# LISAmax: Improving the Low-Frequency Gravitational-Wave Sensitivity by Two
Orders of Magnitude

W. Martens1, M. Khan1, and J.-B. Bayle2 1Mission Analysis Section, European
Space Agency, Darmstadt, Germany  2University of Glasgow, Glasgow G12 8QQ,
United Kingdom<EMAIL_ADDRESS>
###### Abstract

Within its Voyage 2050 planning cycle, the European Space Agency (ESA) is
considering long-term large class science mission themes. Gravitational-wave
astronomy is among the topics under study. Building on previous work by other
authors [1, 2, 3], this paper studies a gravitational-wave interferometer
concept, dubbed “LISAmax”, consisting of three spacecraft, each located close
to one of the Sun-Earth libration points L3, L4 and L5, forming a triangular
constellation with an arm length of 259 million kilometers (to be compared to
LISA’s 2.5 million kilometer arms). We argue that this is the largest
triangular formation that can be reached from Earth without a major leap in
mission complexity and cost (hence the name). The sensitivity curve of such a
detector is at least two orders of magnitude lower in amplitude than that of
LISA, at frequencies below $1\text{\,}\mathrm{mHz}$. This makes the
observatory sensitive to gravitational waves in the
$\mathrm{\SIUnitSymbolMicro Hz}$ range and opens a new window for
gravitational-wave astronomy, not covered by any other planned detector
concept. We analyze in detail the constellation stability for a 10-year
mission in the full numerical model including insertion, dispersion, and self-
gravity-induced accelerations. We compute the orbit transfers using a European
launcher and chemical propulsion. Different orbit options, such as precessing,
inclined orbits, the use of flybys for the transfer, and the launch strategy,
are discussed. The payload design parameters are assessed, and the expected
sensitivity curve is compared with a number of potential gravitational-wave
sources. No show stoppers are identified at this point of the analysis.

††: Class. Quantum Grav.

Keywords: gravitational-wave detector, LISA, Voyage 2050

##  1 Introduction

In June 2021, the European Space Agency (ESA) has chosen the future science
mission themes for the time frame 2035-2050. These are summarized under the
so-called Voyage 2050 program [4]. Gravitational waves are among the chosen
science themes covered by the L6 slot “New Physical Probes of the Early
Universe” [5]. This has triggered some interest in studying successor space
missions to the Laser Interferometer Space Antenna (LISA) currently planned to
be launched in 2035 [6, 7]. Four White Papers related to gravitational-wave
astronomy have been submitted to the Voyage 2050 call [3, 8, 9, 10] targeting
different frequency ranges. Moreover, there are several space-based
gravitational-wave observatory proposals and projects at different stages of
development [1, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Each of them aims
at either a better sensitivity or a different frequency band compared to LISA.
All of these use one or more triangular spacecraft constellations located at
different regions of space.

The current work studies a potential LISA successor targeting the
$\mathrm{\SIUnitSymbolMicro Hz}$ band. The study is driven by the question:
how large can a post-LISA space-based gravitational-wave observatory be, while
still remaining in the budget of an L-class mission (i.e., around one billion
Euros at today’s costing)? The constellation size, or arm length, mainly
determines the sensitive frequency band. Larger arm lengths imply lower
frequencies, while also improving the sensitivity in general. Such an
observatory would, for instance, allow the observation of super-massive black
hole inspirals long before they enter the LISA band. For a more in-depth
survey of the science case of a $\mathrm{\SIUnitSymbolMicro Hz}$ observatory,
refer to [3].

To target a low frequency range, one can start by placing three spacecraft at
the triangular liberation points L3, L4, and L5 of the Sun-Earth system as
proposed in [19, 20, 21]. These are equilibrium points located at
$1\text{\,}\mathrm{ua}$ from the Sun, which span an equilateral triangle. The
downside of such a constellation is, however, that L3 is located behind the
Sun as seen from Earth. This would preclude direct communications from Earth
and require one of the other satellites to act as a relay, increasing the
mission complexity. A more robust solution to this problem is to rotate the
full constellation by $5\text{\,}\mathrm{\SIUnitSymbolDegree}$ to
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$, allowing a direct link to Earth for
all three satellites. It turns out that such a rotation does not degrade the
constellation stability in a significant way. Such an offset triangular
constellation was first used in reference [1, fig. 2]. However, the aim in
that proposal was not to target lower frequencies but to recover part of the
LISA performance by a simpler, non-drag-free satellite concept. An
$\mathrm{ua}$-sized drag-free detector consisting of two constellations that
aims at the $\mathrm{\SIUnitSymbolMicro Hz}$ range was later proposed in [3].
This reference does not provide many details on the orbital mechanics aspects
but focuses on the science case of a $\mathrm{\SIUnitSymbolMicro Hz}$
detector. We note, however, that the out-of-plane constellation of that
proposal is not reachable with today’s propulsion systems due to the high
$\Delta v$ demands. The present paper is an attempt to condense the proposed
concepts into an operationally feasible and realistic mission scenario.

