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\shorttitle{Rings and Close Encounters}
\shortauthors{Wood et al.}
\title[MEASURING THE SEVERITY OF CLOSE ENCOUNTERS BETWEEN RINGED SMALL BODIES AND PLANETS]{MEASURING THE SEVERITY OF CLOSE ENCOUNTERS BETWEEN RINGED SMALL BODIES AND PLANETS}
\author[Jeremy Wood]{Jeremy Wood$^{1}$, Tobias C Hinse$^{2}$, Jonti Horner$^{3}$, Awaiting Activation$^{4}$\\
$^{1}$Affiliation not available
\\
$^{2}$Affiliation not available\\
$^{3}$Affiliation not available\\
$^{4}$University College of Southern Queensland}
\begin{document}
%\date{Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11}
%\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2002}
\maketitle
\label{firstpage}
\selectlanguage{english}
\begin{abstract}
The field of ringed Centaurs is only a few years old. Since Centaurs are known to regularly encounter the giant planets, it is of interest to explore the effect of a close encounter between a ringed Centaur and a giant planet on the ring structure. The severity of such an encounter depends on quantities such as the small body mass; velocity at infinity, $v_{inf}$; ring orbital radius, $r$; and encounter distance. In this work, we derive a formula for a critical distance at which the radial force is zero on a collinear ring particle in the four-body, circular restricted, planar problem. Numerical simulations of close encounters with Jupiter or Uranus in the three-body planar problem are made to experimentally determine the largest encounter distance, $R$, at which the effect on the ring is "noticeable" using different values of small body mass, $v_{inf}$, and $r$. $R$ values are compared to the critical distance. We find that $R$ lies inside the critical distance for Centaurs with masses $\ll$ the mass of Pluto but can lie beyond it for Centaurs with the mass of Pluto and ring structure analogous to Chariklo's. Changing the mass by a factor of almost 4 changed $R$ by $\le 0.2$ tidal disruption distance, $R_{td}$. Effects on $R$ due to changes in $v_{inf}$, or $r$ are found to be $\le$ 1.5$R_{td}$. $R$ values found using a four-body problem suggest that the critical distance might be useful as a first approximation of the constraint on $R$.%
\end{abstract}%
%\begin{keywords}
%circumstellar matter -- infrared: stars.
%\end{keywords}
\correspondingauthor{Jeremy Wood}
\email{jeremy.wood@kctcs.edu}
\keywords{minor planets, asteroids: individual: 10199 Chariklo, planets and satellites: dynamical evolution and stability, planets and satellites: rings}
\section{INTRODUCTION}
Since the discovery of the small body Chiron in 1977\citep{KowaiCT:1979} small icy-rock bodies between Jupiter and Neptune continue to be found and are known collectively as Centaurs. The exact definition of a Centaur varies with the researcher, but this work will define Centaurs as objects with semi-major axes between the orbits of Jupiter and Neptune, and perihelia beyond Jupiter\citep[e.g.][]{SheppardS:2000}. Using this definition, more than 130 of these objects have been discovered\footnote{http://www.minorplanetcenter.net/iau/lists/Unusual.html (accessed 15th January 2016)}.\par
Dynamically, Centaurs are indeed an ephemeral class of object with dynamical lifetimes on the order of 10 Myr\citep{TiscarenoM:2003} which is much less than the age of the Solar system. It is the general consensus that Centaurs were perturbed into their present orbits via gravitational interactions with the giant planets and are former members of other more stable small body populations such as the Kuiper Belt objects\citep[e.g.][]{LevisonHF:1997,HornerJ:2004b}, Scattered Disk Objects\citep[e.g.][]{DiSistoRP:2007, VolkK:2008} Oort Cloud objects\citep[e.g.][]{Emel'yanenkoVV:2005, BrasserR:2012, FouchardM:2014}, Trojan asteroids of Jupiter\citep[e.g.][]{HornerJ:2006, HornerJ:2012b}, and Trojan asteroids of Neptune\citep{HornerJ:2010a, HornerJ:2010b, HornerJ:2012a}.\par
While existing as Centaurs, frequent close encounters with the four giant planets further perturb the orbits of these bodies, chaotically redistributing them throughout the Centaur region and finally to other small body populations of the Solar system such as the short period comet population or other fates such as a planetary collision, collision with the Sun, or ejection from the Solar system\citep[e.g.][]{HornerJ:2004a,HornerJ:2004b, BaileyBL:2009}.\par
