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:

Zoran Knezevi c, Bojan Novakovi c and Alberto Cellino

1

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?

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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.

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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 subjectivemethods 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 numberof objects in the data catalog is a critical parameter. Two sets of data have to becombined: dynamical classification (some form of orbit similarity with the propertyof lasting over a very long time, 107

∼ 108 years) and spectral classification (inthe space of color parameters, related to the mineralogy). You need a large set ofasteroids for which you have both information available, and then there is a problemin the quality of the data.

Proper elements are quasi-integrals of motion, that is quantities stable over verylong (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 wasneeded. The taxonomic classes were ways to represent similarity of spectra andthus presumably of composition, but data were even less numerous and less accu-rate than for the dynamics, with controversies among the different authors.

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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 resultswas to establish the collaboration between Z. Knezevic and myself to produce anaccurate algorithm for analytic proper elements, which could be computed for all theasteroids with a good orbit. Others were involved in this, in particular the Namurgroup, but also Schubart and others in Latin America.

As soon as proper elements catalogs with better understood quality control andwith 5000 ∼ 6000 asteroids were available, dynamical classification into families,obtained with an objective mathematical taxonomy method, started making sensein the asteroid main belt (for moderate eccentricities and inclinations) (Zappala etal. 1994). The spectral classifications, although available only for smaller samples,did not show anymore statistically significant contradictions. Collisional evolutiontheories could use the proposed families as starting points.

This was a success, but was the job finished? NO! There are many groups ofasteroids outside the area covered by the 1994 work. They were considered in laterwork, including Hildas (Schubart 1982), Trojans (Milani 1992-1993, later Beaugeand Roig), high inclination MBA (Lemaitre and Morbidelli), other resonant groups(Moons and Morbidelli). Invariably, a good catalog of proper elements led to theidentification of asteroid families and was the starting point for the understandingof the dynamical and collisional evolution of these regions.

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3. The Main Belt of Asteroids and its marginal groups

2.2 2.4 2.6 2.8 3 3.2 3.4 3.60

5

10

15

20

25

30

35

40

Proper semimajor axis (AU)

Pro

per

incl

inat

ion

(DE

G)

red=Tlyap<20,000; green=rms(e,sinI)>0.003

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?

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4. The Hungaria group

1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 212

14

16

18

20

22

24

26

28

30

semimajor axis (AU)

incl

inat

ion

(deg

)

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).

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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.

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4. The Hungaria population: completeness

10 11 12 13 14 15 16 17 18 19 200

200

400

600

800

1000

1200

Absolute magnitude

Num

ber

of H

unga

ria

Numbered and multiopposition asteroids

The absolute magnitude (≃ size) distribution of the Hungaria known population(numbered and multiopposition orbits only). The search for Hungaria is completeup to a magnitude ≃ 16.5, corresponding to ≃ 900 m of diameter, assuming thealbedo estimated for (434) Hungaria by radar (0.38) is applicable to all.

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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 Knezevic and Milani) should be used for proper

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, Knezevic and myself switched to the com-

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. Novakovic

promptly computed synthetic proper elements for 4424 Hungaria.

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5. The Hungaria proper elements

1.75 1.8 1.85 1.9 1.95 20.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Proper semimajor axis (AU)

Pro

per

ecce

ntric

ity

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 secularresonances; the red points positive Lyapounov exponents. The green line corre-sponding to ap (1− ep) = 1.65 AU, the current aphelion distance of Mars.

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5. Hungaria region surrounded by resonances

1

2

2

s−s6

Dynamical boundaries

Proper semimajor axis (AU)

Pro

per

sine

of i

nclin

atio

n

0 0 0 0

1 1 1 1

2 2 2 2g−g5

g−g3

g−g4

s−s4

1.8 1.85 1.9 1.95

0.2

0.25

0.3

0.35

0.4

0.45

0.5

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.

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6. Dynamical families (histogram)

0 1000 2000 3000

0

50

100

150

200

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.

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6. The Hungaria family: internal structure

1.8 1.85 1.9 1.95 20

20

40

60

80

100

120

140

160

180

200

Proper semimajor axis (AU)

Num

ber

of H

>15

.8 H

unga

ria

Histogram of the proper semimajor axis ap of the Hungaria asteroids. The vertical

line marks the ap value for (434) Hungaria. The asimmetry is much larger than

what would be justified by observational selection effects.

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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.

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7. The two types of Yarkovsky effect

There are two main effects: the seasonal Yarkovsky effect is a result of the averageillumination of different parts of the surface, with stronger emission of thermal radi-ation from the summer emisphere with respect to the winter emisphere; the seculareffect on semimajor axis can only be positive, that is the asteroid is pushed awayfrom 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 thanmornings. Thus the secular effect in semimajor axis is positive for a spin alignedwith the orbital angoular momentum, or anyway with obliquity < 90◦, negative forretrograde rotation. The seasonal efect is significantly less important, typically oneorder 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 somedependence upon thermal properties like conductivity. Thus what is the collectivebehaviour 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 axiswith time is a function of the inverse size and the cosine of the obliquity of the spinaxis, with values from −1 to +1.

