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data-v-da72eda5><span class="visually-hidden" id="sidebar-aria-label" data-v-da72eda5> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-da72eda5><section class="VPSidebarItem level-0 collapsible has-active" data-v-da72eda5 data-v-18b5e57d><div class="item" role="button" tabindex="0" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><h2 class="text" data-v-18b5e57d>IPhO Formulas: JP Ver.</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-18b5e57d><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-18b5e57d><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-18b5e57d><!--[--><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div 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data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" data-v-18b5e57d></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-18b5e57d data-v-add0466d><!--[--><p class="text" data-v-18b5e57d>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-18b5e57d data-v-18b5e57d><div class="item" data-v-18b5e57d><div class="indicator" 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data-v-6d510875><div><h1 id="formulas-for-ipho-日本語版-section-12" tabindex="-1">Formulas for IPhO 日本語版: Section 12 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-12" aria-label="Permalink to "Formulas for IPhO 日本語版: Section 12""></a></h1><h2 id="_12-kepler-の法則" tabindex="-1">12: Kepler の法則 <a class="header-anchor" href="#_12-kepler-の法則" aria-label="Permalink to "12: Kepler の法則""></a></h2><h3 id="_12-1-f-u" tabindex="-1">12.1: F & U <a class="header-anchor" href="#_12-1-f-u" aria-label="Permalink to "12.1: F & U""></a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>U</mi><mo>=</mo><mo>−</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><mi>r</mi><mtext>. </mtext></mrow><annotation encoding="application/x-tex">F=G M m / r^2, U=-G M m / r \text {. }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><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="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord text"><span class="mord">. </span></span></span></span></span></li></ol><h3 id="_12-2-kepler-の第一法則" tabindex="-1">12.2: Kepler の第一法則 <a class="header-anchor" href="#_12-2-kepler-の第一法則" aria-label="Permalink to "12.2: Kepler の第一法則""></a></h3><ol start="2"><li>Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).</li></ol><h3 id="_12-3-kepler-の第二法則" tabindex="-1">12.3: Kepler の第二法則 <a class="header-anchor" href="#_12-3-kepler-の第二法則" aria-label="Permalink to "12.3: Kepler の第二法則""></a></h3><ol start="3"><li>Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.</li></ol><h3 id="_12-4-kepler-の第三法則" tabindex="-1">12.4: Kepler の第三法則 <a class="header-anchor" href="#_12-4-kepler-の第三法則" aria-label="Permalink to "12.4: Kepler の第三法則""></a></h3><ol start="4"><li>Kepler の第三法則 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">: r^{-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><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"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{3}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><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:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></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">3</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></span><span class="mclose nulldelimiter"></span></span></span></span></span> 乗 に比例する :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>T</mi><mn>1</mn><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><msubsup><mi>T</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>a</mi><mn>1</mn><mn>3</mn></msubsup><mi mathvariant="normal">/</mi><msubsup><mi>a</mi><mn>2</mn><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">T_1^2 / T_2^2=a_1^3 / a_2^3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;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 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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;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 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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;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 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">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;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 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">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_12-5-楕円軌道" tabindex="-1">12.5: 楕円軌道 <a class="header-anchor" href="#_12-5-楕円軌道" aria-label="Permalink to "12.5: 楕円軌道""></a></h3><ol start="5"><li>重力場中で楕円軌道を描く質点の全エネルギー <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>+</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K+U)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">)</span></span></span></span> :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo>=</mo><mo>−</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><mn>2</mn><mi>a</mi></mrow><annotation encoding="application/x-tex">E=-G M m / 2 a </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/2</span><span class="mord mathnormal">a</span></span></span></span></span></p></li></ol><h3 id="_12-6-楕円率" tabindex="-1">12.6: 楕円率 <a class="header-anchor" href="#_12-6-楕円率" aria-label="Permalink to "12.6: 楕円率""></a></h3><ol><li>楕円率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi><mo>=</mo><mi>d</mi><mi mathvariant="normal">/</mi><mi>a</mi><mo>≪</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\varepsilon=d / a \ll 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord">/</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> の場合, 軌道は焦点をずらし た円形をしていると考えられる.</li></ol><h3 id="_12-7-楕円の性質" tabindex="-1">12.7: 楕円の性質 <a class="header-anchor" href="#_12-7-楕円の性質" aria-label="Permalink to "12.7: 楕円の性質""></a></h3><ol start="2"><li>楕円の性質 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msub><mi>l</mi><mn>1</mn></msub><mo>+</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mtext>（</mtext><msub><mi>l</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">: l_1+l_2=2 a （ l_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;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></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;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 class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mord cjk_fallback">（</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;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></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">l_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;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></span> は焦点までの距 離). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;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></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;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></span> (一方の焦点から出た光は他方の焦点に 反射する). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">S=\pi a b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">πab</span></span></span></span>.</li></ol><h3 id="_12-8-円の中心" tabindex="-1">12.8: 円の中心 <a class="header-anchor" href="#_12-8-円の中心" aria-label="Permalink to "12.8: 円の中心""></a></h3><ol start="3"><li>円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.</li></ol><h3 id="_12-9-nge-lenz-ベクトル" tabindex="-1">12.9: nge-Lenz ベクトル <!----> <a class="header-anchor" href="#_12-9-nge-lenz-ベクトル" aria-label="Permalink to "12.9: nge-Lenz ベクトル <Badge type="tip" text="supplemental" />""></a></h3><ol start="4"><li>Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> に一致する.]：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">e</mi><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow></mfrac><mo>−</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">\boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0491em;vertical-align:-0.686em;"></span><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:1.3631em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</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></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal 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></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p></li></ol></div></div></main><footer class="VPDocFooter" data-v-6d510875 data-v-dd3264f6><!--[--><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><!--]--><div class="edit-info" data-v-dd3264f6><div class="edit-link" data-v-dd3264f6><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/12.md" target="_blank" rel="noreferrer" data-v-dd3264f6 data-v-add0466d><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" aria-label="edit icon" data-v-dd3264f6><path 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