{"id":19000,"date":"2024-12-28T18:10:52","date_gmt":"2024-12-28T18:10:52","guid":{"rendered":"\/?p=19000"},"modified":"2024-12-28T18:12:49","modified_gmt":"2024-12-28T18:12:49","slug":"many-groups-reaching-towards-each-other-but-not-filling-all-the-gaps-and-not-doing-it-for-all-humans","status":"publish","type":"post","link":"\/?p=19000","title":{"rendered":"Many groups reaching towards each other but not filling all the gaps and not doing it for all humans"},"content":{"rendered":"
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Many groups reaching towards each other but not filling all the gaps and not doing it for all humans<\/strong><\/p>\n<\/div>\n

Richard Collins:\u00a0<\/strong><\/span><\/div>\n
If one monitors the motion of an object in space, how does one determine if the motion is parabolic, elliptical or hyperbolic? Are there cases where that is difficult to determine? Assume perfect ability to observe and record.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/article>\n
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OpenAI ChatGPT 01:<\/span><\/strong><\/span><\/h5>\n
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Thought about determining orbital motion type for a couple of seconds<\/div>\n<\/div>\n
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In a two-body problem (one dominant gravitating body and a much less massive orbiting object), the trajectory of the smaller object is conic\u2014either elliptical, parabolic, or hyperbolic\u2014depending on its total energy and corresponding eccentricity. In mathematical terms:<\/p>\n

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  1. Elliptical trajectory<\/strong>:\n
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    • Eccentricity e<1e < 1<\/span>e<\/span><<\/span><\/span>1<\/span><\/span><\/span><\/span><\/li>\n
    • Negative total orbital energy<\/li>\n<\/ul>\n<\/li>\n
    • Parabolic trajectory<\/strong>:\n
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      • Eccentricity e=1e = 1<\/span>e<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span><\/li>\n
      • Zero total orbital energy<\/li>\n<\/ul>\n<\/li>\n
      • Hyperbolic trajectory<\/strong>:\n
          \n
        • Eccentricity e>1e > 1<\/span>e<\/span>><\/span><\/span>1<\/span><\/span><\/span><\/span><\/li>\n
        • Positive total orbital energy<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

          Below is a concise way to see how you can distinguish these cases and under which conditions it might be difficult to do so.<\/p>\n

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          How to determine the orbit type<\/h2>\n
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          1. Measure position and velocity vectors<\/strong> r\u20d7\\vec{r}<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and v\u20d7\\vec{v}<\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at a given instant in time (with respect to the main gravitational body).<\/li>\n
          2. Compute the specific angular momentum<\/strong>h\u20d7=r\u20d7\u00d7v\u20d7.\\vec{h} = \\vec{r} \\times \\vec{v}.<\/span>h<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>v<\/span><\/span><\/span><\/span><\/span>.<\/span><\/span><\/span><\/span><\/span>(In computer syntax with explicit multiplication, that might be h_vec = cross(r_vec,v_vec)<\/code>. Note that all vector magnitudes should be carefully computed in your favorite programming language.)<\/li>\n
          3. Compute the eccentricity vector<\/strong>e\u20d7=v\u20d7\u00d7h\u20d7\u03bc\u2212r\u20d7\u2223r\u20d7\u2223,\\vec{e} = \\frac{\\vec{v} \\times \\vec{h}}{\\mu} – \\frac{\\vec{r}}{|\\vec{r}|},<\/span>e<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03bc<\/span>v<\/span><\/span><\/span>\u00d7<\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u2223r<\/span><\/span><\/span>\u2223r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span><\/span>where \u03bc=G\u2217M\\mu = G * M<\/span>\u03bc<\/span>=<\/span><\/span>G<\/span>\u2217<\/span><\/span>M<\/span><\/span><\/span><\/span> is the standard gravitational parameter (with GG<\/span>G<\/span><\/span><\/span><\/span> as the universal gravitational constant and MM<\/span>M<\/span><\/span><\/span><\/span> as the mass of the central body).<\/li>\n
          4. Take the magnitude of the eccentricity vector<\/strong> to gete=\u2223e\u20d7\u2223.e = |\\vec{e}|.<\/span>e<\/span>=<\/span><\/span>\u2223<\/span>e<\/span><\/span><\/span><\/span><\/span>\u2223.<\/span><\/span><\/span><\/span><\/span><\/li>\n
          5. Identify orbit type<\/strong> from ee<\/span>e<\/span><\/span><\/span><\/span>:\n
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            • e<1e < 1<\/span>e<\/span><<\/span><\/span>1<\/span><\/span><\/span><\/span> \u2192 Elliptical<\/li>\n
            • e=1e = 1<\/span>e<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span> \u2192 Parabolic<\/li>\n
            • e>1e > 1<\/span>e<\/span>><\/span><\/span>1<\/span><\/span><\/span><\/span> \u2192 Hyperbolic<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

