Do the asteroid families dynamically evolve?
(in collaboration with Alessandro Morbidelli, Bill Bottke, David Vokrouhlicky and Mira Broz)
This is intended to be a twin page to the Prague yarko-page (by Mira Broz and David Vokrouhlicky) . Refer to the Prague page if interested in *cool* results on Dora, Eos, Themis and Eunomia families. Bill's `smoking gun' animation of the Koronis family is at http://www.boulder.swri.edu/~bottke/Animation/koronis.gif
Here we show results of the simulations of the Flora, Maria, Eunomia, and Koronis families. The purpose of these simulation was to gain some understanding on how the asteroid families dynamically evolve over long time intervals. There are three effects which can change proper orbital elements of family members: (i) Yarkovsky effect, (ii) resonances, and (iii) gravitational effects by large asteroids like Ceres, Pallas, and Vesta. We estimated that (iii) is not a large effect comparing to (i) nad (ii). Thus, we have neglected (iii) in our simulations, and accounted only for (i) and (ii).
We started by creating synthetic families by this algorithm . The synthetic families were assumed to be initially more tightly clustered in the orbital elements space than the observed families. The reason in doing so was that we do not believe that the observed dispersion of asteroid families is completely due to collisional dynamics. Indeed, the hydrocode simulations suggest that families were created by disruptions events where the ejection velocities did not largely exceed the effective escape velocity. We typically assume that the initial dispersion of a family is consistent with an isotropic ejection field of velocities not exceeding 100 m/s `in infinity'. This gives a dispersion in the proper orbital elements space which is usually a factor of few smaller than the observed families' dispersions. We want to see if resonances and Yarkovsky effect can account for the difference!
We also had to assume sizes and thermal parameters of our synthetic families.
We collect here the results obtained on the Flora, Maria, and Eunomia families (Morby and David N), and on the Koronis family (Bill).
Flora family (`living in kingdom of chaos')
The following two plots show the initial and t=500 My distributions of the synthetic Flora family. Each white dot represents one body. Many motion resonances with Mars and Jupiter are shown in the background. Diagonal white lines represent q=1.92 and q=1.84 AU. Initially, our synthetic family is tightly clustered around (8) Flora being a factor of >3 smaller than the observed Flora family. In later evolution, each body leaves a track denoted in red (if Mars-crosser) or blue (if non-Mars-crosser). The proper elements shown in figures were computed as averages of orbital elements over the 10 My window, shifting it by 0.1 My between two consecutive outputs. The proper elements thus defined would be constant if the evolution of asteroids were quasi--periodic.
Synthetic Flora family (initial instant). Click here to enlarge
Synthetic Flora family( t = 531 My ). Click here to enlarge
Due to the effects of mean motion resonances (mainly with Mars) and the Yarkovsky force, the synthetic family increases in size
( !!! SEE MOVIE HERE !!! )
and at t ~ 500 My resembles the orbital distribution of the observed Flora family:
Observed Flora family (proper elements as 10My average)
In general, one sees a good agreement of orbital distributions of the synthetic at ~500 My and observed Flora families. The dispersions in proper elements are similar (see later). The synthetic Flora family is located at a bit smaller semimajor axis than the observed one. This is a consequence of (8) Flora itself being displaced to smaller semimajor axis with respect to the rest of the family. It is curious to note how the core of the observed Flora family is bracketed by 2.215 AU (4:7 mean motion resonances with Mars) and 2.255 AU (5:9 with Mars and 7:2 with Jupiter). This is due to instabilities generated by mean motion resonances (much larger voids in the asteroid distribution are known to exist at strong mean motion resonances with Jupiter -Kirkwood gaps). The main secular resonance present in the region of the Flora family is z2 = 2(g-g6) +s-s6. Several bodies interact with this resonance.
