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Asteroid Mass Estimates from Synchronized Flybys

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Asteroid Mass Estimates from Synchronized Flybys JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Engineering Notes Asteroid Mass Estimates from flight ranging and Doppler shift, measurements that are enabled through the existing systems of most spacecraft, as well as optical Synchronized Flybys of Multiple navigation to assess the flyby velocity. An overview of the expected Spacecraft sensing required is described in Table 1. This method has been constructed as an analytic problem to prove that unique solutions exist for the multiple-spacecraft flyby case. It has thus been compared William Crowe,∗ John Olsen,† and John R. Page‡ to analytic approximations for the current state of the art, although the University of New South Wales, Sydney, authors recognize that all current mass estimates conducted today consist of numerical analyses that converge on a solution and account New South Wales 2052, Australia for the many other perturbations that exist in deep space. DOI: 10.2514/1.G002907 To allow this method to be solvable mathematically, the problem has been restricted in how the multiple spacecraft are oriented before flyby. However, it is envisioned that a more robust setting could be allowed with accommodation for small changes in velocity and position with linearized approximations. A further consideration to I. Introduction acknowledge is that single measurements are required by this OME of the most accurate estimates of asteroid masses made to solution before and after flyby, although there is likely to be a large S date have come from their gravitational perturbation acting on quantity of noisy data produced, requiring nonlinear filters to account passing spacecraft [1–3]. These are only second in accuracy to for uncertainty and a least-squares regression of the data. measurements of perturbations to spacecraft in closed orbits around asteroids [4]. The perturbation has the effect of deflecting the spacecraft into a hyperbolic trajectory. To measure this perturbation, II. Problem Definition a ground station on Earth detects the Doppler shift that occurs in the A. Hyperbolic Trajectory communications frequency of the spacecraft as the orbit of the A hyperbolic trajectory is one where a secondary body flies past a spacecraft is perturbed by the gravitational pull of the asteroid. This primary body in a nonclosed orbit and has a positive velocity at provides a change in velocity of the spacecraft relative to the ground infinity. The motion can then be approximated to analytic equations, station. This can then be used to find the hyperbolic trajectory shift in which have been applied and derived in the past for other flyby and ’ asymptote, which, when combined with the spacecraft s minimum planetary capture studies [7–10]. The orientation of a hyperbolic passing distance from the asteroid, can allow for a mass estimate. trajectory has been displayed in Fig. 1, similarly described in One disadvantage of this strategy is that there must be a direct introductory astrodynamics texts [11]. measurement using a link with an Earth-based ground station. In Figure 1 shows a hyperbolic trajectory, where the secondary body addition to this, the passing distance from the asteroid center of mass (the spacecraft, denoted s/c) is assumed to have negligible mass when must be estimated using ephemeris data, which can lead to errors of compared to the primary body. In this work, the gravitational effects kilometers [1]. Last, there is a need for the spacecraft to remain a safe of other bodies are ignored, and so the secondary body will initially distance away from an asteroid as it passes, which in turn reduces the fly at a velocity V∞ along a straight line (asymptote a) toward point C, perturbation on the spacecraft and so decreases the accuracy of the which is located on the apse line from the center of gravity of the mass estimate. asteroid by a distance of rp − a, where rp is the radius at the periapsis, This Note presents a novel method to measure an asteroid gravity and a is the semimajor axis (which is negative for hyperbolas). The field by using several spacecraft making hyperbolic flybys. This can B-plane target B and the “true anomaly of the asymptote” angle θ∞ be done by using relative measurements of range and velocity are defined in Fig. 1. The eccentricity e is related to B and a through between spacecraft, rather than through a communications link Eq. (1) and to θ∞ through Eq. (2) [Eqs. (1–5) have been adapted from Earth, meaning that regular contact with Earth is not required. from [11]]: When three or more spacecraft make a flyby, the passing distance p can also be estimated, eliminating the need for estimates based on B −a e2 − ephemeris data. 1 (1) Other studies have suggested that