The (153591) 2001 SN 263 asteroid system, target of the first Brazilian interplanetary space mission, is one of the known three triple systems within the population of NEAs. One of the mission objectives is to collect data about the formation of this system. The analysis of these data will help in the investigation of the physical and dynamical structures of the components (Alpha, Beta and Gamma) of this system, in order to find vestiges related to its origin. In this work, we assume the irregular shape of the 2001 SN 263 system components as uniform density polyhedra and computationally investigate the gravitational field generated by these bodies. The goal is to explore the dynamical characteristics of the surface and environment around each component. Then, taking into account the rotational speed, we analyze their topographic features through the quantities geometric altitude, tilt, geopotential, slope, surface accelerations, among others. Additionally, the investigation of the environment around the bodies made it possible to construct zero-velocity curves, which delimit the location of equilibrium points. The Alpha component has a peculiar number of 12 equilibrium points, all of them located very close to its surface. In the cases of Beta and Gamma, we found four equilibrium points not so close to their surfaces. Then, performing numerical experiments around their equilibrium points, we identified the location and size of just one stable region, which is associated with an equilibrium point around Beta. Finally, we integrated a spherical cloud of particles around Alpha and identified the location on the surface of Alpha were the particles have fallen.

Radar observations show that (16) Psyche is one of the largest and most massive asteroids of the M-class located in the main belt, with a diameter of approximately 230 km. This fact makes Psyche a unique object since observations indicated an ironnickel composition. It is believed that this body may be what was left of a metal core of an early planet that would have been fragmented over millions of years due to violent collisions. In this work we study a variety of dynamical aspects related to the surface, as well as, the environment around this asteroid. We use computational tools to explore the gravitational field generated by this body, assuming constant values for its density and rotation period. We then determine a set of physical and dynamical characteristics over its entire surface. The results include the geometric altitude, geopotential altitude, tilt, slope, among others. We also explore the neighborhood around the asteroid (16) Psyche, so that the location and linear stability of the equilibrium points were found. We found four external equilibrium points, two of them linearly stable. We confirmed the stability of these points by performing numerical simulations of massless particles around the asteroid, which also showed an asymmetry in the size of the stable regions. In addition, we integrate a cloud of particles in the vicinity of (16) Psyche in order to verify in which regions of its surface the particles are most likely to collide.

The 99942 Apophis close encounter with Earth in 2029 may provide information about asteroid’s physical characteristics and measurements of Earth’s effects on the asteroid surface. In this work, we analysed the surface and the nearby dynamics of Apophis. The possible effects of its 2029 encounter on the surface and environment vicinity are also analysed. We consider a 340 metres polyhedron with a uniform density (1.29 g·cm−3, 2.2 g·cm−3 and 3.5 g·cm−3). The slope angles are computed, as well their variation that arises during the close approach. Such variation reaches 4○ when low densities are used in our simulations and reaches 2○ when the density is high. The zero-velocity curves, the equilibrium points, and their topological classification are obtained. We found four external equilibrium points and two of them are linearly stable. We also perform numerical simulations of bodies orbiting the asteroid, taking into account the irregular gravitational field of Apophis and two extra scenarios of perturbations: the solar radiation pressure and the Earth’s perturbation during the close approach. The radiation pressure plays an important role in the vicinity of the asteroid, only cm-sized particles survived for the time of integration. For densities of 2.2 g·cm−3 and 3.5 g·cm−3, a region of 5 cm radius particles survived for 30 years of the simulation, and for 1.29 g·cm−3, only particles with 15 cm of radius survived. The ejections and collisions are about 30-50 times larger when the close encounter effect is added, but around 56-59% of particles still survive the encounter.