A detailed comparison between last two figures (Synthetic at 531 My and observed Flora families) must account for:
To avoid small number statistics as much as possible, we base the following analysis on bodies having diameters between 2 and 4 km. There are 201 observed Flora family members (IRAS albedo and assuming 0.21 albedo otherwise) and 125 synthetic bodies in this range, which means a reasonably good `overlap' for statistics. We did not integrate any bodies at smaller diameters and at larger diameters, our sampling of the diameter bins is sparse. The following histograms show the orbital distributions (in proper elements) of the synthetic (green: initial , t=250 My, and t=500 My) and real Flora family members (red) in the 2<D<4km range (make your browser window sufficiently large -if possible- to see them side by side) :
- the `observed' Flora family was indentified by a hierarchical clustering method (Zappala et al.) on a dense background of asteroids in inner belt while for the synthetic family, we show all asteroids ejected in the breakup. This difference is twofold:
- Objects attaining Mars-crossing orbits in the simulation of our synthethic family evolve so much in proper elements that they would not be recognizable as family members by hierarchical clustering method.
- The observed Flora family contains many background object, which have nothing to do with the original breakup. These objects are not part of our synthetic family.
- There 385 Flora family members of the observed family (all numbered ones, from Zapalla et al.) and only 210 synthetic family members shown in figures.
Alternatively, you may consult a movie:
!!! SEE MOVIE HERE !!!
We do not show in these histograms synthetic bodies which were eliminated from the simulation because an their impact on the Sun or planets (or ejection from the solar system). We also do not show bodies on Mars-crossing trajectories: by this we apriori assume that such bodies would NOT be identified as family members by the clustering algorithm.
Although starting with a very peaked distribution in all elements, after ~0.5 Gy, the orbital distribution of the synthetic family resembles the observed Flora family. The agreement is especially striking in the eccentricty and inclination distributions. Although our integration does not extend much over 0.5 Gy, we may guess that at later times, the synthetic Flora family woudl disperse even more, and the fit between the synthetic and observed orbital distributions would deteriorate. We can conclude that the age of the Flora family may be roughly near 0.5 My (possibly somwhat larger). This conclusion is relatively insensitive to the initial family dispersion.
The number of bodies in the 2<D<4 km range remaining in the classic asteroid belt (non-Mars-crossers) is:
14 bodies were directly injected to Mars-crossing space, from the remaining ones: 45 became Mars-crossers in later evolution. In total, about one half of the bodies in the 2<D<4 km range became Mars-crossers within 0.5 Gy. There are 200 family members in this range, but this number may be some two orders in magnitude incomplete due to the observational bias (according to Jedicke and Metcalfe). We may roughly estimate within the frame of our scenario of the Flora family origin that ~10^4 fragments of 2<D<4Km reached Mars-crossing orbits within 0.5 Gy after the collision. This have in turn probably significantly increased the number of Earth-crossers objects in this period. (*** these numbers should be worked out with care, its worth of effort!!!***) . We may also infer from the above figure, and under the assumption of the 0.5Gy Flora family age, that the number of Mars (Earth) - crossers at t=-0.5Gy could have been a factor of two larger than now. Is this supported by an observational evidence???
To make the comparison more appropriate, we choose a background population and run the hierarchical clustering algorithm on the synthetic family. Than we identify all asteroids which are linked with largest surviving body using 1st metrics of Zappala et. al. (eq. 1). We have some preliminary results showing that the assymetry in the semimajor axis distribution which is not well reproduced in the above plot may be due to the clustering algorithm. The results are however sensitive to several parameters (as the density of the background objects, on level at which we group the family members, etc.) and should be considered with care.
Maria family (`on the age')
Next, we analyzed Maria family. This family is located on the age of the 3:1 mean motion resonance with Jupiter. This is how look like our synthetic family orbital distributions initially and at t=300 My:
Synthetic Maria family (initial instant). Click here to enlarge.
Synthetic Maria family ( t = 306 My ). Click here to enalrge.
