Kepleri seadused – Vikipeedia

<!doctype html> <html class="client-nojs skin-theme-clientpref-day mf-expand-sections-clientpref-0 mf-font-size-clientpref-small mw-mf-amc-clientpref-0" lang="et" dir="ltr"> <head> <base href="https://et.m.wikipedia.org/wiki/Kepleri_seadused"> <meta charset="UTF-8"> <title>Kepleri seadused – Vikipeedia</title> <script>(function(){var className="client-js skin-theme-clientpref-day mf-expand-sections-clientpref-0 mf-font-size-clientpref-small mw-mf-amc-clientpref-0";var cookie=document.cookie.match(/(?:^|; )etwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","jaanuar","veebruar","märts","aprill","mai","juuni","juuli","august","september","oktoober","november","detsember"],"wgRequestId":"1badea79-4e01-4ac1-8dfa-66012cc5ce78","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Kepleri_seadused","wgTitle":"Kepleri seadused","wgCurRevisionId":6270518,"wgRevisionId":6270518, "wgArticleId":50893,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgPageViewLanguage":"et","wgPageContentLanguage":"et","wgPageContentModel":"wikitext","wgRelevantPageName":"Kepleri_seadused","wgRelevantArticleId":50893,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"et","pageLanguageDir":"ltr","pageVariantFallbacks":"et"},"wgMFMode":"stable","wgMFAmc":false,"wgMFAmcOutreachActive":false,"wgMFAmcOutreachUserEligible":false,"wgMFLazyLoadImages":true,"wgMFEditNoticesFeatureConflict":false,"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgMFIsSupportedEditRequest":true,"wgMFScriptPath":"","wgWMESchemaEditAttemptStepOversample":false, "wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":true,"wgEditSubmitButtonLabelPublish":true,"wgSectionTranslationMissingLanguages":[{"lang":"ace","autonym":"Acèh","dir":"ltr"},{"lang":"ady","autonym":"адыгабзэ","dir":"ltr"},{"lang":"alt","autonym":"алтай тил","dir":"ltr"},{"lang":"am","autonym":"አማርኛ","dir":"ltr"},{"lang":"ami","autonym":"Pangcah","dir":"ltr"},{"lang":"an","autonym":"aragonés","dir":"ltr"},{"lang":"ang","autonym":"Ænglisc","dir":"ltr"},{"lang":"ann","autonym":"Obolo","dir":"ltr"},{"lang":"anp","autonym":"अंगिका","dir":"ltr"},{"lang":"ary","autonym":"الدارجة","dir":"rtl"},{"lang":"arz","autonym":"مصرى","dir":"rtl"},{"lang":"as","autonym":"অসমীয়া","dir":"ltr"},{"lang":"av","autonym":"авар","dir":"ltr"},{"lang":"avk","autonym":"Kotava","dir":"ltr"},{"lang":"awa","autonym":"अवधी","dir":"ltr"},{"lang":"ay","autonym":"Aymar aru","dir":"ltr"},{"lang":"azb","autonym": "تۆرکجه","dir":"rtl"},{"lang":"ba","autonym":"башҡортса","dir":"ltr"},{"lang":"ban","autonym":"Basa Bali","dir":"ltr"},{"lang":"bar","autonym":"Boarisch","dir":"ltr"},{"lang":"bbc","autonym":"Batak Toba","dir":"ltr"},{"lang":"bcl","autonym":"Bikol Central","dir":"ltr"},{"lang":"bdr","autonym":"Bajau Sama","dir":"ltr"},{"lang":"bew","autonym":"Betawi","dir":"ltr"},{"lang":"bho","autonym":"भोजपुरी","dir":"ltr"},{"lang":"bi","autonym":"Bislama","dir":"ltr"},{"lang":"bjn","autonym":"Banjar","dir":"ltr"},{"lang":"blk","autonym":"ပအိုဝ်ႏဘာႏသာႏ","dir":"ltr"},{"lang":"bm","autonym":"bamanankan","dir":"ltr"},{"lang":"bo","autonym":"བོད་ཡིག","dir":"ltr"},{"lang":"bpy","autonym":"বিষ্ণুপ্রিয়া মণিপুরী","dir":"ltr"},{"lang":"br","autonym":"brezhoneg","dir":"ltr"},{"lang":"btm","autonym":"Batak Mandailing","dir":"ltr"},{"lang":"bug","autonym":"Basa Ugi","dir":"ltr"},{"lang":"cdo","autonym": "閩東語 / Mìng-dĕ̤ng-ngṳ̄","dir":"ltr"},{"lang":"ce","autonym":"нохчийн","dir":"ltr"},{"lang":"ceb","autonym":"Cebuano","dir":"ltr"},{"lang":"ch","autonym":"Chamoru","dir":"ltr"},{"lang":"chr","autonym":"ᏣᎳᎩ","dir":"ltr"},{"lang":"ckb","autonym":"کوردی","dir":"rtl"},{"lang":"co","autonym":"corsu","dir":"ltr"},{"lang":"cr","autonym":"Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ","dir":"ltr"},{"lang":"crh","autonym":"qırımtatarca","dir":"ltr"},{"lang":"cu","autonym":"словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ","dir":"ltr"},{"lang":"dag","autonym":"dagbanli","dir":"ltr"},{"lang":"dga","autonym":"Dagaare","dir":"ltr"},{"lang":"din","autonym":"Thuɔŋjäŋ","dir":"ltr"},{"lang":"diq","autonym":"Zazaki","dir":"ltr"},{"lang":"dsb","autonym":"dolnoserbski","dir":"ltr"},{"lang":"dtp","autonym":"Kadazandusun","dir":"ltr"},{"lang":"dv","autonym":"ދިވެހިބަސް","dir":"rtl"},{"lang":"dz","autonym":"ཇོང་ཁ","dir":"ltr"},{"lang":"ee","autonym": "eʋegbe","dir":"ltr"},{"lang":"eml","autonym":"emiliàn e rumagnòl","dir":"ltr"},{"lang":"fat","autonym":"mfantse","dir":"ltr"},{"lang":"ff","autonym":"Fulfulde","dir":"ltr"},{"lang":"fj","autonym":"Na Vosa Vakaviti","dir":"ltr"},{"lang":"fo","autonym":"føroyskt","dir":"ltr"},{"lang":"fon","autonym":"fɔ̀ngbè","dir":"ltr"},{"lang":"frp","autonym":"arpetan","dir":"ltr"},{"lang":"frr","autonym":"Nordfriisk","dir":"ltr"},{"lang":"fur","autonym":"furlan","dir":"ltr"},{"lang":"fy","autonym":"Frysk","dir":"ltr"},{"lang":"gag","autonym":"Gagauz","dir":"ltr"},{"lang":"gan","autonym":"贛語","dir":"ltr"},{"lang":"gcr","autonym":"kriyòl gwiyannen","dir":"ltr"},{"lang":"glk","autonym":"گیلکی","dir":"rtl"},{"lang":"gn","autonym":"Avañe'ẽ","dir":"ltr"},{"lang":"gom","autonym":"गोंयची कोंकणी / Gõychi Konknni","dir":"ltr"},{"lang":"gor","autonym":"Bahasa Hulontalo","dir":"ltr"},{"lang":"gpe","autonym":"Ghanaian Pidgin","dir":"ltr"},{"lang":"gu","autonym": "ગુજરાતી","dir":"ltr"},{"lang":"guc","autonym":"wayuunaiki","dir":"ltr"},{"lang":"gur","autonym":"farefare","dir":"ltr"},{"lang":"guw","autonym":"gungbe","dir":"ltr"},{"lang":"gv","autonym":"Gaelg","dir":"ltr"},{"lang":"ha","autonym":"Hausa","dir":"ltr"},{"lang":"hak","autonym":"客家語 / Hak-kâ-ngî","dir":"ltr"},{"lang":"haw","autonym":"Hawaiʻi","dir":"ltr"},{"lang":"hif","autonym":"Fiji Hindi","dir":"ltr"},{"lang":"hsb","autonym":"hornjoserbsce","dir":"ltr"},{"lang":"ht","autonym":"Kreyòl ayisyen","dir":"ltr"},{"lang":"hyw","autonym":"Արեւմտահայերէն","dir":"ltr"},{"lang":"ia","autonym":"interlingua","dir":"ltr"},{"lang":"iba","autonym":"Jaku Iban","dir":"ltr"},{"lang":"ie","autonym":"Interlingue","dir":"ltr"},{"lang":"ig","autonym":"Igbo","dir":"ltr"},{"lang":"igl","autonym":"Igala","dir":"ltr"},{"lang":"ilo","autonym":"Ilokano","dir":"ltr"},{"lang":"io","autonym":"Ido","dir":"ltr"},{"lang":"iu","autonym":"ᐃᓄᒃᑎᑐᑦ / inuktitut","dir":"ltr"} ,{"lang":"jam","autonym":"Patois","dir":"ltr"},{"lang":"jv","autonym":"Jawa","dir":"ltr"},{"lang":"kaa","autonym":"Qaraqalpaqsha","dir":"ltr"},{"lang":"kab","autonym":"Taqbaylit","dir":"ltr"},{"lang":"kbd","autonym":"адыгэбзэ","dir":"ltr"},{"lang":"kbp","autonym":"Kabɩyɛ","dir":"ltr"},{"lang":"kcg","autonym":"Tyap","dir":"ltr"},{"lang":"kg","autonym":"Kongo","dir":"ltr"},{"lang":"kge","autonym":"Kumoring","dir":"ltr"},{"lang":"ki","autonym":"Gĩkũyũ","dir":"ltr"},{"lang":"kl","autonym":"kalaallisut","dir":"ltr"},{"lang":"km","autonym":"ភាសាខ្មែរ","dir":"ltr"},{"lang":"kn","autonym":"ಕನ್ನಡ","dir":"ltr"},{"lang":"koi","autonym":"перем коми","dir":"ltr"},{"lang":"krc","autonym":"къарачай-малкъар","dir":"ltr"},{"lang":"ks","autonym":"कॉशुर / کٲشُر","dir":"rtl"},{"lang":"ku","autonym":"kurdî","dir":"ltr"},{"lang":"kus","autonym":"Kʋsaal","dir":"ltr"},{"lang":"kv","autonym":"коми","dir":"ltr"},{"lang":"kw", "autonym":"kernowek","dir":"ltr"},{"lang":"lad","autonym":"Ladino","dir":"ltr"},{"lang":"lez","autonym":"лезги","dir":"ltr"},{"lang":"lg","autonym":"Luganda","dir":"ltr"},{"lang":"li","autonym":"Limburgs","dir":"ltr"},{"lang":"lij","autonym":"Ligure","dir":"ltr"},{"lang":"lld","autonym":"Ladin","dir":"ltr"},{"lang":"lmo","autonym":"lombard","dir":"ltr"},{"lang":"ln","autonym":"lingála","dir":"ltr"},{"lang":"lo","autonym":"ລາວ","dir":"ltr"},{"lang":"ltg","autonym":"latgaļu","dir":"ltr"},{"lang":"mad","autonym":"Madhurâ","dir":"ltr"},{"lang":"mai","autonym":"मैथिली","dir":"ltr"},{"lang":"map-bms","autonym":"Basa Banyumasan","dir":"ltr"},{"lang":"mdf","autonym":"мокшень","dir":"ltr"},{"lang":"mg","autonym":"Malagasy","dir":"ltr"},{"lang":"mhr","autonym":"олык марий","dir":"ltr"},{"lang":"mi","autonym":"Māori","dir":"ltr"},{"lang":"min","autonym":"Minangkabau","dir":"ltr"},{"lang":"mn","autonym":"монгол","dir":"ltr"},{"lang":"mni","autonym": "ꯃꯤꯇꯩ ꯂꯣꯟ","dir":"ltr"},{"lang":"mos","autonym":"moore","dir":"ltr"},{"lang":"mr","autonym":"मराठी","dir":"ltr"},{"lang":"mrj","autonym":"кырык мары","dir":"ltr"},{"lang":"mt","autonym":"Malti","dir":"ltr"},{"lang":"mwl","autonym":"Mirandés","dir":"ltr"},{"lang":"my","autonym":"မြန်မာဘာသာ","dir":"ltr"},{"lang":"myv","autonym":"эрзянь","dir":"ltr"},{"lang":"mzn","autonym":"مازِرونی","dir":"rtl"},{"lang":"nah","autonym":"Nāhuatl","dir":"ltr"},{"lang":"nan","autonym":"閩南語 / Bân-lâm-gú","dir":"ltr"},{"lang":"nap","autonym":"Napulitano","dir":"ltr"},{"lang":"nb","autonym":"norsk bokmål","dir":"ltr"},{"lang":"nds","autonym":"Plattdüütsch","dir":"ltr"},{"lang":"nds-nl","autonym":"Nedersaksies","dir":"ltr"},{"lang":"ne","autonym":"नेपाली","dir":"ltr"},{"lang":"new","autonym":"नेपाल भाषा","dir":"ltr"},{"lang":"nia","autonym":"Li Niha","dir":"ltr"},{"lang":"nqo","autonym":"ߒߞߏ","dir" :"rtl"},{"lang":"nr","autonym":"isiNdebele seSewula","dir":"ltr"},{"lang":"nso","autonym":"Sesotho sa Leboa","dir":"ltr"},{"lang":"ny","autonym":"Chi-Chewa","dir":"ltr"},{"lang":"om","autonym":"Oromoo","dir":"ltr"},{"lang":"or","autonym":"ଓଡ଼ିଆ","dir":"ltr"},{"lang":"pag","autonym":"Pangasinan","dir":"ltr"},{"lang":"pam","autonym":"Kapampangan","dir":"ltr"},{"lang":"pap","autonym":"Papiamentu","dir":"ltr"},{"lang":"pcd","autonym":"Picard","dir":"ltr"},{"lang":"pcm","autonym":"Naijá","dir":"ltr"},{"lang":"pdc","autonym":"Deitsch","dir":"ltr"},{"lang":"pnb","autonym":"پنجابی","dir":"rtl"},{"lang":"ps","autonym":"پښتو","dir":"rtl"},{"lang":"pwn","autonym":"pinayuanan","dir":"ltr"},{"lang":"qu","autonym":"Runa Simi","dir":"ltr"},{"lang":"rm","autonym":"rumantsch","dir":"ltr"},{"lang":"rn","autonym":"ikirundi","dir":"ltr"},{"lang":"rsk","autonym":"руски","dir":"ltr"},{"lang":"rup","autonym":"armãneashti","dir":"ltr"},{"lang":"rw","autonym":"Ikinyarwanda","dir": "ltr"},{"lang":"sa","autonym":"संस्कृतम्","dir":"ltr"},{"lang":"sah","autonym":"саха тыла","dir":"ltr"},{"lang":"sat","autonym":"ᱥᱟᱱᱛᱟᱲᱤ","dir":"ltr"},{"lang":"sc","autonym":"sardu","dir":"ltr"},{"lang":"sco","autonym":"Scots","dir":"ltr"},{"lang":"sd","autonym":"سنڌي","dir":"rtl"},{"lang":"se","autonym":"davvisámegiella","dir":"ltr"},{"lang":"sg","autonym":"Sängö","dir":"ltr"},{"lang":"sgs","autonym":"žemaitėška","dir":"ltr"},{"lang":"shi","autonym":"Taclḥit","dir":"ltr"},{"lang":"shn","autonym":"ၽႃႇသႃႇတႆး ","dir":"ltr"},{"lang":"si","autonym":"සිංහල","dir":"ltr"},{"lang":"skr","autonym":"سرائیکی","dir":"rtl"},{"lang":"sm","autonym":"Gagana Samoa","dir":"ltr"},{"lang":"smn","autonym":"anarâškielâ","dir":"ltr"},{"lang":"sn","autonym":"chiShona","dir":"ltr"},{"lang":"so","autonym":"Soomaaliga","dir":"ltr"},{"lang":"srn","autonym":"Sranantongo","dir":"ltr"},{"lang":"ss","autonym":"SiSwati","dir": "ltr"},{"lang":"st","autonym":"Sesotho","dir":"ltr"},{"lang":"stq","autonym":"Seeltersk","dir":"ltr"},{"lang":"su","autonym":"Sunda","dir":"ltr"},{"lang":"sw","autonym":"Kiswahili","dir":"ltr"},{"lang":"szl","autonym":"ślůnski","dir":"ltr"},{"lang":"tay","autonym":"Tayal","dir":"ltr"},{"lang":"tcy","autonym":"ತುಳು","dir":"ltr"},{"lang":"tdd","autonym":"ᥖᥭᥰ ᥖᥬᥲ ᥑᥨᥒᥰ","dir":"ltr"},{"lang":"tet","autonym":"tetun","dir":"ltr"},{"lang":"tg","autonym":"тоҷикӣ","dir":"ltr"},{"lang":"ti","autonym":"ትግርኛ","dir":"ltr"},{"lang":"tk","autonym":"Türkmençe","dir":"ltr"},{"lang":"tly","autonym":"tolışi","dir":"ltr"},{"lang":"tn","autonym":"Setswana","dir":"ltr"},{"lang":"to","autonym":"lea faka-Tonga","dir":"ltr"},{"lang":"tpi","autonym":"Tok Pisin","dir":"ltr"},{"lang":"trv","autonym":"Seediq","dir":"ltr"},{"lang":"ts","autonym":"Xitsonga","dir":"ltr"},{"lang":"tum","autonym":"chiTumbuka","dir":"ltr"},{"lang":"tw","autonym":"Twi","dir":"ltr"},{"lang": "ty","autonym":"reo tahiti","dir":"ltr"},{"lang":"tyv","autonym":"тыва дыл","dir":"ltr"},{"lang":"udm","autonym":"удмурт","dir":"ltr"},{"lang":"ve","autonym":"Tshivenda","dir":"ltr"},{"lang":"vec","autonym":"vèneto","dir":"ltr"},{"lang":"vep","autonym":"vepsän