This paper investigates the possible application of periodic orbit control maneuvers for so-called evolved-LISA (eLISA) missions, i.e., missions for which the constellation arm lengths and mean distance from the Earth are substantially reduced. We find that for missions with arm lengths of 106 km and Earth-trailing distance ranging from approx. 12deg to 20deg over the science lifetime, the occasional use of the spacecraft micro-Newton thrusters for constellation configuration maintenance should be able to essentially eliminate constellation distortion caused by Earth-induced tidal forces at a cost to science time of only a few percent. With interior angle variation kept to approx. +/-0:1deg, the required changes in the angles between the laser beam pointing directions for the two arms from any spacecraft could be kept quite small. This would considerably simplify the apparatus necessary for changing the transmitted beam directions

Bender, Peter L.

Welter, Gary L.

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**Volume Title**
ASP Conference Series, Vol. **Volume Number**
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c©**Copyright Year** Astronomical Society of the Pacific
Possible Periodic Orbit Control Maneuvers for an eLISA Mission
Peter L. Bender1 and Gary L. Welter2
1JILA, Univ. of Colorado and Nat. Inst. of Standards and Technology, USA
2National Aeronautics and Space Administration/Goddard Space Flight
Center, Software Engineering Division, USA
Abstract.
This paper investigates the possible application of periodic orbit control maneu-
vers for so-called evolved-LISA (eLISA) missions, i.e., missions for which the con-
stellation arm lengths and mean distance from the Earth are substantially reduced. We
find that for missions with arm lengths of ∼ 106 km and Earth-trailing distance rang-
ing from ∼ 12◦ to 20◦ over the science lifetime, the occasional use of the spacecraft
micro-Newton thrusters for constellation configuration maintenance should be able to
essentially eliminate constellation distortion caused by Earth-induced tidal forces at a
cost to science time of only a few percent. With interior angle variation kept to ∼ ±0.1◦,
the required changes in the angles between the laser beam pointing directions for the
two arms from any spacecraft could be kept quite small. This would considerably sim-
plify the apparatus necessary for changing the transmitted beam directions.
1. Introduction
A gravitational wave mission called the Laser Interferometer Space Antenna (LISA)
was under study for some time as a proposed joint mission of ESA and NASA (ref Sal-
lusti et al. (2009); ref ESA (2011)). It consisted of a nearly equilateral triangle formed
by three spacecraft, with 5-million-km long baselines between them. Changes in the
distances between reference test masses in the different spacecraft due to gravitational
waves would be measured by laser interferometry along the baselines. The triangle
was to be located about 20◦ behind the Earth, in a one year period nearly circular orbit
around the Sun.
In order to minimize variations in the lengths of the baselines, each spacecraft
would be given an eccentricity e of about 0.01 and an inclination of
√
3 times this
size (ref Sweetser (2005a); Povoleri & Kemble (2006)). With proper initial conditions
for the orbits, the plane of the triangle would be tipped at 60◦ to the ecliptic. From
Kepler’s laws, neglecting planetary perturbations, the amplitude of the time variations
in the distances Di j between the spacecraft pairs (i j = “12”, “23”, “31”) along the
interferometer arms could be made as small as e/2 times the mean arm length D, or
about 0.5% (ref Sweetser (2005b), eq. 30):
δDi j = (15/32)eD cos[L + φi j] + (1/32)eD cos[3(L + φi j)] (1)
Here L = nt is the mean longitude of the centre of the triangle, n is the mean
motion, and φi j is a phase angle of 0◦, 120◦, or 240◦. The corresponding variations in
the angles of the triangle would be about 0.5◦.
1
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2 Bender and Welter
The preceding description was without consideration of the Earth’s gravitational
perturbations. However, even 20◦ away from the Earth, the gravitational perturbations
would cause substantial secular variations in the triangle geometry. By choosing the
orbit initial conditions carefully, the amplitude of the variations in the angles of the
triangle for the original LISA mission could be kept down to about 0.8◦ over a roughly
8-year total mission lifetime. In order to accommodate such changes in the angles
between the directions in which the laser beams from a given spacecraft had to be
pointed, each transmitting telescope and its associated optical system was mounted in
an “optical assembly” that could be rotated by small amounts with respect to the main
spacecraft structure.
We will take the main characteristic of evolved LISA (eLISA) type missions (ref
Danzmann (2012)) to be 1-million-km arm lengths. To understand the situation better,
we first consider what would happen if there were no perturbations due to the Earth or
planets. In this case, the natural Kepler variations in the angles of the triangle would be
reduced by a factor of 5 with respect to those for LISA, to an amplitude of 0.090◦.
