Dynamical evolution of asteroid 2009 HE60, a quasi-satellite of Earth.

Transcription

1 Dynamical evolution of asteroid 2009 HE60, a quasi-satellite of Earth. P. Caspari ª ª Swinburne Astronomy Online, Student Not yet received Not yet available online Abstract This paper announces that recently discovered asteroid 2009 HE60 is currently in a quasi orbit of the Earth. From an Earth bound perspective it appears to be in a retrograde orbit with a period of approximately 1 year. Computations suggest this quasi satellite dynamic will continue until 2036 when it will transition into a horseshoe orbit. Its chaotic orbital dynamics, largely resulting from frequent approaches to Venus, suggest a transient object. This study also uncovered a dynamic previously unknown in both nature and theory which also relates to its interactions with Venus. This dynamic produces a quasi stable cyclic change in the semi-major axis resulting in a quasi satellite type dynamic except unrelated to a major body. This analysis will study the three-body orbital interactions and perturbations of asteroid 2009 HE60 to understand its current and future dynamics and stability as a companion to our Earth. Its will also explore this bodies unique interactions with Venus. Keywords: Co-orbital; Asteroid; Dynamics; Earth; Venus; 1. Introduction Three-body motion can explain stable companions moving in remarkably similar orbits to a parent planet while still orbiting the Sun. Until relatively recently the only known interactions of these types of objects was that of Jupiter s Trojans. A Trojan or tadpole orbit (TP) is a type of co-orbital threebody interaction that occurs around one of two Lagrange points of stability, L4 and L5, which are 60 ahead and behind the primary body. Prior to 1991 only Jupiter was known to have Trojans which is where the name originated. However in 1991 the first Trojan of Mars was discovered (Innanen, 1991) called asteroid 5261 Eureka. A Trojan asteroid of Neptune has also been found (Brasser et al., 2004). More recently two other complex co-orbital dynamics have been found to exist in the form of the quasi satellite (QS) orbit and the horseshoe (HS) orbit. The first asteroid to be determined to exhibit co-orbital dynamics beyond the TP was 3753 Cruithne in 1997 (Wiegert et al, 1998). Since then other examples have been 1 discovered and studied including 2002 AA29, 2003 YN107, 2000 PH5 and 2001 GO2. Of this population of co-orbitals only 3 are known to currently be in a QS phase and they are 1998 UP1, 2008 KT and 2009 BD. Asteroid 2009 HE60 is part of a small but increasing population of Earth co-orbital asteroids. This paper will demonstrate how, in addition to its current QS status 2009 HE60, within current uncertainties, is also likely to exhibit HS dynamics. 2. Background This paper will use the standard notation a.e.i,ω,ω and M for the semi-major axis, eccentricity, inclination, argument of perihelion, longitude of the ascending node and mean anomaly. Due to this body having an e its orbit does not resemble that of the Earth. Wiegert et al. (1998) suggested that the term co-orbital only applied to bodies that share the same orbit however the term co-orbital is often used for bodies a with very similar a. For the purposes of this paper the term co-orbital will be used for 2009 HE60 as no clear alternative term for this type of object exists.

