N.21 campo bagatin-the-legacy-of-paolofarinella-in-the-devel

The legacy of Paolo Farinella in the development of
collisional evolution models

Adriano Campo Bagatin
Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal.
Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías.
Universidad de Alicante (Spain)

INTERNATIONAL WORKSHOP ON PAOLO FARINELLA (1953-2000):
THE SCIENTIST AND THE MAN
Pisa, 14th to 16th of June, 2010

The legacy of Paolo Farinella in the development of collisional evolution models



Scaling laws in the strength regime

A fragmentation and cratering model

Collisional evolution of small solar system
bodies.




Q* : specific energy necessary to produce fragmentation
S

Shattering experiments in various materials

(Hartmann, 1969; Fujiwara et al., 1977; Fujiwara and Tsukamoto,
1980,1981; Lange and Ahrens, 1981; Matsui et al., 1982, 1984;
Kawakami et al., 1983; Fujiwara and Asada, 1983; Takagi et al.,
1984; Cintala and Hörz, 1984; Cintala et al, 1985; Smrekar et al.,
1985; Hartmann, 1985; …)

Strain-rate scaling + Self-compression

Scaling laws:

Q* ∝ aD−α + bDβ
S




Fujiwara (1980).
Moore et al. (1965) and Gault et al. (1972): Fragmentation
governed by the growth and coalescence of cracks.
Griffith: Cracks of lenght L begin to grow when the stress
exceeds a threshold value ~ L-1/2
Then: Largest cracks control failure of target:
Fracture stress ~ R-1/2 and also does Q*S.
Q*S ~ Tensile strenght (assumed ~ R-1/2)

Farinella et al. (1982), Paolicchi et al. (1983):
Energy required to fragment a body depends on the area
of new created surfaces, not on target’s volume!
They showd that:
Q*S~ R-1/2.




Holsapple and Housen (1986).
The era of dimensional analysis begins.
They adopt a rate-dependent material model:
fracture strength ~ (strain rate)1/4.
Q*S~ R-0.25

Housen and Holsapple (1990).
Assumption: fracture strength is strain-rate and target-size
dependence + energy and velotciy of impactor (V).
Q*S~ V0.35R-0.24.

Holsapple (1994). The duration of the loading is important.
In a large scale event, large flaws have time to coalesce and can
be activated at low stresses.
Q*S~ R-0.33

Housen and Holsapple (1999).
Fragmentation is accomplished through the growth and
coalescence of pre-existing flaws.
Q*S~ V0.35R-0.55 (Q*S~ R-0.667)



Benz and Asphaug (1999).
The era of Smooth Particle Hydrocodes.
Lagrangian approach.
Based on solving conservation equations (mass, momentum,
energy) + Hooke’s law + material e.o.s. (Tillotson).

Leinhardt and Stewart (2009).
CTH + N-body approach to study collisions in the asteroid size
range. Same power-law dependendence in the strength regime,
but Q*S 20 times smaller than in B&A.

Scaling laws of porous bodies (Jutzi et al., 2010)




Scaling laws:

Q* ∝ aD−α + bDβ
S

0.24 ≤ α ≤ 0.667
1.8 ≤ β ≤ 3.5



A fragmentation and cratering model

1993: Jean-Marc Petit, Paolo Farinella.
(Celestial mechanics and dynamical astronomy)

The most complete algorithm –to date- for:

fragmentation,
gravitational re-accumulation,
cratering

taking into account laboratory results of hiper-velocity
impacts and scaling-laws.





Collisional evolution of small solar system bodies.
What do we mean?

Collisional Systems
The Main Asteroid Belt ∼ 10−3 M⊕
The Trojan Asteroids ∼ 10−4 M⊕
The Trans-Neptunian Objects ∼ 10−1 M⊕



Theoretical studies

• Pietrowski (1953): dN(m) ∝ m −5 / 3dm
• Hellyer (1970), Dohnanyi (1969):
rate of change of the number of particles
∂f (m, t )
= per unit volume and unit time in mass
∂t range m to m+dm due to erosion of these
particles by collisions with smaller ones

rate of loss, because of ‘catastrophic
−
dN(m) ∝ m−11 / 6dm
collisions, in the number of particles per
unit volume and unit time in the mass
range m to dm

number of particles in the mass range m

+ to m+dm, created per unit time and unit
volume by erosive and catastrophic colli-
sional crushing of larger objects



Theoretical studies

Dohnanyi’s assumptions:

i) Asteroids are spheres of equal density.
ii) All the collisional response parameters
are size independent.
iii) The population has an upper cutoff in
mass, but no lower cutoff.