Since the orbits of Centaurs are frequently perturbed, it came as a complete surprise when two narrow rings were discovered around the largest Centaur Chariklo after it occulted a star in 2013\citep{Braga-RibasF:2014}. Since this discovery it has been suggested that rings may also exist or have existed around satellites of Saturn and Uranus; and even the dwarf plant Pluto\citep{2011IJFPS...1....6R,2016arXiv161203321S}.\par
But the best candidate for the next ringed small body is the Centaur Chiron as extra nuclear material has been observed. A reanalysis of star occultation data for Chiron showed that one possible interpretation of the data was that Chiron could have rings with a mean orbital radius of 324 km\citep[e.g.][]{OrtizJL:2015, PanM:2016}.\par
Given the frequency with which Centaurs encounter the giant planets, a natural question is what effect a close encounter with a giant planet would have on the rings. This has been investigated for the case of Chariklo by\citet{AraujoRAN:2016} in the four-body non-planar problem who stated that the effect on the rings was "noticeable" if the maximum change in eccentricity, $e_{max}$, of the orbit of any ring particle was $\ge 0.01$ given that the orbits of the ring particles were initially circular.\par
The goals of this work are to use numerical integration in the three-body problem(Sun, small body, ring particle) to experimentally determine: the distance of closest approach, $R$, of a close encounter between a Chariklo-like body and a planet for which $e_{max}=0.01$ (or in other words a close encounter for which the effect on the outermost ring is just noticeable); the dependence of $R$ on small body and planet mass, velocity at infinity, and ring orbital radius; the dependence of the distance of closest approach on $e_{max}$; and the relationship between $e_{max}$ and the largest change in semi-major axis of any ring particle, $a_{max}$.\par
This paper is partitioned as follows: in section 1 we introduce our topic of study, in section 2 we describe the properties of Chariklo and its rings, in section 3 we present the theory of close encounters, in section 4 we present our experimental method, in section 5 we present our results and finally summarize conclusions in section 6.
\section{THE PROPERTIES OF CHARIKLO AND ITS RINGS}
The Spacewatch program\footnote{http:\/\/spacewatch.lpl.arizona.edu\/discovery.html accessed 29th October, 2016} discovered Chariklo in 1997. Though this was decades ago, even today many important properties of Chariklo remain largely unknown. For example, it has been suggested that the density of Chariklo could be anywhere from 800 kgm$^{-3}$ to 3,000 kgm$^{-3}$, and its mass anywhere in the range $6 -- 30\times 10^{18}$ kg\citet{2016arXiv161203321S}.\par
The exact size of Chariklo also remains elusive. Various values of its effective physical radius have been reported ranging from 118 km - 151 km. A few are shown in Table~\ref{chariklo_radius}.\par
Table~\ref{chariklo_orbit} lists the orbital properties of Chariklo. Its current orbit places it between Saturn and Uranus where it crosses the orbit of Uranus but not Saturn.\par\selectlanguage{english}
\begin{table}
\begin{center}
\caption{{Orbital elements of Chariklo taken from the Asteroids Dynamic Site (http://hamilton.dm.unipi.it/astdys/index.php?pc=1.1.0\&n=Chariklo accessed 1 February, 2017) for epoch 57800.0 MJD based on an observational arc of 9,684.35 days.}}\label{chariklo_orbit}
\begin{tabular}{|c|c|}
\hline
Element&Value $\pm$ Uncertainty (1-sigma)\\
\hline
$a$&15.8116$\pm$3.783$\times 10^{-5}$ au\\
$e$&0.172363$\pm$1.809$\times 10^{-6}$\\
$i$&23.391$\pm$1.525$\times 10^{-5}$ deg\\
$\Omega$&300.408$\pm$2.887$\times 10^{-5}$ deg\\
$\omega$&242.671$\pm$0.0001575 deg\\
$M$&74.735$\pm$0.0003251 deg\\
\hline
\end{tabular}
\end{center}
\end{table}\selectlanguage{english}
\begin{table}
\begin{center}
\caption{{Possible values for the physical radius of Chariklo. [1]\citep{GroussinO:2004}, [2]\citep{FornasierS:2014}, [3]\citep{Braga-RibasF:2014}, [4]\citep{JewittD:1998}.}}\label{chariklo_radius}
\begin{tabular}{|c|c|}
\hline
Radius (km)&Uncertainty\\
\hline
118$^1$&$\pm 6$\\
119$^2$&$\pm 5$\\
124$^3$&$\pm 9$\\
151$^4$&$\pm 15$\\