With the perturbations from the Earth turned on, if the antenna were at 20◦ from
the Earth, the differential accelerations due to the Earth would be reduced by a factor
of 5 compared with those for LISA. Thus it is of interest to consider the possibility of
scheduling periodic orbit correction maneuvers in order to compensate for the differ-
ential accelerations due to the Earth, and keep the angle variation amplitudes down to
about 0.090◦ over the whole mission lifetime.
It will be assumed here that the normal operation of the mission would be inter-
rupted every six days in order to reorient the high gain communications antennas on the
three spacecraft and do checks on the optical systems. The necessary orbit corrections
would be done at these times using the micro-Newton thrusters. The main drawback
would be the loss of a few percent of the science data during the period of the thrusting.
The requirements to achieve this will be discussed in the rest of this article.
2. Periodic Thrust Requirements
The orbits currently being considered for an eLISA type mission (ref Danzmann (2012);
Stebbins (2012)) are drift-away orbits, where the distance from Earth is increasing with
time at a rate of up to 6◦/yr. The initial distance from Earth at the beginning of the
commissioning period for the mission would be 9◦ or more. As an example, an initial
separation of 12◦ and a drift rate of 5◦/yr over a 6 yr total mission lifetime will be
assumed here. For this case, in comparison with starting 9◦ from the Earth and drifting
away at 6◦/yr, the extra δV required would be ∼60 m/s.
From eq. 30 of Sweetser (ref Sweetser (2005b)), the second time derivatives of
the arm lengths between spacecraft due to the Earth are given approximately by:
D¨23 =
√
3K [cos(2L) − 0.5] , D¨12 =
√
3K [cos 2(L − 120◦) − 0.5] ,
D¨31 =
√
3K [cos 2(L + 120◦) − 0.5] . (2)
Here K = GMD/[(
√
3)(S 3)], where G is the gravitational constant, M is the mass of
the Earth, D is the arm length, and S is the mean distance from the Earth. The eccen-
tricity of the Earth’s orbit is neglected. For 12◦ from the Earth, K = 7.5 × 10−9 m/s2.
The geometry for periodic orbit corrections using the micro-Newton thrusters on each
spacecraft is shown in Fig. 1. It is assumed that radial and tangential accelerations
Orbit Control Maneuvers 3
Figure 1. LISA periodic orbit control concept
(Ri,Ti) of the i-th spacecraft with respect to the centre of the triangle can be produced
by the thrusters. For the LISA mission, the numbers that have been considered pre-
viously for the maximum thrust in each direction range from roughly 30 µN to 150
µN, and we will assume 80 µN here. If the spacecraft mass is roughly 660 kg, this
would give a maximum spacecraft acceleration in the radial or transverse direction of
1.2 × 10−7 m/s2.
If the thrusters could be used continuously to cancel out the time-varying part of
the differential accelerations due to the Earth, the following combination of accelera-
tions due to the thrusters would do what is needed:
R1 = K cos(2L), R2 = K cos 2(L + 120◦), R3 = K cos 2(L − 120◦) (3)
T1 = K sin(2L), T2 = K sin 2(L + 120◦), T3 = K sin 2(L − 120◦). (4)
However, continuous thrusting cannot be used because the resulting acceleration
of each spacecraft would have to be applied to the test masses inside it also to keep them
centered in their housings. Thus noise in the required forces during regular science
operations would be a serious concern, and periodic orbit corrections would be used
instead.
Some of the thrust from the micro-Newton thrusters on each spacecraft will be
required to cancel out the effect of the solar radiation pressure. Thus it will be assumed
here that 70 µN is the maximum thrust available in the radial and transverse directions
for use in the periodic orbit correction maneuvers. With 660 kg for the spacecraft mass
and orbit corrections every six days, the time necessary for the corrections would be
10.2 hr initially at 12◦ from the Earth. However, this would be reduced to 6.9 hr at the
4 Bender and Welter
end of the commissioning period, about four months later. The result would be the loss
of 4.8% of the science data. One year later, the required time would be further reduced
to 2.7 hr, corresponding to 1.9% loss of the science data.
