Orbit Optimization for a Space-Based Gravitational Wave Detector

Transcription

1 LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY - LIGO - CALIFORNIA INSTITUTE OF TECHNOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY Technical Note LIGO-T v1 2015/09/26 Orbit Optimization for a Space-Based Gravitational Wave Detector Elvira Kinzina, Rana Adhikari, Thomas Callister, Curt Cutler California Institute of Technology LIGO Project, MS Pasadena, CA Cambridge, MA Phone (626) Phone (617) Massachusetts Institute of Technology LIGO Project, Room NW Fax (626) Fax (617) info@ligo.caltech.edu info@ligo.mit.edu LIGO Hanford Observatory LIGO Livingston Observatory Route 10, Mile Marker LIGO Lane Richland, WA Livingston, LA Phone (509) Phone (225) Fax (509) Fax (225) info@ligo.caltech.edu info@ligo.caltech.edu

2 1 Abstract Gravitational waves (GWs), oscillations of spacetime that propagate at the speed of light, may soon become a very helpful tool for exploring the universe from an absolutely novel point of view. All gravitational wave detectors existing at the moment are ground-based and therefore are limited by seismic noise at frequencies below approximately 10 Hz. Thus new concepts of space detectors have been presented. One possibility is the Laser Interferometer Space Antenna (LISA), a constellation of three satellites which forms an equilateral triangle with million kilometer long arms. But due to its large cost it will likely not be launched until the mid-2030s. Thus other new simplier and cheaper LISA-like missions have to be proposed. One alternative to LISA is a smaller mission, using 100 kilometer arms and fewer lasers. In order to measure gravitational waves, this satellite formation must remain stable over a several year period. Here we find the most stable orbital configuration by determining the optimal parameters of the satellites such as initial positions of the spacecraft and their velocities, the semimajor axis of a desired orbit, inclination and arm length. 2 Introduction Gravitational waves were already implied in Albert Einstein s works a century ago. In 1916 he showed that accelerating massive bodies generate time-dependent gravitational fields that propagate at the speed of light as perturbations of spacetime [1]. Half a century afterwards the existence of GWs was indirectly proved by Hulse and Taylor [5], who were awarded the 1993 Nobel Prize in Physics for the discovery of a binary pulsar. Their high-precision measurements of the orbital period of this binary star system revealed that the pulsar s orbit is gradually contracting at just the rate predicted by general relativity if the binary was emitting GWs [7]. However, GWs have not been directly observed yet due to the weakness of gravitational interaction. Therefore, new interferometry techniques, filters for different types of noise sources and upgrades in sensitivity of detectors are essential. 3 GW detectors Several gravitational wave detectors have already been or are being constructed. The ground-based detectors instruments like LIGO, Virgo, GEO600 and TAMA300 perform measurements in the high-frequency band where 10 f 10 4 Hz. They are limited at low frequencies by seismic and gravity-gradient noise. Space detectors: LISA, elisa, BBO, DECIGO and ALIA and others are intended to work in the low-frequency band with 10 5 f 1Hz. Some of these detectors will likely not be launched for many more decades. The most optimistic date for at least one of these detectors to be launched is around Therefore, new concepts for a space GW detector that could be lauched earlier are needed. page 1

3 3.1 LIGO The Laser Interferometer Gravitational-wave Observatory (LIGO) is a pair of L-shaped Michelson laser interferometers with 4 km long arms: one in Hanford, Washington, the other in Livingston, Louisiana. They are about 3000 km apart. When gravitational waves pass they distort the space and change the distance between the mirrors. This leads to accumulation of the phase which can be observed on the photodetector. The need for the second interferometer is based on, first of all, the ability to filter out local noise. Secondly, by measuring a time delay between signals from different detectors with the use of triangulation techniques we can determine sky location of the object emitting GWs as well. 3.2 LISA The Laser Interferometer Space Antenna (LISA) and its evolved and cheaper version elisa have three spacecraft which form an equilateral triangle with million kilometre arms (5 million km for classic LISA, 1 million km for elisa) and share Earth s orbit, trailing the Earth by 20. Each of LISA s three spacecraft contains two telescopes, two lasers and two test masses, arranged in two optical assemblies pointed at the other two spacecraft (Figure 1). From the central satellite, a laser beam is sent to the others using a large beam expanding telescope. [3] Because of the extreme lengths of the interferometers arms, Fabry Perot interferometry as in the ground-based detectors is not possible: diffraction spreads the laser beam over a diameter of about 20 km as it propagates from one spacecraft to the other. A portion of that 20 km wavefront is sampled with the telescope. That light is then interfered with a sample of light from the on-board laser. Each spacecraft thus generates two interference data streams. Thus six signals are totally generated by the LISA (the new elisa concept has only two interferometric links between spacecraft). From these signals, we can construct the time variations of armlengths. Figure 1: LISA s scheme and orbit The LISA configuration also undergoes complex motion as it orbits the Sun. Let us suppose page 2

