In the case of the Sagnac variables , however, this is not the case as it is easy
to understand on simple physical grounds. In the case of
for instance, light originating
from spacecraft 1 is simultaneously sent around the array on clockwise and counter-clockwise
loops, and the two returning beams are then recombined. If the array is rotating, the two beams
experience a different delay (the Sagnac effect), preventing the noise
from cancelling in the
combination.
In order to find the solution to this problem let us first rewrite in such a way to explicitly emphasize
what it does: attempts to remove the same fluctuations affecting two beams that have been made to
propagated clockwise and counter-clockwise around the array,
In the case of , however, the rotation of the array breaks the symmetry and therefore its uniqueness.
However, there still exist three generalized TDI laser-noise-free data combinations that have
properties very similar to
, and which can be used for the same scientific purposes [30]. These
combinations, which we call
, can be derived by applying again our time-delay operator
approach.
Let us consider the following combination of the ,
measurements, each being delayed only once [1
]:
If the delay-times also change with time, the perfect cancellation of the laser noises is no longer achieved
in the combinations. However, it has been shown in [34
] that the magnitude of the residual
laser noises in these combinations are significantly smaller than the LISA secondary system noises, making
their effects entirely negligible.
The expressions for the Monitor, Beacon, and Relay combinations, accounting for the rotation and
flexing of the LISA array, have been derived in the literature [34] by applying the time-delay
iterative procedure highlighted in this section. The interested reader is referred to that paper for
details.
A mathematical formulation of the “second generation” TDI, which generalizes the one presented in
Section 4 for the stationary LISA, still needs to be derived. In the case when only the Sagnac effect is
considered (and the delay-times remain constant in time) the mathematical formulation of Section 4 can be
extended in a straight-forward way where now the six time-delays and
must be taken into
account. The polynomial ring is now in these six variables and the corresponding module of syzygies can be
constructed over this enlarged polynomial ring [22]. However, when the arms are allowed to flex, that is, the
operators themselves are functions of time, the operators no longer commute. One must then resort to
non-commutative Gröbner basis methods. We will investigate this mathematical problem in the near
future.
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