1. International Workshop On Paolo Farinella:
The Scientist and the Man
Pisa 14-16 June, 2010
Asteroid families, new ideas and reuse of old ones:
the unfinished business with the Hungaria
Andrea Milani Comparetti
`
Dipartimento di Matematica, Universita di Pisa
Reporting work in collaboration with:
z ´ ´
Zoran Kneˇ evic, Bojan Novakovic and Alberto Cellino
1
2. PLAN
1. Unfinished business
2. Asteroid families and proper elements
3. The Main Belt of Asteroids and its marginal groups
4. The Hungaria, why they are important and why they were forgotten
5. Proper elements and dynamical structure of the Hungaria region
6. The Hungaria group and the Hungaria Family
7. Yarkovsky effect in the Hungaria family
8. The family membership problem
9. The Hungaria couples
10. The PF theory of couples rediscovered
11. What is the origin of the asteroid couples?
2
3. 1. Unfinished business
Nostalgia is OK, recognition of the previous contribution is essential. Newton: we
are dwarfs on the shoulders of giants. Without acknowledging what we have got
from the past, we may be unable to understand what we are doing in our current
research.
The problem is that we should not give for granted that the present is better than
the past. Are we keeping up with the past? Are we seeing beyond the horizon for
the giant?
Referring this argument specifically to the legacy of Paolo Farinella, do we still have
to complete some of the research programs which where proposed in his time,
with his prominent contribution? This talk is dedicated to examples of unfinished
business with some of Paolo’s fundamental ideas, in particular the ones on asteroid
families, non-gravitational perturbations and binaries.
3
4. 2. Asteroid families and proper elements
In the second half of the 80s, PF and his coworkers found that the current knowl-
edge on asteroid families was soft science. The main reason for this was insuffi-
cient data, both in quantity and in quality; another reason was in the too subjective
methods of analysis.
Asteroid families are statistical entities; if the number of family members is small,
the conclusions are weak (either little content or low reliability). Thus the number
of objects in the data catalog is a critical parameter. Two sets of data have to be
combined: dynamical classification (some form of orbit similarity with the property
of lasting over a very long time, 107 ∼ 108 years) and spectral classification (in
the space of color parameters, related to the mineralogy). You need a large set of
asteroids for which you have both information available, and then there is a problem
in the quality of the data.
Proper elements are quasi-integrals of motion, that is quantities stable over very
long (not infinite) times. In the late 80s they were of uneven and not well doc-
umented accuracy, and computed with a cumbersome method. The work of J.
Williams had advanced this field a great deal, but was not yet up to what was
needed. The taxonomic classes were ways to represent similarity of spectra and
thus presumably of composition, but data were even less numerous and less accu-
rate than for the dynamics, with controversies among the different authors.
4
5. 3. The Main Belt of Asteroids and its marginal groups
In 1987 a small workshop, organized by PF, was held in Pisa. One of the results
was to establish the collaboration between Z. Kneˇ evi´ and myself to produce an
z c
accurate algorithm for analytic proper elements, which could be computed for all the
asteroids with a good orbit. Others were involved in this, in particular the Namur
group, but also Schubart and others in Latin America.
As soon as proper elements catalogs with better understood quality control and
with 5 000 ∼ 6 000 asteroids were available, dynamical classification into families,
obtained with an objective mathematical taxonomy method, started making sense
`
in the asteroid main belt (for moderate eccentricities and inclinations) (Zappala et
al. 1994). The spectral classifications, although available only for smaller samples,
did not show anymore statistically significant contradictions. Collisional evolution
theories could use the proposed families as starting points.
This was a success, but was the job finished? NO! There are many groups of
asteroids outside the area covered by the 1994 work. They were considered in later
work, including Hildas (Schubart 1982), Trojans (Milani 1992-1993, later Beauge ´
and Roig), high inclination MBA (Lemaitre and Morbidelli), other resonant groups
(Moons and Morbidelli). Invariably, a good catalog of proper elements led to the
identification of asteroid families and was the starting point for the understanding
of the dynamical and collisional evolution of these regions.
5
6. 3. The Main Belt of Asteroids and its marginal groups
red=Tlyap<20,000; green=rms(e,sinI)>0.003
40
35
30
Proper inclination (DEG)
25
20
15
10
5
0
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6
Proper semimajor axis (AU)
Red = positive Lyapounov exponents (due to mean motion resonances). Green =
reduced stability proper elemets (due to secular resonances).
Proper elements catalog from the AstDyS site (http://hamilton.dm.unipi.it/astdys);
Trojan proper elements are not on the same plane. What is missing in this figure?