\hline
\end{tabular}
\end{center}
\end{table}
Properties of the rings of Chariklo are shown in Table~\ref{ring_data}. Analysis of occultation data showed that the rings have radii 391 km and 405 km. For a mass of Chariklo of $8\times 10^{18}$ kg these distances correspond to 0.00151 and 0.00156 Hill radius respectively. The rings were separated by a gap of about 9 km and had widths of about 7 km for the inner ring and 3 km for the outer ring. The rings are assumed to orbit Chariklo in its equatorial plane in circular orbits. The rings are believed to be composed of a little amorphous carbon, silicates, water ice, and tholins\citep{DuffardR:2014}.\par\selectlanguage{english}
\begin{table}
\begin{center}
\caption{{Ring data for Chariklo\citep{Braga-RibasF:2014}.}}\label{ring_data}
\begin{tabular}{|c|c|}
\hline
Ring Property&Chariklo\\
\hline
Inner Radius (km)&390.6$\pm$3.3\\
Outer Radius (km)&404.8$\pm$3.3\\
Inner Width (km)&7.17$\pm$0.14\\
Outer Width (km)&3.4+1.1-1.4\\
Radial Separation (km)&14.2$\pm$0.2\\
Gap between rings (km)&8.7$\pm$0.4\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{THE THEORY OF CLOSE ENCOUNTERS}
The severity of a close encounter between a small body and a planet in the three-body(Sun, planet, small body) planar problem depends on the mass of the small body, the mass of the planet, the velocity at infinity, $v_{\infty}$, of the small body relative to the planet and the distance of closest approach, $d_{min}$, as these influence the tidal effects on the small body.\par
Hyodo et al.(2016) show the derivation of the relative velocity at infinity of a small body in the case in which the planet is restricted to a circular orbit about the Sun in this planar three-body problem. The reader is referred to that work for details. The resulting equations are:
\begin{equation}
v^2_{\infty}=v^2_r+(v_{\omega}-v_K)^2\label{infinity_velocity}
\end{equation}
where $v_r$ and $v_{\omega}$ are the radial and azimuthal velocities respectively of the small body at the orbital distance of the planet from the Sun; and $v_K$ is the Keplerian velocity of the planet in its orbit about the Sun. $v_r$ and $v_{\omega}$ are given by:
\begin{equation}
v_r=v_K\sqrt{2 - \frac{a_o}{a}-\frac{a(1-e^2)}{a_o}}\label{radial_velocity}
\end{equation}
\begin{equation}
v_{\omega}=v_K\sqrt{\frac{a(1-e^2)}{a_o}}\label{azimuthal_velocity}
\end{equation}
where $a$ is the semi-major axis of the orbit of the small body, $a_o$ is the semi-major axis of the orbit of the planet, and $e$ is the eccentricity of the orbit of the small body. For any hyperbolic orbit the following relation applies:
\begin{equation}
d_{min}v_{\infty}^2=G(M_p + m_s)(e-1)\label{r_p_v_inf_eqn}
\end{equation}
where G is the gravitational constant, $M_p$ the mass of the planet and $m_s$ the mass of the small body. The equation for a hyperbola is:
\begin{equation}
r_{radial}=\frac{a(1-e^2)}{1+e\textnormal{cos}\theta}\label{hyperbola_eqn}
\end{equation}
where $r_{radial}$ is the radial distance from the two bodies and $\theta$ is the true anomaly. If $c$ is the distance from the planet to the origin, and $a$ the distance from the origin to the point of closest approach between the two bodies, then the eccentricity of the orbit is given by:
\begin{equation}
e=\frac{c}{a}\label{e_eqn}
\end{equation}
and:
\begin{equation}
$a$ = c - $d_{min}$\label{a_d_min_eqn}
\end{equation}
The true anomaly is related to the eccentric anomaly $E$ via:
\begin{equation}
\textnormal{cosh}E = \frac{\textnormal{cos}\theta -e}{1-e\textnormal{cos}\theta}\label{coshE_eqn}
\end{equation}
The mean anomaly $M_o$ is related to the eccentric anomaly via:
\begin{equation}
M_o = e\textnormal{sinh}E - E\label{M_o_eqn}
\end{equation}
\subsection{CRITICAL DISTANCES AND THE RING LIMIT}
Since the effect of a close encounter on the outermost ring around a small body is noticeable if $d_{min}\le R$, then $R$ defines a boundary between a noticeable and non-noticeable effect. Thus, $R$ is in the same class as other known critical distances such as the Hill radius, $R_H$, tidal disruption distance, $R_{td}$, and Roche limit, $R_{roche}$, as each defines a boundary for a dynamical effect. We will refer to the distance $R$ as the "ring limit". It is beneficial to discuss these other critical distances in more detail.\par
