toshiki-notebook/assets/academic_physics_ipho-formulas-jpn_12.md.f0861b99.lean.js

import{_ as e}from"./chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js";import{_ as m,C as i,o as r,c as p,H as l,k as s,a,Q as c}from"./chunks/framework.b7580407.js";import"./chunks/commonjsHelpers.725317a4.js";const E=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 12","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/12.md","filePath":"academic/physics/ipho-formulas-jpn/12.md","lastUpdated":1710267506000}'),h={name:"academic/physics/ipho-formulas-jpn/12.md"},o=s("h1",{id:"formulas-for-ipho-日本語版-section-12",tabindex:"-1"},[a("Formulas for IPhO 日本語版: Section 12 "),s("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"'},"​")],-1),g=s("h2",{id:"_12-kepler-の法則",tabindex:"-1"},[a("12: Kepler の法則 "),s("a",{class:"header-anchor",href:"#_12-kepler-の法則","aria-label":'Permalink to "12: Kepler の法則"'},"​")],-1),u=s("h3",{id:"_12-1-f-u",tabindex:"-1"},[a("12.1: F & U "),s("a",{class:"header-anchor",href:"#_12-1-f-u","aria-label":'Permalink to "12.1: F & U"'},"​")],-1),d=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"F=G M m / r^2, U=-G M m / r \\text {. }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])],-1),y=c("",5),v=s("ol",{start:"4"},[s("li",null,[a("Kepler の第三法則 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},": r^{-2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(" 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 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mn",null,"3"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{3}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(" 乗 に比例する :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"T"),s("mn",null,"1"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"T"),s("mn",null,"2"),s("mn",null,"2")]),s("mo",null,"="),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"1"),s("mn",null,"3")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"2"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"T_1^2 / T_2^2=a_1^3 / a_2^3 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])])])])])])])])],-1),b=s("h3",{id:"_12-5-楕円軌道",tabindex:"-1"},[a("12.5: 楕円軌道 "),s("a",{class:"header-anchor",href:"#_12-5-楕円軌道","aria-label":'Permalink to "12.5: 楕円軌道"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[a("重力場中で楕円軌道を描く質点の全エネルギー "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"K"),s("mo",null,"+"),s("mi",null,"U"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(K+U)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mclose"},")")])])]),a(" :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"E=-G M m / 2 a ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a")])])])])])])],-1),_=s("h3",{id:"_12-6-楕円率",tabindex:"-1"},[a("12.6: 楕円率 "),s("a",{class:"header-anchor",href:"#_12-6-楕円率","aria-label":'Permalink to "12.6: 楕円率"'},"​")],-1),k=s("ol",null,[s("li",null,[a("楕円率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",null,"="),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon=d / a \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" 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.")])],-1),w=s("h3",{id:"_12-7-楕円の性質",tabindex:"-1"},[a("12.7: 楕円の性質 "),s("a",{class:"header-anchor",href:"#_12-7-楕円の性質","aria-label":'Permalink to "12.7: 楕円の性質"'},"​")],-1),z=s("ol",{start:"2"},[s("li",null,[a("楕円の性質 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"l"),s("mn",null,"2")]),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,"（"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},": l_1+l_2=2 a （ l_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord cjk_fallback"},"（"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"l"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"l_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は焦点までの距 離). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\alpha_1=\\alpha_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" (一方の焦点から出た光は他方の焦点に 反射する). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"S"),s("mo",null,"="),s("mi",null,"π"),s("mi",null,"a"),s("mi",null,"b")]),s("annotation",{encoding:"application/x-tex"},"S=\\pi a b")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"πab")])])]),a(".")])],-1),f=s("h3",{id:"_12-8-円の中心",tabindex:"-1"},[a("12.8: 円の中心 "),s("a",{class:"header-anchor",href:"#_12-8-円の中心","aria-label":'Permalink to "12.8: 円の中心"'},"​")],-1),M=s("ol",{start:"3"},[s("li",null,"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.")],-1),P={id:"_12-9-nge-lenz-ベクトル",tabindex:"-1"},L=s("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" />"'},"​",-1),K=s("ol",{start:"4"},[s("li",null,[a("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 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"e")]),s("annotation",{encoding:"application/x-tex"},"e")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"e")])])]),a(" に一致する.]："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"M")]),s("mrow",null,[s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m")])]),s("mo",null,"−"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{e}=\\frac{\\boldsymbol{v} \\times \\boldsymbol{M}}{G M m}-\\boldsymbol{e}_r=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0491em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3631em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])])])],-1);function T(G,S,U,F,q,B){const t=e,n=i("Badge");return r(),p("div",null,[o,l(t,{readTime:"1",words:"332"}),g,u,d,y,v,b,x,_,k,w,z,f,M,s("h3",P,[a("12.9: nge-Lenz ベクトル "),l(n,{type:"tip",text:"supplemental"}),a(),L]),K])}const $=m(h,[["render",T]]);export{E as __pageData,$ as default};