Methods for orbit optimization for the LISA gravitational wave observatory
ABSTRACT The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA mission for detecting low-frequency gravitational waves in the frequency range from 0.1 mHz to 1 Hz, by using accurate distance measurements with laser interferometry between three spacecraft, which will be launched around 2015 and one year later reach their orbits around the Sun. In order to operate successfully, it is crucial for the constellation of the three spacecraft to have extremely high stability. In this paper, several problems of the orbit optimization of the LISA constellation are discussed by using numerical and analytical methods for satisfying the requirements of accuracy. On the basis of the coorbital restricted problem, analytical expressions of the heliocentric distance and the trailing angle to the Earth of the constellation's barycenter are deduced, with the result that the approximate analytical solution of first order will meet the accuracy requirement of the spacecraft orbit design. It is proved that there is a value of the inclination of the constellation plane that will make the variation of the arm-length a minimum. The principle for selecting the optimum starting elements of orbits at any epoch is proposed. The method and programming principles of finding the optimized orbits are also presented together with examples of the optimization design.
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ABSTRACT: The Laser Interferometer Space Antenna (LISA) is a joint ESA-NASA mission for detecting low-frequency gravitational waves in the frequency range from 0.1 mHz to 1 Hz, by using accurate laser interferometry between three spacecrafts, which will be launched around 2018 and one year later reach their operational orbits around the Sun. In order to operate successfully, it is crucial for the constellation of the three spacecrafts to have extremely high stability. Based on the study of operational orbits for a 2015 launch, we design the operational orbits of beginning epoch on 2019-03-01, and introduce the method of orbit design and optimization. We design the orbits of the transfer from Earth to the operational orbits, including launch phase and separation phase; furthermore, the relationship between energy requirement and flight time of these two orbit phases is investigated. Finally, an example of the whole orbit design is presented.Science China: Physics, Mechanics and Astronomy 04/2012; 53(1):179-186. · 1.17 Impact Factor
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ABSTRACT: Based on large quantities of co-orbital phenomena in the motion of natural bodies and spacecraft, a model of the co-orbital restricted three-body problem is put forward. The fundamental results for the planar co-orbital circular restricted three-body problem are given, which include the selection of variables and equations of motion, a set of approximation formulas, and an approximate semi-analytical solution. They are applied to the motion of the barycenter of the planned gravitational observatory LISA constellation, which agrees very well with the solution of precise numerical integration.Science China: Physics, Mechanics and Astronomy 05/2012; 53(1):171-178. · 1.17 Impact Factor
International Journal of Modern Physics D
Vol. 17, No. 7 (2008) 1021–1042
c ? World Scientific Publishing Company
METHODS FOR ORBIT OPTIMIZATION FOR THE LISA
GRAVITATIONAL WAVE OBSERVATORY
GUANGYU LI∗,?, ZHAOHUA YI∗,†, GERHARD HEINZEL‡
ALBRECHT R¨UDIGER‡,OLIVER JENNRICH§, LI WANG¶, YAN XIA∗
FEI ZENG∗and HAIBIN ZHAO∗
∗Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, P. R. China
†Department of Astronomy, Nanjing University, Nanjing 210008, P. R. China
‡Max Planck Institute for Gravitational Physics, D-30167 Hannover, Germany
§European Space Research and Technology Center, 2200 AG Noordwijk (The Netherlands)
¶China Academy of Space Science, Beijing 100094, P. R. China
Received 31 December 2006
Revised 26 May 2007
Communicated by W.-T. Ni
The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA mission
for detecting low-frequency gravitational waves in the frequency range from 0.1mHz to
1Hz, by using accurate distance measurements with laser interferometry between three
spacecraft, which will be launched around 2015 and one year later reach their orbits
around the Sun. In order to operate successfully, it is crucial for the constellation of
the three spacecraft to have extremely high stability. In this paper, several problems
of the orbit optimization of the LISA constellation are discussed by using numerical
and analytical methods for satisfying the requirements of accuracy. On the basis of
the coorbital restricted problem, analytical expressions of the heliocentric distance and
the trailing angle to the Earth of the constellation’s barycenter are deduced, with the
result that the approximate analytical solution of first order will meet the accuracy
requirement of the spacecraft orbit design. It is proved that there is a value of the
inclination of the constellation plane that will make the variation of the arm-length a
minimum. The principle for selecting the optimum starting elements of orbits at any
epoch is proposed. The method and programming principles of finding the optimized
orbits are also presented together with examples of the optimization design.