6
7. 4. The Hungaria group
30
28
26
24
inclination (deg)
22
20
18
16
14
12
1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2
semimajor axis (AU)
The distribution of the Hungaria region asteroids in semimajor axis and inclination.
The inclination is large, but also with a large spread. This group is surrounded by
a large, almost empty gap separating from the main belt. On the left this is due to
Mars perturbations, on the right there must be unstable perturbations from Mars
(2/1 resonance), Jupiter and Saturn (g = g6 secular resonance).
7
8. 4. Why the Hungaria should become important?
In 2008 two events forced us to pay attention to the unfinished business with the
Hungaria. R. Matson, in a MPML message of January 9, 2008 with subject Asteroid
pairs: extremely close pair found gave a list of striking cases, prominently two
couples of Hungaria.
At about the same time, while working to the Pan-STARRS asteroid survey sub-
project, I realized one unintended consequence of the PS survey was to make the
Hungaria the best known population of small bodies of the Solar System.
Unlike the Near Earth Asteroids, Hungaria are observable at each opposition. The
current orbit catalog of Hungaria corresponds to observations done up to an ap-
parent magnitude ∼ 19.5; observations are sparse because Hungaria can be at an
ecliptic latitude up to 45 ∼ 50◦. With the current limiting magnitude of PS1 (∼ 22.5),
and the survey area extending to the North pole, the completeness of the Hungaria
population should be down to 200 ∼ 250 m diameter.
8
9. 4. The Hungaria population: completeness
Numbered and multiopposition asteroids
1200
1000
800
Number of Hungaria
600
400
200
0
10 11 12 13 14 15 16 17 18 19 20
Absolute magnitude
The absolute magnitude (≃ size) distribution of the Hungaria known population
(numbered and multiopposition orbits only). The search for Hungaria is complete
up to a magnitude ≃ 16.5, corresponding to ≃ 900 m of diameter, assuming the
albedo estimated for (434) Hungaria by radar (0.38) is applicable to all.
9
10. 4. Why the Hungaria have been forgotten?
In a paper published in the proceedings of the Belgirate ACM (1993) the main
groups computing proper elements agreed on a partition of zones of influence,
e.g., analytical proper elements (by Kneˇ evi´ and Milani) should be used for proper
z c
I < 15◦, semianalytic ones (by Lemaitre and Morbidelli) should be used for proper
I > 17◦, each type where they are more stable.
However, this agreement apparently forgot the Hungaria; with the methods avail-
able at that time, K&M could not compute proper elements. Thus the software of
both groups apparently selected for proper elements computations only asteroids
with a > 2 AU.
In the meantime, after the death of PF, Kneˇ evi´ and myself switched to the com-
z c
putation of proper elements by a synthetic method, for which there is no limit in
the inclination, thus the Hungaria (with moderate e) would have been perfectly suit-
able, but we simply forgot to extend the catalog to this region. In 2008 B. Novakovi´
c
promptly computed synthetic proper elements for 4 424 Hungaria.
10
11. 5. The Hungaria proper elements
0.18
0.16
0.14
0.12
Proper eccentricity
0.1
0.08
0.06
0.04
0.02
1.75 1.8 1.85 1.9 1.95 2
Proper semimajor axis (AU)
The Hungaria asteroids projected on the proper semimajor axis/proper eccentric-
ity plane. The green points indicate instability in proper e and/or I , due to secular
resonances; the red points positive Lyapounov exponents. The green line corre-
sponding to a p (1 − e p) = 1.65 AU, the current aphelion distance of Mars.
11
12. 5. Hungaria region surrounded by resonances
Dynamical boundaries
0.5 0 0 0
0 1
1 g−g5 1
1 2
2 2
2
0.45 s−s4
0.4
Proper sine of inclination
0.35
2
0.3
0.25 g−g3
1
g−g4
2
0.2
s−s6
1.8 1.85 1.9 1.95
Proper semimajor axis (AU)
The Hungaria asteroids projected on the proper semimajor axis/proper sine of in-
clination plane. Contour lines (labels in arcsec/y) are drawn for the small divisors
associated to the secular resonances g − g6 and g − g5, and contour lines for the
values −0.5, 0, +0.5 arcsec/y for the weaker resonances g − g3, g − g4 and s − s4.
12
13. 6. Dynamical families (histogram)
200
150
100
50
0
0 1000 2000 3000
`
The stalactite diagram (following Zappala et al. 1994) for the Hungaria family has
been computed by A. Cellino. There is no evidence for more than one large family.
The boundary is not known, for lack of statistical control on the background, but not
all the Hungaria belong to the family. There might be much smaller subfamilies.
