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% Yarkovsky effect and the Dynamics of the Solar System (abstract of the PhD thesis)
% Miroslav Broz (miroslav.broz@email.cz), May 20th 2006

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{\large\bf Charles University, Faculty of Mathematics and Physics}

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{\large\bf Astronomical Institute}

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Abstract of the PhD Thesis

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{\Large\bf Yarkovsky effect and the Dynamics of the Solar System}

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{\large\bf Miroslav Bro\v z}

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{\large supervisor: Doc.~RNDr. {\bf David Vokrouhlick\'y}, DrSc.}

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Prague, 2006

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{\large\bf Univerzita Karlova, Matematicko-fyzik�ln� fakulta}

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{\large\bf Astronomick� �stav}

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autorefer�t doktorsk� diserta�n� pr�ce

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{\Large\bf Jarkovsk�ho jev a dynamika slune�n� soustavy}

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{\large\bf Miroslav Bro�}

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{\large �kolitel: Doc.~RNDr. {\bf David Vokrouhlick�}, DrSc.}

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Praha, 2006

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V�sledky diserta�n� pr�ce byly z�sk�ny b�hem kombinovan�ho doktorsk�ho studia
na MFF UK v~Praze v~letech 2001 a� 2006.

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\begin{tabular}{@{}p{6cm}l}
{\bf Uchaze�:}	& Mgr.~Miroslav Bro�		\\
		& Astronomick� �stav MFF~UK	\\
		& V~Hole�ovi�k�ch~2		\\
		& 180\,00 Praha~8		\\[24pt]
{\bf �kolitel:}	& Doc.~RNDr.~David Vokrouhlick�, DrSc.	\\
		& Astronomick� �stav MFF~UK		\\
		& V~Hole�ovi�k�ch~2			\\
		& 180\,00 Praha~8			\\[24pt]
{\bf �kol�c� pracovi�t�:}	& Astronomick� �stav MFF~UK	\\
				& V~Hole�ovi�k�ch~2		\\
				& 180\,00 Praha~8		\\
\end{tabular}

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\begin{tabular}{@{}p{6cm}l}
{\bf Oponenti:}	& \\[12pt]
Prof.~Joseph Burns	& RNDr.~Milo� �idlichovsk�, DrSc.	\\
Cornell University	& Astronomick� �stav AV �R		\\
Space Sciences		& Bo�n� II 1401				\\
NY 14853 Ithaca		& CZ 141\,31 Praha 4			\\
United States		& �esk� republika			\\
\end{tabular}

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Autorefer�t byl rozesl�n dne 26.~5.~2006.

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Obhajoba se kon� dne 28.~6.~2006 ve~13~hod. p�ed komis� obhajoby
doktorsk�ch diserta�n�ch prac� oboru F1 --- Teoretick� fyzika,
astronomie a~astrofyzika na MFF~UK, Ke Karlovu~3, 121\,16 Praha~2, v~m�stnosti �.~105.

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S disertac� je mo�no se sezn�mit na studijn�m odd�len� ---
doktorsk� studium MFF~UK, Ke Karlovu~3, 121\,16 Praha~2.

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\begin{tabular}{@{}p{6cm}l}
{P�edseda oborov� rady F1:}	& Prof.~RNDr.~Ji�� Bi��k, DrSc.		\\
				& �stav teoretick� fyziky MFF UK	\\
				& V~Hole�ovi�k�ch~2			\\
				& 180\,00 Praha~8			\\
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\section*{Acknowledgements}

The work of MB was supported by the Grant Agency of the Czech Republic
by grant 205/05/2737
and by the Observatory and Planetarium in Hradec Kr\'alov\'e.

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\chapter{Introduction}

We see an increasing interest in the non-gravitational phenomenon
called the {\em Yarkovsky/YORP effect\/} during the last 15~years.
This is motivated by many observations of small asteroids
(and their groups), which properties can be elegantly
interpreted with help of the Yarkovsky effect
--- an electromagnetic recoil force arising due to
anisotropic thermal emission from the surface of a celestial body.
The implications of this phenomenon on the evolution
of small Solar System bodies are the main topic of this thesis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Non-gravitational forces in the Solar System}

\section{A brief history of non-gravitational phenomena}

The Yarkovsky effect is called after \name{Ivan Osipovich Yarkovsky}\index{Yarkovsky, I.O.} (Figure~\ref{fig:ioyarkovsky}),
a person mostly unknown to present-day scientists.
His work fell into oblivion, but `his' effect (described in \cite{yarkovsky1901})
was rediscovered in 1950's by \"Opik and Radzievskii.