The Gravitational Reference Sensors on the spacecraft are assumed to be essen-
tially the same as those that have been developed for use on the LISA Pathfinder mission
(ref Weise et al. (2008)). To keep each test mass centered in its housing during the orbit
correction periods, the maximum voltage needed on the capacitor plates surrounding
the test mass would be under 40 volts rms. This is larger than the voltage range pro-
vided when the “fine control mode” is being used for the test masses, but well within the
range for the “coarse control mode”, which is required during initial release of the test
masses. The extra acceleration noise that is present during the use of the coarse mode
would not be sufficient to interfere with keeping the interferometric links between the
spacecraft in nearly normal operation.
3. Advantages of Keeping the Angle Variations Small
For the proposed NGO mission, the expected amplitude of the angle variations at each
spacecraft over the six years of nominal plus extended mission lifetime was about 0.8◦.
This amplitude was similar to that for the LISA mission. The effect of the shorter
baselines was compensated for by the smaller initial distance from the Earth. The mass
of the optical assemblies that had to be rotated with respect to a spacecraft was less
than for LISA because of the telescope diameter being reduced from 40 cm to 20 cm.
However, the optical assembly still had to be locked in place during launch, and to have
very small pointing jitter during operation on orbit.
An approach that had been considered earlier for LISA was to keep each telescope
fixed with respect to the main spacecraft structure, but to accomodate the required angu-
lar motion of the transmitted laser beam by having a small rotating mirror on the optical
bench before the telescope that would change the direction of the beam (ref McNamara
(2012)). This was called “in-field guiding”. It was regarded as feasible, but requiring
careful design and testing in order to prevent slight translational jitter in the effective
mirror position from causing noise in measuring changes in the arm length differences
between spacecraft.
With the considerably smaller angle variations that would be needed if periodic
orbit corrections are planned on and made, it appears that the use of in-field guiding
would be quite attractive for an eLISA mission. Careful design and testing of the mirror
rotation system still would be needed, but having a considerably smaller rate of angular
change to correct for would be a substantial advantage. With in-field guiding plus the
expected smaller telescope diameter for eLISA, the reduction in payload complexity
would be expected to be significant. And the extra fuel needed for the micro-Newton
thrusters used in periodic orbit corrections would be small because of the small duty
cycle of the extra thrusting.
One issue to consider, if the in-field guiding range is restricted to plus and minus
0.1◦, or perhaps 0.15◦, is what would happen if the thrusters on one spacecraft lost the
capability of providing the assumed level of thrust. This has not been investigated, but
it seems likely that other spacecraft could take over the necessary thrusting responsibil-
ities with only a moderate increase in the loss of observing time.
The possibility of applying orbit corrections to the LISA spacecraft in order to
keep the effects of the Earth on the triangle geometry from increasing with time was
Orbit Control Maneuvers 5
discussed some time ago (ref Bender (2006)) by one of us. However, at the time it
was assumed to be sufficient to cancel out the annual averages of the perturbations
due to the Earth, and this was incorrect. Canceling out the semi-annual variations in
the differential accelerations actually is necessary, so that much higher compensating
accelerations would be needed than calculated in Bender (ref Bender (2006)).
4. Conclusions
It appears that planning on including periodic orbit correction maneuvers is worth con-
sidering for any eLISA type mission. No modifications to, or additional requirements
on, the micro-Newton thrusters are required, and the main expected impact is a reduc-
tion of science observation time on the order of a few percent. This approach would
make the use of in-field guiding for the spacecraft optical systems considerably sim-
pler, and probably would contribute to reducing the payload mass. Simulations of the
expected orbit correction strategy certainly are needed, but preliminary estimates of
what would be required are encouraging.
It should be emphasized that the required thrust estimates in this paper are based
on a simplified model where the line between the eLISA triangle of spacecraft and the
Earth is assumed to be perpendicular to the direction toward the Sun, and the Earth is
in a circular orbit. Thus there will be some small components of the differential forces
that are perpendicular to the plane of the triangle. However, it is expected that enough
out-of-plane thrust capability will be available to counteract such forces.
Acknowledgements: Discussions with Bill Weber at Trento and Dennis Weise at
Astrium on various aspects of the suggested approach are appreciated. This work was
supported in part under grant NNX11A940G from NASA to the University of Colorado,
and by funding from Physics of the Cosmos (PCOS) Office of the NASA Science Mis-
sions Directorate (SMD). One of us (PLB) would like to thank Angelo Povoleri from
EADS Astrium/UK for pointing out the error in Bender (ref Bender (2006)).
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Possible Periodic Orbit Control Maneuvers for an eLISA Mission

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