Hereafter, we discuss so-called "triangular plots"; this means
relationship between the absolute magnitude `H` and the (proper) orbital
elements `a`, `e`, sin `i`.
We compare the results of our numerical simulation with the observed
distribution of Eos family asteroids (their proper
elements are taken from AstDys catalogue; membership identifications are done by means of HCM method).
Moreover, we briefly comment on possible dynamical processes, which might have
affected the overall shape of Eos family, with emphasis to what could
be learnt from these "triangular plots" (remember that each asteroid family
seem to be unique and particular with this respect!).

The main page on asteroids families, with a detailed description of the model, is available here.

**Initial conditions of the numerical simulations:**

`NTP` = 210;
osculating elements,
*thermal parameters:*
radii from 1.0 upto 32.3 km
(see data file),
bulk density 2500 km/m^{3},
surface density 1500 km/m^{3},
thermal conductivity 0.001 W/K/m,
thermal capacity 680 W/kg/K,
albedo 0.10,
infrared emissivity 0.90,
random rotation periods between 4 - 12 hr,
random (and *fixed*) spin axes orientations.

Please notice, that the initial dispersion in
`a`, `e`, sin `i` space
does NOT resemble the current, observed shape of the Eos family.
Mainly, we did not took into account the dependence
of the dispersion on asteroids' radii (with larger fragments
closer to the family barycenter and smaller ones farther).
This could be clearly seen at Figure 1.
This may seem as a drawback, but note that big bodies basically
did not move during the simulation, so that putting them more
tightly around the center of the family they would stay there.

**List of figures:**

- Figure 1 -
Sequences of
`H`(`a`),`H`(`e`),`H`(sin`i`) plots for times 0 upto 600 Myr; triangular shape of the family. - Figure 2 -
Proper
`a`(`t`) and dispersion`sigma`_{a}. - Figure 3 -
Proper (
`a`,`e`) plot with evolutionary paths of "synthetic" bodies and comparison with family members at 90 m/s and at 60 m/s. - Figure 4 -
The same for proper (
`a`,`i`). - Figure 5 -
`H`(`a`) plot of observed Eos family members at three different cut-off velocities (60, 80 and 90 m/s) and also including resonance borders. - Figure 6 -
`H`(`e`) plot at the three cut-offs. - Figure 7 -
`H`(`e`) plot with asteroids on right/left side of 9:4 mean motion resonance with Jupiter. - Figure 8 -
`H`(sin`i`) plot displaying the same two groups. - Figure 9 -
`H`(`e`) distribution of asteroids in`z`_{1}resonance (with |`z`_{1}frequency| < 1") - Figure 10 -
`H`(sin`i`) plot with asteroids in`z`_{1}resonance.

**See also:**

- Eos family and J9/4 resonance,
- Spectroscopic observations in the Eos region
(a discussion of interlopers in the family; a group of asteroids
in
`z`_{1}resonance)

initial state | after 600 Myr | |

H(a) |
||

H(e) |
||

H(sin i) |

Figure 1 -
**"Triangular plots"**,
sequences of `H`(`a`), `H`(`e`),
`H`(sin `i`) plots for times 2.5 Myr and 600 Myr.
According to these figures, the initial "box-like" structure
of the swarm slowly disperses (in all the elements) and forms something,
which can roughly resemble "triangular" cloud of observed asteroids.
The data (proper elements of our bodies) are avaiable here.
Click on the left images to obtain a whole sequence/animation
with time-step 100 Myr.

The smallest (yet unobservable) asteroids (`H` > 16 mag)
might be after 2 Gyr evolution even strongly depleted from this family
(by their interactions with J9/4, J7/3 and J11/5 resonances).
This certainly depends on the timescale of spin axis reorientations
(which might be affected by collisions, YORP effect and possible spin-orbital
resonances). Note also, that the extension of the corresponding "triangle"
of the observed family depends on the HCM cutoff velocity as we shall
discuss below.

Figure 2 -
Here is a plot of the proper `a`(`t`) and dispersion
of semimajor axes `sigma`_{a} during
our 600 Myr simulation. Note the "traffic" across the J9/4 resonance and
how only a fraction of asteroids get through it.

Figure 3 and 4 -
Proper (`a`, `e`) and (`a`, `i`)
plots with evolutionary paths of "synthetic" bodies and comparison with
family members in the background (at the HCM cut-off velocity equal to
90 m/s).

Figure 5 -
`H`(`a`)
plot of **observed Eos family members** at three different cut-off velocities
(60, 80 and 90 m/s) and also including resonance borders.
The marked width of resonances is calculated for the mean eccentricity
of the Eos family (ie. resonances could be narrower/wider
for asteroids with lower/higher eccentricities).

There is a significant number of asteroids behind the right branch of "Yarkovsky envelope". They cannot reach this location from the family center (at 3.02 AU) within 2 Gyr by Yarkovsky drift (even in case of most favorable spin axis orientation). Certainly, the furthermost may be interlopers in the family: we actually already known about few of them aaand if everything goes well during this spring, we should have a much clearer idea about these objects "far from the Yarko envelope".

Additionally the drift could be effectively speeded up by jumping through resonances (the widest "en route" are J9/4, J11/5 and 3J-2S-1). Notice that behind the J11/5 resonance the overlap is even more increased (upto 0.02 AU), which could possibly support this hypothesis. However, the sum of effective widths of all these resonances seem to be smaller than 0.02 AU, and this would be the maximum gain as compared to the Yarko drift.

Figure 6 -
`H`(`e`) plot at the three cut-offs.
Only at 90 m/s cut-off we include both groups of low and high
eccentricity bodies, thus we use this cut-off in the following plots.

Figure 7 -
`H`(`e`)
plot with asteroids on right/left side of 9:4 mean motion resonance with Jupiter.
Practically all bodies with eccentricities higher than 0.105 - and
forming thus the bulk of the "right wing" in the `H`(`e`) triangle at
higher HCM cutoffs - are also
located on the right side of J9/4 resonace, which may indicate,
that crossing through J9/4 is responsible for the eccentricity increase.

Figure 8 -
`H`(sin `i`) plot displaying bodies on the left/right
side of J9/4 resonance. Here we do not see a strong difference between
inclination distributions (as J9/4 does NOT strongly affect the value of
inclination).

Figure 9 -
`H`(`e`)
distribution of asteroids in `z`_{1} secular
resonance (with |`z`_{1} frequency| < 1").
Most of asteroids with `e` < 0.047 are likely to be affected
by `z`_{1} resonance; they could reach this
low-eccentricity region by "sliding along" the resonance borders,
while drifting by Yarkovsky effect.

Figure 10 -
`H`(sin `i`)
plot with asteroids in `z`_{1} resonance.
The triangular shape does not seem to be somewhat correlated
with the distribution of these asteroids.

Miroslav Broz (miroslav.broz@email.cz), Jan 16th 2002