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PhysikLernen
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| <h1 data-line="0" class="code-line" id="gravitation">Gravitation</h1> | |
| <h2 data-line="2" class="code-line" id="keplersche-gesetze">Kepler'sche Gesetze</h2> | |
| <ol> | |
| <li data-line="3" class="code-line"> | |
| <p data-line="3" class="code-line">Alle Planeten bewegen sich auf Ellipsenbahnen um die Sonne. In einem der Beiden Brennpunkten steht die Sonne.</p> | |
| </li> | |
| <li data-line="5" class="code-line"> | |
| <p data-line="5" class="code-line">Die gerade Verbindung zwischen Sonne und Planeten überstreicht in gleicher Zeit gleiche Flächen.<br> | |
| <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>A</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac></mrow><mo>=</mo><mi>k</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">{\Delta A \over \Delta t} = konst.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.872331em;"></span><span class="strut bottom" style="height:1.217331em;vertical-align:-0.345em;"></span><span class="base"><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathit mtight">t</span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathit mtight">A</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mord mathit">o</span><span class="mord mathit">n</span><span class="mord mathit">s</span><span class="mord mathit">t</span><span class="mord">.</span></span></span></span></eq> Wobei <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Delta A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base"><span class="mord">Δ</span><span class="mord mathit">A</span></span></span></span></eq> die überstrichene Fläche ist.</p> | |
| </li> | |
| <li data-line="8" class="code-line"> | |
| <p data-line="8" class="code-line">In einem Planetensystem ist <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mn>3</mn></msup><mi mathvariant="normal">/</mi><msup><mi>T</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">a^3 / T^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></eq> konstant dabei ist <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit">a</span></span></span></span></eq> die große Halbachse und <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span></eq> die Umlaufzeit.</p> | |
| </li> | |
| </ol> | |
| <h2 data-line="10" class="code-line" id="aphel-perihel-und-die-elipse">Aphel, Perihel und die Elipse</h2> | |
| <ul> | |
| <li data-line="11" class="code-line">Perihel: Sonne am nächsten</li> | |
| <li data-line="12" class="code-line">Perigäum: Erde am nächsten</li> | |
| <li data-line="13" class="code-line">Aphel: Sonne am fernsten</li> | |
| <li data-line="14" class="code-line">Apogäum: Erde am fernsten</li> | |
| <li data-line="15" class="code-line">Ellipse | |
| <ul> | |
| <li data-line="16" class="code-line">Grße Halbachse: <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit">a</span></span></span></span></eq> (längster Radius)</li> | |
| <li data-line="17" class="code-line">Lineare Exzentrizität: <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit">e</span></span></span></span></eq> (Mitte zu Brennpunkt)</li> | |
| <li data-line="18" class="code-line">Numerische Exzentrizität: <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ϵ</mi><mo>=</mo><mi>e</mi><mi mathvariant="normal">/</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\epsilon = e/a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord mathit">ϵ</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit">e</span><span class="mord">/</span><span class="mord mathit">a</span></span></span></span></eq></li> | |
| </ul> | |
| </li> | |
| </ul> | |
| <h2 data-line="19" class="code-line" id="geostation%C3%A4rit%C3%A4t">Geostationärität</h2> | |
| <ul> | |
| <li data-line="20" class="code-line">Immer über einem Punkt am Equator</li> | |
| <li data-line="21" class="code-line">Geschwindigkeit entspricht Erdrotation</li> | |
| <li data-line="22" class="code-line">Winkelgeschwindigkeit <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">\omega = 2 \pi / T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="mord">/</span><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span></eq></li> | |
| <li data-line="23" class="code-line">Zentripetalkraft <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>r</mi></msub><mo>=</mo><mi>m</mi><msup><mi>ω</mi><mn>2</mn></msup><mi>r</mi><mo>=</mo><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">F_r = m \omega^2 r = m v^2 / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span></span></eq></li> | |
| <li data-line="24" class="code-line">Geostationär: | |
| <ul> | |
| <li data-line="25" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">\omega = 2 \pi / T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="mord">/</span><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span></span></eq></li> | |