The most constraining aspect of the studied detector, dubbed “LISAmax”, is
that the constellation is bound to the ecliptic plane. Even though the three
satellites rotate around the Sun once per year, this limits the antenna
pattern and thus possibly the directional sensitivity of the observatory.
Placing the constellation out of the ecliptic plane would require prohibitive
amounts of $\Delta v$ during the transfer. Higher inclinations can be reached
using planetary flybys. However, reaching such inclinations and then
accurately maintaining the triangular configuration would imply a major leap
in mission complexity and cost.

The rest of the paper is organized as follows: the design of the science orbit
in the full dynamical model and the analysis of its stability under
dispersions are presented in sections 2 and 3. Section 4 follows with a
discussion of the achievable sensitivity curve. In section 5, the
interplanetary transfer to the science orbit is analyzed. Various mission
options of using inclined, precessing orbits or planetary flybys are discussed
in section 6. Section 7 concludes the paper.

##  2 LISAmax science orbit

For the design of the LISAmax science orbit, we assume similar requirements as
for LISA [22] extending into LISAmax frequency range.

  1. 1.

The total duration of the science phase is 10 years including extension, i.e.
the orbits need to be stable for this duration.

  2. 2.

No station keeping maneuvers are assumed during the science phase. Such
maneuvers would interrupt the science operations and are followed by a re-
acquisition of the constellation, which is not desirable.

  3. 3.

The corner angles of the constellation during 10 years shall not deviate from
the nominal value of $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ by more than
$\pm 1.0\text{\,}\mathrm{\SIUnitSymbolDegree}$. This constraint comes mainly
from the limitation of the optical assembly tracking mechanism (OATM), which
ensures that the laser beams point to the correct direction at all times.

  4. 4.

The relative velocity (arm length rate of change) shall not exceed
$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ during the 10-year science
phase duration. This is due to the bandwidth limitation of the phasemeter that
detects the beat notes in the interferometric laser signal [23].

It is convenient to represent the orbits of LISAmax in the Sun-Earth rotating
frame, with the $x$-axis along the Sun-Earth line; the $z$-axis along the
Earth orbit angular momentum vector; and the $y$-axis completing the right-
handed system. The fully optimized science orbit in this frame is depicted in
fig. 1. The gravitational accelerations of all relevant celestial bodies (Sun,
Earth, Moon, Venus, Mars, Jupiter, Saturn) have been taken into account in the
numerical propagation using a Runge-Kutta integrator. Solar radiation pressure
is not affecting the spacecraft trajectories because its effect is assumed to
be compensated by the Drag Free and Attitude Control System (DFACS), similarly
to LISA [24, 25, 26]. The problem was set up as a constraint satisfaction
problem using GODOT [27] for the trajectory propagation, PyGMO [28] and the
NLP solver WORHP [29] for optimization. As an initial guess, Keplerian orbits
in a triangular formation, offset by $10\text{\,}\mathrm{\SIUnitSymbolDegree}$
from the Sun-Earth triangular libration points, have been used. Constraints
have been imposed on the maximum deviation of all three corner angles, arm
lengths, and arm length rates. The maximum values for these constraints have
been manually adjusted to be as small as possible while still allowing for
convergence of the NLP solver. The final values of the maximum deviations are
$60\pm 0.06\text{\,}\mathrm{\SIUnitSymbolDegree}$ for the corner angles and
$\pm 5.0\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ for the arm length
rates. The full evolution of both quantities is shown in fig. 2. These
variations are significantly smaller that what is achievable for LISA [22]
($60\pm 1.0\text{\,}\mathrm{\SIUnitSymbolDegree}$ for the corner angles and
$\pm 10.0\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ for the arm length
rates). This is due to the larger distance from perturbing sources, mainly the
Earth.

Figure 1: The optimized science orbit of LISAmax over a 10-year duration in
the Sun-Earth rotating frame. The elongated shape of the Earth and spacecraft
trajectories in the plot indicate the variation of their locations over the
science phase duration.

(a)

(b)

Figure 2: Corner angles (a) and arm length rates (b) evolution over the
10-year science phase duration.