4 that the spacecraft A is at its apastron at initial moment. This means that the distance from the spacecraft A to the sun at this moment (let it be r A ) is the maximum of distances from any of the spacecraft to the sun at any moment of the mission. At the same starting time the distances between the spacecraft B, C and the sun are equal to each other and are shorter than r A. These spacecraft will move further from the sun within a half a year until their distance from the sun will be equal to r A (while the spacecraft A will move closer to the sun) and after that they will move closer to the sun for another half a year until they reach their initial positions. This will lead to the rotation of the triangle around its center of mass with a period of one year as shown in the Figure 2. This rotation helps the detector to localize sources of GWs in the sky. Figure 2: Rotation of LISA around its center of mass 3.3 UNGO The goal of this project is to come up with a new smaller and cheaper LISA-like detector. Its preliminary name is UNGO. The sensitivity of this detector is expected to be between one of the LISA and Advanced LIGO (Figure 3). According to this sensitivity plot UNGO could provide an early warning system for smaller mass binaries before they enter the band of ground-based detectors. The calculated sensitivity curve for UNGO is represented in the Figure 4. 4 Objectives How might one design a detector which would be cheaper than LISA? One option is to use shorter arms. Another is to place the detector closer to Earth in order to save on fuel for delivery to an initial position. But if UNGO is too close to Earth the attraction of the planet will make an orbit of the spacecraft unstable so that it will not be able to detect any GWs. page 3

5 Figure 3: A plot of the square root of power spectral density against frequency for a variety of detectors and sources. Figure 4: The calculated sensitivity curve for UNGO. Here balance between cost and stability should be found. Fewer laser links would also make the detector cheaper. The plan for UNGO is to have only two pair of links instead of three as in LISA (Figure 5). The characteristics which are directly related to the cost of the mission page 4

6 are presented in the Table 1. Figure 5: LISA s (left) and UNGO s (right) laser links Table 1: Comparison of LISA and UNGO Characteristic LISA UNGO initial arm length, km trailing angle, 20 < 20 number of lasers Requirements for the orbit Since spacecraft orbits have non-zero eccentricity and due to gravitational forces from the planets in the solar system arm lengths, internal angles and relative velocities among the spacecraft vary continuously. In order for the orbit of the spacecraft to be stable it is essential that the arm lengths of the detector do not change significantly. The length change should not exceed a couple percent of the initial arm length. And relative velocities have to be less than 15 m/s so that it would be possible to reduce the Doppler effect. [4] 6 Methods In order to understand the optimal orbital parameters for UNGO, we need to numerically simulate the satellites orbits around the sun. 6.1 Determination of initial parameters To perform the numerical integration of the satellites orbits we must first determine the satellites initial positions and velocities. Let us call vertices of the triangle A, B and C. In order to uniquely identify a specific orbit of a spacecraft it is necessary to know an initial page 5

