File:Lagrangian points equipotential.gif
Summary
Animation showing the relationship between the five Lagrangian points (red) of a planet (blue) orbiting a star (yellow), and the gravitational potential in the plane containing the orbit (grey surface with purple contours of equal potential). The potential was computed in POV-Ray using
<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e08dba9e12ab76355ff5260d29d5603731296e50" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -3.338ex; width:88.285ex; height:6.843ex;" alt="{\displaystyle \Psi (x,y,z)={\Big (}x-{\frac {q}{1+q}}{\Big )}^{2}+y^{2}+{\frac {2}{(1+q){\sqrt {x^{2}+y^{2}+z^{2}}}}}+{\frac {2q}{(1+q){\sqrt {(x-1)^{2}+y^{2}+z^{2}}}}}}">
for q = 0.1 and z = 0.<a href="#cite_note-1">[1]</a>
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 08:25, 15 January 2017 | | 256 × 192 (709 KB) | 127.0.0.1 (talk) | Animation showing the relationship between the five Lagrangian points (red) of a planet (blue) orbiting a star (yellow), and the gravitational potential in the plane containing the orbit (grey surface with purple contours of equal potential). The potential was computed in POV-Ray using <p><span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi mathvariant="normal">Ψ<!-- Ψ --></mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.623em" minsize="1.623em">(</mo></mrow></mrow><mi>x</mi><mo>−<!-- − --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>q</mi><mrow><mn>1</mn><mo>+</mo><mi>q</mi></mrow></mfrac></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.623em" minsize="1.623em">)</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></msqrt></mrow></mrow></mfrac></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mn>2</mn><mi>q</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><mo stretchy="false">(</mo><mi>x</mi><mo>−<!-- − --></mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></msqrt></mrow></mrow></mfrac></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Psi (x,y,z)={\Big (}x-{\frac {q}{1+q}}{\Big )}^{2}+y^{2}+{\frac {2}{(1+q){\sqrt {x^{2}+y^{2}+z^{2}}}}}+{\frac {2q}{(1+q){\sqrt {(x-1)^{2}+y^{2}+z^{2}}}}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e08dba9e12ab76355ff5260d29d5603731296e50" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -3.338ex; width:88.285ex; height:6.843ex;" alt="{\displaystyle \Psi (x,y,z)={\Big (}x-{\frac {q}{1+q}}{\Big )}^{2}+y^{2}+{\frac {2}{(1+q){\sqrt {x^{2}+y^{2}+z^{2}}}}}+{\frac {2q}{(1+q){\sqrt {(x-1)^{2}+y^{2}+z^{2}}}}}}"></span> </p> for <i>q</i> = 0.1 and <i>z</i> = 0.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> |
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