kel’","dir":"ltr"},{"lang":"vls","autonym":"West-Vlams","dir":"ltr"},{"lang":"vo","autonym":"Volapük","dir":"ltr"},{"lang":"vro","autonym":"võro","dir":"ltr"},{"lang":"wa","autonym":"walon","dir":"ltr"},{"lang":"wo","autonym":"Wolof","dir":"ltr"},{"lang":"xal","autonym":"хальмг","dir":"ltr"},{"lang":"xh","autonym":"isiXhosa","dir":"ltr"},{"lang":"xmf","autonym":"მარგალური","dir":"ltr"},{"lang":"yo","autonym":"Yorùbá","dir":"ltr"},{"lang":"yue","autonym":"粵語","dir":"ltr"},{"lang":"za","autonym":"Vahcuengh","dir":"ltr"},{"lang":"zgh","autonym":"ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ","dir":"ltr"},{"lang":"zu","autonym":"isiZulu","dir":"ltr"}], "wgSectionTranslationTargetLanguages":["ace","ady","alt","am","ami","an","ang","ann","anp","ar","ary","arz","as","ast","av","avk","awa","ay","az","azb","ba","ban","bar","bbc","bcl","bdr","be","bew","bg","bho","bi","bjn","blk","bm","bn","bo","bpy","br","bs","btm","bug","ca","cdo","ce","ceb","ch","chr","ckb","co","cr","crh","cs","cu","cy","da","dag","de","dga","din","diq","dsb","dtp","dv","dz","ee","el","eml","eo","es","et","eu","fa","fat","ff","fi","fj","fo","fon","fr","frp","frr","fur","fy","gag","gan","gcr","gl","glk","gn","gom","gor","gpe","gu","guc","gur","guw","gv","ha","hak","haw","he","hi","hif","hr","hsb","ht","hu","hy","hyw","ia","iba","ie","ig","igl","ilo","io","is","it","iu","ja","jam","jv","ka","kaa","kab","kbd","kbp","kcg","kg","kge","ki","kk","kl","km","kn","ko","koi","krc","ks","ku","kus","kv","kw","ky","lad","lb","lez","lg","li","lij","lld","lmo","ln","lo","lt","ltg","lv","mad","mai","map-bms","mdf","mg","mhr","mi","min","mk","ml","mn","mni","mnw","mos","mr","mrj","ms", "mt","mwl","my","myv","mzn","nah","nan","nap","nb","nds","nds-nl","ne","new","nia","nl","nn","nqo","nr","nso","ny","oc","om","or","os","pa","pag","pam","pap","pcd","pcm","pdc","pl","pms","pnb","ps","pt","pwn","qu","rm","rn","ro","rsk","rue","rup","rw","sa","sah","sat","sc","scn","sco","sd","se","sg","sgs","sh","shi","shn","si","sk","skr","sl","sm","smn","sn","so","sq","sr","srn","ss","st","stq","su","sv","sw","szl","ta","tay","tcy","tdd","te","tet","tg","th","ti","tk","tl","tly","tn","to","tpi","tr","trv","ts","tt","tum","tw","ty","tyv","udm","ur","uz","ve","vec","vep","vi","vls","vo","vro","wa","war","wo","wuu","xal","xh","xmf","yi","yo","yue","za","zgh","zh","zu"],"isLanguageSearcherCXEntrypointEnabled":true,"mintEntrypointLanguages":["ace","ast","azb","bcl","bjn","bh","crh","ff","fon","ig","is","ki","ks","lmo","min","sat","ss","tn","vec"],"wgWikibaseItemId":"Q83219","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform", "platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgMinervaPermissions":{"watchable":true,"watch":false},"wgMinervaFeatures":{"beta":false,"donate":true,"mobileOptionsLink":true,"categories":false,"pageIssues":true,"talkAtTop":false,"historyInPageActions":false,"overflowSubmenu":false,"tabsOnSpecials":true,"personalMenu":false,"mainMenuExpanded":false,"echo":true,"nightMode":true},"wgMinervaDownloadNamespaces":[0],"wgSiteNoticeId":"2.9"};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.minerva.styles":"ready","skins.minerva.content.styles.images":"ready","mediawiki.hlist":"ready","skins.minerva.codex.styles":"ready","skins.minerva.icons":"ready","ext.wikimediamessages.styles" :"ready","mobile.init.styles":"ready","ext.relatedArticles.styles":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready","ext.dismissableSiteNotice.styles":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","skins.minerva.scripts","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.OpenStreetMap","ext.urlShortener.toolbar","ext.centralauth.centralautologin","ext.popups","mobile.init","ext.echo.centralauth","ext.relatedArticles.readMore.bootstrap","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.cx.eventlogging.campaigns","ext.cx.entrypoints.mffrequentlanguages","ext.cx.entrypoints.languagesearcher.init","mw.externalguidance.init","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking","ext.dismissableSiteNotice"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=et&modules=ext.cite.styles%7Cext.dismissableSiteNotice.styles%7Cext.math.styles%7Cext.relatedArticles.styles%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.hlist%7Cmobile.init.styles%7Cskins.minerva.codex.styles%7Cskins.minerva.content.styles.images%7Cskins.minerva.icons%2Cstyles%7Cwikibase.client.init&only=styles&skin=minerva"> <script async src="/w/load.php?lang=et&modules=startup&only=scripts&raw=1&skin=minerva"></script> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="theme-color" content="#eaecf0"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/1200px-Kepler_laws_diagram.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1287"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/800px-Kepler_laws_diagram.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="858"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/640px-Kepler_laws_diagram.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="686"> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes, minimum-scale=0.25, maximum-scale=5.0"> <meta property="og:title" content="Kepleri seadused – Vikipeedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="manifest" href="/w/api.php?action=webapp-manifest"> <link rel="alternate" type="application/x-wiki" title="Muuda" href="/w/index.php?title=Kepleri_seadused&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Vikipeedia (et)"> <link rel="EditURI" type="application/rsd+xml" href="//et.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://et.wikipedia.org/wiki/Kepleri_seadused"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.et"> <link rel="dns-prefetch" href="//meta.wikimedia.org"> <link rel="dns-prefetch" href="//login.wikimedia.org"> <meta http-equiv="X-Translated-By" content="Google"> <meta http-equiv="X-Translated-To" content="en"> <script type="text/javascript" src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.WgGHrg8C9fE.O/am=DgY/d=1/rs=AN8SPfpNfjzpGCAsUUJ5X-GCaxSfec_Eng/m=corsproxy" data-sourceurl="https://et.m.wikipedia.org/wiki/Kepleri_seadused"></script> <link href="https://fonts.googleapis.com/css2?family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20..48,100..700,0..1,-50..200" rel="stylesheet"> <script type="text/javascript" src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.WgGHrg8C9fE.O/am=DgY/d=1/exm=corsproxy/ed=1/rs=AN8SPfpNfjzpGCAsUUJ5X-GCaxSfec_Eng/m=phishing_protection" data-phishing-protection-enabled="false" data-forms-warning-enabled="true" data-source-url="https://et.m.wikipedia.org/wiki/Kepleri_seadused"></script> <meta name="robots" content="none"> </head> <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Kepleri_seadused rootpage-Kepleri_seadused stable issues-group-B skin-minerva action-view skin--responsive mw-mf-amc-disabled mw-mf"> <script type="text/javascript" src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.WgGHrg8C9fE.O/am=DgY/d=1/exm=corsproxy,phishing_protection/ed=1/rs=AN8SPfpNfjzpGCAsUUJ5X-GCaxSfec_Eng/m=navigationui" data-environment="prod" data-proxy-url="https://et-m-wikipedia-org.translate.goog" data-proxy-full-url="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-source-url="https://et.m.wikipedia.org/wiki/Kepleri_seadused" data-source-language="auto" data-target-language="en" data-display-language="en-GB" data-detected-source-language="et" data-is-source-untranslated="false" data-source-untranslated-url="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.m.wikipedia.org/wiki/Kepleri_seadused&anno=2" data-client="tr"></script> <div id="mw-mf-viewport"> <div id="mw-mf-page-center"><a class="mw-mf-page-center__mask" href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#"></a> <header class="header-container header-chrome"> <div class="minerva-header"> <nav class="navigation-drawer toggle-list view-border-box"><input type="checkbox" id="main-menu-input" class="toggle-list__checkbox" role="button" aria-haspopup="true" aria-expanded="false" aria-labelledby="mw-mf-main-menu-button"> <label role="button" for="main-menu-input" id="mw-mf-main-menu-button" aria-hidden="true" data-event-name="ui.mainmenu" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet toggle-list__toggle"> </label> <div id="mw-mf-page-left" class="menu view-border-box"> <ul id="p-navigation" class="toggle-list__list"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--home" href="https://et-m-wikipedia-org.translate.goog/wiki/Vikipeedia:Esileht?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-mw="interface"> Esileht </a></li> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--random" href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Juhuslik_artikkel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-mw="interface"> Juhuslik </a></li> <li class="toggle-list-item skin-minerva-list-item-jsonly"><a class="toggle-list-item__anchor menu__item--nearby" href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:L%C3%A4hikond?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.nearby" data-mw="interface"> Lähikond </a></li> </ul> <ul id="p-personal" class="toggle-list__list"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--login" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Eri:Sisselogimine&returnto=Kepleri+seadused&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.login" data-mw="interface"> Logi sisse </a></li> </ul> <ul id="pt-preferences" class="toggle-list__list"> <li class="toggle-list-item skin-minerva-list-item-jsonly"><a class="toggle-list-item__anchor menu__item--settings" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Eri:Mobiili_suvandid&returnto=Kepleri+seadused&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.settings" data-mw="interface"> Sätted </a></li> </ul> <ul id="p-donation" class="toggle-list__list"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--donate" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source%3Ddonate%26utm_medium%3Dsidebar%26utm_campaign%3DC13_et.wikipedia.org%26uselang%3Det%26utm_key%3Dminerva" data-event-name="menu.donate" data-mw="interface"> Annetused </a></li> </ul> <ul class="hlist"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--about" href="https://et-m-wikipedia-org.translate.goog/wiki/Vikipeedia:Tiitelandmed?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-mw="interface"> Tiitelandmed </a></li> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--disclaimers" href="https://et-m-wikipedia-org.translate.goog/wiki/Vikipeedia:%C3%9Cldine_lahti%C3%BCtlus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-mw="interface"> Lahtiütlused </a></li> </ul> </div><label class="main-menu-mask" for="main-menu-input"></label> </nav> <div class="branding-box"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Vikipeedia:Esileht?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-et.svg" alt="Vikipeedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"> </a> </div> <form action="/w/index.php" method="get" class="minerva-search-form"> <div class="search-box"><input type="hidden" name="title" value="Eri:Otsimine"> <input class="search skin-minerva-search-trigger" id="searchInput" type="search" name="search" placeholder="Otsi Vikipeediast" aria-label="Otsi Vikipeediast" autocapitalize="sentences" title="Otsi vikist [f]" accesskey="f"> </div><button id="searchIcon" class="cdx-button cdx-button--size-large cdx-button--icon-only cdx-button--weight-quiet skin-minerva-search-trigger"> Otsi </button> </form> <nav class="minerva-user-navigation" aria-label="Kasutaja navigatsioon"> </nav> </div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Kepleri seadused</h1> <div class="tagline"></div> </div> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Keel" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Keel </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a role="button" id="ca-watch" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Eri:Sisselogimine&returnto=Kepleri+seadused&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Jälgi </a></li> <li id="page-actions-edit" class="page-actions-menu__list-item"><a role="button" id="ca-edit" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Muuda </a></li> </ul> </nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="et" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <table class="metadata plainlinks ambox ambox-content" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> See artikkel ootab <a href="https://et-m-wikipedia-org.translate.goog/wiki/Keeletoimetamine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Keeletoimetamine">keeletoimetamist</a>. Kui oskad, siis palun aita <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/w/index.php?title%3DKepleri_seadused%26action%3Dedit">artiklit keeleliselt parandada</a>. (<a href="https://et-m-wikipedia-org.translate.goog/wiki/Juhend:Toimetusm%C3%A4rkuste_eemaldamine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Juhend:Toimetusmärkuste eemaldamine">Kuidas ja millal see märkus eemaldada?</a>) </div></td> </tr> </tbody> </table> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Johannes_Kepler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Johannes Kepler">Kepleri</a> seadused kirjeldavad <a href="https://et-m-wikipedia-org.translate.goog/wiki/Planeet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Planeet">planeetide</a> liikumist ümber <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päikese</a>. Kolm Kepleri seadust on järgmised: <ol> <li>Iga planeedi <a href="https://et-m-wikipedia-org.translate.goog/wiki/Orbiit?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orbiit">orbiit</a> on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ellips_(geomeetria)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ellips (geomeetria)">ellips</a>, mille ühes <a href="https://et-m-wikipedia-org.translate.goog/wiki/Fookus_(geomeetria)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fookus (geomeetria)">fookuses</a> on <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päike</a>.