7 position of the spacecraft (x, y, z) and its initial velocity (v x, v y, v z ) or another six parameters called Keplerian elements: a, e, i, ω, Ω, f, these are a semimajor axis, eccentricity, inclination, argument of periapsis, longitude of the ascending node and true anomaly respectively. The first five parameters determine the exact orbit and the last one determines the position of the spacecraft on this orbit. The initial steps of choosing the orbits for the spacecraft are the same as for LISA. These two papers have been used for determing initial parameters in this project [4, 6]. The only difference is up to now the arm length. The Sun is placed in the origin and the base plane is the ecliptic plane. For the center of mass of the triangle we have COM = (AU, 0, 0), (1) where AU is the astronomical unit. The direction from the origin to the projection of A on the ecliptic plane is put as the x-axis. The initial positions of three spacecraft A, B and C are determined accodring to following formulae. A = (COM x + R, COM 2 y, COM z + 3 R) 2 B = (COM x r 2, COM y + l, COM 2 z 3 r) (2) 2 C = (COM x r, COM 2 y l, COM 2 z 3 r) 2 where l is the arm length, the radius of the circumscribed circle R = the inscribed circle r = R. 2 l 3, and the radius of The spacecraft A is supposed to be at aphelion at the initial moment. The semimajor axis for the orbit is assumed to be equal to one astronomical unit (AU). Since apocenter distance r ap = (1 + e)a = r A = a = AU (3) ( a + R ) 2 + ( R) 2 = a2 + ar + R 2, (4) the eccentricity is equal to e = r ap a 1 = 1 + R a + ( R a ) 2 1 (5) The inclination can be determined as i = arcsin A 3 z = arcsin R 2 AU + A x AU + R 2 (6) According to the initial position of the spacecraft A being at its aphelion, the argument of periapsis: ω = 3π 2 page 6 (7)

8 These four parameters (a, e, i, w) are the same for all orbits in the formation. The longitude of the ascending node, Ω, and the mean anomaly, M are different for each orbit and given by: A: Ω, M B: Ω + 2π 3, M 2π (8) 3 C: Ω 2π, M + 2π 3 3 Since the vertice A is situated at aphelion at t = 0 and the projection of its radius-vector on the ecliptic plane is x-axis, Ω = 3π and M = π. 2 Such a choice of parameters makes the orbits of the three spacecraft equal ellipses with different positions in space (Figure 6). They coincide with each other when rotating around the z-axis by 2π 3. Figure 6: Orbits of spacecraft Using these parameters it is possible to calculate initial velocities of A, B and C. To do this it is convenient to convert Keplerian elements to Cartesian coordinates. A great number of papers on conversion of Keplerian elements to Cartesian elements have been published. For instance this paper is very straightforward [2]. 6.2 The orbit calculation without Earth With the initial positions and velocities of three spacecraft we can calculate the orbits taking into account only the attraction force from the Sun by solving a system of differential page 7

9 equations: ẋ = v x ẏ = v y ż = v z v x = GMsx r 3 v y = GMsy r 3 v z = GMsz where M s is a mass of the Sun and r is the distance of a spacecraft from the Sun. A numerical solution of this one-body problem is obtained with Dormand Prince 5 method which is a member of the Runge-Kutta family of Ordinary Differential Equations (ODE) solvers and calculatee fifth-order accurate solutions. The convergence graphs for Dormand Prince 5 method in the case without Earth are represented in the Figure 7. r 3, (9) Figure 7: Convergence of Dormand Prince 5 method for the arms AB and BC respectively without Earth The number of steps taken to obtain the resultes represented in the subsection 7.1 was The orbit calculation with Earth The initial parameters are the same as if it was with only the Sun but the system of ODE is slightly different: ẋ = v x ẏ = v y ż = v z v x = GMsx GMe(x xe) r 3 re 3 v y = GMsy GMe(y ye) r 3 re 3 v z = GMsz, r 3 GMez r 3 e, (10) where M e is a mass of Earth, r e is the distance of a spacecraft from Earth and x e and y e are coordinates of Earth at given moment of time. For simplicity the orbit of Earth is considered to be circular. page 8

10 The convergence graphs for Dormand Prince 5 method in the presence of Earth are represented in the Figure 8. Figure 8: Convergence of Dormand Prince 5 method for the arms AB and BC respectively in the presence of Earth The number of steps taken to obtain the resultes represented in the subsection 7.2 was Results 7.1 The orbits without Earth For detailed description of the orbit determination look at the subsection 6.2. In the case without Earth the maximum relative arm length change is less than 1% (Figure 9) and the maximum relative velocities are about 0.02 m/s (Figure 10). In other words the orbits meet the requirements. Here Figure 9 shows the change in detector arm length as a function of time, and Figure 10 shows the relative velocities of the satellites. Figure 9: The change of the arm length for the arms AB and BC respectively without Earth The arm length change for the arm AB is much larger than for the arm BC since vertices B and C are symmetric with respect to the x-axis, i.e. they are symmetrically closer to each other than to A. In order to know how the formation would behave if this ideal case did not take place it is necessary to break this symmetry which should be done in future work. page 9