##  3 Science orbit insertion accuracy and self-gravity-induced accelerations

The science orbit stability analysis in section 2 assumed a perfect insertion
of the three satellites into the desired orbits. In reality, insertion
disturbances of the final thruster burns are expected to disperse the initial
spacecraft states. This will lead to a slight deterioration of the achievable
corner angles and arm length rates. To quantify the insertion dispersion, a
full simulation of the ground-based Range/Doppler tracking and guidance
strategy during the transfer would be required; however, this is beyond the
scope of this paper. Instead, we take some conservative assumptions on the
initial state dispersion of the satellites based on experience. We perform a
Monte-Carlo analysis propagating $10\,000$ samples, each with slightly
different initial conditions within normal distributions detailed in table 1.
These distributions are realistically achievable with standard Range/Doppler
observations and a sequence of gradually decreasing clean-up maneuvers after
the nominal science orbit insertion. For each sample, the maximum deviation of
the corner angles and arm length rates is recorded. As shown in fig. 3 (a),
the resulting dispersion on these parameters is rather limited despite the
conservative assumptions, owing to the quiet environment in the vicinity of
the libration points. Note that for LISA the resulting disturbance is more
significant [22].

Table 1: Assumed science orbit insertion uncertainties and self-gravity-
induced accelerations.

  * Parameter | Distribution | Uncertainty
---|---|---
Position dispersion, $1-\sigma$ |  |
(along-track, cross-track, radial) | Normal |  $50\text{\,}\mathrm{k}\mathrm{m}$, $200\text{\,}\mathrm{k}\mathrm{m}$, $50\text{\,}\mathrm{k}\mathrm{m}$
Velocity dispersion, $1-\sigma$ |  |
(along-track, cross-track, radial) | Normal |  $10\text{\,}\mathrm{c}\mathrm{m}\mathrm{/}\mathrm{s}$, $20\text{\,}\mathrm{c}\mathrm{m}\mathrm{/}\mathrm{s}$, $10\text{\,}\mathrm{c}\mathrm{m}\mathrm{/}\mathrm{s}$
Self-gravity-induced accelerations |  |
(constant in spacecraft-fixed frame) | Uniform |  $2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ width, spherical

There exists a further effect, specific to drag-free spacecraft, that can
destabilize the constellation due to its stochastic nature: the so-called
self-gravity-induced accelerations. This is an effective acceleration
component acting on the spacecraft resulting from the DFACS following the
motion of the two internal test masses, if these are not located precisely at
their nominal positions. To understand the effect of self-gravity-induced
accelerations, first consider a single test mass that is offset from its
nominal position, where the gravity acceleration induced by the spacecraft
mass vanishes. Due to the gravitational force exerted by the spacecraft on the
test mass, the latter will be accelerated towards its nominal location. This
motion will be detected by the test-mass interferometer and the DFACS will
command the propulsion system to accelerate the spacecraft into the opposite
direction, such that the test mass effectively stays at rest relative to the
spacecraft. This will also happen if the test mass remains at its nominal
position, but the spacecraft mass distribution does not exactly cancel at this
location.

This acceleration produced by the DFACS has an impact on the spacecraft
trajectory. Since LISAmax is assumed to have two test masses (as LISA), the
situation is more complicated in reality. For more details, refer to [22]. For
the purpose of the current analysis, the relevant result is that it is only
possible to predict the magnitude and direction of a part of the self-gravity-
induced accelerations. This known part can be taken into account in the
optimization of the nominal orbit achieving similar stability. However, there
is a remaining stochastic component that will act on the spacecraft and
potentially destabilize the constellation. To assess its effect in the Monte-
Carlo analysis, a similar assumption as for LISA has been taken on the
magnitude of the acceleration (see table 1): a uniform distribution of the
three acceleration vector components defined in the spacecraft-fixed frame.
The result is shown in fig. 3 (b). Again, the overall effect on the
constellation stability is significantly less than for LISA. A comparison with
the results obtained for LISA in reference [22] is shown in table 2.

To summarize, neither the expected size of spacecraft insertion maneuver
dispersions nor the expected levels of stochastic self-gravity-induced
accelerations are sufficient to deteriorate significantly the stability of the
constellation. The assumed requirements of $\pm
1.0\text{\,}\mathrm{\SIUnitSymbolDegree}$ for the corner angles and $\pm
10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ for the arm length rates are
still fulfilled with significant margins when taking into account these two
disturbances for the science orbit.

(a)

(b)

Figure 3: Sample clouds ($10\,000$ samples) showing the science orbit
stability due to insertion dispersion without (a) and with (b) stochastic
self-gravity-induced accelerations. The shown parameter space represents the
allowed region due to the assumed requirements ($60\pm
1.0\text{\,}\mathrm{\SIUnitSymbolDegree}$ for the corner angles and $\pm
10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ for the arm length rates).
Table 2: Results for the 99% confidence intervals of orbit stability metrics
under dispersions for LISA [22] and LISAmax. Both cases include self-gravity-
induced accelerations.

