Framework and conventions

Note

Most of the model description sections are based on the publication [1].

You can find a list of acronyms in the LISA Acronym Dictionary.

Constellation overview

LISA is an almost equilateral triangle, composed of 3 identical spacecraft, which we label 1, 2, 3 clockwise when looking down at their solar panels. These spacecraft exchange laser beams, which are combined on optical benches inside MOSAs.

To uniquely label these MOSAs, we use two indices. The first one is that of the spacecraft the MOSA is mounted on, while the second index is that of the spacecraft the MOSA is pointing to. Most components of interest (such as the optical benches, test masses, etc.) can be uniquely associated with one of the MOSAs, in which case we use the same two indices. Elements that exist only once onboard a spacecraft, such as the USOs, are indexed by that spacecraft index. These labeling conventions, which are largely based on the proposed unified conventions of the LISA consortium [2], are summarized in Fig. 1.

Labeling conventions

Fig. 1 Labeling conventions used for spacecraft, light travel times, lasers, MOSAs, and interferometric measurements (from [3]).

Quantities that describe a process that involves the propagation between two spacecraft will be interpreted as being associated with the spacecraft in which the quantity is measured. For example, the gravitational-wave signal observed in the interferometer on MOSA 12 will be indexed with the same indices 12. The same convention applies to the propagation delay of a beam arriving on spacecraft 1 from spacecraft 2, which will be labeled by the indices 12.

In this paper, we often derive equations for a specific spacecraft or MOSA. The expressions for the other 2 spacecraft or the other 5 MOSAs can then be deduced by combining cyclic permutations \(\{ 1 \rightarrow 2 \rightarrow 3 \rightarrow 1 \}\), and reflections \(\{1 \leftrightarrow 1, 2 \leftrightarrow 3 \}\). See [4], Appendix A, for more details.

Time coordinate frames

The instrumental simulation mostly concerns itself with the physics inside a spacecraft (e.g., the evolution of laser beam phases and their interferometric beatnotes), which is best modeled in the three TPSs (spacecraft proper times). These time frames are defined as the times shown by perfect clocks comoving with the spacecraft centers of mass. We denote them with \(\tau_1\), \(\tau_2\), and \(\tau_3\).

The TPSs are idealized timescales, which cannot be realized in practice. All measurements instead refer to an imperfect on-board timer, which represents an approximation of the associated TPS. We denote these three onboard clock time frames as \(\hat{\tau}_1\), \(\hat{\tau}_2\), and \(\hat{\tau}_3\).

Finally, processes on the Solar-system scale are modeled according to a global time frame, such as the TCB, denoted \(t\). This is the case for the spacecraft orbits or the gravitational waveforms. Our instrumental simulation does not make a direct use of the TCB. Instead, we rely on external tools, such as LISA Orbits [5], to directly compute quantities expressed in the TPSs.

In general, signals are expressed in their natural time coordinate. E.g., laser beam phases and beatnotes are expressed in the TPS of the spacecraft housing the laser. It is sometimes useful to express a signal in a different time coordinate. To prevent confusion, we will use the same symbol but add a superscript denoting the time coordinate. For example, a phase \(\phi\) could be expressed as a function of the TPS 1, writing \(\phi^{\tau_1}(x)\), or as a function of the clock time of that spacecraft, writing \(\phi^{\hat\tau_1}(x)\). Note that the symbol used for the function argument is arbitrary, and does not specify the reference frame. We will often use \(\tau\) without subscripts as a generic function argument.

Conversions between time coordinates can easily be expressed with these conventions. For example, \(\tau_1^{\hat\tau_1}(\tau)\) is the TPS as a function of the clock time onboard spacecraft 1. Trivially,

(1)\[t^t(\tau) = \tau_1^{\tau_1}(\tau) = \hat\tau_1^{\hat\tau_1}(\tau) = \tau.\]

It is often useful to model the deviation of the onboard clock time with respect to the associated TPS. We adopt the notation

(2)\[ \begin{align}\begin{aligned}\tau_1^{\hat\tau_1}(\tau) &= \tau + \delta \tau_1^{\hat\tau_1}(\tau),\\\hat\tau_1^{\tau_1}(\tau) &= \tau + \delta \hat\tau_1^{\tau_1}(\tau).\end{aligned}\end{align} \]

One important class of signals we study are phases \(\phi\) of electromagnetic waves. As scalar quantities, these are invariant under coordinate transformations, such that they transform from one time frame to another using a simple time shift,

(3)\[ \begin{align}\begin{aligned}\phi^{\tau_1}(\tau) &= \phi^{\hat\tau_1}(\hat\tau_1^{\tau_1}(\tau)),\\\phi^{\hat\tau_1}(\tau) &= \phi^{\tau_1}(\tau_1^{\hat\tau_1}(\tau)).\end{aligned}\end{align} \]

This is used in section Signal sampling.