Didymos and Dimorphos are primary and secondary, respectively, asteroids who compose a binary system that make up the set of Near Earth Asteroids (NEAs). They are targets of the Double Asteroid Redirection Test (DART), the first test mission dedicated to study of planetary defense, for which the main goal is to measure the changes caused after the secondary body is hit by a kinect impactor. The present work intends to conduct a study, through numerical integrations, on the dynamics of massless particles distributed in the vicinity of the two bodies. An approximate shape for the primary body was considered as a model of mass concentrations (mascons) and the secondary was considered as a massive point. Our results show the location and size of stable regions, and also their lifetime.

No abstract

The first proposed Brazilian mission to deep space, the ASTER mission, has the triple asteroid system (153591) 2001 SN263 as a target. One of the mission’s main goals is to analyse the physical and dynamical structures of the system to understand its origin and evolution. The present work aims to analyse how the asteroid’s irregular shape interferes with the stability around the system. The results show that the irregular shape of the bodies plays an important role in the dynamics nearby the system. For instance, the perturbation due to the (153591) 2001 SN263 Alpha’s shape affects the stability in the (153591) 2001 SN263 Gamma’s vicinity. Similarly, the (153591) 2001 SN263 Beta’s irregularity causes a significant instability in its nearby environment. As expected, the prograde case is the most unstable, while the retrograde scenario presents more stability. Additionally, we investigate how the solar radiation pressure perturbs particles of different sizes orbiting the triple system. We found that particles with a 10-50 cm radius could survive the radiation pressure for the retrograde case. Meanwhile, to resist solar radiation, the particles in prograde orbit must be larger than the particles in retrograde orbits, at least one order of magnitude.

<p lang="en-US" align="justify"><strong><span lang="en-US">Introduction</span></strong></p> <p class="western" lang="en-US"><span lang="en-US">Asteroids are reminiscent bodies of the Solar System that are irregularly shaped, though some of them are nearly spheroidal. Despite of their irregular shape, some asteroids have similar characteristics. The spinning-top asteroids are characterized as oblate shapes with a pronounced equatorial bulge, as well as their fast rotation.</span></p> <p class="western" lang="en-US"><span lang="en-US">To understand the dynamics around these asteroids, we calculate the equilibrium points [1] of some spinning-top bodies, considering their irregular shape. The equilibrium points are points where there is a balance between the gravitational and centrifugal forces. Thus, when the body is spinning faster, the equilibrium points may be closer to the body, and some of them could touch the body's surface or even be inside the body. Besides the location of the equilibrium points, we also examined their topological structure according to the eigenvalues of the characteristic equation in the complex plane.</span></p> <p lang="en-US" align="justify"><strong><span lang="en-US">Location and classification of the equilibrium points</span></strong></p> <p class="western" lang="en-US"><span lang="en-US">We considered a set of spinning-top asteroids to compute their respective equilibrium points: the Alpha bodies of the triples systems 2001 SN263 [2] and 1994 CC [3], the Alpha body of the binary system 1999 KW4 [4], the asteroids 2008 EV5 [5,6], Ryugu [7] and the retrograde model of the asteroid 1950 DA [8,9].</span></p> <p lang="en-US" align="justify"><span lang="en-US">In this study, we identified that the Alpha body from the triple system 2001 SN</span><sub><span lang="en-US">263 </span></sub><span lang="en-US">has the </span><span lang="en-US">larger</span><span lang="en-US"> number </span><span lang="en-US">of </span><span lang="en-US">equilibrium points, with 13 points. Among these points, the </span><span lang="en-US"><em>E</em></span><sub><span lang="en-US"><em>13</em></span></sub><span lang="en-US"> point is the central one, the point </span><span lang="en-US"><em>E</em></span><sub><span lang="en-US"><em>5</em></span></sub><span lang="en-US"> is an inner point and the </span><span lang="en-US">remaining</span><span lang="en-US"> are </span><span lang="en-US">close</span><span lang="en-US"> to the surface of the body, as if they were contouring the surface (Fig. 1).