The Hill radius of a less massive body with respect to a more massive body can be defined as the distance from the less massive body within which a satellite may orbit. If the satellite orbit is within the Hill radius then the less massive body-satellite binary cannot be broken up by tidal forces due to the more massive body. In the case for which the more massive body is a planet and the less massive body a small body of the Solar system, the Hill radius of the small body with respect to the planet is approximately given by:
\begin{equation}
R_{H}\approx R_{radial}(\frac{m_{s}}{{3M_p}})^{\frac{1}{3}}\label{hilleqn}
\end{equation}
\citep[e.g.][]{MurrayCD:1999} where $m_{s}$ is the mass of a small body, $M_p$ the mass of the planet and $R_{radial}$ the radial distance between the small body and the planet. Thus, during a close encounter between a ringed small body and a planet, the distance of orbiting ring particles from the small body must be less than that small body's Hill radius with respect to the planet in order for the ring particles to remain in orbit. Analogously the satellite of a planet must remain within the planet's Hill radius with respect to the Sun.\par
If a ringed small body is at a distance just within the tidal disruption distance from a planet, tidal forces can disrupt a small body-ring particle binary pair instantaneously. The tidal disruption distance for a binary consisting of a small body and a massless ring particle in a circular orbit of radius $r$ about the small body is given by:
\begin{equation}
R_{td}\approx r(\frac{3M_p}{m_{s}})^{\frac{1}{3}}\label{tidal_disrupteqn}
\end{equation}
\citep{AgnorCB:2006, PhilpottCM:2010}. As an example, we calculated the tidal disruption distances of Chariklo for each of the giant planets using a Chariklo mass of $8\times 10^{18}$ kg and ring orbital radius set to that of the outer ring of Chariklo of 405 km(see Table~\ref{ring_data}). The masses of the four giant planets were taken from the NASA JPL HORIZON ephemeris\footnote{http://ssd.jpl.nasa.gov/horizons.cgi?s\_body=1$\#$top (accessed 31st December 2015) for epoch Jan 1, 2000, at 0:00 UT}. The results are shown in Table~\ref{chariklo_possible_tidal}.\par\selectlanguage{english}
\begin{table}
\begin{center}
\caption{{Possible tidal disruption distances of Chariklo for the four giant planets using a Chariklo mass of $8\times 10^{18}$ kg and a ring orbital radius of 405 km.}}
\label{chariklo_possible_tidal}
\begin{tabular} {|c|c|}
\hline
Planet&$R_{td}$ (Mm)\\
\hline
J&362\\
S&242\\
U&129\\
N&137\\
\hline
\end{tabular}
\end{center}
\end{table}
When Chariklo is just within the tidal disruption distance to a planet, an outermost ring particle is just outside Chariklo's Hill radius with respect to the planet.\par
There is one more critical distance to consider. At an even closer distance to a planet is the Roche limit - the distance from a more massive body(the primary body) within which a much less massive body(the secondary body) held together only by gravity can be torn apart by tidal forces. For a rigid secondary body, the equation for the Roche limit with respect to a primary body is approximately:
\begin{equation}
R_{roche}\approx 1.44R_{p}(\frac{\rho_p}{\rho_{s}})^{\frac{1}{3}}\label{rocheeqn}
\end{equation}
where $R_{p}$ is the physical radius of the primary body, $\rho_p$ is the density of the primary body and $\rho_{s}$ is the density of the secondary body\citep[e.g.][]{1928asco.book.....J,1947MNRAS.107..260J,MurrayCD:1999}. As an example, we calculated the Roche limit for a rigid secondary body with respect to Chariklo(the primary body) using Chariklo densities of 800 kgm$^{-3}$ and 3,000 kgm$^{-3}$\citep{2016arXiv161203321S}; three possible physical radii of Chariklo taken from Table~\ref{chariklo_radius}; and a ring particle density of 500 kgm$^{-3}$\citep{2016arXiv161203321S}. The results are shown in Table~\ref{chariklo_roche}. The Roche limit ranges from 200 km - 395 km. The orbital radii of both of Chariklo's rings(see Table~\ref{ring_data}) lie outside of all these Roche limit values except one.\selectlanguage{english}
\begin{table*}
\begin{center}
\caption{{Possible Roche limits for a rigid secondary body with respect to Chariklo using two density ratios and three physical radii of Chariklo. Ring orbital radii lie outside of all Roche limit values except one.}}