Theoretical studies
Dohnanyi’s result confirmed by other researchers:
• Paolicchi (1994)
• Williams and Wetherill (1994)
−4
The -11/6 exponent changes less than 10 when
the relative importance of cratering and catastrophic breakup
events, the mass distribution of fragments from a single impact, etc.
are varied in a substantial way.

• Tanaka (1996)
The resulting power--law distribution is
independent on the details of collisional outcomes
as long as the fragmentation model is self--similar,
and the value of the exponent itself is determined
only by the mass-dependence of the collisional rate

• Martins (1999)
Non steady-state: dN(m,t+dt) can be described
( )
5/3 −q n
as a power series of m



Observables: size distribution

Dohnany
SK A D S

Waves in the
Main Asteroid Belt?
Where is the
Dohnanyi’s slope?



Collisional evolution of asteroids.

Fragmentation algorithm (Petit and Farinella, 1993)
+ Evolution algorithm





Models results.

Waves in the
Main Asteroid Belt?

Release of Dohnani’s
iii) assumption:
Introducing a sharp
lower cutoff in the
size distribution

Campo Bagatin et al. (1994)


Models results.
Waves in the
Main Asteroid Belt?

Release of Dohnani’s
ii) assumption:
NON self-similarity in
fragmentation physics

Discontinuities (Campo Bagatin
et al., 1994, Durda et al., 1997)
and size dependence of Q*
(Durda et al., 1997) trigger
wavy behaviour in distributions.

Durda et al. (1998)


Our model upgraded

First estimations of the abundances of
gravitational aggregates in the M.A.B.
Depending on scaling-law, most 10<D<100 km
bodies should be G.A.




From asteroids to TNOs (Davis & Farinella, 1997):

First simulations of the EKB collisional evolution.

Prediction of a roll over in size distribution around
50-100 km.
Larger bodies keep their primordial size distribution
> Dohnanyi’s slope.

Small (<50 km) bodies are fragments rather than primordial.

Estimation of fragment production as a source of JFC.





The EKB, 15 years after.

Observables
Thousands of TNOs observed.
4 dwarf planets.
Roll-over of size distribution confirmed around
50-150 km.
High slopes in size distributions of large TNOs
(dN~D-adD, a~4.5-4.8).
Number of cold classical objects constrained by
CFEPS (45000-55000).

A dynamical framework: the Nice model.




Asteroid LIke Collisional ANd Dynamical Evolution Package
(ALICANDEP) (Campo Bagatin & Benavidez):

Mutually interacting 3-zone collisional evolution.
Maxwellinan distributions for relative velocities.
Zones evolve in time according to the NM (positions,
orbital elements, volumes)
Dynamical depletion statistically implemented in
different phases (pre-LHB, LHB, post-LHB)
Migration of bodies according to NM.
Algorithm for keeping track of
- Gravitational aggregates
- Dynamically “cold” bodies
- Primordial populations.


Observables met.
4 dwarf planets.
Roll-over of size distribution confirmed around 50-150 km.
High slope in s.d. (dN~D-adD, a~4.5-4.8) of large TNOs
confirmed.
Number of cold classical objects (45000-55000) and ratio of
Plutinos/Classical objects (CFEPS)
Model’s implication on initial conditions.
M0~60 ME if size distribution for small objects was shallower
than dN~D-bdD, b~3.0.
Scaling law not much “weaker” than B&A 1999 for ice.
Initial distributions compatible with surf. density ~ r-3/2.
Other model predictions.
Present mass: 0.15-0.18 ME
~50% prob. of existence of more bodies > 2000 km
5-10% Plutinos, 25-30 % Classical objects are primordial
A few Mars-size objects survived the LHB. Lately scattered by
Neptune’s perturbations.


Summary

Paolo’s intuition and work had a key role in the
development of fragmentation models.

His pioneering work in collisional evolution has
widely improved our understanding of the
evolution of small bodies populations.

His legacy is still inspiring current research in this
area.



Conclusion

We definitely miss Paolo,
both the amazing great person
and the outstanding scientist.

Thank you, once again, for your
human and scientific legacy!


The legacy of Paolo Farinella in the development of
collisional evolution models

Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal.
Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías.
Universidad de Alicante (Spain)

INTERNATIONAL WORKSHOP ON PAOLO FARINELLA (1953-2000):
THE SCIENTIST AND THE MAN
Pisa, 14th to 16th of June, 2010

N.21 campo bagatin-the-legacy-of-paolofarinella-in-the-devel

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