Only a very recent publication by \cite{Beekman06}
unveiled some information about his life and work
and we present an excerpt in Section~2.1 of the thesis.
We also reprint Yarkovsky's `lost' publication in Section~9.3.

\begin{figure}[h]
\begin{center}
\includegraphics[width=.35\textwidth]{figs/ioyarkovsky.eps}
\caption{Ivan Osipovich Yarkovsky. From \cite{Beekman06}.\label{fig:ioyarkovsky}}
\end{center}
\vskip-\baselineskip
\end{figure}


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\section{Non-gravitational forces acting on small bodies}

We review recent advances in the studies of non-gravitational forces.
We focus on meteoroids and small asteroids
in the 10\,cm--10\,km size range, for which the principal
force and torque arise from an anisotropic thermal emission
of the absorbed solar radiation energy. Related perturbations
of the orbital and rotational motion are called the Yarkovsky
and YORP effects.
%
This section is an extended version of the reviews published in~\cite{Brozetal06_iaus229}
and in the Triennial report 2003--2006 of the IAU Commission~7.

The \idx{Yarkovsky force} is related to the orbital dynamics (\cite[Rubincam 1995]{Rubincam95}; \cite[Vokrouhlick\'y 1998]{Vokrouhlicky98}, \cite[1999]{Vokrouhlicky99}).
Its diurnal\index{diurnal Yarkovsky effect} variant (Figure~\ref{fig:yyp}), driven by the rotational frequency,
dominates for bodies with low thermal conductivity
(e.g., with regolith on the surface).
It can either increase or decrease semimajor axis $a$
and the change $\Delta a$ is proportional to the cosine of the \idx{obliquity}~$\gamma$.
%
In case of the seasonal\index{seasonal Yarkovsky effect} variant, the changes of temperature on the surface
are mainly driven by the orbital frequency. It is a usual situation
for bodies with higher thermal conductivity (regolith-free surface).
The semimajor axis $a$ steadily decreases and $\Delta a \propto -\sin^2\gamma$.

The \idx{YORP torque} (\cite[Rubincam 2000]{Rubincam00}; \cite[Vokrouhlick\'y \& \v{C}apek 2002]{VokrouhlickyandCapek02})
works for non-spherical bodies only.
It has an asymptotic behaviour --- it pushes the obliquity towards $0$ or $180^\circ$
and the rotation period towards 0 or $\infty$.

The most important applications of the Yarkovsky/YORP models include:

\begin{publications}
\item precise ephemeris calculations
(\cite{Chesleyetal03});

\item explanation of the bi-modal obliquity distribution of the Koronis family members
(\cite{Vokrouhlickyetal03});

\item meteorite transport from the Main Belt
(e.g., \cite{Farinellaetal98}, \cite{VokrouhlickyandFarinella00});

\item delivery of near-Earth asteroids from the Main Belt
(\cite{MorbidelliandVokrouhlicky03});

\item resupply of unstable resonant populations
(\cite{Tsiganisetal03}, \cite{Brozetal05_mnras});

\item explanation of asteroid families distributions in the proper element space
(\cite{Bottkeetal01}, \cite{Vokrouhlickyetal06_eos}, \cite{Vokrouhlickyetal06_ahchron});

\item measurement of the Yarkovsky semimajor axis drift for the youngest clusters
(e.g., \cite{NesvornyandBottke04}, \cite{Nesvornyetal06_datura}).

\end{publications}


\begin{figure}[t]
\begin{center}
%\includegraphics[width=10cm]{figs/poster2.1}
%\includegraphics[width=10cm]{figs/iaus229_fig01.eps}
\includegraphics[width=10cm]{figs/iaus229_fig01.1}
\end{center}
\caption{An illustration of the Yarkovsky/YORP effect principle.
As an asteroid absorbs the solar radiation,
its part facing the Sun becomes hotter than the reverse one.
The infrared emission from the surface is then anisotropic,
what gives rise to the Yarkovsky force, affecting the orbital motion of the asteroid,
and the YORP torque, modifying the spin state.
A non-zero transverse component of the force, caused by
a thermal lag and the rotational or orbital motion,
gives rise to a secular change of the semimajor axis.}
\label{fig:yyp}
\end{figure}


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\section{A mathematical formulation of the Yarkovsky/YORP effect}

\label{sec:ymath}

We present, at first, a very simple analytical solution
of the 1-dimensional heat diffusion equation,
which allows us to quantitatively estimate the Yarkovsky acceleration.
This solution, thought being simple and clear, holds basic
properties of the Yarkovsky effect, such as its dependence on material,
the rotational or orbital frequency;
we also discuss the dependence on size and obliquity.