\label{chariklo_roche}
\begin{tabular} {|c|c|c|}
\hline
Chariklo Radius (km)&Density Ratio&Roche Limit (km)\\
\hline
119&1.6&200\\
119&6&311\\
124&1.6&209\\
124&6&324\\
151&1.6&254\\
151&6&395\\
\hline
\end{tabular}
\end{center}
\end{table*}
Table~\ref{chariklo_roche2} shows Roche limits of Chariklo with respect to each giant planet for the case in which the density ratio is one. Planetary radii were obtained from NASA\footnote{http://solarsystem.nasa.gov/planets}.\selectlanguage{english}
\begin{table*}
\begin{center}
\caption{{Possible Roche limits of Chariklo with respect to each giant planet for the case in which the density ratio is one.}}
\label{chariklo_roche2}
\begin{tabular} {|c|c|}
\hline
Planet&Roche Limit (Mm)\\
\hline
J&101\\
S&84\\
U&37\\
N&35\\
\hline
\end{tabular}
\end{center}
\end{table*}
We now present in Table~\ref{CE_severity} a severity scale for a close encounter between a ringed small body and a planet based on the value of $d_{min}$ relative to the Hill radius of the planet with respect to the Sun, ring limit, tidal disruption distance and Roche limit. This scale is an improvement over our previously published scale in which the ring limit was given only as a constant ten tidal disruption distances\citep{Wood et al......2017}.\selectlanguage{english}
\begin{table}
\begin{center}
\caption{{A scale ranking the close encounter severity between a ringed small body and a planet based on the minimum distance obtained between the small body and the planet, $d_{min}$, during the close encounter. $R_H,R,R_{td}$ and $R_{roche}$ are the Hill radius of the planet with respect to the Sun, ring limit, tidal disruption distance and Roche limit respectively.}}\label{CE_severity}
\begin{tabular} {|c|c|}
\hline
$d_{min}$ Range&Severity\\
\hline
$R\le d_{min}< R_H$&Low\\
$R_{td}\le d_{min}< R$&Moderate\\
$R_{roche}\le d_{min}< R_{td}$&Severe\\
$d_{min}$1,000,000 km and 0.0001 otherwise. The use of these different tolerances made $R$ accurate to within 1,000 km in each case. For each value of $r$, 46 measurements of $R$ were made each using a different value of velocity at infinity chosen from the range 0 - 9 km/s.\par
For each planet, $R$ was measured over 900 times using different values of $r$ and $v_{\infty}$ in the ranges previously discussed.
\subsection{Observing the Relationship between the Ring Limit and Ring Orbital Radius}
To observe the linearity between $R$ and $r$, a set of results of close encounter simulations used to measure $R$ with a given planet for which equation~\ref{cnst_factor} was approximately true was examined. Another way to say this is that was a set of close encounter simulations for which the relationship between $R$ and $r$ is approximately linear.\par
For this set of simulations, $M_o$ was found using equations~\ref{hyperbola_eqn}, \ref{coshE_eqn} and \ref{M_o_eqn} using an initial value for the radial distance between the small body and the planet. $c$ was set to the initial distance between the planet and the origin.\par
$\Delta t$ was found by recording the time at the instant of closest approach between the two bodies. The eccentricity was found using a 2 step process. First, $a$ was calculated using equation~\ref{a_d_min_eqn} with $d_{min}=R$. Second, $e$ was found using equation~\ref{e_eqn}. \par
$n_{int}$ was found by solving equation~\ref{Delta_t_eqn} for $n_{int}$ and truncating the result using the Floor() function. The values of $R$ and $r$ for the set were fit to a power law, and the results checked for linearity. This entire process was repeated for each planet yielding a total of four obervations of the linearity between $R$ and $r$.\par
For each planet, A 3D graph was created in $R-r-v_{\infty}$ space using data from all ($\sim$900) close encounters between the Pluto-like body and the planet in question. 2D slices of each 3D graph were made using a constant value of $v_{\infty}$ for each slice. For each slice, $R$ and $r$ were fit to a power law, and the best fit exponent recorded. Then the relationship between the best fit exponent and $v_{\infty}$ was examined. \par
2D slices of each 3D graph were also made using a constant value of $r$ for each slice. For each slice, the relationship between $R$ and $v_{\infty}$ was examined.\par
