Purpose of this analysis is to estimate the systematic error on the 1st order calculation of dL, neglecting the higher order terms of the beam displacement. This is done in one particular case: considering the reference orbit as the central orbit. This is the case that one wants to study because, in the general case, one has to consider the variation of the displacement of the reference orbit along s (position around the ring). This quantity is difficult to measure in practice because the BPM system is not "absolutly" calibrated.

Considering the central orbit as reference orbit, it is possible to exactly
calculate the orbit length for a general orbit as

**L = cyclic int (sqrt ( f(s) ) ) ds **, where

**f(s) = (1+X/ro)^2 + (dX/ds)^2 ** (for bent sections) and

**f(s) = 1 + (dX/ds)^2** (for straight sections)

(X: beam displacement at the position s, with respect to the reference orbit;

ro: average bending radius of the bent sections)

Because the reference orbit is the central orbit, it is NOT possible to use this "exact" calculation to get physical results. ALL OF THIS IS A TEST OF THE 1st ORDER ALGORITHM TO CALCULATE dL.

It has been compared the "exact" dL with the 1st ord. dL for the stable run

run stack res 704 3 chi2 706 3 chi2 707 3 chi2 708 3 chi2 709 3 chi2 839 6 psip 1011 8 psip 1313 19 1p1 1314 19 1p1 1315 19 1p1 1424 21 chi1 1425 21 chi1 1431 21 chi1 2015 22 psip 2202 38 chi2 2218 39 psip 2399 50 1p1 2421 52 1p1 2422 52 1p1 2440 54 1p1 2450 55 1p1 2451 55 1p1 2452 55 1p1 3014 56 1p1 3022 57 1p1 3023 57 1p1 3025 57 1p1 3026 57 1p1 3047 58 1p1 3050 58 1p1 3079 60 jpsi 3091 61 1p1 3092 61 1p1 3149 63 1p1 3150 63 1p1 3151 63 1p1 3194 65 chi1 3288 74 chi1in three different configurations of BPM saturation:

The results are summarized in this plot:

In average, the 1st order dL understimate the "real" dL for 80 um
(Sat. OK value). Furthermore, the error slightly depend upon dL: in the case
of Sat. OK (slope = 1.002), this means that the 1st order calculation of
dL = 10 mm is underestimated by 60 um and of dL = -10 mm by 100 um.
This is an error of /e835/people/15 KeV in energy in the worst case.

These are the table of data:

correct saturation pattern for the given run

BPMs considered NOT saturated

BPMs considered saturated like in run 2369

E835 run info table: alphanumerical ;
only numerical (useful for paw)

Offline energy calculation stack per stack

Golden Data Stream and DSTs Page

Luminosity Home Page

ACNET run number stack per stack

BPM saturation history

Single Value Decomposition Threshold Plots

Beam Position and Angle at the target from BPMs

Systematic error in the energy calculation neglecting 2nd order terms in the calculation of dL

BPM position vs time stack per stack

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