</li> <li>Planeedi <a href="https://et-m-wikipedia-org.translate.goog/wiki/Raadiusvektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Raadiusvektor">raadiusvektor</a> katab võrdsete ajavahemike jooksul võrdsed pindalad.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Wolfram2nd-1">[1]</a></li> <li>Planeetide <a href="https://et-m-wikipedia-org.translate.goog/wiki/Tiirlemisperiood?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tiirlemisperiood">tiirlemisperioodide</a> ruudud suhtuvad nagu nende orbiitide pikemate <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Pooltelg&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Pooltelg (pole veel kirjutatud)">pooltelgede</a> kuubid.</li> </ol> <figure typeof="mw:File/Thumb"> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Fail:Kepler_laws_diagram.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/300px-Kepler_laws_diagram.svg.png" decoding="async" width="300" height="322" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/450px-Kepler_laws_diagram.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/600px-Kepler_laws_diagram.svg.png 2x" data-file-width="400" data-file-height="429"></a> <figcaption> Kepleri kolme seaduse illustratsioon. (1) Orbiidid on ellipsid, kus esimese planeedi fookusteks on ƒ1 ja ƒ2 ning teise planeedi fookusteks ƒ1 ja ƒ3. Päike asub fookuses ƒ1. (2) Kaks tumedamat sektorit A1 ja A2 on võrdsete pindaladega. Aeg, mis kulub planeedil 1, et katta sektorit A1, on võrdne ajaga, mis kulub, et katta sektor A2. (3) Orbitaalperioodide suhe planeedi 1 ja planeedi 2 jaoks on a13/2 : a23/2. </figcaption> </figure> Seaduste tuletamisel ei arvestata planeetidevahelise interaktsiooniga ja eeldatakse, et piirjuhul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m_{planeet}}{m_{P{\ddot {a}}ike}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> <mi> l </mi> <mi> a </mi> <mi> n </mi> <mi> e </mi> <mi> e </mi> <mi> t </mi> </mrow> </msub> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo> ¨ </mo> </mover> </mrow> </mrow> <mi> i </mi> <mi> k </mi> <mi> e </mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {m_{planeet}}{m_{P{\ddot {a}}ike}}}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359a072209f5899268544a4102a1b99d6314f7f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.408ex; height:5.676ex;" alt="{\displaystyle {\frac {m_{planeet}}{m_{P{\ddot {a}}ike}}}}"> → 0. Kepleri seadused moodustavad hea mudeli, millega arvutada nende planeetide orbiite, mis ei erine liialt nendest piirangutest. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="et" dir="ltr"> <h2 id="mw-toc-heading">Sisukord</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ajalugu">1 Ajalugu</a></li> <li class="toclevel-1 tocsection-2"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Esimene_seadus">2 Esimene seadus</a></li> <li class="toclevel-1 tocsection-3"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Teine_seadus">3 Teine seadus</a></li> <li class="toclevel-1 tocsection-4"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kolmas_seadus">4 Kolmas seadus</a></li> <li class="toclevel-1 tocsection-5"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%C3%9Cldistus">5 Üldistus</a></li> <li class="toclevel-1 tocsection-6"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Nullekstsentrilisus">6 Nullekstsentrilisus</a></li> <li class="toclevel-1 tocsection-7"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Vaata_ka">7 Vaata ka</a></li> <li class="toclevel-1 tocsection-8"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Viited">8 Viited</a></li> <li class="toclevel-1 tocsection-9"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kirjandus">9 Kirjandus</a></li> <li class="toclevel-1 tocsection-10"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#V%C3%A4lislingid">10 Välislingid</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Ajalugu">Ajalugu</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Ajalugu"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Johannes_Kepler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Johannes Kepler">Johannes Kepler</a> avastas kaks esimest seadust, analüüsides <a href="https://et-m-wikipedia-org.translate.goog/wiki/Tycho_Brahe?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tycho Brahe">Tycho Brahe</a><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Holton-2">[2]</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Astronoomia?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Astronoomia">astronoomilisi</a> vaatlusi. Oma töö avaldas ta 1609. aastal. Kolmanda seaduse avastas Kepler aastaid hiljem ja avaldas 1619. aastal. 17. sajandi algul olid Kepleri seadused radikaalsed: valdavalt usuti, et <a href="https://et-m-wikipedia-org.translate.goog/wiki/Taevakeha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Taevakeha">taevakehade</a> orbiidid on ideaalsed <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ring?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ring">ringid</a>. Orbiitide elliptilisus ei olnud vaatlusel tihti arusaadav ja seega on lihtne pidada neid ringikujulisteks. Esmalt arvutas Kepler planeedi <a href="https://et-m-wikipedia-org.translate.goog/wiki/Marss?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Marss">Marss</a> orbiiti, mis vihjas elliptilisele kujule. Sellest järeldas ta, et ka teistel taevakehadel, sealhulgas ka neil, mis asuvad Päikesest kaugemal, peavad olema elliptilised orbiidid. Kepleri seadused esitasid tõsise väljakutse senisele üldtunnustatud geotsentrilisele <a href="https://et-m-wikipedia-org.translate.goog/wiki/Aristoteles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aristoteles">Aristotelese</a> ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ptolemaios?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ptolemaios">Ptolemaiose</a> maailmasüsteemile ning toetasid üldjoontes <a href="https://et-m-wikipedia-org.translate.goog/wiki/Miko%C5%82aj_Kopernik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mikołaj Kopernik">Mikołaj Koperniku</a> heliotsentrilist teooriat, ehkki Koperinku teoorias olid planeetide orbiidid ikkagi ringikujulised.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Holton-2">[2]</a> Peaaegu sajand hiljem tõestas <a href="https://et-m-wikipedia-org.translate.goog/wiki/Isaac_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Isaac Newton">Isaac Newton</a>, et Kepleri seadused kehtivad kindlatel ideaalsetel tingimustel, mis piisavalt hea lähendusega esinevad <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ikeses%C3%BCsteem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päikesesüsteem">Päikesesüsteemis</a>, ning tulenevad Newtoni kolmest seadusest ja gravitatsiooniseadusest.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-smith-sep-3">[3]</a><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-newt-p-4">[4]</a> Kuna planeetidel on mass ja massist tulenev liikumise <a href="https://et-m-wikipedia-org.translate.goog/wiki/Perturbatsioon?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Perturbatsioon">perturbatsioon</a>, kehtivad Kepleri seadused ainult ligikaudselt ja ei kirjelda täpselt <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ikeses%C3%BCsteem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päikesesüsteem">Päikesesüsteemi</a> sisemist liikumist.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-smith-sep-3">[3]</a><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-plummr-5">[5]</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Voltaire?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Voltaire">Voltaire</a>'i 1738. aastal avaldatud "Eléments de la philosophie de Newton" oli esimene teos, kus nimetati Kepleri seadusi seadusteks.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Wilson_1994-6">[6]</a> Koos Newtoni matemaatiliste teooriatega moodustavad Kepleri seadused osa tänapäeva <a href="https://et-m-wikipedia-org.translate.goog/wiki/Astronoomia?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Astronoomia">astronoomiast</a> ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/F%C3%BC%C3%BCsika?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Füüsika">füüsikast</a>.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-smith-sep-3">[3]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Esimene_seadus">Esimene seadus</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Esimene seadus"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Fail:Kepler-first-law-math.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler-first-law-math.svg/220px-Kepler-first-law-math.svg.png" decoding="async" width="220" height="153" class="mw-file-element" data-file-width="575" data-file-height="400"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 153px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler-first-law-math.svg/220px-Kepler-first-law-math.svg.png" data-width="220" data-height="153" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler-first-law-math.svg/330px-Kepler-first-law-math.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler-first-law-math.svg/440px-Kepler-first-law-math.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> Kepleri esimest seadust kujutav joonis, kus Päike (M) asub ellipsi, mis on planeedi (m) orbiidiks, ühes fookuses </figcaption> </figure> <dl> <dd> Iga planeedi orbiit on ellips, mille ühes fookuses on Päike. </dd> </dl> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ellips?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellips">Ellips</a> on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Matemaatika?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matemaatika">matemaatiline</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kujund?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kujund">kujund</a>, mis meenutab kujult välja venitatud ringjoont. <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päike</a> ei asu ellipsi keskpunktis, vaid ühes fookustest. Ringjoon on ellipsi erijuht, kui mõlemad fookused asuvad ühessamas punktis, mis langeb kokku ellipsi keskpunktiga. Ellipsi kuju kirjeldatakse parameetriga, mida kutsutakse ekstsentrilisuseks. Ekstsentrilisus on parameeter, mis võib muutuda nullist (tavaline ringjoon) üheni (ellips, mis on fookuste kauguse tõttu nii välja venitatud, et moodustab <a href="https://et-m-wikipedia-org.translate.goog/wiki/Parabool?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parabool">parabooli</a>). <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ikeses%C3%BCsteem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päikesesüsteem">Päikesesüsteemi</a> planeetide orbiitide ekstsentrilisus on väikseim <a href="https://et-m-wikipedia-org.translate.goog/wiki/Veenus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Veenus">Veenusel</a> (0,007)<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-veenus-7">[7]</a><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-merkuur-8">[8]</a>. Ometi ei erine isegi <a href="https://et-m-wikipedia-org.translate.goog/wiki/Merkuur?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Merkuur">Merkuuri</a> orbiit kuigi palju ringjoonest. Samas on avastatud taevakehi, mille ekstsentrilisus on väga suur. Nende seas on palju <a href="https://et-m-wikipedia-org.translate.goog/wiki/Komeet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Komeet">komeete</a> ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/Asteroid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Asteroid">asteroide</a>. On vaadeldud ka taevakehasid, mille orbiit on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Parabool?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parabool">paraboolne</a> või <a href="https://et-m-wikipedia-org.translate.goog/wiki/H%C3%BCperbool?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hüperbool">hüperboolne</a>.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-nasa-9">[9]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Fail:Ellipse_latus_rectum.PNG?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Ellipse_latus_rectum.PNG/220px-Ellipse_latus_rectum.PNG" decoding="async" width="220" height="219" class="mw-file-element" data-file-width="786" data-file-height="781"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 219px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Ellipse_latus_rectum.PNG/220px-Ellipse_latus_rectum.PNG" data-width="220" data-height="219" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Ellipse_latus_rectum.PNG/330px-Ellipse_latus_rectum.