</span></p> <p lang="en-US" align="justify"><span lang="en-US">The Alpha bodies from the systems 1994 CC and 1999 KW4, the asteroids 2008 EV5, and Ryugu have seven equilibrium points. </span><span lang="en-US">In the same way as </span><span lang="en-US">the equilibrium points of the 2001 SN</span><sub><span lang="en-US">263</span></sub><span lang="en-US"> Alpha, the equilibrium points of the Alpha bodies from the systems 1994 CC and 1999 KW4 are </span><span lang="en-US">close to</span><span lang="en-US"> or inner to their surfaces. For the Alpha 1994 CC, only one equilibrium point is outside of the body. </span></p> <p lang="en-US" align="justify"><span lang="en-US">The asteroids Ryugu and 2008 EV5 also have points close to the surfaces. However, these points are not so close as the other bodies analyzed in this study. Since the rotation period of Ryugu is slower (7.63262 hours)</span><span lang="en-US"> it </span><span lang="en-US">is</span><span lang="en-US"> expected that its points are not </span><span lang="en-US">so close</span><span lang="en-US"> to the surface. </span></p> <p lang="en-US" align="justify"><span lang="en-US">F</span>or the asteroid 1950 DA we calculated the equilibrium points for different bulk densities (3.5 g cm⁻³ to 1.0 g cm⁻³). We find nine equilibrium points for the larger bulk density and <span lang="en-US">in this case all points are close to the surface</span>. However, as we decrease the bulk density, the equilibrium points move closer to the body's surface, <span lang="en-US">even going inside the body for the smaller densities. </span><span lang="en-US">Consequently, the number of points decreases, reaching the case of having only a single equilibrium point. In this case, the cohesive forces are important to keep the body's shape [9].</span></p> <p lang="en-US" align="justify"><span lang="en-US">In terms of topological classification, we </span><span lang="en-US">verified</span><span lang="en-US"> some similarities among all the points we calculated. All the odd points are classified as Saddle-Centre-Centre, while the even points are divided into two classifications: Sink-Source-Centre and Centre-Centre-Centre, being the Centre-Centre-Centre </span><span lang="en-US">the only</span><span lang="en-US"> linearly stable points.</span></p> <p lang="en-US" align="justify">  <img src="data:image/png;base64, iVBORw0KGgoAAAANSUhEUgAAAXMAAAFQCAYAAABNgp4KAAAgAElEQVR4Xu2dCbxN1fv/H0OmTClDmRORsdCkUJqTBqmU0CTNo1IqKZE0S5EhJTQPZFZUImUeS2SITMWXyMz/9V799v0f17n37r3POufss8+zXq/7uvees/faa33WWp/9rGc9Q66DBw8eFC2KgCKgCCgCKY1ALiXzlB4/bbwioAgoAgYBJXOdCIqAIqAIhAABJfMQDKJ2QRFIRwR2794tixYtkqpVq0qRIkWiQrB161aZOXOm7Ny5U4499lipVauW5M+fP5RwKZmHcli1U4pAeBE4cOCA/Pbbb/Lmm2/Kd999JyNGjJDq1asf1uGNGzfKww8/LMWLF5fatWvLyJEjpXHjxnLfffdJvnz5QgeQknnohlQ7pAiEG4F//vlHJk+eLH/99Zd07dpVxo0bJzVr1jyk09h18PkjjzwiH3zwgfke8h8yZIiMHj1aSpYsGTqQlMxDN6TaIUUg3AhA1PwsWLBAWrRoIWPGjDmMzEHg66+/lgcffFDef/99I5n36dNHPvzwQyOhlyhRInQgWSHz9evXy4ABA0IJUOhGXDukCAQMgf379ws/bdq08SQxz5s3L1sy37Vrl9xzzz2ycuVKOeWUU2Ts2LHywAMPSNu2bSVPnjwBQyH25lgh8zVr1sisWbPk8ssvj71FWoMioAikFQJ79uyRCRMmSP369c0hpduSHZk7kjtqmKZNm8rxxx9v1CuQeI8ePaRYsWJuH5My1ymZp8xQaUMVgXAiEA8y37t3r/Tq1UumTp0qH3/8sbF2+fbbb+Xee++V/v37y+mnnx46MJXMQzek2iFFILUQsEnm27Ztky1btkipUqWke/fuMnfuXEPmhQoVkunTp8utt95qDkKbNGmSWiC5aK2SuQuQ9BJFQBGIHwJeyRzTxB07dsiMGTOkXbt2MmzYMGnQoIEh7HfeecdYrGCuCJEjnXfu3FmqVKliDj9/+OEHGTx4sJQrVy5+HUpSzUrmSQJeH6sIKAL/IeCVzJG+UZ8sWbJEVqxYYYgaa5VGjRrJjz/+aMwWOegsXLiw0cUjkTuOQpzr4TiUO3fu0MGvZB66IdUOKQKphYBXMk+t3iWutUrmicNan6QIKAJREIDMx48fb1QlXqxZFMxDEVAy1xmhCCgCSUVAJXM78CuZ28FRa1EEFAGfCCiZ+wQu021K5nZw1FoUAUXAJwJK5j6BUzK3A5zWoggoAnYQUDK3g6NK5nZw1FoUAUXAJwJK5j6BU8ncDnBaiyKgCNhBQMncDo4qmdvBUWtRBBQBnwioaaJP4FQytwOc1qIIKAJ2EFDJ3A6OKpnbwVFrUQQUAZ8IKJn7BE4lczvAaS2KgCJgBwElczs4qmRuB0etRRFQBHwioGTuEziVzO0Ap7UoAoqAHQSUzO3gqJK5HRy1FkVAEfCJgJK5T+BUMrcDnNaiCCgCdhBQ00Q7OKpkbgdHrUURUAR8IqCSuU/gVDK3A5zWEhsCY8eONem+9u/fL9WqVROyqf/7778mIwxZYo455pjYHqB3pwwCSuZ2hkolczs4ai0eEfjnn3+kdevWcuKJJ8pjjz1myPzPP/+UKVOmSNu2beWoo47yWKNenqoIKJnbGTklczs4ai0eEdi4caOcd9550qlTJ7nxxhuNhL5z507ZsGGDVKxYUfLmzeuxRr08VRFQMrczckrmdnDUWjwiMG/ePLnuuuukT58+0rRpU/n555+lfPnyocya7hGatLtcydzOkCuZ28FRa/GIwFdffSUdO3aUFi1amLyPZFnv0qWLybQeWXbs2GH+PfLII40qZuvWrfK///3PZFfnviOOOMLjk/XyoCGgZG5nRJTM7eCotXhE4MUXXxQOQV999VUpXry4TJo0Sa666iopVqyYqQmVy8KFC+Xll182apiLL75Ytm3bJm+99ZZRw6xfv17++usv6dy5sxQpUsTj0/XyICGgpol2RkPJ3A6OWotHBG655RbJlSuXIevChQsboub/ffv2ydFHH23+/+2336R79+7yxBNPSP369Y21yx9//GEOTXfv3i3NmjWTESNGGPWMltRFQCVzO2OnZG4HR63FAwJI3RdddJHRlT/66KOGxFGhTJw4UUqUKCFnn322qQ3rlttvv13efvtto1KhcB1l1qxZMmDAAOndu7cULVrUw9P10qAhoGRuZ0SUzO3gqLW4RGDLli0ybtw4ee2116R27dpSt25dcycLGr05NubHH3+8+eynn34yZD18+PBDdONI5wMHDpRrr71WatSoYV4GWlIXASVzO2OnZG4HR63FJQKoRzBLRGXCISY/jsTNYWbZsmUzzBLfffddWbp0qVG1QNgHDhyQTZs2yRdffGFeBGeccYYSuUvcg3yZkrmd0VEyt4Oj1pIDAtiRQ8gOeecEGLpznIkaNmwo11xzjdGRT5s2TRYsWGDs00uXLm0OS9UePSckg/+9krmdMVIyt4Oj1hIFAQ4x161bZ/Tc27dvN6oS3PWdwsEnpJzZvHDz5s0ycuRIGTRokNGfX3/99ZIvXz7zm5eBY/HSr1+/w0wZdSBSDwElcztjpmRuB8eUrgWyRRLetWuXMQl0YqRklqL5v1ChQoZ8uR4bcFQfLEbMBPmNBI1OG0nca6H+UqVKmbgslSpVMiaL2Uny/+7ZL6OXbJSF67bJ5bXKSL2yRSW36s+9wp7069U00c4QKJnbwTEla4HE0UHjhIMem3gpSNAQNSTqWJk4ncuTJ4+x6Ua6ZgHiwAOZU8/evXszLE1iBYPnYtVStWpVE4QrqzJ/3TZpM2yOLN20Xdo3LC+9mp8kxQpoGIBY8U/0/SqZ20FcydwOjoGvBUkZqRmpGyn677//NioQiJvvHFIOUkd4eRx33HFGSsf2nF0BnznWK79u2i4dP54vC9b/I7edVkG6nFdVCudXMg/SGLppi5K5G5RyvkbJPGeMUv4KgletWbPGSOHosVk8kHeqFA45CxYsKGXKlJGSJUsaYkcFs/+AyLw/t8nyv3fImZVLSNliBUSNFFNlVP9/O5XM7YyZkrkdHANRC+oOR+UBaePyjvTN3+jDk0ngtMuGPbhziIqqBxUMRA/B85v6nd/8beN5gRjYkDdCydzOACuZ28ExqbVAlKhNIG0sQfCcRJ3CIklmcQjcFpFH6wuEXaBAAaPjx+IF6R1C5zPnABWpHl2/knsyZ0PWz1YytzMuSuZ2cExKLUjbeE0igXOIiU482QQeDYh4kjnPi/bSQLeO9A6B8xtzRnTvSPEQvZbgIKBkbmcslMzt4JiwWiAurE4gcX6QxlOt+CF3JyZLZF/dStrO8xwpHi9TDlbJZkRcF7f1pBrOqdJeP6aJhIUYM2aMfPrpp9KrVy9j+ZS5EDOf6JwcnHPQz0v9rrvukpNOOimUY65kniozXsSYDa5atUpWr15t1Cqx6MCjEWokYXohuGhE68AarR4/ZG57mFDJoHrBph1i5zfqGC2JR8CrZI4wM3r0aJkzZ44MHTrUBGirWbPmYQ2fPHmyie+Dsxmmsx06dDBxfwi7DMGHrSiZB3xEIT6kb6xRli9fbv6OhcTddjcnws3pe7fPsXmdmzZFu4ZDVZJf4LBEcgxUMpC9lsQg4JXMMaXlTOjXX381MfCJix+NzFkrXOvkkyWI28yZM4XEKI4XcWJ6mJinKJknBmfPT3Fc4NeuXSvLli0zOvFEkLjnhobsBiQ2DlGR4NCvayYjuwPMjpLdZeRuDql50aJFcskll2SEOnbzVNQoZKpC3RKNzDPX0bJlS0Piffv2DeUuTMnczaxJ4DVMciQVJjwRA/Gy9OMaH63JftUoCex+YB4FqWPPji4WiR1S96J6CkxHAtYQHNbYYWYm819++UWaN28eNzLnZXHHHXeY1IQXXHBBKMdSyTxAkx3SxjKFic1vNySeWW2QnarBjRrCLRxZ1WXzGW7bYuO6rNqNySM69QoVKpjwvKpXt4H2oXV4VbM4d7uRzBlXhCL05I0bNxak87Cq0JTM7c9NXzWiAyRNGlILB51eSyJsur22KUzXY7uO+gWdOuSu6hd7oxtPMv/999+lT58+0qBBA5PMJMwhk5XM7c1JXzVhG46XJlIGKhU3JZ2J27bk76c+1C7oaEllF2ZycDMXbVzjxzSRceMwkwPQUaNGSb169UxTSAxOJiti4aOqfOqpp+Scc86RU0891ahW8DGoVatWKH0NlMxtzEYfdTAZsZVdsmSJCRmbnbOPH8Lx0SS9xQMC6NTRp5NcWp2QPAAX5VKvkjlWKlOnTpUff/xRPvvsM2nVqpU0atTI/GCq+M4775hE319++aVJLxipGmNX9frrrxu1WdhKWpI5npPffPONeYOzEAm3SnEyw7dp08bYIMerEAccqQHdOH9Hs9OOJPAwkLnTRxuHiMnAI9oz8TJFSieFHb/dZlGK17xK1Xq9kjlWXaglsYLhXIlxQA8OaVMXMflxBnPi80fi4iQ34Z6wlbQkcwYR9UaTJk3MyXb79u3NQuSkHZvV++67Ly4Z35l4RC6cP3++iR8eL1PDZJBd2BZGtP5khSuehSSmrlixoh6Q+pgIXsncxyPS4pa0JXO8yOrUqSPPP/+82aY5BA/R1q9f36ouFBJA6kcSx2acZ9sqXojby7Ve2ue3Xr/3eWlbdtf6tfyJdh8HomzhmVPYMquU7n6UlMzdY5XdlWlL5nPnzjWHJ+jYOBxZuHChVK5c2UTas1mQvoknvnjxYsEByEsJgvlfsgnXC16JvDYrXNjqo3YhqJfq0t2NiJK5O5xyuiptyfyDDz4Q3HvRj6MzR/3x4IMPSrly5XLCzPX3EDk2rhxyYm4YTTfuurKI6IBe7gn7tUF42WRuA1I6HqSnnHKKmjC6mIBK5i5AcnFJWpI5Byddu3aV2bNnG2cC9OezZs2SK6+8MiOOAweTCxYskP79+8vtt98up512mpGwJ0yYINOmTZPevXtneUiKbhz9O9I/L4l46cZdjG9CL7FBrDbqyKnTiXgGB70E74LQS5cunVOT0vp7P6aJaQ1YFp1PSzLHnptQmFgg9OzZ0+g3MRMk2BJ/s1UmhgT6c0JoQvg4i2CBQsjNlStXGkeEaJYZ6MZx/EE/7tVuPFETNBFklqi+2HpOLJhkZXnE/EB/3rRp07gcqNvqe7LrUcnczgikJZlDyoTFxIqlXbt2hsCRpj/66CMjgZOOjAX6888/y1tvvSUvv/yykdj5rFu3buagi3CamQvmUt9//72Ryt244tsZwv+SM1BsmP1l1SaHsGIhPVv9TWY9fvrPOQyOKtg2q5PR4aOnZG5nRqcdmSNxc+gJQV922WVSo0YNQ4bYpiKJv/HGG+YQlM+QwqdPn26C37MIsVvt2LGj3HzzzSbOQ2TBQgW1DS+KWIofsojleXpv/BCIHEvsoPFSRFBQS5dDMVcytzMH047MIWRig3MgyQJznAfQa0PYqFMcSf2FF14wxE60NQrWKBySopo5vnIl2fvrEtn/+3LZU7W6zFi5yujUHf24H1IOgvWKnWmVuFoSsSuJpTeRY4o9OrbohAIoXLhwXHdSsbQ50fcqmdtBPO3I3C1skP7dd98tN9xwg4ntQCGzCUQ+ZMgQyf/PNtncoY3sWzRfNtc7VWa1uCajaj9Enl27vBCW7We7xSss18ULv8h6Ca1L6rLy5cur2kXEeG1iWIB/B/FutPhDQMk8Cm4chv7www9GFYNe/fzzzzcZfshQMn78eEPoVQsXkl2Q+crf5e8Tqsucdv9J70Eo8SKkIPQtldqQeRwi/8cGHTInml+6R2BUMrczq5XMo+CI/hx3eyxTUL1gh86Ec1K2Ye2SSw7KshHDRBbNk0016sj20qkpUYSJ+L32xev1NpZc5meSzYjDUWKlp6suXU0TbcwsESVzHzhC9Khc+O2oQHxUk5K3JJIAE/mseA5GTv1Af04ExurVq6ellK6SuZ3Zp2TuAUcWJU5AhN7EhjwrIteDTA+g6qUGASf5BaEAcDaKp5lp0CBXMrczIkrmHnDEKxTvT5JJpFPJSbJMJyyy66sNnPBnwFwWq5d00aUrmdtZQUrmLnFkoZKC6qeffjJxlG0Xt0SQlWWL2/tttztI9UXDxgsubq91e51fbDBhRIfO4Wg6BOtSMvc7Uw69LxRkTlwV3OfZmpJUgsVGvBXc8y+55BIrSHEYOmPGDOOqH6bixewxqP12Q65urgla//B5gNAh9zAXJXM7oxsKMkeHTQREYkkTLIuFu2LFCiNJE1vFRsFcceLEicYLVEv8EAgC6bptQ07XxfqixKnt5JNPNoejYbZ0UTK3s55CQebECse5p3v37nLppZcaZFavXm0CYmV2u/cL26JFi0xkxcwl1gXrtz3R2pGMQ7OcCM1W/7zU46VNQT+sJtcoc5igcGEtappoZ2RDQeaTJ082URCJUY50ToJkzL1sZXzB7pxnEEBLiyJgCwG3Lx1i7JOsOKzqFpXM7cyoUJA52bhfeuklo1LBHXj06NEmGiKHSDYKLwesWNDDRxa3izFeUrTf59vAJFodQWtPvPrpt97M+LjFC7NF4rnwE8aoi0rmfmfUofeFgsyfeuopGTNmjIlmyALhMJTkEZGWAJgV8jku1M6WFUmbdHHowbHvJbRtZlUFlitYsGQ++HS7EO0Mk9aS7gjgdUx4ZsLohq0omdsZ0VCQ+TXXXCO4RRPlkCiIJE0+8cQTDTEjyeDoQzhb4qo8++yzxn0aKRtpngMmHIBI7fbQQw9JkSJFZc/+A5Irl0i+PLnNd1OmTDks0YQfMndzj5tr7Ax9etaSyvgSVqJZs2YmeUqYipK5ndFMeTInCUTdunVNmFp+nPC1BMrCk47odEjev/32myFvDklRv0Ds6MEhdw5Kn376aXm+Vy/Znu8oee/nP+ToI/NJ2wblZOPKpcZ130uJRhipTCJe+p5O1yZ6TBFOTjjhBJOKLkz6cyVzO6sm5ckcIm7SpIncf//9JrohC+x///ufTJo0yVi4kAyAgvnigAEDjF4dyQbyZtt6xRVXGPUL//fq/aIMW7pLnpv0mxTOn0cGtKojBdbOl/Xr16eVe7WdqfVfBqRkWNh4bX+qtJN+oTps2LChVK5c2Ws3A3u9krmdoUlZMndsyb/99lshXVvmQnZ04pA7evP333/fkDb6dcieJM0QO44Z2I9/9tln0v25HjJmxQ7pNn6plCiUV+6tXVByb/jNRE/UoggkC4HM5q9I5ZjgYrEVhqKmiXZGMWXJ3Gv3e/ToYULZ3nbbbSYjEHHKP/zwQ5Pb88033zRp4+655x7ZdSC3zF67VfLs3yP/rFgof21IrzgsXnF1c70bydfNNW6eFdZrMuODugXvUByLUr2oZG5nBENP5ujUsWQh3dvll18uF198sYlLjscoB56QORI6+nZCkDoFPTmOQk4aODtway1+EUgnsnfTV4Jw4VNBUK5U9w5VMve7Kg69L/RkjvkhnpuoUjg4QvWCaSJ245A1sVzQP55++ukZi4LkzGPHjg2U676bBW5nSmgtqYIAapamTZuaHWcqFyVzO6MXejLHTpzkzUjoSDAE30LniMSN2SGfQeiOdMP106dPNxYu0YqSqp2JF5Za4hUOwO08w+4cQSSVoysqmdtZDaEncy8wsYAIzjVz5szDvD291OP1WrcL12u9er0dBLIbH7/f2WnZf0kt8JvAOxQfi1QsSuZ2Rk3JPAJHDkG//vpr2bx5sx10c6gls5VCQh7q4iFBfLm4bZPb61zAkJRL/LSfYFzEbknVzPZK5nammpL5/+GIemX27Nny66+/ZqSDSwUbaTvTQGsJCgIOmXsldTygia6IGjHVipom2hkxJXMRoz8n/vnPP/8sTCy/xesC9PqceNfvtT1Bvz4MeLntA2c+qFv4SbVgXCqZ21lJSuYiRq2C8xFWLFkVt4vKzrCEv5agqpgSjbyteUU9eDafeeaZQsjcVCpK5nZGK+3JnEXAgSfeoQ7B2IFWa1EEEo8A1i2oW1LJ9twrmTve38RXQj3Ky4v0kNEsevAxIbTHqlWrMq4lhEeYYts4syztyRw7dAJucfgZjxL5ggizDt6WhBmPMchcp9u2+tVfx7sP2bUfi5bzzjtPSpcuHe9mWKvfK5mTX6BLly5Sr149Ex110KBBJsZS69atBWcqpxCGo2/fvsbPpGXLlsZZkL+feeYZKVq0qLX2B6WitCZzoikSq5y3tkrl7qakWyJ0V1tyrrLVB1v12EaBmP3YnqdKqFyvZD5ixAiTr2DChAly9NFHy1tvvSWff/65DB482OQrcMq8efOkQ4cO0qtXL+NcxXMI5YHVT6qdK7iZI2lN5mvWrDGeoOmUpDmoBORmsqbSNcnEmXgtkHnFihVTImqlVzInFMfSpUsFUkfCxrv7zjvvlGHDhsmpp55qpglOgu+++66RzAcOHGhSSLJrweoHU84wlrQlc7ZgvNlRsyRz4SVyUqVLP91iGmY8iOVP6IqgSeeY/n7zzTeHxDxiLWJ8cMstt7iylb/xxhvNmn377bcNMc+fP9/EXBo+fLgJh01Bl/7cc8/JkCFDTGhsLNbIaUA0VWIxgU/YSlqSORMBd/3vv/8+5cYzzAQUy2AoLoejh3TuxPOPBdt43+tVMm/btq0JS01+AuzqCYp32WWXGTLn8JdCnZA56SRHjRolZcqUMS8RJHheAs518e5bIutPSzLnIGTq1KlGKqckmggiD9Z4fuTBaKLbksjJps/KGQGb44/0efbZZ5vYQ0EuXsm8c+fOxsHvk08+MWoWErgTARU1S/369TPULKhX+vXrJ1988YVROS1evFiuv/566dmzp5Hkw1bSjszRpS1YsMAkctbwtmGbztqfSATQERMpFIuPIJsqeiVzpO3HH39cvvrqK5OEnQNOzIvJS4ApIuoUJG/MjUlCQy4Dcqdi7MD/b7zxhpx11lmhmyyBJnPIFimaA0omJhMSyYXP0QX6kTjIMsR2i0iKNqWg0M2MEHTIz/j6uSceUNloB3Vw4Bd06dwrmWORQppHchEQzx2J/K677jK25n369JH+/fubHL+oVl5++WXBqqVVq1Zm3ZOVDJ15qocNjjbnAk3m2H5zGv3KK6+YgWKrhGTNSTbbKRJOeCkMJIme161LrexBjtlkmO3UnXG0QWJe5kQQrvXa56yuj/w88m9yhkJ6QS1eyZx+sJYhZw5OSdJB1EgKYTmWLVtmXmA4EaFbJ0wHn2PGyDkCli1hLIEmcwDHoYdTbrZUpMpikhLZEPMrnCPcFgYVnRnqlVTL6el1sbvFRK8LHgKxjHVW9yKhnnvuuYG1rfZD5sEbueS3KNBkjjoFW9Fu3bqZOOOoWf7++2+TyBajf7fxm5nkOAb9+OOPMQXSitdwxbKAvbYpDFJ+IvHyim+8rs88bl4wYK2gQw5qzBaNmmhn1gSazBlkDi9wCkAaR6LGUYATaS+hPtmKTZkyRbZs2WIHNR+1eFl8PqrP9hZbz7ZVj+3+pUp9ycSvatWqgroliJ6PKpnbmcGBJnNOpnHHXb58uVGpcGiJVN61a1cTgwHJffXq1RmqE/SCTNrIk3v07sRjQGfmSDd+oUvmYvTbZhv3pWu/bWAXWUekSWqizz/QE2PBgd44aEXJ3M6IBJrMUakQIAeb0Pvvv9+EquXw0/HyQnWCWVKbNm0MGkOHDpUnnnhCypYtm4EOevK5c+emnJ7czvBqLYrAfwjw8uCgsG7duoGDRMnczpAEmsyJjoYr7vPPP2+ioiGJI2njEoxbLoejHIzi0UVB/XLHHXeYoDoUTJhQr+zevdsOWj5qSYRUm4hn+Oh66G6JBedY7rUFJKZ8CEZBU7UomdsZ4cCSOZMfM0LiMBAVDYmCz/78809D0O3atTPf48KLWy8mi+3btzcSPCf3W7dulRkzZsj69eszkArCgrIzbFpLqiEQhLmHdM5BKCa+QSpK5nZGI5BkjgSOuy4xFfDWJL4EAXVYEDgQVa9eXa699lpjY/rOO++YoDp4gn344YfSu2dPqbJsiWyaMFYWVqwif1c43g5SWosnBNzYQnuq0MPFybTYidbvZLYnM2zsaM844wzXlmAeYPd9qZK5b+gOuTGQZO62a1i3cNiJVI60jtXL461ayp5Od8m+Fctl8/HVZPZNd7mtzvN1QZC2PDc6gDfEA8d41BkLdMlsT+SzsQJr1KiR8Y4MSlHTRDsjkbJkjjROLHLSZDFZsUdHH9iwYnnZ2vl++Xf6VFlT/3RZevGVdpDSWlIOgWQSaCLA8tM/5yC0du3agYnXopK5ndkSaDJHjYJteGaTQqRxIsIR+XDJkiXG5pyAQkzQvHnyyP4/18qfixbIT2vXyY48ee0gFYJa4qH68EMoQYcyjH2KxJyDUAJPBSVJg5K5nRURaDLH65MQlujNiVfMIsOaBZvxhx9+WCpXrpwlCpg1Eq+cQF1BKWEniaDgrO3IHgE8p4ldwq42CEXJ3M4oBJrMMSl88sknDZl//PHHpsc4Dn355Zdy6aWXHuaefFBEdu3dLwcOiuzevtWQOVYtWRUlVzuTSGs5HIGgzy0sWiD0IITGVTK3s4ICTeYQ90033WSStBLKkoNOoqXxOSEs8+fPfwgKKzf/Ky9O+V227Nwjt9YrIduWzcuWzO1A6L4WW1YNQScK94jolclCANUkZoqEyE12UTK3MwKBJvNNmzYZZyFiEd9zzz3G+xMd+plnnhm19x/PWyftRsyRfQcOyl0NjpFzCmwQ4pcnqijJJgrpcD8nEfMIVQu+GyeddFLSpXMlczvzOdBkjm4c4iYuS5UqVYy65bHHHpMGDRqY3iOp49KPKgUzxXX7C8krs7fKtt375IlzKkvxLcsEL1It/hGwRSy26vHfk//u9NKO7K61tcuKtT+x3I9UjnTuJWhdLM/L6l41TbSDaqDJ/LvvvpNbb71VBg0aZByFxo0bJ82bNzcZRihYsuD9iccnAen/XLdebrjzIcmdr63NT9IAACAASURBVIBUL3WkzPxphkkh5WUB24E1HLUobvbGMYhYkhOA4FvJDo2rkrmdeRZoMidX32effSbvvfeemXAMOj8kYiaY1qeffmpc9l944QWZNGmSjBw50vztpJObPn26IXObJYiL0kv/vLTfy7Ve2pDVtWGQdt3gECuusd4f2UYswjgITWZRMreDfmDJnAF+4IEHjM6btHFsBZnEJGXduHGjXH311SbZBHFbCMZFaij06/ztlHiQuR3Yo9dic5HGs51ad3IRsDlPCCmN2S8hpZNVlMztIB9IMsdZCPPDwYMHGwKvV69ehts+9uNk2MZJCCKH5Mn/R+TEjh07GldlCrHQCcRFoC2bk98O7OGqRfGN73jaxDdzXahaiDKaTPd+JXM78yeQZO6ma+jLn3vuOaNWKV26tDz11FMmLC75QjkYnT9/vsn5yd82F4Obtuk14UIgiPPHb5sy30c4XPTmkQ5ErBmc7fiduZAkGWneZlEyt4NmypI5CScg8pdeekmKFi1qVDJEUsRNGfNF0sxhk07xO/HtQJwatQRVX61jF9/5A5kTRTHSmxo/ji+++MI46nE2hdUL1mKYCp966qnGKMFmUTK3g2bSyZxJQsAsyJcgQEjZLGC8P/mf+OTRCqFwUcMQLhcvNtJhOQksiKDIxHNKEAghCG3IbsoEvX1up3tY+uG2vzauO+GEE+S00047JCwu51G33XabUV9C9hRiIaECJaCdzaKmiXbQTDqZs/iIwYK+G903b30Iet26dcZCpXfv3ln2FMJHNw6ZozvHEeKXX37RNHF25obWkiYIIJXjz8H6cXayn3zyiTz99NMyYcKEjDSMnD9xjW2vUZXM7Uy0pJM53YCAySj0+OOPy5VXXmnIHEcgPD6RGNwWXgBI+RB8OhSVQu2McrrjSLgMyNwJjwG5Ej5jzJgx8tFHHxlBiVwBHJQSRsN2UTK3g2ggyPzbb781uTuHDRsmJ598sixfvtxMLPR1brOYo3aByNesWWMHmYDXku4ElNPwKD7ZIxSJT7G8eeSMkiXk6