There is a description of the spherically symmetric solution
by \cite{Vokrouhlicky98}, \cite{VokrouhlickyandFarinella99} and \cite{Vokrouhlicky99}
in the second part, supplemented by notes on its implementation
in the \swiftrmvsy\ numerical integrator package
(i.e., the \idx{SWIFT} by \cite{LevisonandDuncan94}
modified by \cite{Broz99} to account for the Yarkovsky diurnal and seasonal accelerations).
An example, how the semimajor axis drift rates depend on size
is depicted in Figure~\ref{fig:yarko_prdrag_dadt}.

Finally, we mention 1-D non-linear numerical methods,
which \cite{CapekandVokrouhlicky04} use to calculate
Yarkovsky/YORP effect for realistic shapes of asteroids.




\begin{figure}
\begin{center}
\includegraphics[width=.6\textwidth]{figs/yarko_prdrag_dadt.eps}
\caption{The sum of absolute values $|\d a/\d t|$
of the diurnal and the seasonal semimajor axis drift rates\index{drift rate}
vs. size, calculated for spherical bodies with a moderate value
of obliquity $\gamma = 135^\circ$ and consisting of two materials:
bare basalt or regolith covered,
i.e., with high or low thermal conductivity.
%
The sum plotted here shows clearly
an approximate maximum total drift (per 1\,My) one can expect.
%
The mean collisional lifetime is roughly 50\,My for a 10-m stony meteoroid
and 500\,My for a 1-km asteroid.
%
Note, that we do not expect very small regolith-covered bodies to exist,
thus the drift rates larger than $10^{-2}\,{\rm AU}/{\rm My}$ are not realistic.
%
The drift rate caused by the \idx{Poynting-Robertson drag} is also plotted;
it prevails for sizes smaller than $\lesssim 10\,{\rm cm}$.}
\label{fig:yarko_prdrag_dadt}
\end{center}
\vskip-\baselineskip
\end{figure}


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\chapter{Transport of meteoroids ejected from (6)~Hebe, (170)~Maria or~(8)~Flora}

\looseness=-1
The Yarkovsky effect and gravitational resonances are the most important transport mechanisms
of meteorite precursors, meteoroids or asteroidal fragments in the size-range 1\,m to 1\,km.
We simulate the orbital evolution of putative particles
ejected from three parent bodies: (6)~Hebe, (170)~Maria and (8)~Flora.
Some results of this study were summarised in the review \cite{Bottkeetal02_a3}.
%
We have verified that meteoroids can be delivered to Mars- and Earth-crossing orbits
by the Yarkovsky effect and gravitational resonances on the timescale
of the order $10\,{\rm My}$.
%
Larger asteroids can escape from the Main Belt due to chaotic diffusion
in weaker resonances, which are very common in the inner Main Belt.
%
Transport times to the EC- orbits might be generally consistent
with the observed long CRE ages of meteorites.

We present examples of various processes involved in the meteoroid transport:
%
  i)~Yarkovsky drift and captures in low-order mean motion resonances (e.g., J3/1, J8/3)
     or the powerful $\nu_6$~secular resonance;
 ii)~crossing of resonances (which occurs commonly in the J3/1 case and
     rarely in the $\nu_6$ case for the fastest drifting meteoroids);
iii)~interaction with weaker high-order resonances (e.g., J10/3);
 iv)~exterior resonances with Mars (M1/2);
  v)~3-body resonances (4J$-$2S$-$1);
 vi)~capture in high-order secular resonances ($z_2$);
vii)~frequent collisional reorientations, which cause a random walk behaviour
     for low-conductivity bodies and effectively decrease the diurnal Yarkovsky drift.
%
All these processes affect delivery rates to Mars- and Earth-crossing orbits,
they can both increase or decrease the rate as compared to a simple
Yarkovsky semimajor axis drift towards the nominal position of gravitational
resonances; it depends on individual source regions and meteoroid properties.
We compute several statistical parameters, e.g., the rate of delivery to Mars-crossing orbits
(Figure~\ref{fig:met_mc_ec_time}),
the probabilities that particles of a given size are captured by a given resonance.