In the background, some important resonances are shown: Jupiter 3:1 MMR is the green region at 2.5 AU, three-body resonances are white vertical lines, for secular resonances: the center and +-0.5"/yr contours are shown in red. There are two stricking features related to interaction with the secular resonances in these plots:
(1) the bodies which drift to lower semimajor axis interact with g+s-g6-s6 at a = 2.53 AU (the passage through this resonance usually increases the eccentricity by ~0.005) and with g-2g6+g5 at a = 2.52 AU. The interaction with the latter resonance is surprizingly strong. Fast evolution in eccentricity happens at the age of the 3:1 resonance with Jupiter (shown in green), and this was the reason why we have initially
assigned this evolution to the effect of the 3:1 MMR itself.
( !!! SEE MOVIE HERE !!! )
In fact, the width of the 3:1 MMR is a bit exaggerated here -I took the maximum width (should correct this) - , so that even if tracks seem to enter the green region, the simulated bodies do not interact with the 3:1 (otherwise their proper elements should be equal to 2.5 AU, which is the resonant center). What happens at this place, is that the secular resonances become almost vertical in (a,e) plane near the 3:1 resonant border so that a body that becomes captured in the g-2g6+g5 , and is pushed by the Yarkovsky effect, dramatically evolves in e. Small semimajor axis evolution due to Yarkovsky effect translates to huge changes in eccentricity (and inclination). Was this noted before??? The evolution is so substantial that some bodies become Mars-crossers! Here we get a grip on a previously unknown source of Mars-crossers. The orbital distribution of bodies escaping from the asteroid belt by this mechanism should mimic the orbital distribution of bodies escaping through the 3:1 MMR with Jupiter. This is surely worth of future study!!! Moreover, as Bill pointed out, don't we have here a promising source for Eros-like bodies???
(2) the bodies drifting to larger semimajor axis interact with the 2(g-g6) + s-s6 secular resonance. Some of them are captured and in later evolution displaced to large distances from the family center. Such bodies would not be indetified by clustering algorithms but yet originated (in our scenario) in the parent body breakup. May be that it would be worth to pick up from the asteroid database a couple of bodies in this resonance (not family members) and suggest a spectroscopic study with similar goals as Zapalla et al. (2000). Fugitives from Maria family??? Moreover, several bodies which shortly evolved in the 2(g-g6) + s-s6 were afterwards released with eccentricities ~0.12. This is nicely seen in the movie. A similar trend may be seen in the real Maria family:
Observed Maria family (corrected M&K prop elements)
Indeed, there is a group of asteroids at 2.556<a<2.62 AU, 0.1<e<0.12, 15<i<15.5 deg , which separates from the rest of the family at 10m/s level. While the offset in eccentricities and inclinations of this group from the rest of the family may be due to the evolution in the 2(g-g6)+s-s6 (as suggested by movie), the gap seen in orbital distribution in (a,e) diagonally separating this sub-grouping from main grouping may be due to the fast Yarkovsky-driven evolution in the secular resonance, i.e. due to the short residence times in the corresponding region. We believe that in contrary to what happens in mean motion resonances, a weak secular resonance should NOT chaotically change the proper elements if not assisted by Yarkovsky drift. This demonstrates that the Yarkovky effect acts in the asteroid belt.
*** There is reasonable, but not a *very* good match between the secular resonances computed from Milani and Knezevic secular theory and the tracks left by bodies evolving in the secular resonances. This is because the secular resonances plotted in figures were computed for fixed values of eccentricity (for inclinations plots) and inclination (for eccentricity plots). The secular resonances are, unlike the mean motion resonances, three-dimensional structures. The bodies evolving in secular resonances thus change (a,e,i) and the correspondence to the secular resonances plots at fixed e or i is only approximate. Moreover, the Maria family is close to the 3:1 MMR with Jupiter, where the secular theory of Milani and Knezevic may give small errors in g and s frequencies. ***
The plots of the Eunomia synthetic family at t=0 and t=300 My are the following:
Synthetic Eunomia family (initial instant). Click here to enlarge
Synthetic Eunomia family ( t = 300 My ). Click here to enlarge.