multiple spacecraft can be used Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 23, 2017 | http://arc.aiaa.org DOI: 10.2514/1.G002907 to make gravity measurements to estimate asteroid mass [5,6]. The θ − 1 method presented in this Note requires only a small suite of sensors to cos ∞ e (2) make the necessary measurements, making the approach practical for less sophisticated spacecraft. In this case, it will be using time-of- The gravitational field of the primary body will cause the secondary body to accelerate toward it, increasing its velocity and Received 23 May 2017; revision received 14 August 2017; accepted for changing its direction as it passes the primary body. At positive publication 31 August 2017; published online 27 September 2017. Copyright infinity, the velocity of the secondary body will slow to approach V∞ © 2017 by William Crowe, John Olsen, and John R. Page. Published by the and will approach the outgoing asymptote (asymptote b). After American Institute of Aeronautics and Astronautics, Inc., with permission. All passing the asteroid, the secondary body will have turned through an requests for copying and permission to reprint should be submitted to CCC at angle δ, which is equal to 2θ∞ − π radians. www.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884 Through conservation of energy, the instantaneous velocity V at (online) to initiate your request. See also AIAA Rights and Permissions any point along the trajectory is related to the radial distance d of the www.aiaa.org/randp. secondary body to the primary body. The total specific orbital energy *Ph.D. Candidate, School of Mechanical and Manufacturing Engineering. of the hyperbolic system is μ∕2 −a, where μ Gmp is the Member AIAA. G †Lecturer, School of Mechanical and Manufacturing Engineering. Member gravitational parameter of the primary body, is the gravitational AIAA. constant, and mp is the mass of the primary body. The specific orbital ‡Senior Lecturer, School of Mechanical and Manufacturing Engineering. energy is composed of the specific kinetic energy and the specific Member AIAA. potential energy and the specific potential energy, as shown in Eq. (3) Article in Advance / 1 2 Article in Advance / ENGINEERING NOTES Table 1 Instruments for measurements between spacecraft Measurement Sensing type Description −∞ Range (rn ) Radio A beacon sent regularly with timing information ∞ Range rate/Doppler (r_n ) Radio A beacon sent regularly with a specific frequency −∞ Orientation (τn ) Optical Each spacecraft could be illuminated with light-emitting diodes, to be picked up by optical navigation cameras B B-plane target ( 0) Ephemeris To be estimated through knowledge of the asteroid and spacecraft trajectories Velocity (V∞) Optical Optical navigation cameras, as used on previous asteroid encounters μ V2 μ shift of a constant transmission of electromagnetic waves from the − (3) 2 −a 2 d spacecraft. Equation (7) is an analytic equation to describe the phenomenon based on a two-way transmission of the signal from an Earth-based ground station, assuming that the Doppler shift due to Equations (4–6) show how the parameters B and V∞ are related to orbital movements about the sun and the Earth’s rotation has been μ and θ∞ through the intermediate terms e and a. Equation (4) shows μ∕d accounted for. The equation shows how the change in frequency that, at positive and negative infinity, the term of Eq. (3) becomes Δf t → ∞ zero, producing a relationship between the three variables V, a, and μ. measured long after the spacecraft passes the asteroid is dependent on V∞ and rp, as well as the electromagnetic transmission This can be substituted into Eq. (1), displayed in Eq. (5) when f c rearranged. Substituting Eq. (5) into Eq. (3) results in the relationship frequency , the speed of light in a vacuum , and the asteroid gravitational parameter μ. The angle α 0 is defined in Fig. 2, and β is between θ∞, B, μ, and V∞, shown in Eq. (6): the out-of-plane angle toward the Earth [1,4]: μ −a 2 (4) f μ V∞ Δf t → ∞4 sin α 0 cos β (7) c rpV∞ s 2 4 B V∞ The mass of the asteroid 21 Lutetia was measured in this way [1]. e 1 (5) μ2 The spacecraft was tracked almost continuously for over 10 h while it passed the asteroid. Only about 13% of the maximum possible Doppler shift could be measured due to the orientation of the flyby to Earth at 1 the time, providing a more limited resolution to the measurement. cos θ∞ − p (6) 2 4 2 1 B V∞∕μ C. Multiple-Spacecraft Flyby This section contains the outline of how relative Doppler and B. Perturbation Measurement from Earth ranging measurements between two or more spacecraft, combined Measurements of the mass of an asteroid have previously been with estimated data from ephemeris, can be used to estimate a unique made by sending spacecraft along hyperbolic trajectories past mass parameter of an asteroid.
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