\begin{figure}[h]
\begin{center}
\tmpdim=.375\textwidth
\begin{tabular}{@{}cc@{}}
\includegraphics[height=\tmpdim]{figs/mc_time_median.eps} &
\includegraphics[height=\tmpdim]{figs/ec_time_median_wokey.eps} \\
\end{tabular}
\caption{The dependence of the median Mars- and Earth-crossing times on the radius
for regolith and basalt meteoroids originating from three parent bodies.
Typical values are of the order 10\,My, but the meteoroids drifting slowly
or located far from the major escape routes have the median equal to 150\,My
(i.e., the time span of the integration), because less than half of the bodies
ever reached MC (EC) orbits. The dashed curves indicate the characteristic
disruption timescale $\tau_{\rm disr}(R) = 16.78\,{\rm My} \sqrt{[R]_{\rm m}}$.}
\label{fig:met_mc_ec_time}
\end{center}
\vskip-\baselineskip
\end{figure}

Our results are consistent with the results of \cite{Bottkeetal00b} and we extend them in several respects.
Compared to their work, we integrated meteoroids not only 1\,m in radius,
but also larger bodies. The time span of our integration was substantially longer
(150\,My vs.\ 50\,My), which allowed us to calculate population decay rates
for the sizes $R = 1\,{\rm m}$, 10\,m, 50\,m and 100\,m.
We use a more precise Mars-crossing criterion and
we provide a direct comparison to the analytical decay rates
used in the Monte-Carlo model by \cite{VokrouhlickyandFarinella00},
which were calculated from pure Yarkovsky semimajor axis drift rates
and fixed positions of major resonances.
%
In the previous work, the initial velocities with respect to the parent bodies
were zero, but in our case we included the initial spread.
%
Additionally, we reported interactions with high-order secular resonances
and we estimated the crossing probability of the J3/1 resonance.




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\chapter{Yarkovsky driven orbit and spin axis of (2953)~Vysheslavia}

The asteroid (2953)~Vysheslavia and several others, located just above
the 5/2 mean motion resonance with Jupiter, are known to have
unstable orbits. The dynamical lifetime of Vysheslavia is estimated
to be of the order of only $10$\,My. Such a situation poses a problem,
since Vysheslavia is a member of the Koronis family, which is likely more
than $2$\,Gy old. Three main hypotheses were developed to
solve this apparent contradiction: (i) Vysheslavia might be an outcome
of a recent secondary fragmentation event in the family, (ii) Vysheslavia
might have been placed on its peculiar orbit by close encounters with nearby
massive asteroids, or (iii) the asteroid might have been transported
by a slow inward--drift of the semimajor axis due to the Yarkovsky effect.

We present numerical simulations of the orbital evolution,
with the Yarkovsky effect included, for Vysheslavia and several
neighbouring asteroids and we bring evidence that all these asteroids
have been transported to this unstable region from stable regions
(with larger semimajor axes) by the Yarkovsky semimajor axis drift
(\cite{Vokrouhlickyetal01_2953}, \cite{BrozandVokrouhlicky02}; see also Figure~\ref{f2}).

\begin{figure}[h]
\epsfxsize=10truecm
\begin{center}
 \mbox{\epsfbox{figs/I08681_fig2.eps}}
\end{center}
%\plotone{freqmap.1}
\caption{Koronis family members (open circles) projected on the plane of 
 synthetic proper semimajor axis and proper eccentricity;
 the Karin cluster is indicated in the lower right corner. Assuming
 an isotropic velocity field, ejecta from a hypothetical secondary
 collisional disruption near Vysheslavia (full circle) would fall in the
 region indicated by grey dots. Only one of the observed Koronis asteroids falls
 in this zone, notably in its stable part exterior to the 5/2 mean-motion
 resonance with Jupiter. Orbital stability is indicated by grey shading
 indicating a parameter $\sigma =1-n^{(2)}/n^{(1)}$,
 where $n^{(1)}$ and $n^{(2)}$ are proper mean motions
 of test particles calculated in the two consecutive time intervals, each
 spanning $1.2$~My (dark zones of small
 $\sigma$ value imply stability, white areas of high $\sigma$ value, such as
 the 5/2 mean motion resonance with Jupiter, are unstable). The family
 members were identified using the hierarchical clustering method with
 the standard metric and the velocity cutoff $60$~m/s.}
\label{f2}
\end{figure}