2 Asteroid 2009 HE60 was discovered on the 24 th of April 2009 by the Mt. Lemmon Survey (MPECweb) at an approximate magnitude of 21. Based on an absolute magnitude of 25.6 the diameter of the object is estimated to be 22 meters assuming an S type asteroid with an albedo of 0.2. A more comprehensive integration using multiple test particles was performed using the Solar System Dynamics simulator which is a modified version of the SWIFT code (Levison & Duncan 1994). Simulations will be performed on the Swinburne University supercomputer. The SWIFT code simulates 3 distinct types of bodies, a central dominant gravitational body with up to 20 bodies orbiting the central body and up to 50 non random massless test particles per simulation run. The main central and orbiting bodies affect each other and the test particles. The test particles do not affect each other or the main central and orbiting bodies. Figure HE60 at discovery (JPL Horiznons) The basic orbital parameters for 2009 HE60 are similar to the Earth with the notable exception of e. The perihelion distance is just outside that of Venus resulting in close approaches to Venus and these interactions will be considered in this study. For the Earth, co-orbital dynamics are possible for objects within a distance E, based on the Hill s theory radius and a practically massless object. Ε =(m/3m) ⅓ Where m and M are the masses of the Earth and the Sun respectively. This results in an annulus a-a < 0.01 AU where subscript denotes the Earth (Connors at el 2004). A study of the MPCORB data file from the Minor Planet Center (MPCweb) for this range of a revealed a number of recently discovered asteroids HE60 was selected for this study as it currently exhibits QS dynamics and is likely to transition to a HS phase. A search of the literature has not found any reference to this asteroid exhibiting these unusual dynamics. 3. Numerical Experimentation Initial studies were conducted using the JPL Horizons online system (Giorgini, J.D. 1996). This system offers an ephemeris, vectors and Kelparian elements for a period of integration of 600 years from approximately 1600 to Test particles are removed from the simulation if they exceed preset limitations relating to the proximity of the main bodies or distance from the central body. The simulation stops if all the test particles have exceeded these limits The simulation continues until the preset integration time is reached and it can be continued from a previous run. The option to continue a run allows for more data outputs to be generated as there is a data output limitation to a single run (SWIFTCodePageweb). The subset of the SWIFT code used for solar system dynamics employs a Regularized Mixed Variable Symplectic (RMVS) method. This method addresses close approaches between test particles and orbiting bodies (Levison & Duncan 1994). A time step of years was used for all presented simulation results. A time step of was also tested producing qualitatively equivalent results. The SWIFT code automatically reduces the time step in a 2 stage process when a test particle is in the proximity of the orbiting body (Levison & Duncan 1994). All the major planets have been used as part of this analysis however the SWIFT code does not support the Earth s Moon, instead the Earth Moon Barycenter was applied. The data from any test particles subsequent to a close approach within the Hill sphere for the combined Earth Moon system was ignored. No close approaches to Venus were within its Hill sphere. The SWIFT code has a limit of 50 non random test particles thus each integration runs of >50 test particles are broken into subsets. The orbital parameters, uncertainties and initial state vectors for the major planets and 2009 HE60 were obtained from the JPL Horizons system for an Epoch of which is approximately mid point in the data arc span (see Table 1). A measure of how long a dynamical system, such as 3 body orbital interactions, takes to become chaotic is called the Lyapunov time (Wieggert et al. 1998). Once this time frame has been exceeded the dynamics move outside the bounds of precise predictability and enters a chaotic phase.

3 The Lyapunov time for any Earth crossing object is estimated to be in the order of 100 years (Wieggert et al. 1998). Due to the short observing arc, the relatively short Lyapunov time and frequent close approaches to Venus, simulations over longer time frames become speculative. The true evolution of this object cannot be predicted reliably therefore a statistical approach to the analysis is appropriate. The integration period was also kept to a relatively low 1200 years which will cover 4 HS libration cycles of approximately 200 years. The HS libration period was approximated from plots such as Figure 5. A total of 65 test particles were calculated for integration, 1 for the current best fit solution and 64 based on 1 σ uncertainty (Sim 1). For each of the six primary orbital parameters 2 test particles will explore both extremes of the 1 σ distribution and all combination therein (2^6 combinations). These test particles were cross referenced with a basic binary methodology where 0 = -1 σ and 1 = +1 σ allows efficient analysis of the data. This non random approach provides the opportunity for a more detailed statistical analysis. Element Value Uncertainty ( 1 σ ) a (AU) e i ( ) ω( ) Ω( ) M( ) Table 1. The orbital parameters for 2009 HE60 with 1 σ uncertainty. The Yarkovsky effect is a change in the orbit of a body as a result of the re-emission of solar radiation (Bottke et al 2006). This re-emission can cause the object to accelerate or decelerate by the momentum lost due to radiating photons. This effect is depending on the spin direction and is most significant in low mass objects such as small asteroids. It primarily results in changes to the a. The effect of the Yarkovsky mechanism on 2009 HE60 is difficult to determine due to limited observational information. The effect of this mechanism is dependent on many factors including shape, orientation, and albedo. However considering the uncertainty in a that is covered by the test particles the Yarkovsky effect can be considered irrelevant to this study following a similar logic to Connors et al. (2002) All the resulting data from the both the JPL Horizons system and the Swinburne University supercomputer were imported into a purpose built MS Access database application for further analysis (CoOrb). A visualization tool to produce plots of co-rotating reference frames was developed as part of this database. Approximately 10^7 records of state vectors and orbital elements were collected as part of this study. This database can also automatically produce a series of state vectors from a set of Keplarian elements and covering a range of uncertainties as per Sim 1. This series of state vectors can automatically be entered as Planets or test particles into the SWIFT interface on the Swinburne University supercomputer using a web automation tool. 4. Results Table 2 details the orbital elements and results summary for each dynamical class. For each orbital element the binary test particle methodology discussed above was applied. A count of test particles at +1 σ / -1 σ for each orbital element and dynamical class is displayed. A total of test particles that demonstrated each dynamical class is listed in the far right column. The BN dynamic will be discussed below. The final row labeled Exit applies the same methodology as above for test particles that exited the annulus a-a < 0.01 by the end of the 1200 year integration (subscript denotes the Earth). a E i ω Ω M HS 25/20 21/24 23/22 22/23 21/24 24/21 45 QS ¾ 5/2 1/6 4/3 2/5 5/2 7 BN 1/0 0/1 0/1 0/1 1/0 1/0 1 Exit 6/5 8/3 6/5 6/5 3/8 7/4 11 Table 2. Test Particle results summary from CoOrb for Sim Quasi Satellite Dynamics 2009 HE60 is currently in a QS phase however, as Table 2 indicates, only 7 out of the 65 test particles exhibited another QS phase over the next 1200 years. While a QS of the Earth appears to be in a retrograde orbit of the Earth with a period of 1 year it is actually orbiting the Sun. A QS has a very similar a to the Earth however with a different e which is usually greater. A permanently stable QS orbit can be approximated by i < e - e (S. Mikkola et al 2006) where subscript denotes the Earth. For larger inclinations, i > e - e, a temporary capture into the QS phase is possible depending on relative differences in a and orbit orientation. This is further supported by the bias towards test particles with lower i values entering a QS phase. See table 2. A clear trend amongst the test particles indicates i will increase over the period of Sim 1. This is the probable explanation for relatively few test particles exhibiting future 3