In this case, there is not as much disperion in the orbital elements as in the previous cases (Flora and Maria families). On the other hand, the integration time is probably at least by one order in magnitude smaller than the actual age of the Eunomia family. Once again, one clearly sees the interaction with several MMRs (the one at 2.705 AU is 8:3 with Jupiter) and secular resonances (the body which diagonally evolves in (a,i) plane from the rest of the family is in the s-s6-g5+g6).
!!! SEE MOVIE HERE !!!
At 300 My, the dispersion in e and i of the synthetic family created by dynamics is considerably smaller than the real Eunomia family's dispersion:
Observed Eunomia family (corrected M&K prop elements).
Nevertheless, one feature may be noted. The bunch of mean motion resonances at 2.62 AU seems to create a large dispersion of bodies migrating to smaller semimajor axis. Note, for instance, several bodies which tend to decrease inclinations up to ~12 degrees. Similar feature can be seen in the real family. The situation with eccentricities is similar: note several real bodies at e~0.13, which might have diffused by the effect of MMRs at 2.62 AU similarly as one body does in the simulation. Semimajor axis dispersion of the real family is better reproduced, but we must first discuss sizes before concluding on this point, bacause the dispersion in the semimajor axis is obtained via Yarkovsky force which is sensitive to size.
Koronis family (`Bill's smoking gun!')
This is all based on Bill's integrations. See his movie at http://www.boulder.swri.edu/~bottke/Animation/koronis.gif
Initial and t=500 My orbital distributions of the synthetic Koronis family are the following:
Synthetic Koronis family (initial instant). Click here to enlarge.
Synthetic Koronis family ( t = 500 My). Click here to enlarge.
(!!! SEE MOVIE HERE !!!)
The Koronis family is sqeezed between the 5:2 and 7:3 MMRs with Jupiter. The synthetic family strongly interacts with a bunch of MMRs at 2.905 AU and with secular resonances at > 2.92 AU. Although in latter case, the evolution is complex, we may partially assign it to the effect of the 2(g+s-2g7) secular resonance (*** is there other `vertical' secular resonance at 2.92 AU ***) . An example of a particle interacting with this resonance: eccentricity evolution and resonant angle.
The *great* result Bill has obtained, concerns the match between the evolution of the synthetic family and the orbital distribution of the observed Koronis family:
Observed orbital distribution of the Koronis family (M&K pro)
The Koronis family has been a puzzle for a long time because of its shape in proper (a,e) plane: it is tightly clustered in eccentricities at 2.84 - 2.9 AU but rather dispersed at a>2.9 AU. I remember Morby and me integrating the real Koronis members over 1 Gy back in 1998 without the Yarkovsky effect. Nothing important happened because chaotic MMRs are quite sparse at the region of the Koronis family (low e,i). The secular resonances which seem to be acting in this region need the Yarkovsky drift to move thigs around in the eccentricity!!! Notice a couple of particles in the synthetic family run increasing their e at 2.92 AU and drifting to the higher eccentricity region populated by real family members. This is a very important fit, showing that the Koronis family orbital distribution may be explained by the interaction of secular resonances with the Yarkovsky force. Other scenarios, as for example: a strangely asymmetric ejection field in the parent body breakup, seem to be highly improbable.
- Obtaining the large orbital dispersion of the Flora family and explaining several particular features of the Koronis, Maria and Eunomia families seems to indicate that our model is valid: there is a significant mobility in the asteroid belt due to the Yarkovsky force and resonances.
- Refining the modeling can eventually lead to the datation of families (as in the case of the Flora family), and to linking this information with the impact record on terrestrial planets and their Moons.