This scenario predicts the asteroids have retrograde orientations
of the spin axes (in order to drift towards the J5/2 resonance)
and this was confirmed for Vysheslavia by photometric observations
during the years 2000--2005 (\cite{Vokrouhlickyetal05_2953}).
%
We find admissible solutions for ecliptic latitude and longitude of the
rotation pole $P_3$: $\beta_{\rm p} = -64^\circ\pm 10^\circ$ and
$\lambda_{\rm p} = 11^\circ\pm 8^\circ$ or $P_4$: $\beta_{\rm p} =
-68^\circ\pm 8^\circ$ and $\lambda_{\rm p} = 192^\circ\pm 8^\circ$.
These imply obliquity values $\gamma = 154^\circ\pm 14^\circ$ and
$\gamma = 157^\circ\pm 11^\circ$, respectively. The sidereal rotation
period is $P_{\rm sid}= (0.2622722 \pm 0.0000018)$~day.


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\chapter{Yarkovsky origin of the unstable asteroids in the 2/1 resonance with Jupiter}

The 2/1 mean motion resonance with Jupiter, intersecting the main
asteroid belt at $\approx\!3.27\,{\rm AU}$, contains a small population
of objects. Numerical investigations (\cite{Roigetal02}) have classified three groups within
this population: asteroids residing on stable orbits (i.e., Zhongguos),
marginally stable orbits with dynamical lifetimes on the order $100\,
{\rm My}$ (i.e., Griquas) and unstable orbits. We reexamine
the origin, evolution and survivability of objects in the
2/1 population (\cite{Brozetal05_mnras}).

Using recent asteroid survey data, we have identified
one hundred new members since the last search, which increases the
resonant population to $153$.
The most interesting new asteroids are those located
in the theoretically-predicted stable island A, which
until now had though to be empty (\cite{Brozetal05_iauc197}).

Next, we investigated whether the population of objects residing on the unstable orbits
could be resupplied by material from the edges of the 2/1 resonance
by the Yarkovsky/YORP effect. Using \hbox{$N$-body} simulations, we showed
that test particles pushed into the 2/1 resonance by the Yarkovsky effect
visit the same regions occupied by the unstable asteroids (Figure~\ref{3d_all}).
We also found that our test bodies had dynamical lifetimes consistent with the
integrated orbits of the unstable population (Figure~\ref{num_lifetime}).

\begin{figure}[h]
%\includegraphics[width=84mm]{figs/distmap25bw.eps}
\begin{center}
\includegraphics[width=84mm]{figs/ME1454_fig11.eps}
\end{center}
\caption{An~$n_{\rm TP}=500$ iso-surface of the number density
 in the pseudo-proper orbital element space $(a_p, e_p, \sin I_p)$, resulting from our numerical
 simulation of test particles originating in the neighbouring Main Belt.
 $n_{TP}$ is given as a weighted mean of the contributions
 by the three source populations:
 the background population (contributing by $84.5$\,\%),
 the Themis family ($14.2$\,\%)
 and the Hygiea family ($1.3$\,\%).
 $n_{\rm TP}$ reaches its maximum value of $\simeq 3\times 10^4$~particles per cell per 1\,Gy
 inside this zone. Symbols denote positions of the observed  populations
 inside the 2/1 resonance:
   (i)~Zhongguos (filled circles),
  (ii)~Griquas (squares), and
 (iii)~the unstable asteroids (crosses).
%
 The 3-D surface is plotted as semi-transparent and one can distinguish
 the objects, which are in front of, inside or behind the surface,
 because they are gradually more and more gray/hidden.
%
 We see the test particles visit mainly the space, where the unstable
 asteroids are located; on the other hand, they avoid positions
 of long-lived Zhongguos and Griquas.
%
 An illustrative animation with several coloured and
 partially transparent iso-surfaces can be found on \cite{yarkosite}.
 \label{3d_all}}
\end{figure}