\subsection{Observing the Relationship between Ring Limit and Small Body Mass}
To observe the relationship between the ring limit and the mass of the small body shown in equation~\ref{condition_for_m_s_dep}, a set of close encounter simulations was made for each planet using various masses of the small body between the mass of Chariklo and the mass of Pluto. Simulations were made between each small body and a planet using 46 different values of $v_{\infty}$. As before, values for $v_{\infty}$ ranged from 0 -- 9 km/s.\par
For each simulation the ring orbital radius was held at a constant value of 50,000 km for each close encounter. To observe the relationship between $R$ and $m_s$ in equation~\ref{final_ring_limit_eqn}, a set of results of close encounters with a given planet for which equation~\ref{cnst_factor2} was approximately true was examined by fitting to a power law.\par
A 3D graph was created in $R-m_s-v_{\infty}$ space for each planet. 2D slices were made of each 3D graph using a constant value of $m_s$ or $v_{\infty}$ for each slice.
\subsection{More Severe Encounters}
To study the effect of close encounters more severe than those just noticeable, simulations were run with the Pluto-like small body in which $d_{min}$ was recorded for encounters in which $e_{max}$ ranged between 0.01 to 0.3. For each simulation the maximum change in eccentricity and semi-major axis of any ring particle orbit about the small body was recorded. For each planet, the $a_{max}$ and $e_{max}$ values were fit to a power law.\par
$r$ was held constant at 100,000 km. A set of results of close encounters with a given planet for which equation~\ref{cond_for_delta_e_max_power} was approximately true was examined. For each planet, the values of $R$ and $\Delta e_{max}$ in this set were fit to a power law, and the best fit exponent, $\alpha$, obtained.
\section{RESULTS}
\subsection{Ring Limit as a Function of Velocity at Infinity and Ring Orbital Radius in 3D}
The 3D graph of ring limit vs. velocity at infinity and ring orbital radius did not change from planet to planet as can be seen in Figures~\ref{jupiter_3d}, \ref{saturn_3d}, \ref{uranus_3d} and \ref{neptune_3d}. Lower values of velocity at infinity always yielded lower values of the ring limit for a constant ring orbital radius. For constant velocity at infinity the ring limit increased with the ring orbital radius more linearly for lower values of $v_{\infty}$ and as a power law with the exponent on $r$ increasing with higher values of $v_{\infty}$.\selectlanguage{english}
\begin{figure} [H]
\label{jupiter_3d}
\end{figure}\selectlanguage{english}
\begin{figure}\selectlanguage{english}
\begin{figure}
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/saturn-3d/saturn-3d}
\caption{{This is a caption
{\label{818449}}%
}}
\end{center}
\end{figure}
\caption{{A 3D graph of ring limit vs. velocity at infinity and ring orbital radius for Saturn.}}\label{saturn_3d}
\end{figure}
\subsection{Ring Limit as a Function of Velocity at Infinity and Ring Orbital Radius in 2D}
2D slices for constant $r$ all had the same general shape. At velocities near zero the graphs are nearly horizontal, but then at a certain velocity the graphs takes a sharp curve upward, are very linear over a certain range and then curve downward becoming more horizontal with varying degrees of severity based on the planet. The lower the mass of the planet and higher the $r$ the more horizontal the curve became at the highest velocities. Figure~\ref{Jupiter_Pluto_r_100k} shows a typical example for such a graph. This one is for Jupiter with a constant $r$ value of 100,000 km. It can be seen that at a velocity of about 1 km/s the graph turns sharply upward. The linear best fit line is shown. At a velocity of about 8 km/s the graph begins to depart from linearity and starts to curve.\selectlanguage{english}
\begin{figure}\selectlanguage{english}
\begin{figure}
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Jupiter-Pluto-r-100k1/Jupiter-Pluto-r-100k1}
\end{center}
\end{figure}
\label{Jupiter_Pluto_r_100k}