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Ellipse_latus_rectum.PNG/440px-Ellipse_latus_rectum.PNG 2x" data-class="mw-file-element"> </a> <figcaption></figcaption> </figure> Ellipsi võrrandi kuju <a href="https://et-m-wikipedia-org.translate.goog/wiki/Polaarkoordinaadid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polaarkoordinaadid">polaarkoordinaatides</a> on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> p </mi> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ε </mi> <mspace width="thinmathspace"></mspace> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2607beca148191eeb03bfbff7c383622c182a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.079ex; height:5.176ex;" alt="{\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},}"> </noscript><span class="lazy-image-placeholder" style="width: 16.079ex;height: 5.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2607beca148191eeb03bfbff7c383622c182a0c" data-alt="{\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus r ja θ on ellipsi polaarkoordinaadid, p on <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Fokaalparameeter&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Fokaalparameeter (pole veel kirjutatud)">fokaalparameeter</a> ja ε on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Orbiidi_ekstsentrilisus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orbiidi ekstsentrilisus">orbiidi ekstsentrilisus</a>. Kui vaatame ellipsit planeedi orbiidina kujutab r planeedi kaugust Päikesest ja θ on nurk planeedi hetkeasukoha ja Päikesele lähima asukoha vahel. Olukorda, kus θ = 0° nimetatakse <a href="https://et-m-wikipedia-org.translate.goog/wiki/Periheel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Periheel">periheeliks</a>, siis on planeedi kaugus <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=P%C3%A4ikese&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Päikese (pole veel kirjutatud)">Päikesest</a> minimaalne. <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {min} }={\frac {p}{1+\varepsilon }}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> n </mi> </mrow> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> p </mi> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ε </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{\mathrm {min} }={\frac {p}{1+\varepsilon }}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2e45f9daedd34f18619ca93a8bdc1d60662cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.689ex; height:5.009ex;" alt="{\displaystyle r_{\mathrm {min} }={\frac {p}{1+\varepsilon }}.}"> </noscript><span class="lazy-image-placeholder" style="width: 13.689ex;height: 5.009ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2e45f9daedd34f18619ca93a8bdc1d60662cf6" data-alt="{\displaystyle r_{\mathrm {min} }={\frac {p}{1+\varepsilon }}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Kui θ = 90° või θ = 270°, siis kaugus on p. Kui θ = 180°, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Afeel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Afeel">afeel</a>, on kaugus maksimaalne. <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {max} }={\frac {p}{1-\varepsilon }}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> x </mi> </mrow> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> p </mi> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> ε </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{\mathrm {max} }={\frac {p}{1-\varepsilon }}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5557c621aa4890b6a0db86d92c7d2dad37062a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.008ex; height:5.009ex;" alt="{\displaystyle r_{\mathrm {max} }={\frac {p}{1-\varepsilon }}.}"> </noscript><span class="lazy-image-placeholder" style="width: 14.008ex;height: 5.009ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5557c621aa4890b6a0db86d92c7d2dad37062a5" data-alt="{\displaystyle r_{\mathrm {max} }={\frac {p}{1-\varepsilon }}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Pikem <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Pooltelg&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Pooltelg (pole veel kirjutatud)">pooltelg</a> a on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Aritmeetiline_keskmine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aritmeetiline keskmine">aritmeetiline keskmine</a> rmin ja rmax vahel: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,r_{\max }-a=a-r_{\min },}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace"></mspace> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> max </mo> </mrow> </msub> <mo> − </mo> <mi> a </mi> <mo> = </mo> <mi> a </mi> <mo> − </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> min </mo> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \,r_{\max }-a=a-r_{\min },} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed8c276c7a2c7040d46995b95bb18a8affc935c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.634ex; height:2.343ex;" alt="{\displaystyle \,r_{\max }-a=a-r_{\min },}"> </noscript><span class="lazy-image-placeholder" style="width: 20.634ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed8c276c7a2c7040d46995b95bb18a8affc935c" data-alt="{\displaystyle \,r_{\max }-a=a-r_{\min },}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> seega <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {p}{1-\varepsilon ^{2}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> p </mi> <mrow> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a={\frac {p}{1-\varepsilon ^{2}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b355496bfc6532aa78d07f0628f43a75c29d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.952ex; height:5.343ex;" alt="{\displaystyle a={\frac {p}{1-\varepsilon ^{2}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 11.952ex;height: 5.343ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b355496bfc6532aa78d07f0628f43a75c29d80" data-alt="{\displaystyle a={\frac {p}{1-\varepsilon ^{2}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Lühem <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Pooltelg&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Pooltelg (pole veel kirjutatud)">pooltelg</a> b on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Geomeetriline_keskmine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geomeetriline keskmine">geomeetriline keskmine</a> rmin ja rmax vahel: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r_{\max }}{b}}={\frac {b}{r_{\min }}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> max </mo> </mrow> </msub> <mi> b </mi> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> b </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> min </mo> </mrow> </msub> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {r_{\max }}{b}}={\frac {b}{r_{\min }}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a916ef206123978a5ab60a20d934e4f153753e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.778ex; height:5.676ex;" alt="{\displaystyle {\frac {r_{\max }}{b}}={\frac {b}{r_{\min }}},}"> </noscript><span class="lazy-image-placeholder" style="width: 13.778ex;height: 5.676ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a916ef206123978a5ab60a20d934e4f153753e3" data-alt="{\displaystyle {\frac {r_{\max }}{b}}={\frac {b}{r_{\min }}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> seega <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> b </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> p </mi> <msqrt> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle b={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914cb27ec9f7dc1f3421949bad0cbbfd02b2a953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.043ex; height:6.176ex;" alt="{\displaystyle b={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 14.043ex;height: 6.176ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914cb27ec9f7dc1f3421949bad0cbbfd02b2a953" data-alt="{\displaystyle b={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Fokaalparameeter p on <a href="https://et-m-wikipedia-org.translate.goog/wiki/Harmooniline_keskmine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Harmooniline keskmine">harmooniline keskmine</a> rmin ja rmax vahel: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{r_{\min }}}-{\frac {1}{p}}={\frac {1}{p}}-{\frac {1}{r_{\max }}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> min </mo> </mrow> </msub> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> p </mi> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> p </mi> </mfrac> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix"> max </mo> </mrow> </msub> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {1}{r_{\min }}}-{\frac {1}{p}}={\frac {1}{p}}-{\frac {1}{r_{\max }}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234f57bccd1684ea33ad2b6133f9b506e217be3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.47ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{r_{\min }}}-{\frac {1}{p}}={\frac {1}{p}}-{\frac {1}{r_{\max }}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 23.47ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234f57bccd1684ea33ad2b6133f9b506e217be3e" data-alt="{\displaystyle {\frac {1}{r_{\min }}}-{\frac {1}{p}}={\frac {1}{p}}-{\frac {1}{r_{\max }}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ekstsentrilisus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ekstsentrilisus">Ekstsentrilisus</a> ε on <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Variatsioonikoefitsient&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Variatsioonikoefitsient (pole veel kirjutatud)">variatsioonikoefitsient</a> rmin ja rmax vahel: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {r_{\mathrm {max} }-r_{\mathrm {min} }}{r_{\mathrm {max} }+r_{\mathrm {min} }}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ε </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> x </mi> </mrow> </mrow> </msub> <mo> − </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> n </mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> x </mi> </mrow> </mrow> </msub> <mo> + </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> n </mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \varepsilon ={\frac {r_{\mathrm {max} }-r_{\mathrm {min} }}{r_{\mathrm {max} }+r_{\mathrm {min} }}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3f8b1bc0bd90b1e1c36d87f190db9d5d56f27e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.866ex; height:5.343ex;" alt="{\displaystyle \varepsilon ={\frac {r_{\mathrm {max} }-r_{\mathrm {min} }}{r_{\mathrm {max} }+r_{\mathrm {min} }}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 16.866ex;height: 5.343ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3f8b1bc0bd90b1e1c36d87f190db9d5d56f27e" data-alt="{\displaystyle \varepsilon ={\frac {r_{\mathrm {max} }-r_{\mathrm {min} }}{r_{\mathrm {max} }+r_{\mathrm {min} }}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ellips?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ellips">Ellipsi</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Pindala?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pindala">pindala</a> on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\pi ab\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo> = </mo> <mi> π </mi> <mi> a </mi> <mi> b </mi> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S=\pi ab\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459e899f133ee67a1ff01426b2594a588c04a2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.191ex; height:2.176ex;" alt="{\displaystyle S=\pi ab\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 9.191ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459e899f133ee67a1ff01426b2594a588c04a2ae" data-alt="{\displaystyle S=\pi ab\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Erijuhul, kui on tegemist <a href="https://et-m-wikipedia-org.translate.goog/wiki/Ring?