### 5.2 The Sagnac combinations

In the above Section 5.1 we have used the same symbol for the unequal-arm Michelson combination for both the rotating (i.e. constant delay times) and stationary cases. This emphasizes that, for this TDI combination (and, as we will see below, also for all the combinations including only four links) the forms of the equations do not change going from systems at rest to the rotating case. One needs only distinguish between the time-of-flight variations in the clockwise and counter-clockwise senses (primed and unprimed delays).

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,

where we have accounted for clockwise and counter-clockwise light delays. It is straight-forward to verify that this combination no longer cancels the laser and optical bench noises. If, however, we expand the two terms inside the square-brackets on the right-hand side of Equation (56) we find that they are equal to
If we now apply our iterative scheme to the combinations given in Equation (58) we finally get the expression for the Sagnac combination that is unaffected by laser noise in presence of rotation,
If the delay-times are also time-dependent, we find that the residual laser noise remaining into the combination is actually equal to
Fortunately, although first order in the relative velocities, the residual is small, as it involves the difference of the clockwise and counter-clockwise rates of change of the propagation delays on the same circuit. For LISA, the remaining laser phase noises in , , are several orders of magnitude below the secondary noises.

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]:

where we have used the commutativity property of the delay operators in order to cancel the and terms. Since both sides of the two equations above contain only the noise, is found by the following expression:
If the light-times in the arms are equal in the clockwise and counter-clockwise senses (e.g. no rotation) there is no distinction between primed and unprimed delay times. In this case, is related to our original symmetric Sagnac by . Thus for the practical LISA case (arm length difference ), the SNR of will be the same as the SNR of .

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.