\caption{{The ring limit as a function of velocity at infinity for a constant ring orbital radius. The graph is nearly horizontal for the lowest velocities but then curves sharply upward at a velocity of about 1 km/s. The graph remains linear until at a velocity of 8 km/s it curves downward away from the linear trendline towards being more horizontal. Regardless of the planet or ring orbital radius, all $R$ vs. $v_{\infty}$ graphs of constant $r$ had the same general shape. The square regression coefficient of the linear fit is 0.992.}}
\end{figure}
2D slices for constant $v_{\infty}$ were all well fit to a power law. Figure~\ref{Jupiter_Pluto_v_9} shows a typical example. This one is for Jupiter with a constant $v_{\infty}$ of 9 km/s. The best fit curve is shown. The graph appears very linear because the exponent on $r$ is 1.15 which is near 1. For graphs of this type using other values of $v_{\infty}$, it was found that as $v_{\infty}$ approached zero, the best fit exponent approached 1 as can be seen in Figure~\ref{best_fit_exponent_vs_v}.\par
A regression was performed on $R$ vs. $r$ for all close encounters with Jupiter for which $v_{\infty} = 0.25$ km/s which is the smallest applicable velocity used in this work. The exponent on $r$ was found to be only 1.0007 which is linear to three decimal places. Values for the expression in equation~\ref{cnst_factor} which is supposed to be constant for linearity, varied by only 2.7\% across all data points used in this regression.\selectlanguage{english}
\begin{figure}\selectlanguage{english}
\begin{figure}
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Jupiter-Pluto-v-9/Jupiter-Pluto-v-9}
\caption{{This is a caption
{\label{395980}}%
}}
\end{center}
\end{figure}
\label{Jupiter_Pluto_v_9}
\caption{{Ring limit vs. ring orbital radius for constant velocity at infinity of 9 km/s for close encounters with Jupiter. Graphs of this type fit very well to a power law with the exponents on $r$ approaching 1 with the lowest velocities. In this graph the best fit exponent was 1.15, and the square regression coefficient was 0.99998. The best fit curve only appears to be linear because the exponent is close to 1.}}
\end{figure}\selectlanguage{english}
\begin{figure}\selectlanguage{english}
\begin{figure}
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Jupiter-r-dep/Jupiter-r-dep}
\end{center}
\end{figure}
\caption{{Best fit exponents on $r$ for $v_{\infty}$ values from 0.25 - 9 km/s for close encounters with Jupiter. To get a single data point on this graph, $v_{\infty}$ was held constant, and a power regression was performed on $R$ vs. $r$ to obtain the exponent. Notice how as $v_{\infty}$ approaches zero, the best fit exponent approaches 1.}}
\label{best_fit_exponent_vs_v}
\end{figure}\selectlanguage{english}
\begin{figure}
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Jupiter-Pluto-v-025/Jupiter-Pluto-v-025}
\caption{{This is a caption
{\label{581008}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}
\caption{{Ring limit vs. ring orbital radius for a constant velocity at infinity of 0.25 km/s for close encounters with Jupiter. The exponent on $r$ is 1.0007 which we is linear to three decimal places. The expression in equation~\ref{cnst_factor} which is supposed to be constant for linearity varies by only 2.7\% (130.2 - 133.7) across all data points on this graph. The square regression coefficient was 0.999998.}}\label{Jupiter_Pluto_v_025}
\end{figure}
\subsection{Ring Limit as a function of Small Body Mass}\selectlanguage{english}
\begin{figure}
\subsection{More Severe Encounters}
\section{CONCLUSIONS}
The field of ringed small bodies is only a few years old. The one confirmation of rings around a small body is for the Centaur 10199 Chariklo which was found to have two narrow rings. Given that Centaurs frequently have close encounters with the giant planets, it is of interest to study the effect of such encounters on the ring structure. The severity of such an encounter depends on the mass of the small body, the mass of the planet, the relative velocity of the small body at infinity, the ring orbital radius(assume circular orbits) and the encounter distance, $d_{min}$, (or distance of closest approach).\par
In this work we are concerned with measuring the maximum encounter distance, $R$, at which the effect of a close encounter on the rings is ``noticeable''. We state that the effect is ``noticeable'' if the change in eccentricity of any ring particle orbit about the small body $\ge 0.01$.