| <li data-line="26" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>r</mi></msub><mo>=</mo><mi>r</mi><mi>m</mi><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>T</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">F_r = r m 4 \pi^2 /T^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathit">m</span><span class="mord">4</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></eq></li> | |
| <li data-line="27" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>r</mi></msub><mo>=</mo><msub><mi>F</mi><mi>G</mi></msub><mo>=</mo><mi>G</mi><mi>m</mi><msub><mi>m</mi><mi>z</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">F_r = F_G = G m m_z / r^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit">G</span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></eq></li> | |
| <li data-line="28" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>G</mi><mi>m</mi><msub><mi>m</mi><mi>z</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mi>r</mi><mi>m</mi><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>T</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">G m m_z / r^2 = r m 4 \pi^2 /T^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord mathit">G</span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathit">m</span><span class="mord">4</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></eq></li> | |
| <li data-line="29" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>G</mi><msub><mi>m</mi><mi>z</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mi>r</mi></msup><mo>=</mo><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>T</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">G m_z / r^r = 4 \pi^2 /T^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord mathit">G</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></eq></li> | |
| <li data-line="30" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi><mo>=</mo><mroot><mrow><mi>G</mi><msub><mi>m</mi><mi>z</mi></msub><mo>(</mo><mn>1</mn><mi>a</mi><msup><mo>)</mo><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><mrow><mn>3</mn></mrow></mroot></mrow><annotation encoding="application/x-tex">r = \sqrt[3] {G m_z (1a)^2 / 4\pi^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.935em;"></span><span class="strut bottom" style="height:1.24em;vertical-align:-0.30499999999999994em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7002200000000001em;"><span style="top:-2.878em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathit">G</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mord mathit">a</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord">4</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.8950000000000005em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg width='400em' height='1.28em' viewBox='0 0 400000 1296' preserveAspectRatio='xMinYMin slice'><path d='M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167, | |
| 158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067 | |
| c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84, | |
| 175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71 | |
| c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3, | |
| -252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26 | |
| s76,-59,76,-59s76,-60,76,-60z M1001 80H40000v40H1012z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.30499999999999994em;"></span></span></span></span></span></span></span></eq></li> | |
| </ul> | |
| </li> | |
| </ul> | |
| <h2 data-line="31" class="code-line" id="massenbestimmung">Massenbestimmung</h2> | |
| <ul> | |
| <li data-line="32" class="code-line">Orbit | |
| <ul> | |
| <li data-line="33" class="code-line">Satelit (Mond) <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>z</mi></msub><mo>=</mo><msub><mi>F</mi><mi>G</mi></msub></mrow><annotation encoding="application/x-tex">F_z=F_G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span></eq></li> | |
| </ul> | |
| </li> | |
| <li data-line="34" class="code-line">Fallexperiment und Ortskonstante | |
| <ul> | |
| <li data-line="35" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>g</mi></msub><mo>=</mo><msub><mi>F</mi><mi>G</mi></msub></mrow><annotation encoding="application/x-tex">F_g = F_G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span></eq>; <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mi>g</mi><mo>=</mo><mi>G</mi><mi>m</mi><msub><mi>m</mi><mi>z</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">m g = G m m_z / r^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base"><span class="mord mathit">m</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit">G</span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></eq></li> | |
| </ul> | |
| </li> | |
| </ul> | |
| <h2 data-line="36" class="code-line" id="kosmischegeschwindigkeiten">Kosmischegeschwindigkeiten</h2> | |
| <ol> | |
| <li data-line="37" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.03588em;">v</span></span></span></span></eq> bei festem Radius:</li> | |
| </ol> | |