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ring">ringiga</a>, on ε = 0, mis annab r = p = rmin = rmax = a = b ja S = π r2. <div class="mw-collapsible mw-collapsed" style="font-size:95%; padding:2px; clear:both;"> <div style="font-weight:bold; background-color:transparent; text-align:center;"> Esimese seaduse tuletamine </div> <div class="mw-collapsible-content" style="font-weight:normal; background-color:transparent; text-align:left;"> Esimese seaduse tuletamiseks peab esmalt defineerima <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ u=pr^{-1}\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mi> u </mi> <mo> = </mo> <mi> p </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ u=pr^{-1}\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbd411df94830adae72da49b12bcedd3b99e9bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.594ex; height:3.009ex;" alt="{\displaystyle \ u=pr^{-1}\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 10.594ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbd411df94830adae72da49b12bcedd3b99e9bd" data-alt="{\displaystyle \ u=pr^{-1}\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus konstant <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\ell ^{2}G^{-1}M^{-1}\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> = </mo> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p=\ell ^{2}G^{-1}M^{-1}\,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6984fd06f495f5138d3e57121233f7685d55dc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:15.759ex; height:3.009ex;" alt="{\displaystyle p=\ell ^{2}G^{-1}M^{-1}\,}"> </noscript><span class="lazy-image-placeholder" style="width: 15.759ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6984fd06f495f5138d3e57121233f7685d55dc99" data-alt="{\displaystyle p=\ell ^{2}G^{-1}M^{-1}\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> on pikkuse dimensiooniga. Seega <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GMr^{-2}=\ell ^{2}p^{-3}u^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> G </mi> <mi> M </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 3 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle GMr^{-2}=\ell ^{2}p^{-3}u^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db59ed9101fd223c97ea11d5b3269a7792d5eb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.659ex; height:3.009ex;" alt="{\displaystyle GMr^{-2}=\ell ^{2}p^{-3}u^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.659ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db59ed9101fd223c97ea11d5b3269a7792d5eb7f" data-alt="{\displaystyle GMr^{-2}=\ell ^{2}p^{-3}u^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> ja <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\dot {\theta }}=\ell r^{-2}=\ell p^{-2}u^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mi> ℓ </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ {\dot {\theta }}=\ell r^{-2}=\ell p^{-2}u^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9d2fea162e37adb50d2b8d041801930dacffc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.987ex; height:3.176ex;" alt="{\displaystyle \ {\dot {\theta }}=\ell r^{-2}=\ell p^{-2}u^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.987ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9d2fea162e37adb50d2b8d041801930dacffc0" data-alt="{\displaystyle \ {\dot {\theta }}=\ell r^{-2}=\ell p^{-2}u^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> On üle mindud diferentseerimiselt aja järgi diferentseerimisele nurga järgi: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\dot {X}}={\frac {dX}{dt}}={\frac {dX}{d\theta }}\cdot {\frac {d\theta }{dt}}={\frac {dX}{d\theta }}\cdot {\dot {\theta }}={\frac {dX}{d\theta }}\cdot \ell p^{-2}u^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> X </mi> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> X </mi> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> X </mi> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> X </mi> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ {\dot {X}}={\frac {dX}{dt}}={\frac {dX}{d\theta }}\cdot {\frac {d\theta }{dt}}={\frac {dX}{d\theta }}\cdot {\dot {\theta }}={\frac {dX}{d\theta }}\cdot \ell p^{-2}u^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3112e71e03429d783ce0af73cb113e586dd9f5f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.121ex; height:5.509ex;" alt="{\displaystyle \ {\dot {X}}={\frac {dX}{dt}}={\frac {dX}{d\theta }}\cdot {\frac {d\theta }{dt}}={\frac {dX}{d\theta }}\cdot {\dot {\theta }}={\frac {dX}{d\theta }}\cdot \ell p^{-2}u^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 48.121ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3112e71e03429d783ce0af73cb113e586dd9f5f2" data-alt="{\displaystyle \ {\dot {X}}={\frac {dX}{dt}}={\frac {dX}{d\theta }}\cdot {\frac {d\theta }{dt}}={\frac {dX}{d\theta }}\cdot {\dot {\theta }}={\frac {dX}{d\theta }}\cdot \ell p^{-2}u^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Diferentseerides <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ r=pu^{-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mi> r </mi> <mo> = </mo> <mi> p </mi> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ r=pu^{-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98994482db54a2302516d88237f4121379fafebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.56ex; height:3.009ex;" alt="{\displaystyle \ r=pu^{-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.56ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98994482db54a2302516d88237f4121379fafebb" data-alt="{\displaystyle \ r=pu^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kaks korda, on tulemuseks <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {r}}={\frac {d(pu^{-1})}{d\theta }}\cdot \ell p^{-2}u^{2}=-pu^{-2}{\frac {du}{d\theta }}\cdot \ell p^{-2}u^{2}=-\ell p^{-1}{\frac {du}{d\theta }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo> − </mo> <mi> p </mi> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo> − </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\dot {r}}={\frac {d(pu^{-1})}{d\theta }}\cdot \ell p^{-2}u^{2}=-pu^{-2}{\frac {du}{d\theta }}\cdot \ell p^{-2}u^{2}=-\ell p^{-1}{\frac {du}{d\theta }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7451c0b61d5cd0c7cc9c795c751a022c0a7825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.033ex; height:6.009ex;" alt="{\displaystyle {\dot {r}}={\frac {d(pu^{-1})}{d\theta }}\cdot \ell p^{-2}u^{2}=-pu^{-2}{\frac {du}{d\theta }}\cdot \ell p^{-2}u^{2}=-\ell p^{-1}{\frac {du}{d\theta }}}"> </noscript><span class="lazy-image-placeholder" style="width: 56.033ex;height: 6.009ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7451c0b61d5cd0c7cc9c795c751a022c0a7825" data-alt="{\displaystyle {\dot {r}}={\frac {d(pu^{-1})}{d\theta }}\cdot \ell p^{-2}u^{2}=-pu^{-2}{\frac {du}{d\theta }}\cdot \ell p^{-2}u^{2}=-\ell p^{-1}{\frac {du}{d\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {r}}={\frac {d{\dot {r}}}{d\theta }}\cdot \ell p^{-2}u^{2}={\frac {d}{d\theta }}\left(-\ell p^{-1}{\frac {du}{d\theta }}\right)\cdot \ell p^{-2}u^{2}=-\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <mi> θ </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo> − </mo> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 3 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <msup> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\ddot {r}}={\frac {d{\dot {r}}}{d\theta }}\cdot \ell p^{-2}u^{2}={\frac {d}{d\theta }}\left(-\ell p^{-1}{\frac {du}{d\theta }}\right)\cdot \ell p^{-2}u^{2}=-\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ab742c6fc9de0868ca3b85b55742d05cf4a6e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.416ex; height:6.343ex;" alt="{\displaystyle {\ddot {r}}={\frac {d{\dot {r}}}{d\theta }}\cdot \ell p^{-2}u^{2}={\frac {d}{d\theta }}\left(-\ell p^{-1}{\frac {du}{d\theta }}\right)\cdot \ell p^{-2}u^{2}=-\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 62.416ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ab742c6fc9de0868ca3b85b55742d05cf4a6e9" data-alt="{\displaystyle {\ddot {r}}={\frac {d{\dot {r}}}{d\theta }}\cdot \ell p^{-2}u^{2}={\frac {d}{d\theta }}\left(-\ell p^{-1}{\frac {du}{d\theta }}\right)\cdot \ell p^{-2}u^{2}=-\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Asendades radiaalsesse liikumisvõrrandisse <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-GMr^{-2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi> r </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo> − </mo> <mi> G </mi> <mi> M </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-GMr^{-2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0910b547b473eb65dbc46d7fa5a2a849c777c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.148ex; height:3.343ex;" alt="{\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-GMr^{-2}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.148ex;height: 3.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0910b547b473eb65dbc46d7fa5a2a849c777c5" data-alt="{\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-GMr^{-2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> saab <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}-(pu^{-1})(\ell p^{-2}u^{2})^{2}=-\ell ^{2}p^{-3}u^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 3 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <msup> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> p </mi> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> ℓ </mi> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo> − </mo> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 3 </mn> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}-(pu^{-1})(\ell p^{-2}u^{2})^{2}=-\ell ^{2}p^{-3}u^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae8dda6d71cf87862bae0ad7f2a86a58273b20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:46.821ex; height:6.009ex;" alt="{\displaystyle -\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}-(pu^{-1})(\ell p^{-2}u^{2})^{2}=-\ell ^{2}p^{-3}u^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 46.821ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae8dda6d71cf87862bae0ad7f2a86a58273b20f" data-alt="{\displaystyle -\ell ^{2}p^{-3}u^{2}{\frac {d^{2}u}{d\theta ^{2}}}-(pu^{-1})(\ell p^{-2}u^{2})^{2}=-\ell ^{2}p^{-3}u^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Jagades parema poolega saab lihtsa mittehomogeense lineaarse <a href="https://et-m-wikipedia-org.translate.goog/wiki/Diferentsiaalv%C3%B5rrand?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Diferentsiaalvõrrand">diferentsiaalvõrrandi</a>: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <msup> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> u </mi> <mo> = </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=1.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7131da4a52b7aeecc1be2fb8d74acba747011a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.516ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=1.}"> </noscript><span class="lazy-image-placeholder" style="width: 13.516ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7131da4a52b7aeecc1be2fb8d74acba747011a55" data-alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Ilmne lahend sellele võrrandile on ringikujuline orbiit <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ u=1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mi> u </mi> <mo> = </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ u=1.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e808925d55acdbdcfe0fcfd18014c4b8e6122f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle \ u=1.}"> </noscript><span class="lazy-image-placeholder" style="width: 6.818ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e808925d55acdbdcfe0fcfd18014c4b8e6122f" data-alt="{\displaystyle \ u=1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Ülejäänud lahendeid saab, kui otsida lahendit konstantsete kordajatega homogeensele lineaarsele diferentsiaalvõrrandile <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> u </mi> </mrow> <mrow> <mi> d </mi> <msup> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> u </mi> <mo> = </mo> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=0.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721d49894dd50e8aed45292b1aaa6192ec42415d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.516ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=0.}"> </noscript><span class="lazy-image-placeholder" style="width: 13.516ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721d49894dd50e8aed45292b1aaa6192ec42415d" data-alt="{\displaystyle {\frac {d^{2}u}{d\theta ^{2}}}+u=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Vastavad lahendid on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \cdot \cos(\theta -\theta _{0})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ε </mi> <mo> ⋅ </mo> <mi> cos </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo> − </mo> <msub> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \varepsilon \cdot \cos(\theta -\theta _{0})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e0b0247c40250dfbe71f4f627ed07223dce7e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.759ex; height:2.843ex;" alt="{\displaystyle \varepsilon \cdot \cos(\theta -\theta _{0})}"> </noscript><span class="lazy-image-placeholder" style="width: 13.759ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e0b0247c40250dfbe71f4f627ed07223dce7e4" data-alt="{\displaystyle \varepsilon \cdot \cos(\theta -\theta _{0})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus ε ja <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \theta _{0}\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle \theta _{0}\,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fd01e04602beb0336362eef6cb8265cf31f16e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.99ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \theta _{0}\,}"> </noscript><span class="lazy-image-placeholder" style="width: 1.99ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fd01e04602beb0336362eef6cb8265cf31f16e" data-alt="{\displaystyle \scriptstyle \theta _{0}\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  on suvalised integreerimiskonstandid. Seega tulemuseks on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ u=1+\varepsilon \cdot \cos(\theta -\theta _{0}).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mi> u </mi> <mo> = </mo> <mn> 1 </mn> <mo> + </mo> <mi> ε </mi> <mo> ⋅ </mo> <mi> cos </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> θ </mi> <mo> − </mo> <msub> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ u=1+\varepsilon \cdot \cos(\theta -\theta _{0}).