  * Parameter | LISA | LISAmax
---|---|---
Corner angles, 99% C.I. |  [$58.507\text{\,}\mathrm{\SIUnitSymbolDegree}$, $61.527\text{\,}\mathrm{\SIUnitSymbolDegree}$] |  [$59.886\text{\,}\mathrm{\SIUnitSymbolDegree}$, $60.101\text{\,}\mathrm{\SIUnitSymbolDegree}$]
Arm length rates, 99% C.I. |  [-$11.04\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, $10.16\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$] |  [-$6.27\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, $6.25\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$]

##  4 Sensitivity curve

The assumptions that we make to compute the sensitivity curve of LISAmax are
summarized in table 3. A laser power of only $1\text{\,}\mathrm{W}$ is
certainly conservative; a more powerful laser could further improve the
sensitivity. The driving assumption for the instrument and spacecraft design
is the telescope diameter. To fully compensate for the decreased laser power
at reception, one would have to scale up the telescope diameter by a factor of
10 only, compared to LISA. This is because the received power scales with the
telescope diameter to the fourth power, while the travel distance only enters
inversely quadratically [30, eq. 12]:

$P_{r}=P_{t}\left[\frac{\epsilon\pi^{2}\nu_{0}^{2}D^{2}}{c^{2}}\right]\left[\frac{1}{4\pi
r^{2}}\right]\left[\frac{\epsilon\pi D^{2}}{4}\right].$ (1)

$P_{t}$ and $P_{r}$ are the transmitted and received power, respectively. The
quantity in the first square brackets is the directional gain of the
transmitter optics with diameter $D$ and efficiency $\epsilon$ for light of
frequency $\nu_{0}$. The quantity in the second square brackets is the space
loss at a distance $r$ (arm length), and the quantity in the last square
brackets is the effective cross section of the receiving optics. Because the
increased arm lengths yield an additional quadratic gain in sensitivity to
gravitational-wave strain, we have increased the telescope only by about a
factor of three. Nevertheless, this choice is a driver for the spacecraft
design and must be carefully traded against the requirements of the science
case if a LISAmax-like detector is envisioned. An increased laser power could
also partly compensate for a smaller telescope diameter choice.

Table 3: Instrument assumptions for LISAmax. Parameter | LISA | LISAmax |  Note
---|---|---|---
Arm length [$\mathrm{km}$] | 2.5 million | 259 million |  From orbit selection.
Laser wavelength [$\mathrm{nm}$] | $1064$ | $1064$ |
Laser power [$\mathrm{W}$] | $1.0$ | $1.0$ |  Could be increased to $10\text{\,}\mathrm{W}$ for better sensitivity.
Telescope diameter [$\mathrm{m}$] | $0.3$ | $1.0$ |  Scaled up to partly compensate for lower photon density.
Telescope efficiency | $0.3$ | $0.3$ |
Test-mass noise [$\mathrm{m}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{Hz}}^{-0.5}$] | $$3\text{\times}{10}^{-15}$\sqrt{1+\left(\frac{$0.4\text{\,}\mathrm{mHz}$}{f}\right)^{2}}$ | $$3\text{\times}{10}^{-15}$\sqrt{1+\left(\frac{$0.4\text{\,}\mathrm{mHz}$}{f}\right)^{2}}$ |  LISA Pathfinder achieved $1.74\text{\times}{10}^{-15}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{Hz}}^{-0.5}$ at $1\text{\,}\mathrm{mHz}$ [31].

For the sake of simplicity, the $1/f$ component of the test-mass acceleration
noise has been extrapolated down to frequencies of
$1\text{\times}{10}^{-6}\text{\,}\mathrm{Hz}$, even though the LISA Pathfinder
results [31] only provide data down to
$2\text{\times}{10}^{-5}\text{\,}\mathrm{Hz}$. It is likely that additional
effects, like temperature fluctuations, alter the response at lower
frequencies. Applying the same simple model [30] for both LISA and LISAmax
allows for a indicative comparison. Figure 4 shows the sky-averaged
sensitivity curves (Michelson combination) in amplitude spectral density for
both LISA and LISAmax.