4 QS dynamics. The cause of the increase in i will be discussed below. The QS phase can be identified in a plot of a versus time as a sinusoidal wave style cyclic change in a apparent between 2300 and 2600 in Figure Horseshoe Dynamics A HS orbit liberates around a planet repeatedly reversing its direction as a result of changes in a resulting from close approaches. One libration cycle can also be defined as two opposing circulation transitions. A circulation transition (Connors et al 2005) has been defined as a transition between a < a to a > a or visa versa where subscript denotes the Earth. From a co-rotating reference frame this results in a horseshoe shape with the Earth in a gap between the ends of the horseshoe. Since 1997 a number of asteroids with HS dynamics have been studied such as 3753 Cruithne, 2002 AA29 and 2003 YN107. Figure 2. a vs time for a typical QS Phase between 2300 and 2600 Co-orbital dynamics are often visualized from a reference frame that co-rotates with the selected body in this case, the Earth. Figure 3 is a plot of the co-rotating reference frame for the QS phase from 1946 to The vector data for this plot was obtained from JPL Horizons. Fig. 4. The HS phase from 2036 to Vector data sourced from JPL Horizons. Integrations on both the JPL Horizons and SWIFT systems indicate that 2009 HE60 will transition from QS phase to HS phase around Prior to the work of (Wiegert et al, 1998) relating to Asteroid 3753 Cruthnia co-orbital nature the transfer between QS and HS dynamics had not been realized in theory or in nature. Fig. 3. An Earth (0,-1) based co-rotating reference frame for the QS phase from 1946 to Based on JPL Horizons data the current QS phase commenced in 1946 and will continue until 2036 after which it is likely to transition to a HS phase. Figure 5 represents a typical HS phase amongst the test particles. This phase can clearly be identified by the square wave style cyclical change in a. Also an increasing i is apparent and typical amongst the test particles for Sim 1. 4

5 Fig 6: X,Y for all Major Planets excluding Venus, 32 Test Particles from Sim1 Figure 5 i,a and e versus time for a sample test particle from Sim1 5. The effects of Venus on 2009 HE60 The perihelion distance of the asteroid is very close to the a for Venus. The resulting close approaches have a dramatic effect on the orbit of 2009 HE60. This can be clearly recognized in Figure 6 and 7 which are X versus Y plots for 32 of the 65 test particles in Sim1. Figure 7 shows a much broader spread of these test particles than Figure 6 as a result of these approaches. The chaotic effects can further be appreciated by the effect on a in Figures 8 and 9 for 32 of the 65 test particles in Sim1. Figure 8 indicates a relatively stable combination of HS and QS dynamics, however when Venus is introduced the dynamics become dramatically more chaotic. Clearly the effects of Venus insure that this body is destined to be transient. Fig 7: X,Y for all Major Planets including Venus, 32 Test Particles from Sim1 5