\begin{figure}[ht]
%\includegraphics[width=84mm]{figs/r21_tpsobs_loglftime_woxi_bw.eps}
\begin{center}
\includegraphics[width=84mm]{figs/ME1454_fig14.eps}
\end{center}
\caption{Distribution of the residence time inside the J2/1 for: (i)~test
 bodies which were delivered into the J2/1 by the Yarkovsky effect (bold solid
 curve) and (ii)~observed members of the unstable J2/1 population currently
 residing in the resonance (dashed curve).
 In the case (i), the residence time records the time interval from entry into the J2/1
 till escape out of the J2/1, while the case (ii) records the time interval from the present
 day to escape. The number of bodies $dN$ in each logarithmic bin has been
 normalised by their total number $N$.
 \label{num_lifetime}}
\end{figure}


Using a semi-analytical Monte-Carlo model, we computed the steady-state size
distribution of magnitude $H < 14$ asteroids on unstable orbits within the
resonance. Our results provide a good match with the available observational data.
%
The J2/1 objects with extremely short dynamical lifetimes ($\le2\,{\rm My}$)
are probably Jupiter family comets or near-Earth asteroids.
We were not able to unveil the origin of the long-lived
Zhongguos and Griquas.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Yarkovsky footprints in the Eos family}\index{Eos family}

The Eos asteroid family is the third most populous, after Themis and
Koronis, and one of the largest non-random groups of asteroids in the
main belt. It has been known and studied for decades, but its structure
and history still presented difficulties to understand. We first
revise the Eos family identification as a statistical cluster
in the space of proper elements. Using the most to-date catalogue of
proper elements we determine a nominal Eos family, defined by us
using the hierarchical-clustering method with the cut-off velocity of
$55$~m/s, contains some $4400$~members. This unforeseen increase
in known Eos asteroids allows us to perform a much more detailed
study than was possible so far (\cite{Vokrouhlickyetal06_eos}).

We show, in particular, that most
of the previously thought peculiar features are explained
within the following model: (i) collisional disruption of the parent
body leads to formation of a compact family in the proper element
space (with characteristic escape velocities of the observed asteroids
of tens of metres per second, compatible with hydrocode simulations),
and (ii) as time goes, the family dynamically evolves due to a
combination of the thermal effects and planetary
perturbations. This model allows us to explain sharp termination of the
family at the J7/3 mean motion resonance with Jupiter, uneven
distribution of family members about the J9/4 mean motion resonance
with Jupiter, semimajor axis distribution of large vs. small members
in the family and anomalous residence of Eos members inside the
high-order secular resonance $z_1$ (Figure~\ref{fig:eos}).
Our dynamical method (\cite{Vokrouhlickyetal06_ahchron}) also allows us
to estimate Eos family age to $1.3^{+0.15}_{-0.2}$~Gy.

Several formal members of the Eos family are in conflict with our model and these
are suspected interlopers. We use spectroscopic observations, whose results
are also reported here, and results of 5-color wide-band Sloan Digital Sky
Survey photometry to prove some of them are indeed spectrally incompatible
with the family.

\begin{figure}[h]
\begin{center}
%\includegraphics[height=5.7cm]{figs/eos_family4a.eps}
\includegraphics[width=.475\textwidth]{figs/iaus229_fig10a.eps}
\hfill
%\includegraphics[height=5.7cm]{figs/eos_family4b.eps}
\includegraphics[width=.475\textwidth]{figs/iaus229_fig10b.eps}
\end{center}
\caption{The Eos family in the 3-dimensional space of proper elements
$a_p$, $e_p$ and $\sin I_p$. The three resonances, J7/3 and J9/4 (left)
and $z_1$ (right) are plotted together with examples of bodies drifting
due to the Yarkovsky effect from the center of the family
and interacting with these resonances.
An illustrative animation can be downloaded from \cite{yarkosite}.}
\label{fig:eos}
\end{figure}

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\chapter{The Agnia family embedded inside the $z_1$ secular resonance}

The Agnia asteroid family, a cluster of asteroids located near semimajor
axis $a = 2.79$~AU, has experienced significant dynamical evolution over
its lifetime. The family, which was likely created by the breakup of a
diameter $D \sim 50$~km parent body, is almost entirely contained within
the high-order secular resonance $z_1$. This means that unlike other
families, Agnia's full extent in proper eccentricity and inclination is
a byproduct of the large-amplitude resonant oscillations produced by
this resonance. Using numerical integration methods, we found that the
spread in orbital angles observed among Agnia family members (Figure~\ref{agnia_z1a})
would have taken at least $40$~My to create; this sets a lower limit
on the family's age (\cite{Vokrouhlickyetal06_agnia}).