The overall downward shift of the LISAmax sensitivity curve by two orders of
magnitude at low frequencies is a direct consequence of the increased
constellation arm lengths: a given gravitational wave causes a stronger signal
in a larger laser arm. However, the increased arm lengths also shift the right
slope towards lower frequencies as the observatory becomes less sensitive to
smaller wavelengths (arm length penalty). The highest sensitivity point is
determined by the laser shot noise, which is inversely proportional to the
received laser power. With the current assumptions, this point is slightly
lower for LISAmax than for LISA (around ${10}^{-3}\text{\,}\mathrm{Hz}$,
compared to LISA’s ${10}^{-2}\text{\,}\mathrm{Hz}$). It can be further reduced
by either increasing the telescope diameter ($S_{n}\propto D^{-4}$) or by
increasing the transmitted laser power ($S_{n}\propto P_{t}^{-1}$). The slope
on the left is determined by the test-mass acceleration noise. Note that no
Galactic confusion noise is taken into account here. It degrade the
sensitivity of both detectors in the area around and below
$1\text{\,}\mathrm{mHz}$. A recent study [32] showed that a large part of the
Galactic double white dwarf population could not be resolved by a
$\mathrm{\SIUnitSymbolMicro Hz}$ detector, and would produce a confusion noise
above the instrumental noise, potentially down to
$1\text{\times}{10}^{-5}\text{\,}\mathrm{Hz}$.

Figure 4: Sensitivity curves of LISA and LISAmax. Note that for the sake of
simplicity and comparability, the same simple noise model has been used for
both LISA and LISAmax [30]. As a consequence, the LISA sensitivity curve shown
here is not identical to the official one. The dashed line indicates the range
below the LISA curve validity. Traces of massive black hole (MBHB) inspirals
and mergers at redshift $z=3$ are also shown.

For convenience, we also list the analytical formulas for the included noise
power spectral densities in the LISAmax sensitivity curve consisting of shot
noise and test mass acceleration noise:

$\displaystyle S_{n,\text{shot}}$
$\displaystyle=2\frac{4c^{4}h}{\pi^{4}\epsilon^{2}D^{4}P_{t}\nu_{0}^{3}},$
(2a) $\displaystyle P_{\text{acc}}$
$\displaystyle=\left($3\text{\times}{10}^{-15}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-2}$\right)^{2}\left(1+\left(\frac{$0.4\text{\,}\mathrm{mHz}$}{f}\right)^{2}\right)${\mathrm{Hz}}^{-1}$,$
(2b) $\displaystyle S_{n}$ $\displaystyle=S_{n,\text{shot}}+\frac{4}{(2\pi
f)^{4}L^{2}}P_{\text{acc}},$ (2c)

where $L$ is the arm length of the detector and the other variables have been
defined previously.

A note regarding directional sensitivity is in order here. LISA’s
constellation axis precesses around the ecliptic pole with an opening half
angle of $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ once per year, while the
constellation is also rotating around its own axis. This leads to an amplitude
modulation of a long-lived gravitational-wave signal from a fixed location in
the sky over the year (due to LISA’s non-spherical antenna pattern). In the
case of a LISAmax-like detector, the constellation normal does not precess
because the constellation is bound to the ecliptic plane. Therefore,
localizing a gravitational-wave source in the sky cannot rely solely on this
amplitude modulation.

There are two main ways to mitigate this:

  1. 1.

Incline each satellite orbit with respect to the ecliptic, but offset the
three nodes by $120\text{\,}\mathrm{\SIUnitSymbolDegree}$ [33, 3]. This will
cause a precession of the constellation normal around the ecliptic pole in way
similar to LISA. As a consequence, equivalent sky localization methods could
be employed.

  2. 2.

Add a second constellation with a significant relative inclination, as
proposed in reference [3]. Both constellations will simultaneously measure a
given gravitational-wave signal with different amplitude modulations. This
would allow narrowing down the sky location, possibly up to an ambiguity.

Both options will be discussed in more detail in section 6. However, they
would both imply a significant leap in mission complexity and cost compared to
the baseline concept presented here.

In addition to the precession of the constellation plane, LISA’s center moves
around the Sun, causing a seasonal Doppler modulation of its response. This
effect is missing in LISAmax. Note that, however, because the LISAmax
constellation rotates around the Sun once per year, there is still a seasonal
variation of the directional sensitivity, which can be exploited to gain some
information on the source location. Moreover, different time-delay
interferometry (TDI) combinations can be constructed from the raw Doppler
readouts; each of these combinations has a different antenna pattern [34, 35,
36]. This can also help to obtain information about the direction of the
gravitational-wave source. Therefore, we do not expect that LISAmax looses all
resolution power. A more in-depth study of the localization capabilities of
LISAmax-like missions is being prepared.

To illustrate one science case of LISAmax, fig. 4 also shows the traces of
massive black hole inspirals and mergers in the sensitivity plot [37]. While
the merger itself for both ${10}^{5}$ and ${10}^{7}$ solar mass black-hole
binaries is fully within the LISA band, LISAmax can observe such objects
dozens of years before the actual merger.