6 6. Co-orbital stability and evolution We have discussed how changes in a clearly reflect coorbital dynamics. Changes in e are relatively small (<3% for all test particles in Sim 1). However results already indicate that i is increasing over the duration of the integration. A further analysis of 32 of the 65 test particles in Sim1 (Figure 10) confirms that a clear trend for an increasing i exists. Fig 8: a versus time for all Major Planets excluding Venus, 32 Test Particles from Sim1 Fig 10: i versus time for all Major Planets, 32 Test Particles from Sim1 As discussed the increasing i reduces the likelihood of future QS and HS phases. To confirm this another integration was run with the best fit orbital parameters with the exception of i which ranged from 1 to 3.8 with increments of 0.2 (Sim 2). As expected the likelihood of HS or QS dynamics clearly reduced with increasing i Fig 9: a versus time for all Major Planets including Venus, 32 Test Particles from Sim1 The CoOrb database containing the results of Sim 1 was queried to study the relationship between approaches to both Earth and Venus and their effects on i. For the 65 test particles in Sim 1 any period of 36 days where the i changed by more than was cross referenced with all approaches of <0.2 AU to both Earth and Venus. For Venus the mean change to i during these approaches was where as for the Earth the mean change was Clearly it is the interactions with Earth that is increasing i. To understand exactly when the interactions with Earth increase i a plot was produced for i versus time with a versus time superimposed (Figure 11). From this plot it is clear that i is increasing during circulation transitions. 6

7 To confirm the precession and recession of ω and Ω a plot was produced for 32 of the 65 test particles in Sim1 (Figure 14). Figure 11: i versus time with a versus time superimposed for sample test particle. Another observation was that Ω is recessing and ω is precessing at approximately 4.6 per century. In 2043 ω = Ω = 227. A further integration of 36 test particles based on the best fit solution except with 0 Ω 360 at 10 intervals (Sim 3) was conducted. This study indicates that dependant on Ω, 1 < i < 4. Cleary the recession of Ω alters the rate of change in i. Figure 13: ω and Ω in radians versus time for 32 test particles from Sim 1 A possible explanation for the change in i is that the precession and recession of ω and Ω respectively alter the position of nodes in relation to approaches. If a circulation transition occurs as the asteroid approaches the ascent node it appears to kick up the i with an opposite effect at the descent node. 7. The banana orbit A study of the a from the test particles in Sim 1 uncovered an unexpected type of dynamic. Figure 14 indicates 2 HS librations followed by a sinusoidal wave style cyclic change in a. This sinusoidal phase appears similar to the QS dynamic except with a lower amplitude and is not based around the Earth. For the purposes of this paper this dynamic will be referred to as a Banana orbit (BN) due to its appearance on a corotating reference frame plot. Figure 12: The i versus time of 36 Test Particles with 0 Ω 360 at 10 intervals over 1200 years (Sim 3) 7

8 Figure 14: a vs time with the BN phase starting from approximately 2650 It appears that the right set of orbital parameters can produce a low amplitude oscillation in a which in turn results in a quasi-satellite type of dynamic around a non existent object. A search of Minor Planet Center s website found no reference to any asteroids which exhibits this type of dynamic. Fig 16 1 Librations of BN phase, 1 test particle from Sim 1 For the 3 full liberation cycles the points of circulation transition do not overlap. The BN dynamic appears to precess and this can also be seen in Figure 14 by the fact that the center amplitude is offset. Figures 15,16 and 17 demonstrate this precession from a co-orbital reference frame. Fig 17 2 Librations of BN phase, 1 test particle from Sim 1 Fig Librations of BN phase, 1 test particle from Sim 1 8 To further investigate this phenomenon an integration (Sim4) was run for 64 test particles based on orbital elements from the original test particle from Sim 1 that exhibited BN dynamic. The basic methodology is similar to Sim 1 except uncertainty is fixed to 0.01%. This integration produced a further 4 test particles exhibiting BN dynamics. Integrations without Venus produced no such dynamics.

