Dynamical simulation

In this chapter, we derive the models used in the simulation to compute the position and attitude of the mechanical elements of the constellation; namely the spacecraft and the MOSAs. The nominal positions of the spacecraft are given by the orbital motion of the constellation, described in Spacecraft positions. We currently assume that there is no deviation from these nominal spacecraft positions. The nominal attitude of the spacecraft and the MOSAs are deduced from the position of the spacecraft, as described in Spacecraft and MOSA attitude. In addition, we model the fluctuations of the spacecraft and MOSA attitudes around their nominal values (for example due to thruster noise); we call this motion the spacecraft and MOSA angular jitter.

All physics modeled in this chapter is described and simulated in the TCB at the physics sampling rate \(f_s^\text{phy}\).

Warning

The introduction of a more refined dynamical model will require a precise definition of the reference frames used implicitly here. In addition, the position of the spacecraft will fluctuate around the nominal positions, impacting the PPRs.

Spacecraft positions

The positions (and associated quantities, such as velocities and accelerations) of the three LISA spacecraft are solely used in the instrumental simulation for the inter-spacecraft propagation of laser beams as described in Interspacecraft propagation.

If an external orbit file is used as input, then the PPRs are read either from:

• The column ppr, which includes a relativistic treatment of the TPSs; in which case they also contain the time frame transformation between the emitter and receiver TPSs.

• The column tt (standing for light travel time), such that all TPSs are assumed to be identical to the TCB.

Spacecraft and MOSA attitude

Nominal attitude and opening angles

The spacecraft and MOSA nominal attitudes are computed from the nominal spacecraft positions, obtained from the orbital motion of the constellation (see Spacecraft positions). They are given as follows: the spacecraft \(z\)-axes all point towards the Sun, while the spacecraft \(x\)-axes are the bisectors of the constellation triangle. Theoretically, the opening angle \(\omega_{ij}\) of MOSA \(ij\) is adjusted at all times so that the MOSA \(x\)-axis points towards the distant spacecraft \(j\). In the simulation model, we assume that opening angles remain constant during the entire duration of the simulation and are set as parameters of the simulation. This is depicted in Fig. 2.

Warning

Apparent discrepancy to ESA definition of z-axis direction (see [6]) will be updated in the future.

Note that the opening angles are defined as oriented angles, i.e., we use a positive number when the MOSA is rotated counterclockwise from the spacecraft \(x\)-axis, and a negative number for clockwise rotations. This means that left-sided MOSAs possess positive opening angles, while right-sided MOSAs possess negative opening angles.

Nominal attitude

Fig. 2 Schematics of the nominal attitude of a spacecraft and the two associated MOSAs, their opening angles, and the angular jitter in the yaw rotational degree of freedom.

Spacecraft co-moving reference frame

We define the spacecraft co-moving frame as a locally inertial reference frame based on the Spacecraft Mechanical Reference Frame (\(R_{\mathrm{SC}}\)). The \(R_{\mathrm{SC}}\) was defined in [6]. That means the origin and orientation roughly match the definitions of the \(R_{\mathrm{SC}}\). However, the difference is that in the spacecraft co-moving frame, we can measure MOSA and SC jitters.

We work in the the spacecraft co-moving frame that is co-moving with the satellite along a straight line with constant velocity, tangential to the orbit and we ignore the difference to the free falling frame of the test mass on the timescale of 30 minutes. In other words, we call the inertial system on a local geodesic attached to the accelerated orbit the co-moving spacecraft reference frame. Approximately 30 min is a conservative estimate and further refinement is work in progress.

The axes are defined such that its \(X\) axis is the bisector between the local MOSA \(X\) axes in their nominal state, where nominal is considered to be a full opening angle of 30 degrees. The \(XY\) plane is the plane in which both local MOSA are located, the \(Z\) axis is pointing from the center of mass through the solar panel. The origin of this reference frame is the SC barycenter.

The idealized orbit defined above includes all planetary effects but does not include the effect of commanded thrusts and thruster noise. We use this reference frame to describe the MOSA and SC motion for TTL. It can also be used for the line-of-sight alignment noise and common-mode rejection noise, see the internal documentation [7].