A.

Breathing Angle

The variation of the breathing angle is shown in the top left panel of Fig. 1. The breathing angle is defined as the peak-peak difference in the inter-arm angle - nominally 60° - which varies over a year as the formation changes. The value used in the figure is the maximum change in pointing over the 6 year mission duration that was considered. The value is significant because the optical metrology system must actively be adjusted to accommodate this variation, either or by articulating the entire payload assembly or by utilising a small adjustable mirror within a wide-field telescope (in-field pointing) [6], this places limits on how large a variation can be easily accommodated, and is especially significant in the case of in-field pointing.

From the figure it is clear that there is a small dependence on the berating angle with arm length. More interesting is the trend with starting position; there is a clear decrease in the breathing angle in orbits which start farther from Earth, confirming that perturbations arising from proximity of Earth are a major driver of this effect. Also interesting is that 2-arm configurations can exhibit significantly smaller breathing angles, this is due to the optimisation algorithm only having to optimise for a single vertex - as opposed to all three vertexes for a 3-arm formation.

B.

Doppler Shift

Due to the formation breathing, the relative line-of-sight velocity of the S/C change. This leads to a Doppler shift on the light received by each S/C. Limitations of the photo detection and phase measurement system mean that this cannot be arbitrarily large. Current designs for eLISA hardware can cope with heterodyne frequencies up to 20-25 MHz [7]. The variation of the Doppler shift is shown in in the top right panel of Fig. 1. From the figure we observe a clear linear dependency of the Doppler shift on the formation size. It is also clear that formations starting farther form Earth experience a lower Doppler shift (i.e. the Earth is a significant factor in this effect). Two-arm formations may offer lower Doppler shifts but the gain is marginal.

C.

Point Ahead Angle

The rotation of the formation, combined with the finite light travel time between S/C, results in there being a difference between the direction in which laser light is received from the far S/C and the direction in which it must be transmitted to the far S/C to ensure maximum signal to noise, this is called the Point Ahead Angle (PAA). This is typically characterised by a static offset in the in-plane degree of freedom (i.e. in the plane of rotation of the formation) and by a periodic variation in the out-of-plane degree of freedom. This out-of-plane variation is more significant as it potentially requires an active correction (as in the case of the PAAM which featured in the LISA design [3] [5]). The amplitude of the out-of-plane variation in the PAA is shown in the lower left panel of Fig. 1. Here it is clear that there is a highly linear dependence of the variation on the formation size, with no significant discernible dependence on any of the other parameters.

D.

Arm Length Difference

The final orbit performance criteria considered is the peak arm length difference. In operation, a Michelson interferometer will be synthesised from the individual laser links of eLISA in a process known as Time Delay Interferometry (TDI) [8]. In order to assure good performance of this technique, the differences in the absolute lengths of adjacent arms should be minimal. The peak round trip arm-length difference is shown in the lower right panel of Fig. 1. Here a roughly linear dependence on the arm length is observed, and some improvement can also be seen in formations which start further from Earth. The point for the 2 Gm, 2-arm formation starting at 30 Mkm (blue line in the figure) is believed to be erroneous, and a result of the nature of the optimisation algorithm which is based on a Genetic approach and does not explicitly use this parameter as a main optimisation criteria.

A.

Accommodation Principle

With no active correction of the PAA, the transmit beam (TX) would be sent along the same direction as the nominal receive beam - the beam which optimally completes the science interferometer with maximum heterodyne efficiency (RX). In the case where there is a non-zero out-of-plane PAA, θ, there are then three choices as to how to orient the S/C, illustrated in Fig. 2.

Fig. 2.

Diagram illustrating the out-of-plane point ahead angle. If there is some relative velocity between the S/C, then over the light travel time, τ, the far S/C will move. This means the optimum TX and RX directions will no longer coincide, resulting in a non-zero PAA, θ. With a PAAM, TX can be moved with respect to RX, such that perfect pointing is always possible.

The two quantities which are affected - the received power and the heterodyne efficiency - couple to the instrument performance primarily through the shot noise measured in the science interferometer, the main limiting noise source of the metrology. It therefore makes sense to investigate this effect in terms of the overall system metrology noise - nominally 12 pm/√Hz (for a single link) [4].

To investigate which of the three accommodation options outlined above is most suitable, the metrology noise was calculated as a function of S/C tilt. Fig. 3 (left panel) shows the dependency of the received power and heterodyne efficiency on the S/C tilt - in this case for a S/C with a 20 cm telescope and 1 Gm arm length. It can clearly be seen that the received power decreases faster than the heterodyne efficiency with increasing S/C tilt. Extending this calculation, Fig. 3 (right panel) shows the metrology noise as a function of tilt for a variety of arm lengths. This indicates that there is an optimum balanced position, at approximately 0.23·0 from TX. Interestingly, the location of this optimum is roughly independent of the formation arm length, and thus also from θ. This can be explained by considering that both of these quantities - received power and heterodyne efficiency-are essentially calculated as a double integral over the telescope aperture. The variation with angle is therefore roughly independent of the main system design properties (arm length, laser power and telescope radius).

Fig. 3.

Left: Variation of the received power and heterodyne efficiency with increasing system tilt. Right: Metrology noise as a function of offset from the 'TX' vector, for a variety of formation arm lengths. Since the PAA increases linearly with arm length, the offset has been normalised to 1.