11 Figure 10: Relative velocities between A and B and between B and C respectively without Earth 7.2 The orbits with Earth The results including Earth are presented for the case with the trailing angle of 20. The relative arm length change hardly meets the requirements with its maximum value around 4% (Figure 11) while the relative velocities are still within the same range (as without Earth) way below the upper limit (Figure 12). Here again Figure 11 shows the change in detector arm length as a function of time, and Figure 12 shows the relative velocities of the satellites. Figure 11: The change of the arm length for the arms AB, BC and AC respectively The arm length change and velocities still meet the requirement but the maximum relative arm length change goes above 2% with the decreasing trailing angle. The inclination and page 10

12 Figure 12: Relative velocities between A and B and between B and C respectively initial position of the detector can be changed to decrease this arm length change. 7.3 Inclination Inclination of the triangle to the ecliptic can influence the maximum arm length change within the lifetime of the mission. Figure 13 shows the maximum relative arm length change as a function of the inclination. From this figure it is clear that the inclination of 60 minimizes the change in arm length. Such inclination is called optimal. Figure 13: The dependence of maximum arm length change on inclination for AB and BC respectively without Earth In the presence of Earth optimal inclination is not 60 anymore. The exact optimal inclination depends on the trailing angle between the detector and Earth. In the Figure 14 we can see how optimal inclination changes with trailing angles varing from 20 to 10 degrees. The curves for missions with different lifetimes have different colors. 7.4 Trailing angle It is quite obvious that the smaller the trailing angle the larger the arm length change due to stronger tidal forces from Earth to the spacecraft. Here Figure 15 shows the maximum arm page 11

13 Figure 14: The dependence of maximum arm length change on inclination with Earth length change at the optimal inclination as a function of the trailing angle, and Figure 16 shows the relative velocities of the satellites at the optimal inclination as a function of the trailing angle. From these figures we can conclude that with a trailing angle below 10 the orbits become highly unstable. Figure 15: The dependence of maximum arm length change on trailing angle at the optimal inclination with Earth Above 10 the relative velocities remain below the upper limit of 15 m/s. However, to keep arm length change within the limits we have to sacrifice the mission lifetime. 8 Conclusion The modelling of the orbits for a simpler and cheaper LISA-like detector has been conducted. It revealed that the cost of construction and launching of a space detector indeed can be page 12

14 Figure 16: The dependence of maximum relative velocities on trailing angle at the optimal inclination with Earth reduced by the use of shorted arms, fewer lasers and placing the detector closer to Earth. With reasonable inclination, trailing angle and mission lifetime it is possible to keep the orbits of the spacecraft composing the detector stable within this time. References [1] Barry C Barish and Rainer Weiss. Ligo and the detection of gravitational waves. Physics Today, 52:44 50, [2] M Eng and René Schwarz. Keplerian orbit elements cartesian state vectors [3] Éanna É Flanagan and Scott A Hughes. The basics of gravitational wave theory. New Journal of Physics, 7(1):204, [4] Steven P Hughes. Preliminary optimal orbit design for the laser interferometer space antenna(lisa). Advances in the Astronautical Sciences, 111(1):61 78, [5] Russell A Hulse and Joseph H Taylor. Discovery of a pulsar in a binary system. The Astrophysical Journal, 195:L51 L53, [6] Guangyu Li, Zhaohua Yi, Gerhard Heinzel, Albrecht Rüdiger, Oliver Jennrich, Li Wang, Yan Xia, Fei Zeng, and Haibin Zhao. Methods for orbit optimization for the lisa gravitational wave observatory. International Journal of Modern Physics D, 17(07): , [7] Joel M Weisberg and Joseph H Taylor. Relativistic binary pulsar b : Thirty years of observations and analysis. arxiv preprint astro-ph/ , page 13

15 9 Acknowledgments I would like to thanks my mentors Rana Adhikari, Curt Cutler and, of course, Tom Callister for their guidance, my co-worker Jared Goldberg for fruitful discussions, LIGO Scientific Collaboration for giving me the opportunity to participate in this program, Caltech SURF for organising this internship and NSF for funding. page 14