the xsurvey, ysurvey and zsurvey of the center of the TPC:

2.1 TPC rotations

To determine how TPC is rotated with respect to the survey reference frame center we use the following strategy:

Calculate two vectors joining two pairs of targets in one end of the TPC
Calculate the cross product of those two vectors, this vector is then normal to that end of the TPC
Calculate the angle of that vector to the Ysurvey axis (beam axis) in the Xsurvey,Ysurvey and Ysurvey,Zsurvey planes
Histogram those angles for all the pairs of pairs of targets
Do everything independently for the East and West end of the TPC

The next 2 histograms show the angle of the TPC West Zaxis with respect to the Ysurvey axis in the Xsurvey,Ysurvey and Ysurvey,Zsurvey planes

The next 2 histograms show the angle of the TPC East Zaxis with respect to the Ysurvey axis in the Xsurvey,Ysurvey and Ysurvey,Zsurvey planes

3 Magnet in survey coordinates

From the header of the survey file it appears the center of the magnet defines the 0,0,0 of the survey coordinate system. However it was somehow obscure if the survey axis where also defined by the magnet axis. The available survey data allow to test it.

3.1 Magnet center

The magnet center defines the actual survey origin point.

3.2 Magnet rotations

To determine how Magnet is rotated (if rotated at all) with respect to the survey reference frame we can use a similar approach as for the TPC, as some tragets where identified to be on the front ends of the STAR magnet (in each end West and East):

Calculate two vectors joining two pairs of targets in one end of the Magnet
Calculate the cross product of those two vectors, this vector is then normal to that end of the Magnet
Calculate the angle of that vector to the Ysurvey axis (beam axis) in the Xsurvey,Ysurvey and Ysurvey,Zsurvey planes
Histogram those angles for all the pairs of pairs of targets
Do everything independently for the East and West end of the Magnet

The next 2 histograms show the angle of the Magnet West Zaxis with respect to the Ysurvey axis in the Xsurvey,Ysurvey and Ysurvey,Zsurvey planes:

The next 2 histograms show the angle of the Magnet East Zaxis with respect to the Ysurvey axis in the Xsurvey,Ysurvey and Ysurvey,Zsurvey planes:

So the Magnet appear to be tilted with respect to the survey frame, although the observed tilts seem to be too small! Probably the observed tilts just show the limitation of our method. We then will give the TPC tilts unde the two possible assumptions: the Magnet and survey frame are identical or the Magnet and Survey frame are independent.

4 TPC in Magnet coordinates

4.1 Assuming the Magnet and survey frames are identical

The center of the TPC in Magnet coordinates is: (-2.52 mm, -1.75 mm, -2.05 mm)

The Z axis of the West TPC has an angle of -0.321 mrad with respect to the Z axis of the Magnet in the Xmagnet,Zmagnet plane, i.e. a rotation of -0.321 mrad around the Y axis of the Magnet.

The Z axis of the West TPC has an angle of -0.127 mrad with respect to the Z axis of the Magnet in the Ymagnet,Zmagnet plane, i.e. a rotation of -0.127 mrad around the X axis of the Magnet.

The Z axis of the East TPC has an angle of -0.376 mrad with respect to the Z axis of the Magnet in the Xmagnet,Zmagnet plane, i.e. a rotation of -0.376 mrad around the Y axis of the Magnet.

The Z axis of the East TPC has an angle of +0.039 mrad with respect to the Z axis of the Magnet in the Ymagnet,Zmagnet plane, i.e. a rotation of +0.039 mrad around the X axis of the Magnet.

Taking the average of the East and West side:

The Z axis of the TPC has an angle of -0.345 mrad with respect to the Z axis of the Magnet in the Xmagnet,Zmagnet plane, i.e. a rotation of -0.345 mrad around the Y axis of the Magnet.

The Z axis of the TPC has an angle of -0.044 mrad with respect to the Z axis of the Magnet in the Ymagnet,Zmagnet plane, i.e. a rotation of -0.044 mrad around the X axis of the Magnet.

4.2 The Magnet reference frame is tilted with respect to the survey refernce frame although the origins concide

For the Magnet Z axis we will just take the average of that found from the West and East side targets.

The Z axis of the Magnet has an angle of -0.059 mrad with respect to the Y axis of the survey (beam axis) in the Xsurvey,Ysurvey plane, i.e. a rotation of -0.059 mrad around the Z axis of the survey.

The Z axis of the Magnet has an angle of +0.041 mrad with respect to the Y axis of the survey (beam axis) in the Ysurvey,Zsurvey plane, i.e. a rotation of +0.041 mrad around the X axis of the survey.

The center of the TPC in Magnet coordinates is: (-2.52 mm, -1.75 mm, -2.05 mm)

The Z axis of the West TPC has an angle of -0.262 mrad with respect to the Z axis of the Magnet in the Xmagnet,Zmagnet plane, i.e. a rotation of -0.262 mrad around the Y axis of the Magnet.

The Z axis of the West TPC has an angle of -0.168 mrad with respect to the Z axis of the Magnet in the Ymagnet,Zmagnet plane, i.e. a rotation of -0.168 mrad around the X axis of the Magnet.

The Z axis of the East TPC has an angle of -0.317 mrad with respect to the Z axis of the Magnet in the Xmagnet,Zmagnet plane, i.e. a rotation of -0.317 mrad around the Y axis of the Magnet.

The Z axis of the East TPC has an angle of -0.002 mrad with respect to the Z axis of the Magnet in the Ymagnet,Zmagnet plane, i.e. a rotation of -0.002 mrad around the X axis of the Magnet.

Taking the average of the East and West side:

The Z axis of the TPC has an angle of -0.290 mrad with respect to the Z axis of the Magnet in the Xmagnet,Zmagnet plane, i.e. a rotation of -0.290 mrad around the Y axis of the Magnet.

The Z axis of the TPC has an angle of -0.085 mrad with respect to the Z axis of the Magnet in the Ymagnet,Zmagnet plane, i.e. a rotation of -0.085 mrad around the X axis of the Magnet.

By Javier Castillo