B.

Scaling With Arm Length

In order to calculate the scaling of the effect with arm length, it is necessary to make some assumptions about the two other key system design parameters: the telescope radius and the laser power. (Not doing so results in a large parameter space.) For the purposes of this analysis, it has been assumed that the telescope radius scales linearly with arm length from 20 cm at 1 Gm to 40 cm at 5 Gm - i.e. extrapolating between the NGO and LISA configurations. The laser power then scales such that, with no tilt of the S/C, 200 pW of power is received at the far S/C, as per the nominal design [3] [4]. This enables the entire system to be conveniently parameterised as a function of arm length.

Utilising this assumption, the single-link metrology noise can be calculated as a function of arm length. Fig. 4 (left panel) shows the single link metrology noise calculated between 0.5 Gm and 5 Gm, for each of the three accommodation methods outlined above. The resulting strain sensitivity (δL/L) is shown on the right. As expected, pointing to RX to maximise the heterodyne efficiency results in the largest increase in noise, with the first diffraction minima in the transmit beam visible at around 4 Gm.

Fig. 4.

Left: Scaling of single-link metrology noise with arm length for the three passive PAA accommodation methods. Right: Strain sensitivity (δL/L). The nominal curve shows the strain sensitivity assuming an actively compensated PAA.

Never the less, from the figure, it is clear that passive accommodation of the out-of-plane PAA is feasible at arm lengths in the range of 1 to 2 Gm, and even at 3 Gm only a marginal decrease in performance is experienced.

IV.

MISSION PARAMETERS

The three top-level parameters which define the mission are the arm length, the telescope radius and the laser power. All three of these parameters directly affect the ultimate strain-sensitivity of the detector. It is thus very important that, as far as possible, we understand dependency of the mission performance on these three parameters. In particular, it is important to ensure that when the final eLISA mission design parameters are selected they represent a good balance between instrument cost and complexity and science return.

It is still advantageous to simplify the analysis a little due to the size of the parameter space. Specifically, the relationship between laser power and telescope radius has been calculated as a function of arm length under two different system constraints:

Further, in each case, the dependence has been calculated assuming both active (i.e. with a PAAM) and passive accommodation of the point ahead angle, as discussed in Section II. The results are shown in Fig. 5 and Fig 6.

Fig. 5.

Plot showing the balance between laser power and telescope radius, at a range of arm lengths, under the assumption of a fixed metrology-limited strain sensitivity of 1.2 x 10^-20 /√Hz. Curves assuming both active (dashed) and passive accommodation of the PAA are shown. The black circle denotes the NGO configuration.

Fig. 6.

Plot showing the balance between laser power and telescope radius, at a range of arm lengths, under the assumption of a fixed single-link metrology noise of 12 pm/√Hz. Curves assuming both active (dashed) and passive accommodation of the PAA are shown. The black circle denotes the NGO configuration.

It is interesting to note that, from Fig. 5, simply increasing the constellation arm length does not implicitly lead to a significant increase in the metrology-limited strain sensitivity, rather complementary increases in at least one of the laser-power or telescope radius is also required to increase the strain sensitivity. Also, as expected from the analysis in Section II, keeping the metrology noise fixed requires significant increases in laser power or telescope radius in order to utilise passive accommodation of the point ahead angle. It is also interesting to note that, in the case of a 1 Gm formation, the NGO configuration lies close to the knee of the curve, representing a good balance between laser power and telescope radius for a 1 Gm mission at the required strain sensitivity.

Taking the analysis shown in Fig. 5 and Fig. 6, a selection of case studies demonstrating potential eLISA configurations has been considered as a means of highlighting some of the possibilities. These are summarised in Table 1 and the corresponding strain sensitivity curves for each are shown in Fig. 7. The case studies are restricted to be compatible with the simplifications outlined in the introduction and studied here, such that they could realistically be implemented as ESA-only L3 missions. In particular they all include fully passive accommodation of the point ahead angle.

Table 1.

Case studies to illustrate possible eLISA configurations. The original ESA/NASA LISA configuration and the ESA L1 NGO configuration are included for comparison.

Fig. 7.

All sky and polarisation averaged strain sensitivity for a variety of possible eLISA configurations and the LISA and NGO reference designs.

The 'Science Balance' and 'Optimal Science' configurations are especially interesting as they offer a significant improvement in strain sensitivity over the NGO design with only modest increases in the key system design parameters. It is also interesting to visualise the dependency of the strain sensitivity on the laser power and telescope radius. Fig. 8 shows, for the 1.5 Gm 'Science Balance' case on the left and the 2 Gm 'Optimal Science' case on the right, how the strain sensitivity reduces as the laser power and telescope radii are relaxed towards the NGO reference design. The main effect is towards higher frequencies, where the sensitivity is dominated by the metrology system and the antenna pattern. Crucially, the key low-frequency part of the spectrum is not affected.

Fig. 8.

Left: Dependence of the strain sensitivity on the telescope radius and laser power for a 1.5 Gm mission, showing the relaxation of the 'Science Balance' case study towards the NGO reference design. Right: Dependence of the strain sensitivity on the telescope radius and laser power for a 2 Gm mission, showing the relaxation of the 'Optimal Science' case study towards the NGO reference design.