##  5 Interplanetary transfer

A main driver for the system design is the fuel needed for the transfer. This
can be computed from the transfer $\Delta v$. The following assumptions have
been made for the interplanetary transfer optimization:

  * •

Each spacecraft is launched on a dedicated European launcher from Kourou with
an escape velocity of $1\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. This
assumption is a major cost driver and depends mainly on the choice of the
telescope size, and thus the spacecraft mass. With a smaller telescope than
assumed here (cf., table 3), a shared launch of all three satellites, similar
to LISA, would be conceivable.

  * •

At $1\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ escape velocity, the
launcher is assumed to deliver 10 tons of payload. This is in line with the
expected developments of the large European launch vehicle, Ariane 6.4 [38].
It would also be well within the capabilities of Falcon Heavy or Vulcan
Centaur 6. 10 tons should allow for accommodation of a telescope of the size
assumed here.

  * •

The transfer design assumes chemical propulsion with high thrust, which allows
the maneuvers to be approximated by instantaneous velocity increments.

To transfer a spacecraft from Earth to the desired location, close to the
triangular libration points, a simple phasing strategy can be employed: launch
injects the spacecraft into a heliocentric transfer orbit. To reach the
near-L4 region, the transfer is a heading orbit with a semi-major axis below
$1\text{\,}\mathrm{ua}$. To reach the near-L3 and near-L5 regions, the
transfer is a trailing orbit with a semi-major axis above
$1\text{\,}\mathrm{ua}$. When the desired location is reached, a maneuver is
applied to insert into the circular target orbit and stop the drift relative
to Earth. The required transfer $\Delta v$ is a function of the transfer
duration, which can be a multiple of 1 year, minus (for L4) or plus (for L3
and L5) a fixed increment. The longer the transfer and the more heliocentric
revolutions are completed, the lower the $\Delta v$. From the cost
perspective, it is advantageous to have three spacecraft of identical design,
which means that all three have the same $\Delta v$ capacity. This can be
achieved assuming two revolutions around the Sun for spacecraft 1 and 3 (going
to L4 and L5) and four revolutions for spacecraft 2. The fully optimized
transfer trajectories for a launch in August 2040 are shown in fig. 5.

Figure 5: Transfer trajectories using dedicated launches in the Sun-Earth
rotating frame.

The computation of these transfers was set up as a multiple shooting problem
with a Lambert arc as an initial guess. The spacecraft mass placed into final
orbit was used as the objective function of the optimization problem. Due to
the eccentricity of the Earth orbit, the transfer $\Delta v$, and thus the
arrival mass, depends on the launch season. The results including the seasonal
variations over the full year of 2040 are shown in fig. 6. The results,
however, are applicable to any launch year.

(a)

(b)

Figure 6: Full year transfer $\Delta v$ (a) and spacecraft mass in science
orbit (b).

Spacecraft 1 (going to L5) has the lowest $\Delta v$. This is because it has
the shortest travel distance and the target orbit is Earth-trailing, which has
a slight advantage $\Delta v$-wise. Spacecraft 2 (going to L3) has almost as
large a $\Delta v$ as spacecraft 3 (going to L4) in spite of a much longer
transfer duration. This is because spacecraft 2 has the longest travel
distance. Assuming that launch can take place on any day of the year, the
minimum mass inserted into any of the target locations still amounts to
$6500\text{\,}\mathrm{kg}$. This is about three times larger than that of
LISA. It should be sufficient to accommodate a larger telescope, the larger
high-gain antenna and include the necessary support structure.

The TDB hour of the launcher separation as well as the transfer duration for
the full launch period are shown in fig. 7. The different escape directions
targeted by the three spacecraft result in different launch times (and hence
launcher separation times) because the launcher trajectory is fixed in the
Earth-fixed frame. Spacecraft 2 has a transfer duration of around 4.5 years.
If consecutive launches are envisioned, this one should be launched first to
minimize the waiting time of the others before full science operations can
commence. This may also help to spread the construction effort and cost over a
longer time period.

(a)

(b)

Figure 7: Full year launcher separation TDB hour (a) and transfer duration
(b).

##  6 Mission options

One of the main challenges of an $\mathrm{ua}$-sized detector, such as the one
studied here, is to preserve a good gravitational-wave source localization
capability. In the LISA mission, for long-lived sources, this is achieved by
tracking the amplitude of the response to the gravitational-wave signal of a
given source over the course of the year. The precession of the constellation
plane around the ecliptic north pole induces a characteristic amplitude
modulation in the gravitational-wave signal that can be used to estimate the
sky location. Moreover, the motion of the constellation around the Sun leads
to a location-dependent Doppler shift when the detector is moving towards or
away from the source. Both of these effects are missing in LISAmax. The only
seasonal variation is due to the rotation of the full constellation around the
Sun once per year. In this section, several mission options that address the
challenge of sky localization and the possibility of a joint launch are
discussed.

