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problem
When discussing the orbits of three stars of equal mass under the influence of each other's gravity alone, intuition might suggest a co-circular orbit. Recent research has found that the motion of three stars of equal mass \\(m = 10^{30} \\text{ kg} \\) under the influence of each other's gravity can also present an eight-shaped orbit. It is assumed that the position and shape of the orbits do not change over time.
\\(O\\) and \\(P\\) are points on the eight-shaped orbit. Let \\(O\\) be at the centre of the eight-shaped orbit (where the curve intersects itself). Let \\(P\\) be at one of the points on the orbit furthest from \\(O\\).
We enumerate the stars with \\(1\\), \\(2\\) and \\(3\\) in the order they pass a given point. At a given moment when star \\(1\\) passes point \\(O\\), let \\(O_2\\) and \\(O_3\\) denote the positions of stars \\(2\\) and \\(3\\) respectively at this moment. Likewise, at a given moment when star \\(1\\) passes through point \\(P\\), let \\(P_2\\) and \\(P_3\\) denote the positions of stars \\(2\\) and \\(3\\). \
The radius of curvature of the orbit at \\(P\\) is \\(41.7 L\\), where \\(L\\) is an arbitrary unit of length. Furthermore, the distances \\(OO_2 = 145.2 L\\), and \\(PP_2 = 242 L\\) are known, as well as the angle \\(\\angle OPP_2 = 11.6^\\circ\\).
Suppose the speed of a star passing the point \\(O\\) is \\(v_O = 100 \\text{ km s}^{-1} \\). Find a numerical value for the total energy \\(E\\) of the three-body system in Joules (using symbol \\(\\text{J}\\)) rounded correctly to 2 significant figures. You may assume that the arbitrary constants present within the gravitational potential are chosen so that the gravitational potential of the system tends to zero as the separation between the stars tends to infinity.
Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.subject
physics
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