As other critical distances define a dynamical boundary for behavior of the small body, it is reasonable to assume that $R$ is also related to some critical distance as it defines a boundary just within which the effect of a close encounter on the rings becomes noticeable.\par
We hypothesize that $R$ is related to the critical distance between a ringed small body and a planet at which the radial forces acting on a collinear, outermost ring particle in a circular orbit add to zero in the four-body, circular restricted, planar problem (Sun, small body, ring particle, planet). An equation for this critical distance is derived and named the ring limit.
A severity scale for a close encounter between a ringed small body and a planet for a given planet, small body with rings and velocity at infinity is created based on the distances from $d_{min}$ to the Hill radius of the planet, ring limit, tidal disruption distance and Roche limit.\par
Numerical simulations are carried out to experimentally measure $R$ relative to the tidal disruption distance, $R_{td}$, in the three-body planar problem (small body, planet, ring particle) using different masses of Chariklo, velocities at infinity and ring orbital radius. Four masses of Chariklo are used. Three are within the realm of possibility and lie in the range $8 - 30\times 10^{18}$ kg. Other runs are made using a fictional Chariklo with the same mass and size of Pluto to test an extreme case. The velocities at infinity chosen for this study are 14 km/s and 17 km/s for Jupiter and 0.001 km/s and 9 km/s for Uranus. These values are based on possible velocities at infinity for Centaur orbits in the planar three-body problem (Sun, small body, planet).\par
Two different ring orbital radii are used for the lowest mass and the highest mass - one equal to 0.00156 Hill radius(equal to the currently known outer ring radius of Chariklo) and one set equal to a possible Roche limit.\par
In each simulation, a small body is placed in a hyperbolic orbit about either Jupiter or Uranus. Only these two planets are used because it was determined that results from other planets would only yield results intermediate between those of these two. The small body is started at an initial distance from the planet equal to one Hill radius. Ten massless test particles are evenly distributed in the same circular orbit about the small body. The simulation is then run and is terminated when the small body reaches a distance beyond one planet Hill radius.\par
Overall, changes in relative $R$ values due to changes in mass were no more than 0.2$R_{td}$ for Jupiter and 0.3$R_{td}$ for Uranus for the first three masses. Changes in relative $R$ values due to changes in velocity were no more than 1.5$R_{td}$ for Uranus and 0.3$R_{td}$ for Jupiter. It was found that $R$ increases with velocity. In every case, changes in relative $R$ values due to changes in velocity were not great enough to make relative $R$ values exceed the ring limit. Thus, for these lower mass small bodies, the ring limit acts as a constraint on $R$.\par
Changes in relative $R$ values due to changes in orbital radius were no more than 1 tidal disruption distance. As expected, the larger the ring orbital radius the larger the $R$.\par
Results from simulations of the Pluto-like Chariklo with ring structure analogous to Chariklo's were compared to those of a Chariklo of mass 8$\times 10^{18}$ kg. For Jupiter, the relative $R$ values were 0.2$R_{td}$ higher than those of the lower mass Chariklo. For Uranus, there was no measurable change in either $R$ value. Thus, the change in relative $R$ caused by a change in velocity at infinity was the same for both of these masses for both planets.\par
The one major difference is that for the Pluto-like Chariklo, the $R$ value found using the upper bound velocity at infinity for Uranus lies about 1.4$R_{td}$ beyond the ring limit. Thus, for a fictional Chariklo with the mass of Pluto the constraint that $R$ must lie within the ring limit becomes invalid. Changing the ring orbital radius for the Pluto-like Chariklo from 4,557 km to 1,706 km results in a change in $R$ of no more than $0.7R_{td}$.\par
Finally, the ring limit is compared to results of simulations of close encounters between a Chariklo of mass $7.986 \times 10^{18}$ kg and the four giant planets in the non-planar four-body problem(Sun, Chariklo, planet, ring particle). In each case, $R$ lies within 1 - 10 tidal disruption distances. Calculation of ring limits for this case show that they are all $>$ 10$R_{td}$ with those of Uranus and Neptune being the closest to this value lying at 10.3$R_{td}$ and 10.6$R_{td}$ respectively.\par
We conclude that the ring limit does not define a sharp boundary for a change in dynamical behavior of a small body but can be used as a constraint on $R$ for small enough masses or as a first approximation to this constraint for masses as large as Pluto. It will take more research to determine how good this approximation is.
\section*{ACKNOWLEDGMENTS}
This research has made use of NASA's Astrophysics Data System, NASA's JPL Horizons' database and the Asteroids Dynamic Site.
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