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42915d069d2ea25a2870320bceba86d9f86d2973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.417ex; height:2.843ex;" alt="{\displaystyle \ u=1+\varepsilon \cdot \cos(\theta -\theta _{0}).}"> </noscript><span class="lazy-image-placeholder" style="width: 23.417ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42915d069d2ea25a2870320bceba86d9f86d2973" data-alt="{\displaystyle \ u=1+\varepsilon \cdot \cos(\theta -\theta _{0}).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Valides koordinaatsüsteemi telje nii, et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \,\theta _{0}=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mspace width="thinmathspace"></mspace> <msub> <mi> θ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle \,\theta _{0}=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d562de41cfa5f23cc2e4478fc49e8b0169beff2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.09ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \,\theta _{0}=0}"> </noscript><span class="lazy-image-placeholder" style="width: 4.09ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d562de41cfa5f23cc2e4478fc49e8b0169beff2b" data-alt="{\displaystyle \scriptstyle \,\theta _{0}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , ja lisades <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle u=pr^{-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi> u </mi> <mo> = </mo> <mi> p </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle u=pr^{-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf0b70889542f8c48fae34ffaf054ffdff5b33d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.657ex; height:2.176ex;" alt="{\displaystyle \scriptstyle u=pr^{-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.657ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf0b70889542f8c48fae34ffaf054ffdff5b33d" data-alt="{\displaystyle \scriptstyle u=pr^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  saab <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ pr^{-1}=1+\varepsilon \cdot \cos \theta ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mi> p </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> <mo> + </mo> <mi> ε </mi> <mo> ⋅ </mo> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ pr^{-1}=1+\varepsilon \cdot \cos \theta ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7818fc6c954cbc3f7f75df83d34a57756bfa6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.231ex; height:3.009ex;" alt="{\displaystyle \ pr^{-1}=1+\varepsilon \cdot \cos \theta ,}"> </noscript><span class="lazy-image-placeholder" style="width: 20.231ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7818fc6c954cbc3f7f75df83d34a57756bfa6b" data-alt="{\displaystyle \ pr^{-1}=1+\varepsilon \cdot \cos \theta ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ r={\frac {p}{1+\varepsilon \cdot \cos \theta }}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <mi> r </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> p </mi> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ε </mi> <mo> ⋅ </mo> <mi> cos </mi> <mo> ⁡ </mo> <mi> θ </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ r={\frac {p}{1+\varepsilon \cdot \cos \theta }}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e7743e4fb694ea25b33b28fba4ff2ef3cd91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.565ex; height:5.176ex;" alt="{\displaystyle \ r={\frac {p}{1+\varepsilon \cdot \cos \theta }}.}"> </noscript><span class="lazy-image-placeholder" style="width: 17.565ex;height: 5.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e7743e4fb694ea25b33b28fba4ff2ef3cd91a" data-alt="{\displaystyle \ r={\frac {p}{1+\varepsilon \cdot \cos \theta }}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Kui <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle |{\varepsilon }|<1,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> ε </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> < </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \scriptstyle |{\varepsilon }|<1,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9375d5d180b1277c62f7cc5c8466943424747248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.239ex; height:2.176ex;" alt="{\displaystyle \scriptstyle |{\varepsilon }|<1,}"> </noscript><span class="lazy-image-placeholder" style="width: 4.239ex;height: 2.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9375d5d180b1277c62f7cc5c8466943424747248" data-alt="{\displaystyle \scriptstyle |{\varepsilon }|<1,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  on see ellipsi võrrand ja illustreerib Kepleri esimest seadust. </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Teine_seadus">Teine seadus</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Teine seadus"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <dl> <dd> Planeedi raadiusvektor katab võrdsetes ajavahemikes võrdsed pindalad.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Wolfram2nd-1">[1]</a> </dd> </dl> Mõistmaks Kepleri teist seadust eeldame, et planeedil kulub üks päev liikumaks punktist A punkti B. Jooned <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päikeselt</a> punktidesse A ja B koos planeedi orbiidiga määravad piisavalt heas lähenduses <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kolmnurk?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kolmnurk">kolmnurkse</a> ala. Olenemata, kus kohas <a href="https://et-m-wikipedia-org.translate.goog/wiki/Planeet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Planeet">planeet</a> oma orbiidil paikneb, katab tiirleva planeedi liikumisele vastav raadiusvektor iga päev sama suure pindala. Nagu väidab Kepleri esimene seadus peab planeet liikuma mööda elliptilist orbiiti paiknedes seega eri aegadel <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päikesest</a> eri kaugustel. Seega peab planeet liikuma kiiremini paiknedes Päikesele lähemal ja aeglasemalt olles Päikesest kaugemal, et katta sama suur ala. Kepleri teine seadus vastab faktile, et jõukomponent, mis on risti raadiusvektoriga, on null. <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kiirus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kiirus">Kiirus</a>, millega liigub segment, on vastavuses <a href="https://et-m-wikipedia-org.translate.goog/wiki/Impulsimoment?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Impulsimoment">impulsimomendiga</a> ja seega esitab Kepleri teine seadus impulsimomendi jäävust. Kirjutades valemina: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}r^{2}{\dot {\theta }}\right)=0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}r^{2}{\dot {\theta }}\right)=0,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c21cd4a496ecbcefcc9aaeed5df6d97ea32c7363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.065ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}r^{2}{\dot {\theta }}\right)=0,}"> </noscript><span class="lazy-image-placeholder" style="width: 17.065ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c21cd4a496ecbcefcc9aaeed5df6d97ea32c7363" data-alt="{\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}r^{2}{\dot {\theta }}\right)=0,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}r^{2}{\dot {\theta }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tfrac {1}{2}}r^{2}{\dot {\theta }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e554804fbc6d5b619f45e79e1b9939911cf632a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.117ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{2}}r^{2}{\dot {\theta }}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.117ex;height: 3.676ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e554804fbc6d5b619f45e79e1b9939911cf632a2" data-alt="{\displaystyle {\tfrac {1}{2}}r^{2}{\dot {\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  on kiirus, millega pind liigub. Tuntud ka võrdsete pindade reeglina, mis kehtib ka hüperboolsetele- ja paraboolsetele trajektooridele. <div class="mw-collapsible mw-collapsed" style="font-size:95%; padding:2px; clear:both;"> <div style="font-weight:bold; background-color:transparent; text-align:center;"> Teise seaduse tuletamine </div> <div class="mw-collapsible-content" style="font-weight:normal; background-color:transparent; text-align:left;"> Tuletamaks Kepleri teist seadust on vaja tangentsiaalkiirenduse võrrandit. Suurus <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell =r^{2}{\dot {\theta }}\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ℓ </mi> <mo> = </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell =r^{2}{\dot {\theta }}\,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117f88f50758e720f59c54f15b922da06b996a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.914ex; height:2.843ex;" alt="{\displaystyle \ell =r^{2}{\dot {\theta }}\,}"> </noscript><span class="lazy-image-placeholder" style="width: 7.914ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117f88f50758e720f59c54f15b922da06b996a43" data-alt="{\displaystyle \ell =r^{2}{\dot {\theta }}\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> on <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Liikumisintegraal&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Liikumisintegraal (pole veel kirjutatud)">liikumisintegraal</a>, isegi kui nii kaugus r, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Nurkkiirus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nurkkiirus">nurkkiirus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\dot {\theta }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.356ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" data-alt="{\displaystyle {\dot {\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  kui ka <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Tangentsiaalkiirus&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Tangentsiaalkiirus (pole veel kirjutatud)">tangentsiaalkiirus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r{\dot {\theta }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r{\dot {\theta }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e7e2c70d2fe9acf265d2e082acb296df347e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.843ex;" alt="{\displaystyle r{\dot {\theta }}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.405ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e7e2c70d2fe9acf265d2e082acb296df347e57" data-alt="{\displaystyle r{\dot {\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  muutuvad, sest <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\ell }{dt}}={\frac {d(r^{2}{\dot {\theta }})}{dt}}=r^{2}{\ddot {\theta }}+2r{\dot {r}}{\dot {\theta }}=r(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> ℓ </mi> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mo stretchy="false"> ( </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {d\ell }{dt}}={\frac {d(r^{2}{\dot {\theta }})}{dt}}=r^{2}{\ddot {\theta }}+2r{\dot {r}}{\dot {\theta }}=r(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }})=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c417a2acff1de0f7bf1ee0fb186741f132649b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.969ex; height:6.176ex;" alt="{\displaystyle {\frac {d\ell }{dt}}={\frac {d(r^{2}{\dot {\theta }})}{dt}}=r^{2}{\ddot {\theta }}+2r{\dot {r}}{\dot {\theta }}=r(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 46.969ex;height: 6.176ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c417a2acff1de0f7bf1ee0fb186741f132649b" data-alt="{\displaystyle {\frac {d\ell }{dt}}={\frac {d(r^{2}{\dot {\theta }})}{dt}}=r^{2}{\ddot {\theta }}+2r{\dot {r}}{\dot {\theta }}=r(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus avaldis : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> r </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982340fc57d2abbc6ccee5154256e61f37e29c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.055ex; height:3.009ex;" alt="{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.055ex;height: 3.009ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982340fc57d2abbc6ccee5154256e61f37e29c68" data-alt="{\displaystyle r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  kaob vastavalt tangentsiaalkiirenduse võrrandile. Integreerides pinda ajast t1 kuni ajani t2 saab võrrandi <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \int _{t_{1}}^{t_{2}}{\frac {1}{2}}\cdot r\cdot r{\dot {\theta }}\,dt={\frac {1}{2}}\cdot \ell \cdot (t_{2}-t_{1})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>   </mtext> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⋅ </mo> <mi> r </mi> <mo> ⋅ </mo> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> θ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> t </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⋅ </mo> <mi> ℓ </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ \int _{t_{1}}^{t_{2}}{\frac {1}{2}}\cdot r\cdot r{\dot {\theta }}\,dt={\frac {1}{2}}\cdot \ell \cdot (t_{2}-t_{1})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c78691f8072269f29012447b37add7c2b94c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.591ex; height:6.509ex;" alt="{\displaystyle \ \int _{t_{1}}^{t_{2}}{\frac {1}{2}}\cdot r\cdot r{\dot {\theta }}\,dt={\frac {1}{2}}\cdot \ell \cdot (t_{2}-t_{1})}"> </noscript><span class="lazy-image-placeholder" style="width: 34.591ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c78691f8072269f29012447b37add7c2b94c77" data-alt="{\displaystyle \ \int _{t_{1}}^{t_{2}}{\frac {1}{2}}\cdot r\cdot r{\dot {\theta }}\,dt={\frac {1}{2}}\cdot \ell \cdot (t_{2}-t_{1})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> mis sõltub ainult ajaperioodi pikkusest, ehk t2 − t1. See ongi Kepleri teine seadus. </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Kolmas_seadus">Kolmas seadus</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Kolmas seadus"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <dl> <dd> Planeetide tiirlemisperioodide ruudud suhtuvad nagu nende orbiitide pikemate pooltelgede kuubid. </dd> </dl> Kolmas seadus kirjeldab suhet planeedi kauguse Päikesest ja taevakeha tiirlemisperioodi vahel. Olgu näiteks planeet A neli korda Päikesest kaugemal kui planeet B. Seega peab planeet A läbima iga tiiruga neli korda pikema vahemaa kui planeet B. Planeet A liigub kaks korda aeglasemalt kui planeet B, et säiliks tasakaal vähenenud <a href="https://et-m-wikipedia-org.translate.goog/wiki/Tsentripetaalj%C3%B5ud?