13
14. 6. The Hungaria family: internal structure
200
180
160
140
Number of H>15.8 Hungaria
120
100
80
60
40
20
0
1.8 1.85 1.9 1.95 2
Proper semimajor axis (AU)
Histogram of the proper semimajor axis a p of the Hungaria asteroids. The vertical
line marks the a p value for (434) Hungaria. The asimmetry is much larger than
what would be justified by observational selection effects.
14
15. 7. The Yarkovsky effect
Non-gravitational perturbations have been one of the most recurrent theme of PF
research: we begun together with the LAGEOS mistery drag (see slides), then
with a gneral theory for artifical satellites. In this context the Yarkovsky effect was
redisocvered by Rubincam in the 80s.
One of the innovative ideas was to show that non-gravitational perturbations, espe-
cially Yarkovsky, can be relevant for the long term evolution of asteroid dynamics,
as a transport mechanism modifying, over very long times, the dynamical structure
of all the asteroid populations.
I will just remind that the Yarkovsky effect is due to anisotropic thermal emission
from a body with non-uniform surface temperature. This is a result of the way the
energy from sunlight is absorbed by the surface and conducted inside the body.
15
16. 7. The two types of Yarkovsky effect
There are two main effects: the seasonal Yarkovsky effect is a result of the average
illumination of different parts of the surface, with stronger emission of thermal radi-
ation from the summer emisphere with respect to the winter emisphere; the secular
effect on semimajor axis can only be positive, that is the asteroid is pushed away
from the Sun.
The diurnal Yarkovsky effect results from the delayed effect of illumination on sur-
face temperature, like in the terrestrial experience of afternoons being hotter than
mornings. Thus the secular effect in semimajor axis is positive for a spin aligned
with the orbital angoular momentum, or anyway with obliquity < 90◦, negative for
retrograde rotation. The seasonal efect is significantly less important, typically one
order of magnitude smaller.
The size of this effect depends upon: the inverse square distance from the Sun,
from the inverse of the density and the inverse of the diameter; there is also some
dependence upon thermal properties like conductivity. Thus what is the collective
behaviour under Yarkovsky effect of asteroids of the same family? Even if density,
albedo and conductivity are the same for all, the rate of change of semimajor axis
with time is a function of the inverse size and the cosine of the obliquity of the spin
axis, with values from −1 to +1.
16
17. 7. Yarkovsky signature on the Hungaria family
1.5
1
Inverse of Diameter (1/km)
0.5
2035 1103 3447
434
0
−1500 −1000 −500 0 500
dV due to a− a(434) (m/s)
The distance in velocity space between the group asteroids and (434) Hungaria vs.
the inverse of the diameter in km, estimated assuming same albedo as (434) for
all. Note that for some non-family asteroids, known to be of different spectral type,
the diameter can be much larger than this estimate. Conclusion: many Hungaria
asteroids, especially the largest ones, do not belong to the Hungaria family.
17
18. 8. The family membership problem
1
0.5
Second principal component
0
−0.5
−1
−1.5
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1
First principal component
Principal components of the Sloan color photometry; only the data for the 338
Hungaria in the SDSS catalog are shown.
Without using any color taxonomy, we have split in two groups, one compatible with
the spectral properties of (434) Hungaria (green dots, on the left) and one clearly
incompatible (red crosses, on the right), defined simply by PC1 > 0.5.
18
19. 8. The family membership problem
1.8
1.6
1.4
1.2
Inverse of Diameter (1/km)
1
0.8
0.6
0.4
0.2
0
−1400 −1200 −1000 −800 −600 −400 −200 0 200 400 600
dV due to a− a(434) (m/s)
The same projection on the plane: relative velocity (by difference in proper semiajor
axis) vs. inverse of diameter, only for the 338 Hungaria with good SSDS color data.
Green circles: background asteroids because of the dimeter/semimajor axis, red
crosses: because of the color data (some for both reasons). Black dots inside the
red V-shape could be family members, although there could be interlopers.
19
20. 9. Very Close Couples of Hungaria (3-D distance)
no. name name d δa p/a p δe p δ sin I p
1 88259 1999VA117 0.0000144 +0.0000113 -0.0000007 +0.0000011
2 63440 2004TV14 0.0000313 +0.0000013 +0.0000129 +0.0000088
3 92336 143662 0.0001183 +0.0000839 +0.0000185 -0.0000203
4 23998 2001BV47 0.0001501 +0.0001190 +0.0000005 -0.0000099
5 160270 2005UP6 0.0001959 +0.0000833 +0.0000590 -0.0000583
6 84203 2000SS4 0.0002075 +0.0000881 -0.0000122 +0.0000871
7 133936 2006QS137 0.0002316 +0.0000740 -0.0001048 -0.0000169
8 2002SF64 2007AQ6 0.0002446 -0.0001329 +0.0000879 -0.0000182
9 173389 2002KW8 0.0003029 +0.0001382 +0.0001146 +0.0000483
10 27298 58107 0.0003204 -0.0001992 +0.0000071 -0.0001006
11 115216 166913 0.0003301 +0.0002278 +0.0000668 -0.0000501
12 45878 2001CH35 0.0003482 -0.0002590 +0.0000591 -0.0000250
13 25884 48527 0.0004012 +0.0000761 -0.0001089 +0.0001616
The smallest distances, in the 3-dimensional spaces of proper elements, among
Hungaria; the distance d roughly corresponds to a relative velocity, which ranges
between 30 cm/s and 8 m/s in this table.