BNrSJ7yFWX7jh3yyCOPyKxZs4yQhXBFukaSpUcefpJTAC/tSpUqCVYxfouSuV/kDr0v6WTOpBoxYoQ8+uij5gfvznnz5hlydw5lIGoOZZzCVo9DT2dbyGRDiifVHDp4PyVZi9/Pc/3c4weTeNyTym2PBx7xqNMLxrn375cKP34rx8+eIQVrnCTFuj4vm4sUk5tvvllq1aold911l/H1YG1B7E6UxbVr15p1i38HiWGwcvFblMz9IhcwMsf8qVevXmY7h924M7CYGiKdMzGR2L/55hsjpWOhgodn9+7dpVSpUqY3GzZsMFI5hzPxKF4WRzyer3WmFwK251t29eXZu1eqjf5Ujpv9o+QtV0GOeqWf/FGipFx11VXStWtX8xthCYGKdXbsscca4sYLe+zYsSbMNIeksRQl81jQ+//3Jl0yJ9nE3XffbSRq3vBMHN76WLVQkAQ4COVwFDJHj4fe7vLLL5cj8uaV/evWyuzxY2XZfpG9+Q4NiWsHothqsb0wY2uN3p0OCHiZc7kOHpRiq1dI9aULpew5zaTg1dfLzF9+ldatWxvTRJzzKJs3bzb/X3PNNWb3zDodOHCgccpD8IqlKJnHgl6AyByPTg49kQAcM0SIfdSoUYLJFKngKExQDko5XcdkipRX+35fJv975B7ZNXeW/HFaY1ne7BI5kFfTxNmZGlpLuiCAurJ+/fpy4oknyurVq+XDDz80Ujc+G6gzIW5HKsf7mh0zBPzMM89IjRo15IYbbogJKjVNjAm+jJuTKpnj3MMpOWFtmThYrkDaeJ9xCMohDBlRHMmAgFtkGKpatar5bO/8ObLlvg6yb8VyWV+nviy5/FrZf4S/gxgv0owd6A+vJaiOO/Hqr9YbDAQ41CTYFnl1kcDx0XDmotNCdsX4cnANf3OGxRkXghghqmMpKpnHgl5AJHN03CtXrjTkjSda5IEm5lCQNqfkTKzPP//cuBczebjWSOu7dsmuMV/KmgljZVGFKrKtbAWRXLnsIGOhllheENndG0u9FroVUxWp3PaYOh6Hm7PC0ivGCEw4BnmxSEFS79SpkxG4OCiNpSiZx4JeQMjcbRfY+g0ZMsRYuERzWCBpxZw5c9xWl+V1XhdBzA/UChSBKAjEOg+93I8AhTkw6hLHDBg/DXIJEN8IqR3BClULn5NXACGL6KWoPNkpE6k0lngtSuZ2lkFS1SxuusB2jtPyCy64IOMQNPN9xCzHRMqvWaKbdiTqGi8LMVFt0uccikDQx8hL+7BMIbkL6hOnsI4mT55sUjXee++90rBhQ0PmM2fONJYsHII6gbjYJSNgxRJCV8nczgoLPJlj7YLDAmnjsiqYSSFJRNqi24EnmLW4WaxurrHVu8z61ch6HWkvke3J3C837fOCRZhUYEjdxFrBoCCyfPXVV0YnTihqDBEoRCrFuoyDUptFydwOmoEnczfdxCIGMo+XnbmbNgTpmmQSZ5BwSEZbstNj0x63Hs2JajuSNXlAI1PHIZmzG0a1iYFCsWLFjCUZHtmoU2z3IexkTv8QNPmNqoqD5OLFi/saYuz9GZ9oaq3AkTlOQTgJYVtOwHy2f46FC5nE8ULLvKWDxInoFhkp0RdSCb5JSTfBgCfgcV4PJTNfn5VFU7zmSq7ceaRE1bpyRPHScnK5YlLyyHzGMe/JJ580qktc+CF8IpE++OCDGSTENXiGon6BnAoWLOgbXa+miYQRQOXDrgFccB5s167dIaoi342xeCNtw8AD/xnOGo477riMUN0cHPspeMeTE7lt27bGnDTyxRo4MqeDJJW4/vrr5aGHHjKHK0wYOoGDQps2bQ4j8x2798qkqT/Kpj9WSr7cB/1gpPcoAqFBINoLIitpeuu+3NJndVFZszOPPNqsqjzW7ATZvvV/JoopBech9OOoOrFecSzJiKqICtQ5GGW9+pU2vUrm7BLQ5RNhFZXP888/b7KR8VksLxXbEwC+whYfh8cOHToYqbxly5Zy4YUXGisgPwWpHEEXhy2S81SrVi2D0ANJ5kjZxIbAxZ9BYrCZUEjgBPXJPDGfGveLDJ6+QmoV3C5Xlt4lBfMoofuZKHpP4hGIh8Ttpc6/9uSWZ5cVkX/25ZYbG5SXN1vWlo1/rpHrrrvOOOfddNNNJrcApoiOBI7efNy4cSZDEQWiuu++++T000/3BaBXMh8wYIAhM7zBaRN5DVAJ8UO2sXgXdiRY2KGawhsdXuJ/ok/iZOUU2oWW4YMPPjCHxNznkDmCKp7u7HDYWfAdZM/Lkpckv/mc6LF8Tr2YkKKVAC9eXKhaunXrZqyNKIEkc0Do37+/IXM6hD6ct1lWpXbvKbJw/T9SrsB+ubfSdimd/0C8x1PrVwRCgcDegyKztxeSA0dXkjsaV5PaxxaV+fPnGdLBSY/QuBRIh0NRdsq48zsFEiIYF+oYzBtzKkQ1ZZcdeSiN2uSPP/6QVq1amdgvORV2DSR1p30QGsIfO/bhw4dntDenOmL5nvbjCYtVD6onokuCATsEwgRTkKAJOYJUThwpbPgjyZxdRZcuXYwqGaw/++wzeeedd+Scc84xcXBQI5HPGDwwvaa//fr1M2akFLx0e/ToYTJCOQEJA0nmbOd469JJR1/+2muvZeDPZ7y9eEOxzXt71kYZMnON1My7RS4quVPy5z50qLxIKtzp9fpYJobem94IBGGu5StQUM5sdLaUO6607Nm92xALBETMFaRN1hi6XwgXCdyRBCFh9NYQND4gTjz07EYUKRODhcxkjt06ROaGzDk3oyDwYYWDPTwWOZB5kyZN4j6h4B2wWbVqlYkcOX36dOnTp4+88sorGTGl8KTFEQtdPnyGRO2QOR6zSPXkX+ClwAEzfMc5Rd++fY3qCMkbUmcHAu7Uw4vDCZ3AeQYvC16wTvycwJE5HSCXJ1sKGo/eiZjJ6O4coiXcLZ1nUvHdZZddZtQvxEVngmlRBBQB9wigNkGdiYEB5oeQY+Zkzqg2kTLr1KljzBOR1Flv6NKRit0QeVYt8qpmQQWLdQiSLBzAiwDBDzInLEG8C9hwCAmR80MuYnYsken04C2kaIgcYkZtApnjwc4LiHAlqEjYjYAnsXD4n3rBmeCDhDtBTYOKi11A+/btjdqLgpMkvjdDhw6Viy66yHwWODJnO8HAoLPjrQVwjg6JtxtkT8hcBo03HG83JhLXBZ3MgyCFxXuiOy9c5zm2zdgS0f50eoYzJwlo52zhc+o/0jUSIfpi4ilF6olzujfa917JHKkY1QohPpBqUTWgsoD4eCnFu4AZTlUPP/ywIV1egDw/8gDYIXOuYTfjkDnSNGSODp1rUCVzn18yp8+QeiDJHKmArQSETfAtCkRNrHMkdvKFYgOLdxreaOicCJeLWSIAQ/bRijNp04VQ4z2hc6pfcc4JoWB9T3wVZ7ueXcuQFolzzjpzbNNx6cdMzm/xappIsmlIlLM1duRYjCCpwxmQeyLKsmXLjMUdLzJUUpkPfzHYQOBEMCWQYKSahfM/osQiZSPRY9kCd5FzlT5Fk8yR4LnekcxR7SCt8xJwYuMESjL/9ddfjY6It02LFi2MkwL6OrZ0/PTs2dNsa5DA6RyAIpljokPhhJ3T33QvYSZSW32zVU9Y5ppbMmcdzpgx45B1xkugQoUKvqHwKplDlPAE6hX8UJBwIUx2Fk4mJN+NcXkjLzO0B+wEOnfufFimJXgL1TCCJmoYtAcQMAef3INAyqEx6mLuR0iF25DkeQlg7cILirNCLGXoH8QPB5Kch/tRMyHZc64ROMkcEyg6gN6bbQlbdBYdP5wG8xYk0SxAPvDAAwYAfmP6g4kSWz8OHrKTzF2OVUpf5pao3F6XHRg26ggK2G76kt01bu4PSl8zt8Mtmcej/V7JnDbAFZB6JDckgsghaXgJQbJ3795mV4B2IFpBasbiBKsbLE5QF9NmgpvBV5g08nLENp56wQFtAz9O3yBuNBOotpDuUcnwNypoXqJoKJzzikBJ5m4mCgp/SJyDBU50eUPzlgKQ0aNHm5NyLe4QSGXycddDvcotAqlG5m77ZfM6yBjtAZ6cHFQSswYNQVYFLcHrr79uXjyoUmI9W+A5tMGxoHn22WcPeZGkHJmz5cA9Fl0ZJjvoqjBp4m2ZimSuhGpzuaVHXdmd//idT14OQG2j7Ecyt90GN/Wh0kXdgUSNBQlqFiTn7AqEjvkmVjdY3XmJGR+tXmzOUSejwomMp8O1KUfmTFa2GYAEQLztnO3V+PHjjRmPlvAg4JecwoOAt574xUvJPGeccQQiQitqD5wZ3Yb95XpUx6hDYrXuog2oZKK9FFKOzLODfPbs2bJw4UJzid9JnfOQ6hXpgECqzh+/7VY1S+rP6lCROYcS06ZNS/1R0R4oAj4R8EvmyZbM2VU3aNDAlQeoT2hCf1uoyJxoapgraUkMAo5Ldqxbx8S0NnlPyYlgc/reS8v91qWSuReUg3ltqMicQPocUnid0F6vD+ZQaqvcIpDo8U7k8/yYTqL7JTWck1HILY62rkuVA1Bb/Y1XPaEhc06YIXM/qeMSudjiNZBaryLg96wI93ICVEVLlp4IVJXM7aAcGjLHZJF4DZmTOnPy68eZIFEE7/c5biQwv3XbmVrhqsUvln7vSyR6eCkSm9yJhugknOB35oIVhe0EEErmdkY7FGTuZA7HoN/R4zrwpMJisjOUh9cSpL4HqS1+5oabl2e8xjHe9ZIsHX8NPBMp7HIJg0sgq6pVqxqXdNYYQfDQrRO9z2ZRMreDZijIHNUKUjmTLXMJIolkN3RBaG8Q2mBneturJWyYOP2BwAmSRQztyINsDAluueUWE7wKoofMv//+e/Ob4FE2i5K5HTRDQeYE2oHMiXOgJXYEwkZcsSMSvBq8jlFW16M2ady4sXFRj9yxEMAJd3HOoUgYgUs6ERO5ntC3NovXqIk2nx2mukJB5iSAxmEomo4vTIOlfVEEckIgkrSdv7MzISXt2iWXXHJI1D/IlWh9BK4bOXKkSc0GqZM5Jx4hZlUyz2lU3X0fCjIPohu/V8kp6KoXd9NJr0o2AtHmXVZz8Yid/8rRB/ZLsyuvkjylSovk/i/fImpLAkMRspXMONzPrpe0ZhA7hQxDSOsIUBA8UQD9+hsomduZNSlP5kwoMo3w2w+B+rnHDvRaiyKQPATy7Nkjlb6fJBV+mipHNj5Xij3dU/IcW9Y0iLMnUrORIo4Q06hXiGFO0hgswyDfN954w6hfuJaAd8TkJra4n6Jk7ge1w+9JeTJnIk2ZMiUj7rlX6SAoZB6UdtiZVlpL0BE44t8dUmPkR1JqyXzJe0I1Oer1gXJEjZqm2YSYvvrqqw1Bk83GiadNgDtHAsc5j+QxXEtChRdffNFYvvgpSuZ+UAsZmUOAnLpjkhitJJMgk/lsO1NDa7GNQJDmRK4DB6TUmlVSd9NaKdqosRS8rKXkKljQdBn1CinRSPpSt25dIyiRJ+CTTz6Rtm3bmgNQkhOTT4DAdqhZSOrsN7yrkrmdmZbSkjn6um+++cbo7/yWIC0wv32IvC+7w65IG3yvO5jMz4jlfhv91DpiR+DYUqWkUYMGUrBYUcl1RD6j/547d668+eabRnXZvHlzOeqoowyZk2OXbDkkLsZpCDL/4YcfjC060ruTVNhPq5TM/aAWMsmc7mCWOGnSpLS3ZLH5UrJZl51pmlq1eDmEtIW113p4GZM4mHyUXjykUblwPgWhU8eECRNMspghQ4a4ju+deTTVNNHO/E5pyRwImFxk3kh0ujivi8fOcGVdSxDaE4Q2xBtn6rfZTy/Eb7NvJEo47bTTpGLFip6sUFhn8+fPlzPPPNMkWyCp8KpVq+Spp57y9FKI7ItK5nZGNuXJHBjQmc+aNeuwuCx2IAo2gca7j4mqPzv1UCLbkGj1kZcXg5drc8KMdGdNmzY1ahQvBXXLgAEDTFAukq5zKIp+PRZHIiVzLyOQ9bWhIHMyWeMBSkonLYpA0BCwScK2+ka8FfTcXl9e9AU7dOzOUc+QttHJDu+3bUrmfpE79L5QkDkHN2z9OFkPmheo7YVsuz4700hrcYNAIsbO7TNOPvlkIbtQEIqSuZ1RCAWZA0Us8cztQKm1KALZI+CWaOONI8G1ULFgJx6EomRuZxRCQ+bAMXr0aE8HoUFZXHaGUmtRBNwhgL4cFYsTv9zdXfG7SsncDrahJnMlazuTJMi1xDLGsdxrG5NEtqV8+fImGQXp4oJQ1DTRziiEmsxxWS64ZbP8W+Jo2VewkB3EEliLrQUeSz2x3JtAqPRR2SAQaSnEgSf6cmzMvR5+xgtklcztIBtYMue0fPPmzcaOPLJwgo4ZVLQtIs5Djjdovh3bpcrEUVLy10WyoebJ8tuFl8mBI/Jli1rQiCtI7QlSW2Kd+rb6klU9zue2nhNrfyPvJ99no0aNTJCsoBQlczsjEVgyJ5APAfLRg3NYQ/wHiH3lypVy+eWXmwS0mcuKFStMdDcmR6G/N0nNT4ZKsTWr5H8Vj5e5bTrIvgL/xZ6gBHGh2RlSrSXZCHiZW16utdGvUqVKydlnnx0YfTl9UjK3MbIigSVzujdo0CAZPHiwcVKoUKGCMTv88ssvzRbxlFNOOQwB3IwJvAXh59q7R0ovmCOlFs+TDXXqy4aadeVg7v9yHNou8V6Qbup3c02q9DsIDkS2sQpKfdWqVTNp4IJUlMztjEZgyRwp/NFHH5U1a9ZI//79TVB8Bp3YysWLFxe2i9EKapZp06ZZTyFngyxt1GFn2LWWVEUg1jnUrFmzwJgkOmOgZG5nNgaWzLEbv+2226RcuXLSrVs3Y3JI7GT0fdmdwu/bt8+E8ETlEq3EuhjswJ76tSiOyRnDzLh7GQeCY5EiLigmiUrmdudQYMmcpBMEyOeghshuGzZsMMF9iAOB08PevXtNPGVCcVJwTyb2MvEi+IzQuLgda0k/BLwQXDqhg5MQknnQipom2hmRwJL5zJkz5ZZbbjEZwjmwmTdvnlG1NGjQwPQcdcsTTzxhUlVB7tWrV5crrrjCBMh34jIvWrTIHHSmY1FCS+1R9zt+kfdlrqNhw4ZSo0aNwAGjahY7QxJYMv/444+NemXMmDHm8BP1CXp01Cekp1q9erVRp7Rq1coggckipO4UyJ7gW/wOW3F7QOiVELxeHzZcw9afyPFEtYKKskyZMoHrppK5nSEJJJlD3C+//LIxTURdguoEaXv58uXy7bffyl133WWInLC3nM6jYjnppJOkQIECh6CyZMkSo4pRkrIzWdKtFrcvzWi4BG3Ooa6EzLMyHEjm2Polc9SoGDsQLZVQvqhhMWHOXOAOztt+/PFHwyN+w/8mEyM3zw4cmUPkX331lQl6v3bt2gx7cqRy/m/fvr1ceumlJsMQPyyavn37Gv36hRdeeIhX265du0wmFEc69+LxZmsx2qrHzWDqNeFBwOa8Yd6jhkRF6WUNJApNP2QOH5Debvbs2SbODAlqEOwee+yxQ5JkgCOCH8LhOeecYxJSwwn33nuvOYsLUwkcmbsBl7crB6BOHOVnnnnGpLF68MEHD1G1UBdvZByJuN5tiUUiy/yMWBdlrPc77bFVj1sM43FdrOMSVAyy6pet9iKNc+7EDjaIxQ+Z40vSsmVL6d69u1x00UUyceJEufPOO2XKlCnGAs4pWMW1a9fOEDfZkCgks+GsDWIPU0lJMndSV/GmZcJ36NDBDOhVV111mORBWAC2V9ire9kO21pIiZosbtrr5hqb7U3082y2PUx1oYLEyc5Lrs9E9t8PmZNwGnJ+//33pU6dOiafQevWrY3BBDzgFIib/3v27CktWrRIZLcS/qyUJHPULe+9956RNHjzVqpUyZB5VhlPULfwxt64caMBWEkm4fMsaQ9M1bG20W5UKqyNU089NeZsQLYGEHLlHCwyiQyqVdYx1mtuY8b069fPOBOiksXkEmENVSuEft9992U0F5UKqllCguzcudPo1zkE7tGjhzGkCFNJSTJnInD4geoECxYONCItWaINEEln0Z0hBWgJPwI2yDDVUSKl2xlnnBFY9YqDb06SOWE6eAmwy6ZA+Fi5EeZj1KhRGWSOZdt11113CJmPHTtWOnbsKK+99po5f+OcDbWLQ/pB3a34mXspSeZ+Oop0zsl3VuoWP3XqPeFCwNFdR/YqiAeGblDHAowDT6TPoPchJzLHDLlTp06CIyEFA4gqVaoY6Xr48OFSq1YtkzISgu7SpYshdKdgzcbnr7/+uvF+ZYz5jQ4dfXtOQqAbrINyTdqQOYO4dOlSY87Itk7L4QioNOttVgQZL9QrBNTCiS7oJScyZ71GhsPG2IFzM9QqEDoqVsJfI4F//fXXctxxxxkdOtYt1H3llVcagsekGSm/efPm5kAY4lcyzzQ7kHYhSULTBrls3bpVJk+eLNu2bQtyM7VtioBnBCJfLFhqnHbaaSljrZETmUcDA4J/7rnnzE4b9QpOhniIv/jii0ad+tBDD5kfDj+HDh0qn376qTFHJBAf0jwHpZwlBH3X4mUipI1kDijo2tl2oX/LXNxIWW6u8QJ+TvXl9L2XZ8VybVDaEUsfwnxvpGkj6hWkTszzUoWo/JA544lwxgEncZtKliwp559/vklcw/8jR440h56omagfgickCDpy8Kldu3ZgrXv8ztW0InNAYqDHjx+fI15ZEViiiS3Rz8sJmFhtvXOqP/L7oPXdS9ujXRuv/jj1Qt6VK1c2npCpdLDnl8xjHY+w3Z92ZI7ODDNFSN1videi9Nsevc8fAm5fTPEcb1tCA0SONI56JYgu+9mNkEZN9Dd/M9+VdmSOqgU1y9y5cz15hdqBO/tabJOG7foSgUG6PcPrGGV1Pea5Z511lvFsTBX1ijPWKpnbmfVpR+bAhnS+ePFiIURupPOCHUiDV4tXwqAHfu4JXs9Ts0VesYfIsVxx63ATNFSUzO2MSFqSOdAxgXDzx4Y1HQjdznRJrVq8kmIq9c7pG17PuOpjd51KevJIrJXM7cy8tCVz4MNEkahrOCNEcxixA/GhtYSZYOKBV6rVGa/xjVYvNtLEXcFpJrtUikHHUMnczgilNZkDIS7CeIauW7cuYYRuZ+j+qyVe5GGzjUGtyzZ2tuvLDjeIHGkcL0/MEVO5KJnbGb20J3NgJGfoTz/9ZCxcbEjoiVzUdqZB8l8KicYsHs+LR51ZjW+pUqWMCSLxV1K9KJnbGUEl8//DEQmduOd4lDmE7tZ0LZahSAQBJOIZsWDg99549cv2uNtqp1MPB54kYkk1E8SsxllNE/2ugEPvUzKPwIMwnKhcYrFBtzMsWosbBGyRpJtn+b3GVhudelCp1K9fX0488US/TQrcfSqZ2xkSJfNMOKJymTlzponhkIzidvG7vc5WH7JTPzl2zW7a5OYaW20OWz3gTOxuHINI0ByWomRuZySVzKPgiIQ+depUE8jehg7dzlBpLWFBILsXWnbfkawYPXkqOgZlN3ZK5nZmtpJ5FjhC5ESCJDtROhB6KknMtnXadpbSf7X4xTGr+5zPIXIsVwjvmmoenjnhq2SeE0LuvlcyzwYnJHQci8hO4obQ/S5kd0OV3KuCTKDJRcbO07MjcxIPN2zYULBgCRuRg56SuZ05pGSeA45I5oTPJNymzZJI4k/ks2xilCp1xRNfdOO46odRInfGV8nczkxXMs8BRxYq5orYoe/YscMO6iGtJRqpxZPoQgpjRrciJfIw91VNE+2MrpK5CxyJ3UKOQSIthqmo6iS4o4lE3qhRI5OMOYyqlUjkVTK3Mw+VzF3iyIQjS9GKFSs0MJdLzGxflkovn1jaSrYcbMnLlCkTeiJXnbm9VaJk7gFLbNB/+OEHk0w2nYqqShI32litkJsyXYhcydze3FIy94AlpIbJIpEW+b1//34Pd/u7NNWINCu9Ob13aJVjegAAHppJREFUqy6w2WebdfkbQXfmimBTvHhxY36YTkSuZO53Vh1+n5K5Dyx37dolq1atkqVLl8qWLVt81ODtliAQkrcWh/fqzGNha2xQrUDk6aAjzzw7VGduZ70omfvEkUNR1C6koEOPvm/fPp81he82WwQXZGRs9pHIh5gfpiORq2Rub5YrmceIJaSOymX+/PnGWzQaqdtc+E5z41FnZN1e1CIxQpj02+OJZU6dwxGITEH8Tteipol2Rl7J3A6OxgZ97dq1smDBgtDYoyeK5GKx/Mhu+BLVftrg5VmRLvqYH2JP7vY8wdJ0DVQ1qmaxMxxK5nZwNLWwSNGhL1u2zKheSBytJXkIeCHYWFrp9TkQN5J4vXr1jGol3YuSuZ0ZoGRuB8dDasHKBTLngBQzRkfyjMOj0rZKrwQaFKDI1YlrPkSOrjydJXJnTJTM7cxOJXM7OB5WC2RDPJdFixYZy5fsDkhTlZjiBJ0nlUW82hCPeiHvatWqyfHHHy8FChSIxyNSsk4lczvDpmRuB8csa4HEkdB/++23bIN1KaHHeSCSWD3ZgVCr4NWJLblK44cOhpK5ncmpZG4Hx2xrcSxeiI8e5oQXbl5Ibq5xwPRybQKG0dcj8uTJIzVq1JDq1auHJmenLyCyuUnJ3A6iSuZ2cHRVCweiixcvNgekO3fudHWP24vCQHxu+5qI62LBk3tz585tpPC6detKhQoVEtHklH2GmibaGTolczs4uq4FKR0dOrp0LF/0cNQ1dClzIWoUXPIh8pIlS6paJYeRU8ncztRWMreDo6daIHCyGJE4Gtv0sBN6LFKuJ2ADcDHWKhUrVjREHqaky/GEVsncDrpK5nZw9FULjkYzZsyQP//8M5RhdePlDOQL7CxuiqWNkS8pR62CtUrlypUFUtfiDgElc3c45XSVknlOCMX5eyT0efPmycqVKwND6JE7hawsL8IibfvpR+Q9HHBiZsgBZ/ny5aVw4cJGX67FPQJK5u6xyu5KJXM7OMZUy969e01YXRyNmNg5lVgJKKf6w/C9H4xy6ndmEif2OFI4duMFCxbM6Xb9PgsElMztTA0lczs4xlwL1i0civ7yyy+BkdBj7lRIK4C4q1atanTjELpK4rENtJJ5bPg5dyuZ28HRSi1IfpguEqzLjYSe+aHxkEatdCwElaBuwoMTSRyVSr58+ULQq2B0wa9p4ubNm2Xy5MnyxhtvSJs2beSWW245rEOEqW7fvr1JJIMKbP369ea66667TnDmClNRMg/YaDKxkc6R0lG/2C5BJ3w37XNzjU3c8ufPb2zFkcaPOuooQU+uxR4CfiRz5sB3331n1JP9+vWTRx55JEsyf+WVV6Rdu3ZCApDXXntNxo8fL++9956ccMIJ9joRgJqUzAMwCJmbgBSxfPlymTt3rpHQsU23WRJNhm7bnox2ZWXNgiSO9I0bfu3atY0DUNgkObfjYvM6wltk3nXyPxI2CTqOPfZY14/DCW/btm3SvHlz6dChQ1QyZ+2gwixUqJCx9//ggw/kueeek3feecdkdgpTUTIP6GhC6GwJMVv8448/ZPv27dm2NBlEGFDorDQLUkFy47cGxbICqalkzZo1RkiJLBD86tWrpVWrVp7InDpQo1x44YVZknnmlvfs2VMmTJggw4cP9/wseyjEpyYl8/jgaq1WJjr26BA75ovoCfksKI5GsdhpWwPJUkWOXhx1SqVKlYyFigbFsgTu/1WDNI05bmRBMv/hhx/krLPOikqwSNf//vtvxg6VHZPzgvVC5qR47Ny5s3lpoDMP28G1krnduRrX2pj0BOpCilm3bp2R1oNC6nHtuMXKo+1gIGysUpDCa9asqSRuEW83VeWkM0eaf/LJJ828p1x66aVy7733GrWXWzJn3XTv3l2qVKkit956ayhNSZXM3cy2gF3D5IfIkdYJ2kXc9JxIPUwStK3hcLw2UacQS6VIkSJ6uGkLXA/15ETmzPWff/45Ix1juXLlpE6dOkaydkPmvAw4+CTXKvp1xjmMRck8xUeVhUAiaQ5MN2zYILt27cq2R7Z167brczscsbyciJlyzDHHGGcfSFxd792iHp/r/JomMgeY+0jqmBtyCIqlEea9Xbp0kbvvvttI4vwuW7asnHPOOUZthpqGw088dsNUlMxDMpqQOCnqfv/9d3NoGkv+0WQRdLyHAosGTAz5wUxNbcXjjbi7+nOSzKPVgh596tSpMn/+fJk0aZI5rMbq6JprrjG71R49ehh1ClJ4165djRTv6MixN7/jjjvk7LPPDtWZiJK5u/mWMldhm47ahcNSdIyRB0cp0wmfDT0oIjv255L9B3NJ0bwHJJeI0atiG44U5sROUTtxnwDH6TY/ZE5TmNuZfTFw7EIY4TvnkBQDgsiCdM7hdth2ZErmcZqgya6WCc2hD1I68dPd6NWT3eZYn79pT275ZH1B2bEvl9xQ6aCcXvkYc6iJjhWpXEswEfBL5sHsTfJapWSePOwT8mS2ozhWcAiEBQzqFyQVPk+kiWN2qhtbap1Z2/LJm6uOlH0HRB5vUkGeurC64L3Ji+yLL74wHoNI6ViuONIbW3NM1VTlkpDpGPUhSuZ2sFcyt4NjytTCwoHcIXUOj/COw1qA33zGT06WMZk7a4uM/YIIYUPSuwodI/1/2Stbdu6VF5qfJGdVLpFR5ZIlS6R169bywAMPGCcTXmYk2V64cKE5OAvbltsvlsm4T8ncDupK5nZwTNlaIGKHyJHYIXZ+HCme77ySe6LAQHXCQSbBr3C7L1CwkKz7Z7fs2XdAKpUoKLlzoTX/r0ybNs2Q9sCBA43bOAWzNnYs2Ja

The aim of this work is to verify the stability of the proposed orbital solutions for the third moonlet (Delta) taking into account a realistic gravitational potential for the central body of the quadruple system (Alpha). We also aim to estimate the location and size of a stability region inside the orbit of Gamma. First, we created a set of test particles with intervals of semi-major axis, eccentricities, and inclinations that covers the region interior to the orbit of Gamma, including the proposed orbit of Delta and a wide region around it. We considered three different models for the gravitational potential of Alpha: irregular polyhedron, ellipsoidal body and oblate body. For a second scenario, Delta was considered a massive spherical body and Alpha an irregular polyhedron. Beta and Gamma were assumed as spherical massive bodies in both scenarios. The simulations showed that a large region of space is almost fully stable only when Alpha was modeled as simply as an oblate body. For the scenario with Delta as a massive body, the results did not change from those as massless particles. Beta and Gamma do not play any relevant role in the dynamics of particles interior to the orbit of Gamma. Delta’s predicted orbital elements are fully unstable and far from the nearest stable region. The primary instability source is Alpha’s elongated shape. Therefore, in the determination of the orbital elements of Delta, it must be taken into account the gravitational potential of Alpha assuming, at least, an ellipsoidal shape.