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tsentripetaaljõud">tsentripetaaljõuga</a>, mis tuleneb gravitatsioonilisest tõmbest. Kokku kulub planeedil A 4×2=8 korda kauem aega, et teha üks täistiir ümber Päikese kui planeedil B. See on vastavuses Kepleri kolmanda seadusega (82=43). Kolmandat seadust tunti ka harmoonilise seadusena<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Holton3-10">[10]</a>, kuna Kepler kasutas seda katses määrata "kerade muusika" täpseid reegleid ja esitada neid muusikalises kirjaviisis.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Burtt-11">[11]</a> Praegusel ajal kasutatakse kolmandat seadust, et kindlaks teha <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eksoplaneet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eksoplaneet">eksoplaneedi</a> kaugus <a href="https://et-m-wikipedia-org.translate.goog/wiki/T%C3%A4ht_(astronoomia)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Täht (astronoomia)">tähest</a>, mille ümber see tiirleb. Kauguse määramine aitab kindlaks teha, kas planeet sobib eluks.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-astro-12">[12]</a> Valemites väljendades: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{2}\propto a^{3}\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> T </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ∝ </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T^{2}\propto a^{3}\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cea380720fcb62511501f0f5cd1fc0a437b036d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.191ex; height:3.009ex;" alt="{\displaystyle T^{2}\propto a^{3}\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 9.191ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cea380720fcb62511501f0f5cd1fc0a437b036d" data-alt="{\displaystyle T^{2}\propto a^{3}\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus T on planeedi tiirlemisperiood ja a on orbiidi pikem pooltelg. Võrdelisuse konstant on sama iga Päikese ümber tiirleva planeedi jaoks. <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {T_{\rm {planeet}}^{2}}{a_{\rm {planeet}}^{3}}}={\frac {T_{\rm {Maa}}^{2}}{a_{\rm {Maa}}^{3}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi> T </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> p </mi> <mi mathvariant="normal"> l </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> n </mi> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> p </mi> <mi mathvariant="normal"> l </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> n </mi> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> t </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msubsup> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi> T </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> M </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> a </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> M </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> a </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msubsup> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {T_{\rm {planeet}}^{2}}{a_{\rm {planeet}}^{3}}}={\frac {T_{\rm {Maa}}^{2}}{a_{\rm {Maa}}^{3}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1635af538fdbcf30b2738e79d90e28bb7da4391d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.955ex; height:7.843ex;" alt="{\displaystyle {\frac {T_{\rm {planeet}}^{2}}{a_{\rm {planeet}}^{3}}}={\frac {T_{\rm {Maa}}^{2}}{a_{\rm {Maa}}^{3}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 16.955ex;height: 7.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1635af538fdbcf30b2738e79d90e28bb7da4391d" data-alt="{\displaystyle {\frac {T_{\rm {planeet}}^{2}}{a_{\rm {planeet}}^{3}}}={\frac {T_{\rm {Maa}}^{2}}{a_{\rm {Maa}}^{3}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Konstandi väärtuseks tuleb 1(<a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=T%C3%A4heaasta&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Täheaasta (pole veel kirjutatud)">täheaasta</a>)2(<a href="https://et-m-wikipedia-org.translate.goog/wiki/Astronoomiline_%C3%BChik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Astronoomiline ühik">astronoomiline ühik</a>)−3. Arvutades tulemuse välja saab 2.97472505×10–19 s2m−3. <div class="mw-collapsible mw-collapsed" style="font-size:95%; padding:2px; clear:both;"> <div style="font-weight:bold; background-color:transparent; text-align:center;"> Kolmanda seaduse tuletamine </div> <div class="mw-collapsible-content" style="font-weight:normal; background-color:transparent; text-align:left;"> Erijuhul, kui tegu on ringikujuliste orbiitidega, mis on ellipsid ekstsentrilisusega null, on suhet orbiidi raadiuse a ja perioodi T vahel üpriski lihtne tuletada. Ringliikumise <a href="https://et-m-wikipedia-org.translate.goog/wiki/Tsentripetaalj%C3%B5ud?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tsentripetaaljõud">tsentripetaaljõud</a> on võrdeline a/T2, mis omakorda on võrdeline 1/a2. Seega <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{2}\propto a^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ∝ </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P^{2}\propto a^{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e24e585af212f211ce3943ba3a8899262a0895" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.258ex; height:2.676ex;" alt="{\displaystyle P^{2}\propto a^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.258ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e24e585af212f211ce3943ba3a8899262a0895" data-alt="{\displaystyle P^{2}\propto a^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Üldiselt on tegemist aga elliptiliste orbiitidega ja seega on ka tuletuskäik keerukam. Elliptilise planeedi orbiidi pindala on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\pi ab=\pi a(a{\sqrt {1-\varepsilon ^{2}}})=\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo> = </mo> <mi> π </mi> <mi> a </mi> <mi> b </mi> <mo> = </mo> <mi> π </mi> <mi> a </mi> <mo stretchy="false"> ( </mo> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> π </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S=\pi ab=\pi a(a{\sqrt {1-\varepsilon ^{2}}})=\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fff41df23b645809ce13738d0268b50917675c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.534ex; height:3.509ex;" alt="{\displaystyle S=\pi ab=\pi a(a{\sqrt {1-\varepsilon ^{2}}})=\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 41.534ex;height: 3.509ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fff41df23b645809ce13738d0268b50917675c3" data-alt="{\displaystyle S=\pi ab=\pi a(a{\sqrt {1-\varepsilon ^{2}}})=\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Kiirus, millega raadiusvektor katab orbiidi pinda on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}=\ell /2\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> S </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\dot {S}}=\ell /2\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5e6e76080fcad5618ccd693848045f43f79fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9ex; height:3.343ex;" alt="{\displaystyle {\dot {S}}=\ell /2\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 9ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5e6e76080fcad5618ccd693848045f43f79fc0" data-alt="{\displaystyle {\dot {S}}=\ell /2\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> kus <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}=pGM=a(1-\varepsilon ^{2})GM\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mi> p </mi> <mi> G </mi> <mi> M </mi> <mo> = </mo> <mi> a </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mi> G </mi> <mi> M </mi> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell ^{2}=pGM=a(1-\varepsilon ^{2})GM\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7229bd55a0ed076a8d197a164cc64ae7c366fcf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.142ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}=pGM=a(1-\varepsilon ^{2})GM\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 28.142ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7229bd55a0ed076a8d197a164cc64ae7c366fcf7" data-alt="{\displaystyle \ell ^{2}=pGM=a(1-\varepsilon ^{2})GM\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Orbiidi periood on <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {S}{\dot {S}}}={\frac {\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell /2}}=2\pi {\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> S </mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> S </mi> <mo> ˙ </mo> </mover> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> π </mi> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </mfrac> </mrow> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mi> ℓ </mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T={\frac {S}{\dot {S}}}={\frac {\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell /2}}=2\pi {\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6cf189340feaccee4f868c71c8e21688bd3227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.37ex; height:7.009ex;" alt="{\displaystyle T={\frac {S}{\dot {S}}}={\frac {\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell /2}}=2\pi {\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 41.37ex;height: 7.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6cf189340feaccee4f868c71c8e21688bd3227" data-alt="{\displaystyle T={\frac {S}{\dot {S}}}={\frac {\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell /2}}=2\pi {\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> millest tuleneb <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {P}{2\pi }}\right)^{2}=\left({\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\right)^{2}={\frac {a^{4}(1-\varepsilon ^{2})}{\ell ^{2}}}={\frac {a^{4}(1-\varepsilon ^{2})}{a(1-\varepsilon ^{2})GM}}={\frac {a^{3}}{GM}}\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> P </mi> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mi> ℓ </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> a </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <msup> <mi> ε </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mi> G </mi> <mi> M </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mrow> <mi> G </mi> <mi> M </mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left({\frac {P}{2\pi }}\right)^{2}=\left({\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\right)^{2}={\frac {a^{4}(1-\varepsilon ^{2})}{\ell ^{2}}}={\frac {a^{4}(1-\varepsilon ^{2})}{a(1-\varepsilon ^{2})GM}}={\frac {a^{3}}{GM}}\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c1ce212ca9bb9988e857b321da5696af3d0e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.014ex; height:8.009ex;" alt="{\displaystyle \left({\frac {P}{2\pi }}\right)^{2}=\left({\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\right)^{2}={\frac {a^{4}(1-\varepsilon ^{2})}{\ell ^{2}}}={\frac {a^{4}(1-\varepsilon ^{2})}{a(1-\varepsilon ^{2})GM}}={\frac {a^{3}}{GM}}\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 68.014ex;height: 8.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c1ce212ca9bb9988e857b321da5696af3d0e2d" data-alt="{\displaystyle \left({\frac {P}{2\pi }}\right)^{2}=\left({\frac {a^{2}{\sqrt {1-\varepsilon ^{2}}}}{\ell }}\right)^{2}={\frac {a^{4}(1-\varepsilon ^{2})}{\ell ^{2}}}={\frac {a^{4}(1-\varepsilon ^{2})}{a(1-\varepsilon ^{2})GM}}={\frac {a^{3}}{GM}}\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> mis omakorda näitab Kepleri kolmandat seadust <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{2}\propto a^{3}.\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ∝ </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P^{2}\propto a^{3}.