Coorbital Restricted Problem and Its Application in the Design of the Orbits of the Lisa Spacecraft

Sci. China Phys. Mech. Astron.

et al. 2010

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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. co-orbital restricted problem, orbit design, orbit optimization, launch energyThe 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. Three spacecrafts will be launched around 2018, and reach their operational orbits around the Sun after 14 months. Observation will last 5 to 10 years.As shown in Figure 1, the three spacecrafts form an equilateral triangle with an arm-length (side length) of around 5×10 6 km. The center of the constellation moves on an Earth-like orbit around the sun, trailing 20° behind the Earth. The angle between the direction from the constellation center to the Sun and the constellation plane is about 60°. Distance variations between the spacecraft will be measured by laser interferometry to detect gravitational waves. In order to avoid additional complications in the challenging interferometry, in order for LISA to operate successfully, it is crucial that the constellation of the three spacecrafts has extremely high stability. More stable constellation makes simplifications in the hardware realizationDue to the combined effect of the eccentricity of the spacecraft orbit and the gravity of the other bodies in the solar system including the Earth, all of the arm length, l, the internal angles of the constellation, β and the trailing angle behind the Earth, θ vary continuously, causing relative velocities between the spacecraft, v r , as an indicator of the Figure 1 LISA constellation moving around the Sun.

Section: The Results Of Orbit Optimizationmentioning

confidence: 99%

Section: Co-orbital Restricted Problem Orbit Design Orbit Optimizatmentioning

confidence: 99%

Orbit design for the Laser Interferometer Space Antenna (LISA)

Xia

Sci. China Phys. Mech. Astron.

et al. 2010

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Section: Co-orbital Restricted Three-body Problem Lisamentioning

confidence: 99%

“…The planned gravitational wave observatory LISA is allowed to be regarded as approximately co-orbital, of which the barycenter of the three spacecraft co-orbits with the earth and the distance between each spacecraft and the barycenter is very small (less than 0.02 AU). We have taken part in the LISA orbit design and optimization; since the required operating period is merely 10 years, we have simply made use of a rough method [3,4] to deal with it, and obtained relatively ideal results.Although the classical restricted three-body problem has already had a wide range of applications, it has not been fully resolved. In recent years, lots of researches into co-orbital motion appear, which are mainly aimed at the quasi-satellite (QS for short) case, such as the work by Mikkola et al [5].…”

mentioning

confidence: 99%

The co-orbital restricted three-body problem and its application

Sci. China Phys. Mech. Astron.

et al. 2010

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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. co-orbital, restricted three-body problem, LISAThe co-orbital phenomena in the motion of natural bodies were paid attention to by astronomers a long time ago. In the year 1918 the Japanese astronomer Hirayama found the orbital elements of some asteroids close to each other (especially semi-major axis a, eccentricity e, and orbital inclination i), the ones with a close a were later called an asteroid group. He proposed that asteroids with at least two out of a, e, i, close to each other be called an asteroid family, and that with all three elements close to each other be called co-orbital, which is discussed in this paper. To discuss the problem of common origin, he also proposed the concept of proper elements [1]. As the number of minor planets having been found is increasing rapidly in recent years, the study of asteroid families has greatly developed. Many methods of the definition and calculation of proper orbital elements have been put forward, and the deeper understanding of many aspects, which includes the statistics, dynamics, and spectral types, of asteroid sub-families, has been brought forth. And a relatively comprehensive summary was given by Lemaitre [2] in 2005.The most well-known cases of co-orbits in natural objects are Trojan groups of asteroids, of which the number of the registered asteroids already found has exceeded 2900. There also exist co-orbital phenomena both in the motion of stars in the same spiral arm of the Milky Way galaxy, and in the motion of natural and man-made satellites. The planned gravitational wave observatory LISA is allowed to be regarded as approximately co-orbital, of which the barycenter of the three spacecraft co-orbits with the earth and the distance between each spacecraft and the barycenter is very small (less than 0.02 AU). We have taken part in the LISA orbit design and optimization; since the required operating period is merely 10 years, we have simply made use of a rough method [3,4] to deal with it, and obtained relatively ideal results.Although the classical restricted three-body problem has already had a wide range of applications, it has not been fully resolved. In recent years, lots of researches into co-orbital motion appear, which are mainly aimed at the quasi-satellite (QS for short) case, such as the work by Mikkola et al. [5]. On the basis of widespread co-orbital phenomena in the motion of natural and man-made celestial objects we put forward the model of co-orbit...

“…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. 18 In 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, 4,19 respectively.…”

Section: Selection Of the Starting Orbits For Optimizationmentioning

confidence: 99%

Methods for Orbit Optimization for the Lisa Gravitational Wave Observatory

et al. 2008

Int. J. Mod. Phys. D

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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.