To determine the upper bound on Agnia's age, we used a Monte Carlo
model to track how the small members in the family evolve in semimajor
axis by Yarkovsky thermal forces. Our results indicate the family is no
more than $140$~My old, with a best-fit age of $100^{+30}_{-20}$~My.
Using two independent methods, we also determined that the $D \sim 5$~km
fragments were ejected from the family-forming event at a velocity near
$15$~m/s. This velocity is consistent with results from numerical
hydrocode simulations of asteroid impacts and observations of other
similarly-sized asteroid families. Finally, we found that $57$\% of known
Agnia fragments are prograde rotators. The reason for this limited
asymmetry is unknown, though we suspect it is a fluke produced by the
stochastic nature of asteroid disruption events.

\begin{figure}
\begin{center}
 \epsfxsize=9cm \mbox{\epsfbox{figs/agnia_z1_examples_33.eps}}
\end{center}
\caption{Agnia family members (symbols) projected onto the
 plane of critical angle $\sigma$ of the $z_1$ resonance vs. the
 associated frequency ${\dot \sigma}=g+s-g_6-s_6$. Bulk of the
 family asteroids reside inside this resonance and close to its
 separatrix. Filled symbols for librating orbits, open symbols
 for circulating orbits. The solid lines show evolution paths
 as determined from $10$~My numerical integration for three
 characteristic cases: (i) asteroid (3701) Purkyne is the
 most typical evolutionary track in the family (large-amplitude
 libration), (ii) asteroid 2002 AA43 is a rare case of small-amplitude
 libration in the resonance, and (iii) asteroid (847) Agnia
 is located right outside the resonance, but very near to its
 separatrix. Libration period for (3701) Purkyne is about $8$~My,
 circulation of (847) Agnia, very close to the separatrix, is
 even slower.
 Dispersion of the family members along all values of the resonance
 angle $\sigma$ shows the age of the family to be at least several times
 the characteristic libration period.
\label{agnia_z1a}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Conclusions}

We studied the influence of non-gravitational forces on the long-term
evolution of meteoroids and small asteroids. The major results presented in this
thesis can be summarized as follows:

\begin{enumerate}

\item We reviewed the applications of the Yarkovsky/YORP effect
and its mathematical formulation.

\item We described major aspects of the orbital evolution of meteoroids
and small asteroid fragments under the influence of gravitational
resonances and the Yarkovsky effect. We computed several statistics
(e.g., the crossing probability of the J3/1 resonance)
which may be used in future to improve Monte-Carlo models
of meteoroid transport.

\item The short-lived orbits of (2953)~Vysheslavia
and several other asteroids located just above 5/2 mean motion
resonance with Jupiter can be explained as a consequence of the Yarkovsky effect,
which pushes the asteroids from stable regions into the current
unstable positions.
%
We supported this conclusion by photometric observations
of (2953)~Vysheslavia --- they reveal the retrograde spin-axis
orientation, which is in concert with negative Yarkovsky semimajor axis drift.

\item The unstable asteroids inside 2/1 mean motion resonance with Jupiter
have been most probably transported from the neighbouring Main Asteroid Belt;
the Yarkovsky/YORP effect is efficient enough to keep
this transient population in steady-state.
Unfortunately, we were not able to unveil the origin of the long-lived
Zhongguos and Griquas yet.

\item We studied the Eos asteroid family, namely the three dynamical
processes, how the Yarkovsky drifting orbits interact with resonances:
``bracketing'' by strong mean motion resonances,
``crossing'' of weaker resonances, and
``trapping'' in secular resonances.
The Yarkovsky effect seems to be essential to understand
the current observed shape of the family in the space of proper orbital elements.
We were also able to determine the family age to be 1.3\,Gy.