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21. 9. Very Close Couples of Hungaria (5-D distance)
n name1 H1 name2 H2 dH TCA 1 [yr] TCA 2 [yr]
1 88259 14.82 1999VA117 16.99 2.17 -32000 -32588±687
2 63440 14.89 2004TV14 17.25 2.34 too many
3 92336 15.29 143662 16.40 1.11 -348850 -348964±446
4 23998 15.29 2001BV47 16.47 1.18 -406250 -406565±887
5 160270 16.44 2005UP6 17.37 0.93 -1734250 -1646315±163035
6 84203 15.58 2000SS4 16.59 1.01 -119159 -117593±4920
7 133936 16.10 2006QS137 16.60 0.50
8 2002SF64 18.41 2007AQ6 17.39 1.02 -108950 -113396±12938
9 173389 16.84 2002KW8 16.99 0.15
10 27298 15.16 58107 15.49 0.33
11 115216 15.70 166913 16.46 0.76
12 45878 14.29 2001CH35 15.91 1.62
13 25884 14.26 48527 15.75 1.49 -422100 -422733±900
Very close couples selected after filter 2: couples with the nearest times in the past
of close orbit similarities, obtained by the D-criterion. TCA1 is the time of maximum
orbit similarity for the two nominal asteroid orbits; TCA2 is the mean and range of
uncertainty of the same times of similarity obtained with clones. H1, H2 are the
absolute magnitudes.
21
22. 10. The theory on couples by PF
A hypothesis for the interpretation of asteroid couples, very close in proper ele-
ments, has been proposed long ago, see (Milani, 1994), with reference to the cou-
ple of Trojan asteroids (1583) Antilochus and (3801) Thrasimedes (the distance
expressed in velocity was found to be less than 10 m/s, also much less than the
escape velocity ∼ 65 m/s).
The idea, which was proposed by PF, is the following: the pairs could be obtained
after an intermediate stage as binary, terminated by a low velocity escape through
the so-called fuzzy boundary, generated by the heteroclinic tangle at the collinear
Lagrangian points.
This model predicts an escape orbit passing near one of the Lagrangian points L1
or L2 of the 3-body system asteroid-asteroid-Sun, with a very low relative velocity
of escape, which would be extremely unlikely to be obtained from a direct ejection,
whatever the cause of the fission. (For Antilochus-Thrasimedes, the ejection should
be at a velocity between 65 and 65.7 m/s to have a velocity at infinity < 10 m/s; PF
private report, December 4, 1991).
However, for comparatively large Trojans the mechanism to push the satellite to-
wards the weak stability boundary should be tidal friction, which appears to be too
slow.
22
23. 11. What is the origin of very close couples?
For Hungaria, in the most interesting case of (88259) and 1999 VA117, we have
found a close approach 32 500 years ago at the margins of the sphere of influence,
with relative velocity ∼ 10 cm/s. Escape is close to the asteroid orbital plane. The
non-gravitational perturbations should be the main cause of evolution for a binary.
The YORP effect could have an asymptotic state for the spin axis with an obliquity
of 180◦, from which the spin up could continue until rotational fission. We have
found no cases with ∆H < 1 magnitude (a case with ∆H = 0.93 is dubious). Equal
binaries may be rare rare, or maybe less likely to be the source of a couple.
It has been inappropriately reported that fission by rotational instability could lead to
immediate ejection of the satellite. In cases with realistic parameters, the satellite
cannot be placed on an hyperbolic orbit, but on an elongated elliptic orbit which
would later evolve with large scale instability and reach the weak stability boundary.
In conclusion, we do not yet have a self consistent theory of the evolution of an
asteroid spin state taking into account YORP, fission of the primary, tidal and non-
gravitational perturbations on the orbit of the satellite, until the weak stability bound-
ary is reached. Such theory may take quite some time to be developed. In the
meantime, the portion of this evolutive path we do understand is the last one, which
should be as suggested by PF many years ago.
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