###  6.1 Precessing inclined orbits

A precessing motion similar to that of LISA can be introduced to LISAmax by
inclining all three orbits with respect to the ecliptic and spacing the nodes
of the orbital planes by $120\text{\,}\mathrm{\SIUnitSymbolDegree}$ [33, 3].
This is illustrated in fig. 8 where the optimized orbits in the full numerical
model (c.f., section 2) are shown. The constellation normal will precess
around the ecliptic pole with a period of one year and induce a variation in
the gravitational-wave response amplitude over the course of a year similar to
that of LISA.

(a)

(b)

Figure 8: Illustration of precessing inclined orbits (a). (b) shows orbits
with an inclination of $1\text{\,}\mathrm{\SIUnitSymbolDegree}$ relative to
the ecliptic.

One problem with this kind of precessing orbits is that the arm length rate
between the satellites increases with the chosen inclination. This is
illustrated in fig. 9. An inclination of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ already leads to an arm length rate
variation of $\pm$$33\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ (to be
compared with $\pm$$5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ without
inclination). Attempts have been made to reduce the breathing of the triangle
by assuming a non-zero eccentricity, but were unsuccessful as the optimizer
converges to circular orbits in all cases. Future work will focus on obtaining
the optimal eccentricities and arguments of perihelion in the Keplerian model
analytically to confirm that circular orbits are indeed optimal. In the full
numerical model, this appears to be the case.

Therefore, to use precessing orbits with inclinations higher than
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$, one must cope with the ensuing
relative velocities of significantly more than
$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$. This can only be done by
increasing the bandwidth of the phasemeter, which is currently a technological
limitation (large velocities imply large Doppler shifts and therefore a large
variation of spacecraft beatnote frequencies).

(a)

(b)

Figure 9: Arm length rate for precessing inclined orbits with
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$ (a) and
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ (b) ecliptic inclination. The arm
length rates increase strongly, the higher the inclination.

###  6.2 Planetary flybys to reach high inclinations

A mission concept with a combination of two triangular constellations has been
proposed in [3], where one is inclined by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ relative to the ecliptic while the
other one remains in the ecliptic. However, reaching such high inclinations
requires a prohibitive transfer $\Delta v$: to transfer from an initial Earth-
like orbit to one inclined by $90\text{\,}\mathrm{\SIUnitSymbolDegree}$, an
impulsive $\Delta v$ of about $\sqrt{2}\cdot 30$
$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ has to be imparted. This is beyond the
capability of currently known rocket engines.

To achieve a significant relative inclination between two constellation
planes, however, planetary flybys can be exploited. For instance, the
currently operational Solar Orbiter mission uses a series of resonant Venus
flybys to reach a final inclination of
$33\text{\,}\mathrm{\SIUnitSymbolDegree}$ (extended mission) relative to the
Sun equator in order to observe the Sun from its poles [39, 40, 41] (see fig.
10). In principle, a similar strategy with Venus and Earth flybys could be
employed to place two constellations with opposite inclination values, such
that the relative inclination adds up. However, such a mission would have a
vastly higher complexity and cost compared to the baseline concept presented
here. Not only is the transfer for a single spacecraft significantly longer
and more complex, but all six spacecraft would have to follow dedicated flyby
trajectories, increasing also the operational cost. Such a concept could only
be facilitated by a significant increase in spacecraft autonomy compared to
what is customary at the present day.

Figure 10: Solar Orbiter trajectory. High solar latitudes are reached using a
sequence of Venus flybys. Image retrieved from [42].

###  6.3 Constellations beyond 1 au

A simple modification to the baseline LISAmax concept is to increase the arm
length even further by placing the satellites on a circular orbit beyond
$1\text{\,}\mathrm{ua}$, for instance at the altitude of the Mars orbit, as
suggested in [3]. That would further improve the sensitivity at low
frequencies, but would also come with the following downsides:

  * •

The transfer $\Delta v$ increases significantly due to the higher Earth escape
velocity and arrival maneuver required to reach larger orbits.

  * •

The stable geometry relative to the Sun and the Earth would be lost due to the
different orbital periods of the Earth and the satellites around the Sun. This
would require a regular antenna re-orientation to communicate with the Earth,
which induces undesirable noise on the spacecraft. Moreover, all three
spacecraft would have to be designed to cope with the maximum Earth distance.
In the baseline concept, only the satellite located close to L3 needs to
communicate over the large distance of almost $2\text{\,}\mathrm{ua}$.

