Auxiliary measurements
PRN Ranging
In addition to the GHz sideband modulation, each laser beam will also carry an additional modulation with a pre-determined PRN code used for absolute ranging and timing synchronization. The basic measurement principle is to correlate the received PRN code in each SCI with a local copy generated on the receiving spacecraft. PRN ranging is used to measure the pseudorange, which is defined as the difference between the spacecraft elapsed time (SCET) of the receiving SC at the time of reception and the SCET of the emitting spacecraft at the time of emission. To distinguish it from the proper pseudorange (PPR) (the difference in proper time at both the emission and reception events), we here refer to the pseudorange as instrumental pseudorange (IPR). The IPR is not exactly accessible because of ranging noise. We here refer to the PRN ranging measurement as measured pseudorange (MPR) given by the IPR plus ranging noise.
PRN modulation
The PRN modulation is performed at a relatively high frequency of around 2 MHz, far outside our simulation bandwidth. We therefore do not model the actual phase modulation. This modulation also causes a small additional noise in our measurement band, at a level below \(1\,\mathrm{pm}\, \mathrm{Hz}^{-1/2}\) in units of displacement [25], which we do not model. In addition, we only model the PRN measurement in the SCI, and completely ignore the presence of the PRN codes in the other interferometers.
Instead, we model this measurement by directly propagating the time deviations of each spacecraft timer with respect to their TPSs, alongside the laser beams. We set up the IPR as the difference between the received and local timer. Adding ranging noise gives the MPR.
Similarly to the main interferometric measurements and as described in Filtering and downsampling, pseudoranging simulation is performed at \(f_s^\text{phy}\), while the MPRs are ultimately filtered and downsampled to a lower rate \(f_s^\text{meas}\).
Warning
Show that this modulation can be treated independently from other virtual beams. Maybe show a plot for the spectrum, and mention that we do not expect a significant impact on the main science measurement, since the PRN is outside the measurement band.
PRN ranging as clock time difference
We consider in the following paragraphs a beam received by optical bench 12 at the receiver TPS \(\tau\), which was emitted from optical bench 21 at emitter TPS \(\tau - d_{12}(\tau)\). Here, the PPR \(d_{12}(\tau)\) contains the light time of flight, as well as the conversion between the two proper times.
Conceptually, the MPR measures the pseudorange, given as the difference between the time \(\hat\tau_1^{\tau_1}(\tau)\) shown by the local clock of the receiving spacecraft at the event of reception of the beam, and the time \(\hat\tau_2^{\tau_2}(\tau - d_{12}(\tau))\) shown by the local clock of the sending spacecraft at the event of emission of the beam. In reality, the MPR only measures the pseudorange up to the repetition period of the PRN code, which is around 1 ms. The full pseudorange is then recovered by combining the MPR measurements with ground-based observations.
At the moment, we do not simulate this effect and assume that the MPR directly gives the pseudorange without ambiguity. In addition, we assume that the vacuum between the satellites is sufficiently good that we can neglect (or compensate for) any dispersion effects, such that the PRN code suffers exactly the same delay as the carrier and sidebands.
Thus, we can model the MPR as the difference
where \(N^R_{12}(\tau)\) is a ranging noise term modeling imperfections in the overall correlation scheme.
PRN ranging in terms of timer deviations
As explained in Timer model, we do not simulate the total clock time \(\hat\tau_1(\tau)\) for each spacecraft, but only deviations \(\delta\hat\tau_1(\tau)\) from the associated TPS,
Inserting these definitions into (20) yields
Let us define the clock time of the sending spacecraft propagated to the photodiode of the distant interspacecraft interferometer as
We can then express the MPR as the simple difference
In our simulation, we make the additional assumption that \(d_{12} \approx d_{12}^o\) for this measurement. This is valid since the terms contained in \(d_{12}^\epsilon\) (in our simulation model, only \(H_{12}\)) create timing jitters much less than a nanosecond.
Notice that in (22), we compute the MPR as a function of the receiving TPSs, so that formally \(R_{12} = R_{12}^{\tau_1}\). In reality, the MPR is measured according to the clock time of the receiving spacecraft, \(R_{12}^{\hat\tau_1}\). Similarly to all other measurements, we simulate this by first generating \(R_{12}^{\tau_1}\) and then resampling the resulting time series to obtain \(R_{12}^{\hat\tau_1}\), as described in Signal sampling.
DWS measurements
The photodiodes used onboard LISA have four photo-quadrants, each providing an independent beatnote measurement. This allows estimation of the differential inclinations of the wavefronts of the two interfering beams, also called DWS measurements. Thus, there are two DWS measurements per photodiode: one in yaw (\(\phi\) rotational degree of freedom) and one in pitch (\(\eta\) degree of freedom).
In the current model, we only consider the DWS measurements in the inter-spacecraft interferometers. We denote them as \(\text{DWS}^\phi_{\text{isc}_{12}}(\tau)\) and \(\text{DWS}^\eta_{\text{isc}_{12}}(\tau)\).
The following simple model is used in the simulation: the distant wavefronts impinging on the inter-spacecraft photodiodes can be assumed to be spherical, so that angular jitters of the distant MOSA do not enter the measurements[1]. Therefore, only the total local MOSA angular jitter appears in the DWS measurements.
We have
where \(N^{\text{DWS},\phi}_{\text{isc}_{12}}\) and \(N^{\text{DWS},\eta}_{\text{isc}_{12}}\) represent a DWS measurement noise representing errors in the readout and calibration of the DWS system.
Warning
Align with notation in TTL section.