\input slides

\hyphenation{as-te-roid}

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\conference{NATO ASI --- The Restless Universe, Blair Atholl, Jul 23rd -- Aug 5th 2000}

\papertitle{Evolution of Yarkovsky-driven Orbits of Meteoroids}

\authors{M. Bro\v z\anot{1, 2}, D. Vokrouhlick\'y\anot{1, 3}}

\affilation{1}{Institute of Astronomy, Charles University, V Hole\v sovi\v ck\'ach 2, 180 00 Prague, Czech Republic}

\correspondence{2}{miroslav.broz@email.cz}
\correspondence{3}{vokrouhl@mbox.cesnet.cz}

\endtitle

\leavevmode\kern1cm\hbox{\vbox{\advance\hsize by -1.5cm
\ul{Numerical integrations}

\li{{\bf 150 Myr} numerical integration of {\bf 1500} asteroid fragments}

\li{symplectic integrator \swift, time step = 36 days}

\li{gravitational forces, Yarkovsky effect, simple collisional dynamics}

\li{output: mean elements (432 yr) $\sim$ 1 GB of data per 100 particles}

\ul{Initial conditions}

\li{7 planets (excluding Mercury and Pluto), DE 405 ephemeris}

\li{parent asteroids: (8) {\bf Flora}, (6) {\bf Hebe}, (170) {\bf Maria}}

\li{five different sizes: R = 1 m, 10 m, 50 m, 100 m and 500 m}

\li{isotropic power-law velocity distribution}

\li{surface conductivity: K = 0.0015 W/m/K and K = 2.65 W/m/K}

\ul{Data analysis}

\li{temporal decay of the main--belt population}

\li{3:1 crossing probability for different meteoroids' sizes}

\li{$\nu_6$ vs. pre-$\nu_6$ extraction from the main belt}

%\li{interaction with higher--order resonaces --- mixed mean motion
%with Jupiter and Saturn, outer with Mars and secular}

\li{interaction with higher--order, mixed mean motion, secular resonances}

}}

\vfill\eject

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\landscape
\pagetitle{Small, regolith covered meteoroids with high drift--rate}

\li{``100'' ejecta/fragments isotropically released from (6) {\bf Hebe}}

\li{low-conductivity surfaces assumed (``diurnal Yarkovsky effect dominates'')}

\li{``50'' fragments R = 10 m ({\bf dark}) and ``50'' fragments R = 50 m ({\bf light})}

\epsfxsize=\hsize
\figure{\epsfbox{hebe-5-a.ps}}

\portrait
\vfill\eject

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\landscape
\pagetitle{1 km large, slowly drifting fragments}

\li{``100'' ejecta/fragments isotropically released from (6) {\bf Hebe}}

\li{all fragments {\bf R = 500 m}}

\li{``50'' fragments K = 0.0015 W/m/K ({\bf dark}) and ``50'' fragments K = 2.65 W/m/K ({\bf light})}

\li{{\bf slow drift rate}, particles do not reach 3:1 or $\nu_6$ resonance
within 150 Myr}

\smallskip

\lii{{\bf (1)}}{observable captures in $\hbox{7J}-\hbox{2S}-2$
resonance, which do not affect drift rate}

\lii{{\bf (2)}}{captures in unidentified resonances, decrasing drift rate
by $\sim$ 10 \%}

\epsfxsize=\hsize
\figure{\epsfbox{hebe-8-a.ps}}

\portrait
\vfill\eject

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\landscape
\pagetitle{Orbital evolution of a fast drifting particle --- capture and ``jump'' events}

\li{{\bf Hebe} fragment, R = 10 m, regolith covered}

\li{The capture by the M1/2 resonance spans the
same time as a drift through this region.}

\li{The ``jump'' over the $\hbox{4J}-\hbox{2S}-1$ resonance yields a
higher effective drift rate.}

\epsfxsize=\hsize
\figure{\epsfbox{hebe-5-a-002.ps}}

\portrait
\vfill\eject

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\landscape
\pagetitle{Inner and middle belt --- comparsion of slow and fast TPs}

\li{(8) {\em Flora} fragments, $R = 500$~m, high conductivity ---
slow drift in $a$, {\em chaotic diffusion} in $e$ and $i$}

\li{meteoroids released from (170) {\em Maria}, $R = 10$~m, low conductivity ---
fast drift rate, depletion through 3:1, 5:2 and 8:3 resonances,
$\sim$~10~\% of TPs crossed 3:1 resonance}