Keywords: Coorbital restricted problem; LISA constellation; optimized orbit.
The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA
mission for detecting low-frequency gravitational waves in the frequency range from
0.1mHz to 1Hz. Three spacecraft will be launched around 2015, and reach their
orbits around the Sun after one year. Observation will last for 5 to 10 years. In
2009, LISA Pathfinder will be launched to test the technology’s feasibility.
1022G. Li et al.
Fig. 1.LISA constellation moves around the Sun.
Distance variations between the spacecraft are measured with laser interferom-
etry to detect gravitational waves. In order for LISA to operate successfully, it is
crucial that the constellation of the three spacecraft have extremely high stability.
As shown in Fig. 1, the three spacecraft form an equilateral triangle with an arm-
length (side length) of around 5 × 106km. The center of the constellation moves
on the Earth orbit, trailing 20 degrees behind the Earth. The constellation plane is
tilted 60◦out of the ecliptic.1
Due to the effect of the eccentricity of the spacecraft orbit and the gravity of
the other bodies in the solar system including the Earth, and non-gravitational
effects,2,3the side length, the internal angles, the relative velocities among the
spacecraft and the trailing angle between their barycenter and the Earth vary con-
tinuously. In order to ensure the precision of the distance measurements between
the spacecraft, the variation of those parameters mentioned above should be at
least within the given ranges shown in Table 1 according to the LISA requirements.
Smaller variations may make simplifications in the hardware possible and conse-
quently are highly preferred.
Thus, the problem of optimizing the orbits of the LISA constellation can be
summarized as follows: finding a set of orbital elements ai,ei,ωi,Ωi,Ii,M0i, (i =
1,2,3) to make the following cost function take its minimum:
Q(ai,ei,ωi,Ωi,Ii,M0i) = w1?D2+ w2?α2+ w3?v2
The variation ranges of the parameters of the LISA
ParameterAverage valuePermitted variation
Internal angle α
Relative velocity vr
Trailing angle to Earth θ
5 × 106km
?l = ±5 × 104km
?α = ±1.5◦
?vr = ±15m/s
?θ small enough
Methods for Orbit Optimization for the LISA Gravitational Observatory1023
with appropriately chosen weights wj. The orbital elements are semi-major axis,
eccentricity, argument of perihelion, ascending node, inclination and mean anomaly
at the initial epoch, respectively.4
Since Faller5et al. presented the concept of detecting gravitational waves by
using laser ranging in space in 1985, research on the LISA constellation has been
going on for many years. Vincent6et al. (1987), Folkner7et al. (1997), Cut-
ler8(1998), Hughes9(2002), Hechler10et al. (2003), Dhurandhar11et al. (2005),
Sweetser12(2005) and K. Rajesh Nayak13et al. (2006) have investigated the LISA
constellation from the viewpoint of science and spaceflight project. Because of the
complexity of the space environment in which the LISA constellation moves, and the
extremely high stability of the constellation required in the space project, further
advanced research is expected.
In 1993, Ni14et al. proposed the ASTROD mission concept. Some of its science
objectives are similar to those of the LISA project, and they have been discussed at
three international symposia in 2001, 2005 and 2006. Both LISA and ASTROD use
interferometric laser ranging, and the Doppler effects on transmitted and received
frequencies need to be addressed. LISA’s strategy is to minimize arm-length varia-
tion and relative velocity of the spacecraft. For ASTROD, the arm-length changes
of the three spacecraft are of the same order as the distances between the three
spacecraft and the relative velocities go up to 70km/s with line-of-sight velocities
varying from −20 to +20km/s. For 1064 nm (532 nm) laser light, the Doppler fre-
quency change goes up to 40 (80) GHz. For ASTROD, a strategy that relies on a
different technology is used. The recent development of optical clocks and frequency
synthesizers using optical combs makes this heterodyne problem tractable.15
Orbit design and simulation are two important parts of the pre-phase A study
of ASTROD. Wei-Tou Ni, Chien-Jen Tang, Guangyu Li, and Yan Xia took part in
the research successively,16,17which accumulated a wealth of experience for us to
study the LISA orbits. In February 2006, Guangyu Li and Yan Xia were invited to
visit the Max Planck Institute for Gravitational Physics and took part in research
on the optimization of the LISA orbit design in cooperation with Gerhard Heinzel,
Oliver Jennrich and Albrecht R¨ udiger. This paper presents our results within this
work so far.