| <section><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo>=</mo><msqrt><mrow><mfrac><mrow><mi>G</mi><msub><mi>m</mi><mi>z</mi></msub></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msqrt></mrow><annotation encoding="application/x-tex">v_1=\sqrt{Gm_z \over r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.67104em;"></span><span class="strut bottom" style="height:2.44em;vertical-align:-0.7689599999999999em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.67104em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">G</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.6310400000000005em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg width='400em' height='2.48em' viewBox='0 0 400000 2592' preserveAspectRatio='xMinYMin slice'><path d='M424,2478c-1.3,-0.7,-38.5,-172,-111.5,-514c-73, | |
| -342,-109.8,-513.3,-110.5,-514c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5, | |
| 25c-5.7,9.3,-9.8,16,-12.5,20s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13, | |
| -13s76,-122,76,-122s77,-121,77,-121s209,968,209,968c0,-2,84.7,-361.7,254,-1079 | |
| c169.3,-717.3,254.7,-1077.7,256,-1081c4,-6.7,10,-10,18,-10H400000v40H1014.6 | |
| s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185c-2,6,-10,9,-24,9 | |
| c-8,0,-12,-0.7,-12,-2z M1001 80H400000v40H1014z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7689599999999999em;"></span></span></span></span></span></span></span></span></eqn></section><ol start="2"> | |
| <li data-line="41" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.03588em;">v</span></span></span></span></eq> benötigt zur Flucht aus System</li> | |
| </ol> | |
| <section><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mn>2</mn></msub><mo>=</mo><msqrt><mrow><mn>2</mn><mrow><mfrac><mrow><mi>G</mi><msub><mi>m</mi><mi>z</mi></msub></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></msqrt></mrow><annotation encoding="application/x-tex">v_2=\sqrt{2 {Gm_z \over r}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.67104em;"></span><span class="strut bottom" style="height:2.44em;vertical-align:-0.7689599999999999em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.67104em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">2</span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">G</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span style="top:-3.6310400000000005em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg width='400em' height='2.48em' viewBox='0 0 400000 2592' preserveAspectRatio='xMinYMin slice'><path d='M424,2478c-1.3,-0.7,-38.5,-172,-111.5,-514c-73, | |
| -342,-109.8,-513.3,-110.5,-514c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5, | |
| 25c-5.7,9.3,-9.8,16,-12.5,20s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13, | |
| -13s76,-122,76,-122s77,-121,77,-121s209,968,209,968c0,-2,84.7,-361.7,254,-1079 | |
| c169.3,-717.3,254.7,-1077.7,256,-1081c4,-6.7,10,-10,18,-10H400000v40H1014.6 | |
| s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185c-2,6,-10,9,-24,9 | |
| c-8,0,-12,-0.7,-12,-2z M1001 80H400000v40H1014z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7689599999999999em;"></span></span></span></span></span></span></span></span></eqn></section><ol start="3"> | |
| <li data-line="45" class="code-line"><eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.03588em;">v</span></span></span></span></eq> benötigt zur flucht aus Erde und Sonnensystem<br> | |
| Fluchtgeschw. aus Sonnensystem minus Erderevolutionsgeschw. plus Erdfluchtgeschw.</li> | |
| </ol> | |
| <section><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>v</mi><mn>3</mn></msub><mo>=</mo><msqrt><mrow><mo>(</mo><msub><mi>v</mi><mrow><mn>2</mn><mo separator="true">,</mo><mi>S</mi></mrow></msub><mo>−</mo><msub><mi>v</mi><mi>E</mi></msub><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><msubsup><mi>v</mi><mrow><mn>2</mn><mo separator="true">,</mo><mi>S</mi></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow><annotation encoding="application/x-tex">v_3=\sqrt{(v_{2,S}-v_E)^2+v_{2,S}^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.2170095000000003em;"></span><span class="strut bottom" style="height:1.84em;vertical-align:-0.6229904999999998em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2170095000000003em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.795908em;"><span style="top:-2.4064690000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4296389999999999em;"></span></span></span></span></span></span></span><span style="top:-3.1770095em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em;"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7, | |
| 572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7, | |
| -7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39 | |
| c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60 | |
| s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5, | |
| -658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10z | |
| M1001 80H400000v40H1013z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6229904999999998em;"></span></span></span></span></span></span></span></span></eqn></section><h2 data-line="49" class="code-line" id="gesamt-energie-formel">Gesamt Energie Formel</h2> | |
| <ul> | |
| <li data-line="50" class="code-line">Energie auf einer Ellipsenbahn:</li> | |
| </ul> | |