\figure{%
\epsfysize=.5\hsize\epsfbox{flora-bas500-ae.ps}%
\hfill%
\epsfysize=.5\hsize\epsfbox{maria-reg10-ae.ps}%
}

\portrait
\vfill\eject

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\landscape
\pagetitle{Examples of orbits driven by secular resonances}

\li{150 Myr integration of 7+50 particles in the inner belt, $R = 100$~m,
regolith surface --- $(a, e)$ plane, borders of secular resonances
$\nu_6$, $z_2$ (Milani and Kne\v zevi\'c, 1997), dependence on inclination}

\li{50 Myr integration of 7+40 particles in the middle belt, $R = 200$~m,
C-type thermal parameters --- captures in resonances $g_5-2g_6+g$ and $z_1$}

\figure{%
\epsfysize=.45\hsize\epsfbox{flora-reg100_secres.eps}%
\hfill%
\epsfysize=.45\hsize\epsfbox{a25-200-1b_secres.eps}%
}

%\figure{%
%\epsfysize=.45\hsize\epsfbox{a25-200-1b-ae.ps}%
%\leavevmode\put{-9cm}{2.5cm}{particle no. 29}%
%\hfill%
%\epsfysize=.45\hsize\epsfbox{a25-200-1b-sr_029_prn1.ps}%
%}

\portrait
\vfill\eject

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\landscape

\li{run {\tt a25-200-1b}, particle no.~29 ($R = 200$~m, C-type)
 --- transition between $g_5-2g_6+g$ and $g+s-g_6-s_6$}
\li{critical angle $\dot\varpi_5-2\dot\varpi_6+\dot\varpi$
of resonance $g_5-2g_6+g$ --- librations between 0 and 40~Myr}

\figure{%
\epsfysize=.45\hsize\epsfbox{follow_29.eps}%
\hfill%
\epsfysize=.45\hsize\epsfbox{a25-200-1b-sr_029_prn1.ps}%
}

%\figure{%
%\epsfysize=.45\hsize\epsfbox{a25-200-1b-ae.ps}%
%\leavevmode\put{-9cm}{2.5cm}{particle no. 29}%
%\hfill%
%\epsfysize=.45\hsize\epsfbox{a25-200-1b-sr_029_prn1.ps}%
%}

\portrait
\vfill\eject

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\pagetitle{Temporal decay of meteoroids population}

\li{comparsion of {\em numerical integrations} (1, points) and
{\em Monte Carlo model} (Vokrouhlick\' y \and\ Farinella, 2000) (2, solid lines)}

\li{different criteria: (1) Mars--crossing orbit, (2) reaching the ``nominal''
position of $\nu_6$ resonance}

\ul{Maria regolith and basalt fragments}

\epsfxsize=.8\hsize
\figure{\epsfbox{maria_reg_decay_500k.eps}}
%
\leavevmode\vskip-1cm
%
\epsfxsize=.8\hsize
\figure{\epsfbox{maria_bas_decay_500k.eps}}

\vfill\eject

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\ul{Hebe and Flora fragments}

\epsfxsize=.8\hsize
\figure{\epsfbox{hebe_reg_decay_500k.eps}}
%
\leavevmode\vskip-1cm
%
\epsfxsize=.8\hsize
\figure{\epsfbox{flora_bas_decay_500k.eps}}

\li{{\em high order resonances} with Mars, mixed mean motion resonances with
Jupiter and Saturn in the Flora region affect large bodies $\Rightarrow$
diffusion in $e$, extraction without Yarkovsky, differences in decay-plots}

\vfill\eject

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\pagetitle{3:1 resonance crossing and interaction with $\textfont1=\mthtitle\nu_6$ resonance}
%\pagetitle{Statistics of 3:1 resonance crossing}

\li{significant number of 3:1 crossers: $\sim$~60~\% for Maria,
$\sim$~30~\% for Hebe and Flora regolith fragments with $R=1$~m}

\epsfxsize=.8\hsize
\figure{\epsfbox{cross31_bw.eps}}

\li{values of $\dot\varpi$, when TPs become Mars-crossers,
comparsion with $g_6$}

%\vfill\eject

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\separator

\ul{Applications}

\li{improvement of complex collisional models with only simple dynamics
--- resonance crossing, chaotic diffusion in $e$, $I$}

\ul{References}

\bibitem{Vokrouhlick\'y, D., Farinella, P.}
{Delivery of stony meteorites from the asteroid belt: A~dynamical--collisional Monte Carlo model}
{Nature, submitted (2000)}

\li{results also on Inet: \web{http://sirrah.troja.mff.cuni.cz/\~{}mira/mp/}}

\end