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7. Yarkovsky signature on the Hungaria family

−1500 −1000 −500 0 5000

0.5

1

1.5

Inve

rse

of D

iam

eter

(1/

km)

dV due to a− a(434) (m/s)

43411032035 3447

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.

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8. The family membership problem

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

First principal component

Sec

ond

prin

cipa

l com

pone

nt

Principal components of the Sloan color photometry; only the data for the 338Hungaria in the SDSS catalog are shown.

Without using any color taxonomy, we have split in two groups, one compatible withthe spectral properties of (434) Hungaria (green dots, on the left) and one clearlyincompatible (red crosses, on the right), defined simply by PC1 > 0.5.

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8. The family membership problem

−1400 −1200 −1000 −800 −600 −400 −200 0 200 400 6000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Inve

rse

of D

iam

eter

(1/

km)

dV due to a− a(434) (m/s)

The same projection on the plane: relative velocity (by difference in proper semiajoraxis) vs. inverse of diameter, only for the 338 Hungaria with good SSDS color data.

Green circles: background asteroids because of the dimeter/semimajor axis, redcrosses: because of the color data (some for both reasons). Black dots inside thered V-shape could be family members, although there could be interlopers.

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9. Very Close Couples of Hungaria (3-D distance)no. name name d δap/ap δep δsin Ip

1 88259 1999VA117 0.0000144 +0.0000113 -0.0000007 +0.00000112 63440 2004TV14 0.0000313 +0.0000013 +0.0000129 +0.00000883 92336 143662 0.0001183 +0.0000839 +0.0000185 -0.00002034 23998 2001BV47 0.0001501 +0.0001190 +0.0000005 -0.00000995 160270 2005UP6 0.0001959 +0.0000833 +0.0000590 -0.00005836 84203 2000SS4 0.0002075 +0.0000881 -0.0000122 +0.00008717 133936 2006QS137 0.0002316 +0.0000740 -0.0001048 -0.00001698 2002SF64 2007AQ6 0.0002446 -0.0001329 +0.0000879 -0.00001829 173389 2002KW8 0.0003029 +0.0001382 +0.0001146 +0.0000483

10 27298 58107 0.0003204 -0.0001992 +0.0000071 -0.000100611 115216 166913 0.0003301 +0.0002278 +0.0000668 -0.000050112 45878 2001CH35 0.0003482 -0.0002590 +0.0000591 -0.000025013 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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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±6872 63440 14.89 2004TV14 17.25 2.34 too many3 92336 15.29 143662 16.40 1.11 -348850 -348964±4464 23998 15.29 2001BV47 16.47 1.18 -406250 -406565±8875 160270 16.44 2005UP6 17.37 0.93 -1734250 -1646315±1630356 84203 15.58 2000SS4 16.59 1.01 -119159 -117593±49207 133936 16.10 2006QS137 16.60 0.508 2002SF64 18.41 2007AQ6 17.39 1.02 -108950 -113396±129389 173389 16.84 2002KW8 16.99 0.1510 27298 15.16 58107 15.49 0.3311 115216 15.70 166913 16.46 0.7612 45878 14.29 2001CH35 15.91 1.6213 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.

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10. The theory on couples by PFA 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 distanceexpressed in velocity was found to be less than 10 m/s, also much less than theescape velocity ∼ 65 m/s).

The idea, which was proposed by PF, is the following: the pairs could be obtainedafter an intermediate stage as binary, terminated by a low velocity escape throughthe so-called fuzzy boundary, generated by the heteroclinic tangle at the collinearLagrangian points.

This model predicts an escape orbit passing near one of the Lagrangian points L1or L2 of the 3-body system asteroid-asteroid-Sun, with a very low relative velocityof escape, which would be extremely unlikely to be obtained from a direct ejection,whatever the cause of the fission. (For Antilochus-Thrasimedes, the ejection shouldbe at a velocity between 65 and 65.7 m/s to have a velocity at infinity < 10 m/s; PFprivate 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 tooslow.

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11. What is the origin of very close couples?For Hungaria, in the most interesting case of (88259) and 1999 VA117, we havefound a close approach 32500 years ago at the margins of the sphere of influence,with relative velocity ∼ 10 cm/s. Escape is close to the asteroid orbital plane. Thenon-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 obliquityof 180◦, from which the spin up could continue until rotational fission. We havefound no cases with ∆H < 1 magnitude (a case with ∆H = 0.93 is dubious). Equalbinaries 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 toimmediate ejection of the satellite. In cases with realistic parameters, the satellitecannot be placed on an hyperbolic orbit, but on an elongated elliptic orbit whichwould 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 anasteroid 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 themeantime, the portion of this evolutive path we do understand is the last one, whichshould be as suggested by PF many years ago.

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