  * •

Due to the relative drift between the Earth and the spacecraft, regular solar
conjunctions would occur, which would interrupt two-way communications with
the Earth for up to several weeks.

Overall, these significant downsides must be weighed against the relatively
limited gain in sensitivity and no gained advantage in source localization
power. It appears that tuning other key parameters, such as the laser power,
is a more achievable means of increasing the sensitivity.

###  6.4 Joint launch

The dedicated launch scenario (a separate rocket for each satellite)
considered in this paper is driven by the assumption of a large telescope mass
($1\text{\,}\mathrm{m}$ in diameter) and the resulting large spacecraft mass.
A joint launch (all three satellites on the same rocket) would likely be
favorable from the mission cost point of view if a smaller telescope is used.
It would, however, complexify the transfer strategy, because the Earth
departure velocity asymptote can no longer be chosen separately for the three
spacecraft. While spacecraft 1 and 2 require a launch away from the Sun, into
an Earth-trailing orbit, spacecraft 3 requires to be launched towards the Sun
into an Earth-heading orbit. In a joint launch scenario this can be achieved,
for instance, by launching towards the Sun-Earth libration point L2 and
exploiting the heteroclinic connections towards L1 with spacecraft 3 only (see
fig. 11). Each spacecraft would have to perform small correction maneuvers
shortly after launch to acquire the respective escape trajectories. A similar
strategy will be employed by the ESA Vigil mission [43], which takes the
opposite route (from L1 towards L2).

An alternative joint launch strategy would employ lunar flybys to separate the
Earth-leading and Earth-trailing paths of the spacecraft, similar to the NASA
STEREO mission [44]. This is more complex in terms of trajectory design and
spacecraft navigation, but may have a higher achievable dry mass.

Figure 11: An example of a heteroclinic connection between L2 (right diamond)
and L1 (left diamond) in the Sun-Earth rotating frame with the Earth in the
origin. Using such a trajectory can facilitate an escape from Earth towards
the Sun, even though the launch was away from the Sun. Image courtesy of
Lorenzo Bucci.

##  7 Conclusions and outlook

A mission concept for a space-based gravitational-wave observatory with a sky-
averaged sensitivity two orders of magnitude better at low frequencies than
that of LISA has been studied. A LISAmax-like detector could be a potential
successor of LISA as part of ESA’s Voyage 2050 planning cycle. The analysis of
the operational orbit shows a better stability in terms of corner angle
variations and arm length rates over 10 years than that of LISA without
station keeping. Taking into account the effect of insertion dispersions and
self-gravity-induced accelerations does not significantly alter that picture.
With three dedicated launches using a future European launcher from Kourou, a
mass into science orbit of $6500\text{\,}\mathrm{kg}$ per spacecraft (about
three times the LISA mass) could be achieved. Such a large spacecraft mass is
mainly motivated by the assumption of a larger telescope, assumed to be
$1\text{\,}\mathrm{m}$ in diameter (compare this to $30\text{\,}\mathrm{cm}$
for LISA). It is not clear whether such a large telescope is actually needed
for a compelling science case. A smaller telescope would significantly reduce
the spacecraft mass and might permit a cheaper joint launch. It would also
deteriorate the instrument sensitivity by increasing the laser shot noise, but
the loss would mainly be in the region around $1\text{\,}\mathrm{mHz}$, where
Galactic confusion noise is expected to be dominant in any case.

The main challenge of the presented mission concept is to retain the ability
to localize gravitational-wave sources, since the constellation is constrained
to the ecliptic plane. A dedicated in-depth study of the LISAmax sky
localization capability is underway. Several mitigation options have been
preemptively discussed in the paper. The considered mission options also show
that LISAmax is the largest constellation that can be achieved without another
leap in mission complexity and cost. Therefore, it is the best starting point
when considering a space-based gravitational-wave detector for the
$\mathrm{\SIUnitSymbolMicro Hz}$ band.

Overall, no show stoppers are identified so far. A LISAmax-like detector would
be a valuable continuation of the LISA program, by extending the sensitive
frequency band below the LISA band and allowing observations of super-massive
black hole inspirals many of years before the merger, and a vast population of
Galactic binary systems. Studies are ongoing to characterize the full science
case of a LISAmax-like $\mathrm{\SIUnitSymbolMicro Hz}$ gravitational-wave
observatory.

The authors would like to thank Oliver Jennrich for valuable discussions. The
image in fig. 11 was kindly provided by Lorenzo Bucci. Jean-Baptiste Bayle
gratefully acknowledges support from UK Space Agency via STFC [ST/W002825/1].

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