9 a e i ω Ω M BN 2/2 1/3 2/2 4/0 1/3 3/1 4 Table 3. Test Particle results summary from CoOrb for Sim 4. Figure 18 and 19 are examples of the BN phase which covers the period of time approximately from 2730 to Another integration (Sim5) extended the time frame for Sim 4 to 2000 years indicating the BN phase can be quasi-stable for a period of 800 years covering 8 librations. This period of quasi-stability is comparable to the periods of stability of HS and QS phases (Figure 20). Figure 20: BN phase for 1 test particle from Sim 5 over 2000 years Figure 18: BN phase near L3 for 1 test particle from Sim 4. An 8:13 resonance exists between asteroid 2009 HE60 and Venus respectively however this is not a classical mean motion orbital resonance due to the close approaches involved. These close approaches occurring at an interval of approximately 8 years. By studying these approaches with a dynamic orbital plot an explanation for the cyclic nature of a becomes apparent. When the asteroids lags behind Venus during one of these approaches the gravitation attraction of Venus increases the angular velocity near Periapsis which in turn increases the a of the asteroid. As a result subsequent approaches become closer to and abeam Venus eventually causing the asteroid to take over and lead Venus. When the asteroid leads Venus the reciprocal effect decreases the asteroids angular velocity and the a from which a cyclic dynamic results. The Periapsis distance of 2009 HE60 is just outside the orbit of Venus ensuring that these close approaches do not result in an impact. The interactions that occur during these approaches are similar to the QS and HS dynamics described above. Figure 19: BN phase near L5 for 1 test particle from Sim 4. To further understand this dynamic and its range of stability further simulations need to be conducted employing the binary test particle methodology discussed above. Numerous integrations similar to Sim 4 explored a range of uncertainties and other examples were uncovered however all examples of the BN dynamic occur in approximately 700 9

10 years. To exclude the effect of 700 years of perturbations, a valid analysis of the orbital parameters requires the integration to enter the BN phase almost immediately. To this end an Epoch of (1 st of March 2065) was selected as this is 6 months prior to a close approach with Venus while significantly separated from Earth. Unfortunately only 1 integration from this epoch was conducted before access to the Solar System Dynamics Module was discontinued. This integration did not reproduced the BN dynamic however a much broader search is required. Without access to the Solar System Dynamics Module the study has moved to the data available with the JPL Horizons system. This study will look for asteroids with aphelion or perihelion distances within the Hills sphere of the major planets and also with resonance ratios of two small integers. These orbital elements, particularly the a, will be analyzed looking for further examples of the BN dynamic. This work is on going. Clearly the study of the BN orbit is in its infancy. The following are just some of the aspects of BN dynamics that needs to be explored Confirm that the BN dynamic is not an artifact of the integration software or a type of numerical or statistical anomaly. Reproduce the dynamic by using different integration software Study if the BN dynamic can occur at aphelion as well as perihelion Study if the BN dynamic can occur as a result of the interaction with bodies other than Venus. Venus has a very low eccentricity as compared to the other major planets which may play a role. Are there any examples in nature (This study has already commenced using JPL Horizons data). Consider if this dynamic is a just a scientific curiosity or whether it may offer useful trajectories for future space probes. Develop a mathematical understanding of the dynamic Unless this is a known phenomenon further studies of this unique phenomenon are warranted. The Swinburne University supercomputer running the SWIFT code in conjunction with the CoOrb database developed as part of this study provide the necessary computing resources for such an investigation. 8. Conclusion Asteroid 2009 HE60 is a QS of the Earth and numerical studies indicate a high probability this object will then transition into a horse shoe orbit with a libration period of approximately 200 years. The object s short Lyaponvic time and frequent close approaches to Venus imply this objects current orbit is not primordial. An integration of 65 test particles indicated that 11 of the test particles will exit the region of likely co-orbital behavior over the next 1200 years. This is largely due to the effects of Venus This study determined that the asteroid s i is likely to increase due to the precession and recession of ω and Ω respectively and the resulting interactions at the position of the nodes. This increase in i will reduce the likelihood of future quasi-stable co-orbital dynamics. The BN dynamic uncovered by this study has been determined to be the result of interactions with Venus. This unique dynamic is due to a cyclic leading and lagging interaction with Venus. As more of these types of co-orbital asteroids are discovered and studied, statistically trends may develop allowing us to develop our understanding of the dynamics of these intrinsically interesting objects. Asteroid 2009 HE60 s unique dynamics and transfer mechanisms in relation to known co-orbital behavior warrants closer study especially when further astrometrics become available. However the unique BN co-orbital dynamic uncovered as part of this study cries out for further investigation. Acknowledgements This paper was developed as part of my studies with Swinbourne Astronomy Online. I d like to thank Jarrod Hurley and Swinburne University for their invaluable assistance with this effort. References Bottke, W., Vokrouhlický, D., Rubincam, D., Nesvorný, D., The Yarkovsky and Yorp Effects: Implications for Asteroid Dynamics, Annual Review of Earth and Planetary Sciences, vol. 34, p Brasser, R., Innanen, K., Connors, M., Veillet, C., Wiegert, P.,Mikkola, S., Chodas, P.W., 2004b. Transient co-orbital asteroids. Icarus 171,

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