              Equivalently, one may compute the total specific orbital energy<\/strong> (\u03b5\\varepsilon<\/span>\u03b5<\/span><\/span><\/span><\/span>, often in units of J\/kg), which is given by<\/p>\n

              \u03b5=\u2223v\u20d7\u222322\u2212\u03bc\u2223r\u20d7\u2223.\\varepsilon = \\frac{|\\vec{v}|^{2}}{2} – \\frac{\\mu}{|\\vec{r}|}.<\/span>\u03b5<\/span>=<\/span><\/span>2\u2223v<\/span><\/span><\/span>\u22232<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u2223r<\/span><\/span><\/span>\u2223\u03bc<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>.<\/span><\/span><\/span><\/span><\/span><\/p>\n

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              • If \u03b5<0\\varepsilon < 0<\/span>\u03b5<\/span><<\/span><\/span>0<\/span><\/span><\/span><\/span>, the orbit is elliptical.<\/li>\n
              • If \u03b5=0\\varepsilon = 0<\/span>\u03b5<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>, the orbit is parabolic.<\/li>\n
              • If \u03b5>0\\varepsilon > 0<\/span>\u03b5<\/span>><\/span><\/span>0<\/span><\/span><\/span><\/span>, the orbit is hyperbolic.<\/li>\n<\/ul>\n
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                When is it difficult to tell?<\/h2>\n
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                1. Orbit near parabolic<\/strong>:\n
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                  • In practice, a perfectly parabolic orbit (e=1)(e = 1)<\/span>(<\/span>e<\/span>=<\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span> is an idealized case. Minor measurement uncertainties or minute perturbations from other bodies can easily make the orbit slightly elliptical or slightly hyperbolic.<\/li>\n
                  • When ee<\/span>e<\/span><\/span><\/span><\/span> is extremely close to 1 (e.g. e=0.999999999e = 0.999999999<\/span>e<\/span>=<\/span><\/span>0.999999999<\/span><\/span><\/span><\/span>), distinguishing between elliptical and parabolic can be quite difficult.<\/li>\n<\/ul>\n<\/li>\n
                  • Data over a short arc<\/strong>:\n
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                    • If the object has only been observed for a short time relative to its orbital period (for near-elliptical orbits) or relative to its hyperbolic flyby timescale (for hyperbolic encounters), you might not have enough baseline to confidently pin down the trajectory. Even if you have \u201cperfect\u201d measurements of position\/velocity at each observation, the coverage might be too short to differentiate elliptical vs. nearly parabolic (or hyperbolic) motion.<\/li>\n<\/ul>\n<\/li>\n
                    • Perturbations from multiple bodies<\/strong>:\n
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                      • In a complex multi-body environment (e.g., near a planet in the Solar System with gravitational influences from other planets and moons), there can be small but non-negligible accelerations from third bodies. These extra perturbations complicate the standard two-body analysis, making it more difficult to assign a strict conic category without carefully modeling all relevant influences.<\/li>\n<\/ul>\n<\/li>\n
                      • Relativistic corrections<\/strong>:\n
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                        • For extremely massive central bodies (like black holes) or for objects traveling at high velocities, one must include relativistic effects. While for many practical orbital calculations Newtonian gravity is sufficient, at high precision or in extreme gravitational fields, ignoring relativistic corrections can introduce discrepancies when identifying the orbit type.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                          In a purely theoretical two-body scenario with perfect measurements and no external influences, one can directly classify the path from the eccentricity (or orbital energy) as described above. In real-world cases, measurement uncertainties, perturbations, and sampling over limited time intervals may blur the distinction\u2014especially for orbits with eccentricity close to 1.<\/p>\n

                          Richard Collins:\u00a0<\/strong><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/article>\n

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                          We say in artillery, and simple physics classes, that motion is “parabolic”. But you just aid the motion is sort of constantly on the edge or elliptical and hyperbolic, since perfect balance is difficult to maintain.<\/div>\n