In Sec. 2 of this paper, the selection of starting orbits for LISA for optimization
is discussed. In Sec. 3, the motion of the barycenter of the constellation on the
base of the plane coorbital motion is analyzed. In Sec. 4, the optimization algo-
rithm and program structure that are used in this work are introduced briefly. In
Sec. 5, the method to select the optimization orbits at any epoch is presented with
2. Selection of the Starting Orbits for Optimization
The starting orbits is such a set of orbits that determines the starting point of the
optimization trajectory in the orbit space, along which we can get to the targeted
1024G. Li et al.
Fig. 2.Formation of the LISA constellation.
point and obtain the optimization orbits by given optimization algorithms, which
have been discussed in detail in many papers.7,11–13A brief review is given as the
beginning of our discussion.
As is shown in Fig. 2, the Sun is located at point S while the horizontal circle is
the orbit of the Earth, and three spacecraft SC1, SC2 and SC3 compose the LISA
constellation, the barycenter of which, point C, moves along the orbit of the Earth
coorbitally.18In this paper, we assume that the masses of the three spacecraft are
identical, so the barycenter of the constellation coincides with its geometrical center.
The inclined ellipse is the osculating orbit of spacecraft SC1 at the initial epoch
t = 0, with a,e,I and f as semi-major axis, eccentricity, inclination to the ecliptic
plane and true anomaly,4,19respectively. Assuming spacecraft SC1 is located at its
aphelion at the initial epoch, we have f0 = M0 = π here, where f0 and M0 are
the true anomaly and mean anomaly respectively at the time t = 0. Defining φ
as the angle between the constellation plane in which the three spacecraft move
and the ecliptic plane, the important relationship between the eccentricity e, the
inclination I to the ecliptic plane of the spacecraft orbit, the average arm-length l
and the angle φ is expressed by the following equations:
√3cosφ − 1,
√3(1 + e),
√3 + lcosφ
√3(1 + e)
Methods for Orbit Optimization for the LISA Gravitational Observatory1025
Putting the Sun, S, at the origin, the ecliptic plane as the base plane and the
direction from the origin to the projection of SC1 on the ecliptic plane as x-axis,
a coordinate system, which is a non-inertial rotating coordinate system, is set up.
Defining Ω and ω as the ascending node and the argument of perihelion4,19of the
spacecraft orbit in this system, the radius vector of SC1 can be expressed as
r = R3(Ω)R1(I)R3(ω + f)(r,0,0)τ,
the superscript τ denoting the transpose operation of a matrix, Rkis the rotating
matrix around the kth axis (k = 1,3):
, R3(α) =
As I is a small quantity, 1 and I could be regarded as the approximate values of
cosI and sinI, respectively. Therefore
R1(I) = I + I
where I is the unit matrix. Substituting (7) into (5) and simplifying it, we get
r = r(cos(Ω + ω + f), sin(Ω + ω + f), sin(ω + f)I)τ.
From Fig. 2, we can see that both the Ω and ω of the spacecraft SC1 equal to 3π/2.
Then at t = 0,
Ω + ω + M0= 0 (mod 2π).
To form the constellation, it is a necessary condition for all the three spacecraft
that each of their orbits keeps the relationship described in Eq. (8). Numerical
calculation indicates that the better way to select the other two orbits is:
(i) Rotating the orbit of SC1 around the z-axis by π/3 and 2π/3, respectively, to
form the orbits of SC2 and SC3;
(ii) Adjusting the mean anomalies of SC2 and SC3 to keep the relationship (8).
Thus the five intial orbit elements of the three spacecraft for optimization are shown
in Table 2. The unit of the semi-major axis a is the astronomical unit AU.
From Eq. (8), we gain
Ω + ω + f = f − M0.
Substituting it into expression (7), we get
r = r(cos(f − M0), sin(f − M0), − cos(f)I)τ.