| <section><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>E</mi><mrow><mi>g</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>=</mo><mo>−</mo><mrow><mfrac><mrow><mi>G</mi><mi>m</mi><msub><mi>m</mi><mi>z</mi></msub></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mrow><annotation encoding="application/x-tex">E_{ges}=-{G m m_z \over 2a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.36033em;"></span><span class="strut bottom" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathit">a</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">G</span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></eqn></section><ul> | |
| <li data-line="53" class="code-line">Energie auf einer Kreisbahn</li> | |
| </ul> | |
| <section><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>E</mi><mrow><mi>p</mi><mi>o</mi><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>E</mi><mrow><mi>k</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><mo>−</mo><mrow><mfrac><mrow><mi>G</mi><mi>m</mi><msub><mi>m</mi><mi>z</mi></msub></mrow><mrow><mi>r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_{pot} + E_{kin} = -{G m m_z \over r} + {1 \over 2} m v^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.36033em;"></span><span class="strut bottom" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28055599999999997em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">p</span><span class="mord mathit mtight">o</span><span class="mord mathit mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">G</span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></eqn></section><h2 data-line="56" class="code-line" id="hohman-%C3%BCbergang">Hohman übergang</h2> | |
| <ul> | |
| <li data-line="57" class="code-line">Erst Beschleunigung <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span></eq> später noch eine Beschleunigung</li> | |
| <li data-line="58" class="code-line">Manöver Zeit durch drittes kepler Gesetz</li> | |
| <li data-line="59" class="code-line">Energie durch elipsen Bahnenergie</li> | |
| </ul> | |
| <h2 data-line="60" class="code-line" id="bahnen-beschreiben">Bahnen beschreiben</h2> | |
| <ul> | |
| <li data-line="61" class="code-line">durch Energie und Geschwindigkiet | |
| <ul> | |
| <li data-line="62" class="code-line">Wenn <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>E</mi><mrow><mi>g</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E_{ges} > 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span></eq> dann Flucht auf Hyperbelbahn</li> | |
| <li data-line="63" class="code-line">Wenn <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>E</mi><mrow><mi>g</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E_{ges} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span></eq> dann Flucht auf Parabelbahn</li> | |
| <li data-line="64" class="code-line">Wenn <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>E</mi><mrow><mi>g</mi><mi>e</mi><mi>s</mi></mrow></msub><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E_{ges} < 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span></eq> dann bleibt Objekt auf Ellipsenbahn</li> | |
| <li data-line="65" class="code-line">Wenn <eq><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>E</mi><mrow><mi>g</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>G</mi><mi>m</mi><msub><mi>m</mi><mi>z</mi></msub><mi mathvariant="normal">/</mi><mn>2</mn><mi>r</mi></mrow><annotation encoding="application/x-tex">E_{ges} = -G m m_z / 2r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord mathit">G</span><span class="mord mathit">m</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord">/</span><span class="mord">2</span><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span></span></eq> dann Kreisbahn</li> | |
| </ul> | |
| </li> | |
| </ul> | |
| <h2 data-line="66" class="code-line" id="gravitationsgesetz">Gravitationsgesetz</h2> | |
| <section><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>g</mi></msub><mo>=</mo><mi>G</mi><mrow><mfrac><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow></mrow><annotation encoding="application/x-tex">F_g = G {m_1 m_2 \over r^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.1075599999999999em;"></span><span class="strut bottom" style="height:1.7935599999999998em;vertical-align:-0.686em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord mathit">G</span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width='400em' height='0.2em' viewBox='0 0 400000 200' preserveAspectRatio='xMinYMin slice'><path d='M0 80H400000 v40H0z M0 80H400000 v40H0z'/></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></eqn></section><h2 data-line="68" class="code-line" id="himmelsk%C3%B6rper-klasifizieren">Himmelskörper klasifizieren</h2> | |
| <ul> | |
| <li data-line="69" class="code-line">Monde (kreisen um Planeten)</li> | |
| <li data-line="70" class="code-line">Sterne (in Galaxien)</li> | |
| <li data-line="71" class="code-line">Planeten (Kreisen um Sterne mit einer bestimmten Masse)</li> | |
| <li data-line="72" class="code-line">Galaxie (Ansammlung von Sternen)</li> | |
| </ul> | |
| </body></html> |
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