\item The Agnia family is located almost entirely inside
the $z_1$ secular resonance, a lucky coincidence,
which allowed us to disclose the family age
(surely more than 40\,My and less then 140\,My)
and the magnitude of the mutual velocities
(15\,m/s for 5\,km fragments)
gained due to the original disruption event.

\item We implemented computation of proper and resonant orbital
elements into the commonly used SWIFT integration package.

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Appendices}

\section{A catalogue of synthetic proper elements}

\label{sec:filters}

In order to study fine details of orbital dynamics
and small structures in the $(a, e, I)$ elements space,
we have to apply a suitable digital averaging
to the osculating elements produced by numerical integrators.
We use the following approach for regular (non-resonant) orbits:
%
  i)~we sample osculating orbital elements;
 ii)~we apply a Kaiser windows filter (\cite{Quinnetal91}) to remove fast-period oscillations;
iii)~we apply frequency modified Fourier transform (\cite{SidlichovskyandNesvorny97})
     to obtain proper frequencies, amplitudes and phases;
 iv)~we drop the terms with planetary frequencies and the remaining largest
     amplitudes are the required proper elements.

We implemented the filtering outlined above
in the framework of the SWIFT integrator.
%
As a particular test of our algorithms, we prepared
a catalogue of proper elements (\verb|prop_fmft.dat|;
presented at \cite{yarkosite}). The system allows an automated
calculation of proper elements for all numbered asteroids,
listed the Bowell's AstOrb catalogue (\cite{astorb})
of osculating elements,
and a comparison with the already existing AstDyS proper elements catalogue
(\cite{astdys}) by Milani and Kne\v zevi\'c.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The SWIFT-MVS2 integrator, a faster variant of the MVS}

\label{sec:mvs2}

The numerical integrator package SWIFT (\cite{LevisonandDuncan94})
is a well known and commonly used tool for solar system studies.
We have modified the original code in the following manner:
%
i)~We have incorporated Yarkovsky and Poynting-Robertson
dissipative accelerations into the integrators and checked their results
against analytic predictions (see Section~\ref{sec:ymath}).
%
ii)~We have implemented on-line digital filters based on the Kaiser
windows (see Section~\ref{sec:filters}).
%
iii)~We have implemented the $2^{\rm nd}$ order symplectic integration scheme
(MVS2) by \cite{LaskarandRobutel01}, which seems to be at least 2 times
faster than MVS, while keeping the same relative energy error (Figure~\ref{fig:dt_energy_2}).
However, the algorithm is not ``regularised'', i.e., no close encounters
are allowed.
%
iv)~We have parallelised integrators (namely the calculation of the TP accelerations)
according to the OpenMP standard,
which allows to run SWIFT on multiprocessor machines.
%
We present results of various tests involving the new integrators.
The \swiftrmvsy\ package is available on our \cite{yarkosite}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{figs/dt_energy_2.eps}
\end{center}
\caption{The relative timestep $\tau'$ vs. the relative energy error,
computed for the Sun-Jupiter-Saturn system evolving for 0.1\,My.
$\tau'$ is the timestep $\d t$ divided by the method index,
i.e.,~1 for the classical leap-frog MVS and~2~for the new MVS2.
The relative energy error is computed as the standard deviation $\sigma_E$
of the total energy $E(t)$, in course of the integration,
divided by the mean total energy~$\langle E\rangle$.}
\label{fig:dt_energy_2}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A reprint of Yarkovsky's `lost' pamphlet}

The reprint of Yarkovsky's `lost' pamphlet quoted by \cite{Opik51}
is included in the thesis, which can be downloaded from the \cite{yarkosite}.
%
All credit goes to George Beekman (e-mail:\ \url{gbeek@xs4all.nl}),
who rediscovered the publication in the library
of the Sternberg Astronomical Institute in Moscow in 2002.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

\begingroup

\def\mag#1{\textit{#1\/}}
\def\book#1{\textit{#1\/}}
\def\vol#1{{\bf #1}}
\def\and{\&}

\begin{thebibliography}{}

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\bibitem[{\em AstOrb\/}]{astorb}
 {}
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\bibitem[Beekman (2006)]{Beekman06}
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%??? page range?

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\bibitem[Levison \& Duncan (1994)]{LevisonandDuncan94}
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\bibitem[Vokrouhlick\'y \etal\ (2006b)]{Vokrouhlickyetal06_ahchron}
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\end{thebibliography}

\endgroup

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}