\,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfc1730250a57b7571c0c156fc14b4f5dd7b877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.292ex; height:2.676ex;" alt="{\displaystyle P^{2}\propto a^{3}.\,}"> </noscript><span class="lazy-image-placeholder" style="width: 9.292ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfc1730250a57b7571c0c156fc14b4f5dd7b877" data-alt="{\displaystyle P^{2}\propto a^{3}.\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Üldistus">Üldistus</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Üldistus"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> Kepleri seadused kirjeldavad ligikaudselt kahe keha liikumist orbiidil üksteise ümber. Kahe keha massid võivad olla peaaegu võrdsed, näiteks <a href="https://et-m-wikipedia-org.translate.goog/wiki/Charon?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Charon">Charon</a> ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/Pluuto?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pluuto">Pluuto</a> (erinevus ~1:10), väikese erinevusega, näiteks <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kuu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kuu">Kuu</a> ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/Maa_(planeet)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Maa (planeet)">Maa</a> (~1:100), või suure erinevusega, näiteks <a href="https://et-m-wikipedia-org.translate.goog/wiki/Merkuur?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Merkuur">Merkuur</a> ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päike</a> (~1:10 000 000). Kahe keha liikumise korral sõltub tiirlemine süsteemi <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Bar%C3%BCtsenter&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Barütsenter (pole veel kirjutatud)">barütsentrist</a> (punkt, mille ümber kehade süsteem tiirleb), kuna kummagi keha <a href="https://et-m-wikipedia-org.translate.goog/wiki/Massikese?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Massikese">massikese</a> ei asu täpselt ühes ellipsi fookuses. See-eest on mõlemad orbiidid ellipsid, mille üks fookus asub barütsentris. Juhul, kui masside suhe on suur, võib barütsenter olla suurema keha sees massikeskme lähedal. Sellisel juhul on tarvis keerukat täppismõõtmist, et eristada barütsentrit suure keha massikeskmest. Taevakehadest, mis tiirlevad ümber Päikese, on suurima massiga <a href="https://et-m-wikipedia-org.translate.goog/wiki/Jupiter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jupiter">Jupiter</a> (masside suhe 1/1047,3486) ja <a href="https://et-m-wikipedia-org.translate.goog/wiki/Saturn?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Saturn">Saturn</a> (1/3497,898)<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Almanac-13">[13]</a>, seega on juba pikka aega teatud, et <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ikeses%C3%BCsteem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päikesesüsteem">Päikesesüsteemi</a> massikese võib kohati paikneda väljaspool <a href="https://et-m-wikipedia-org.translate.goog/wiki/P%C3%A4ike?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Päike">Päikest</a>.<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Principia-14">[14]</a>, mõnikord kuni Päikese <a href="https://et-m-wikipedia-org.translate.goog/wiki/Diameeter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Diameeter">diameetri</a> kaugusel Päikese keskmest. Seega Kepleri esimene seadus, ehkki ligikaudselt piisavalt täpne, ei kirjelda klassikalise <a href="https://et-m-wikipedia-org.translate.goog/wiki/F%C3%BC%C3%BCsika?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Füüsika">füüsika</a> abil planeetide tiirlemist ümber Päikese. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Nullekstsentrilisus">Nullekstsentrilisus</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Nullekstsentrilisus"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> Kui planeedi orbiidi ekstsentrilisus oleks null, siis Kepleri seaduste järgi: <ol> <li>planeedi orbiit oleks ringjoon;</li> <li>Päike paikneks orbiidi keskkohas;</li> <li>planeedi liikumise kiirus orbiidil oleks konstantne;</li> <li>täheperiood ruudus on võrdeline kaugusega Päikesest kuubis.</li> </ol> Kuue planeedi, mis olid Kopernikule ja Keplerile teada, ekstsentrilisused on üpriski väikesed, mis võimaldab hea täpsusega hinnata planeetide liikumist lähtudes neist punktidest. Kuna ühtlast ringliikumist peeti normaalseks, siis kõrvalekaldeid taolisest liikumisest peeti anomaaliateks. Kepler täiendas Koperniku mudelit, saades uuteks reegliteks: <ol> <li>planeedi orbiit ei ole ringjoon, vaid ellips;</li> <li>Päike ei asu orbiidi keskkohas vaid fookuspunktis;</li> <li>nii planeedi <a href="https://et-m-wikipedia-org.translate.goog/wiki/Joonkiirus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Joonkiirus">joonkiirus</a> kui ka <a href="https://et-m-wikipedia-org.translate.goog/wiki/Nurkkiirus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nurkkiirus">nurkkiirus</a> muutuvad;</li> <li>täheperiood ruudus on võrdeline miinimumkauguse ja maksimumkauguse aritmeetilise keskmisega kuubis.</li> </ol> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Vaata_ka">Vaata ka</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Vaata ka"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul> <li><a href="https://et-m-wikipedia-org.translate.goog/wiki/Johannes_Kepler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Johannes Kepler">Johannes Kepler</a></li> <li><a href="https://et-m-wikipedia-org.translate.goog/wiki/Gravitatsioon?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gravitatsioon">Gravitatsioon</a></li> <li><a href="https://et-m-wikipedia-org.translate.goog/wiki/Ringliikumine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ringliikumine">Ringliikumine</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="Viited">Viited</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Viited"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <div class="reflist" style="list-style-type: decimal;"> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-Wolfram2nd-1">↑ <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Wolfram2nd_1-0">1,0</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Wolfram2nd_1-1">1,1</a> Bryant, Jeff; Pavlyk, Oleksandr. "<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://demonstrations.wolfram.com/KeplersSecondLaw/">Kepler's Second Law</a>", <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Wolfram_Demonstrations_Project">"Wolfram Demonstrations Project"</a>. Retrieved December 27, 2009.</li> <li id="cite_note-Holton-2">↑ <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Holton_2-0">2,0</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Holton_2-1">2,1</a> <style data-mw-deduplicate="TemplateStyles:r6066747">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFHolton,_Gerald_James2001" class="citation book cs1">Holton, Gerald James (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/?id%3DczaGZzR0XOUC%26pg%3DPA40">Physics, the Human Adventure: From Copernicus to Einstein and Beyond</a> (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. Lk 40–41. <a href="https://et-m-wikipedia-org.translate.goog/wiki/Rahvusvaheline_raamatu_standardnumber?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rahvusvaheline raamatu standardnumber">ISBN</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Raamatuotsimine/0-8135-2908-5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eri:Raamatuotsimine/0-8135-2908-5"><bdi>0-8135-2908-5</bdi></a>. Vaadatud 27. detsembril 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics%2C+the+Human+Adventure%3A+From+Copernicus+to+Einstein+and+Beyond&rft.place=Piscataway%2C+NJ&rft.pages=40-41&rft.edition=3rd+paperback&rft.pub=Rutgers+University+Press&rft.date=2001&rft.isbn=0-8135-2908-5&rft.au=Holton%2C+Gerald+James&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DczaGZzR0XOUC%26pg%3DPA40&rfr_id=info%3Asid%2Fet.wikipedia.org%3AKepleri+seadused" class="Z3988"> <code class="cs1-code">{{<a href="https://et-m-wikipedia-org.translate.goog/wiki/Mall:Cite_book?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mall:Cite book">cite book</a>}}</code>: eiran tundmatut parameetrit <code class="cs1-code">|coauthor=</code>, kasuta parameetrit (<code class="cs1-code">|author=</code>) (<a href="https://et-m-wikipedia-org.translate.goog/wiki/Juhend:Viitamismallide_vead?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#parameter_ignored_suggest" title="Juhend:Viitamismallide vead">juhend</a>)</li> <li id="cite_note-smith-sep-3">↑ <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-smith-sep_3-0">3,0</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-smith-sep_3-1">3,1</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-smith-sep_3-2">3,2</a> G E Smith, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://plato.stanford.edu/archives/win2008/entries/newton-principia/">"Newton's Philosophiae Naturalis Principia Mathematica"</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://plato.stanford.edu/archives/win2008/entries/newton-principia/%23HisConPri">Historical context ...</a> in The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.).</li> <li id="cite_note-newt-p-4"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-newt-p_4-0">↑</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/books?id%3DTm0FAAAAQAAJ%26pg%3DPA85">Book 1, Proposition 13, Corollary 1</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/books?id%3DTm0FAAAAQAAJ%26pg%3DPA231">Book 1, Proposition 65</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/books?id%3DTm0FAAAAQAAJ%26pg%3DPA232">(Proposition 65, Case 1)</a>, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/books?id%3D6EqxPav3vIsC%26pg%3DPA234">Book 3, Proposition 13</a>.</li> <li id="cite_note-plummr-5"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-plummr_5-0">↑</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.archive.org/stream/introductorytrea00plumuoft%23page/n19/mode/2up">page 1</a> H C Plummer (1918), An introductory treatise on dynamical astronomy, Cambridge, 1918.</li> <li id="cite_note-Wilson_1994-6"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Wilson_1994_6-0">↑</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r6066747"><cite id="CITEREFWilson1994" class="citation journal cs1">Wilson, Curtis (mai 1994). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20110817151556/http://had.aas.org/hadnews/HADN31.pdf">"Kepler's Laws, So-Called"</a> (PDF). HAD News. Washington, DC: Historical Astronomy Division, <a href="https://et-m-wikipedia-org.translate.goog/wiki/American_Astronomical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="American Astronomical Society">American Astronomical Society</a> (31): 1–2. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://had.aas.org/hadnews/HADN31.pdf">Originaali</a> (PDF) arhiivikoopia seisuga 17. august 2011. Vaadatud 27. detsembril 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=HAD+News&rft.atitle=Kepler%27s+Laws%2C+So-Called&rft.issue=31&rft.pages=1-2&rft.date=1994-05&rft.aulast=Wilson&rft.aufirst=Curtis&rft_id=http%3A%2F%2Fhad.aas.org%2Fhadnews%2FHADN31.pdf&rfr_id=info%3Asid%2Fet.wikipedia.org%3AKepleri+seadused" class="Z3988"></li> <li id="cite_note-veenus-7"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-veenus_7-0">↑</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r6066747"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20150529073832/http://solarsystem.nasa.gov/planets/profile.cfm?Object%3DVenus%26Display%3DFacts">""NASA Venus: Facts & Figures""</a>. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://solarsystem.nasa.gov/planets/profile.cfm?Object%3DVenus%26Display%3DFacts">Originaali</a> arhiivikoopia seisuga 29. mai 2015. Vaadatud 6. novembril 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%22NASA+Venus%3A+Facts+%26+Figures%22&rft_id=http%3A%2F%2Fsolarsystem.nasa.gov%2Fplanets%2Fprofile.cfm%3FObject%3DVenus%26Display%3DFacts&rfr_id=info%3Asid%2Fet.wikipedia.org%3AKepleri+seadused" class="Z3988"></li> <li id="cite_note-merkuur-8"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-merkuur_8-0">↑</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r6066747"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20140408125609/http://solarsystem.nasa.gov/planets/profile.cfm?Object%3DMercury%26Display%3DFacts">""NASA Mercury: Facts & Figures""</a>. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://solarsystem.nasa.gov/planets/profile.cfm?Object%3DMercury%26Display%3DFacts">Originaali</a> arhiivikoopia seisuga 8. aprill 2014. Vaadatud 6. novembril 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%22NASA+Mercury%3A+Facts+%26+Figures%22&rft_id=http%3A%2F%2Fsolarsystem.nasa.gov%2Fplanets%2Fprofile.cfm%3FObject%3DMercury%26Display%3DFacts&rfr_id=info%3Asid%2Fet.wikipedia.org%3AKepleri+seadused" class="Z3988"></li> <li id="cite_note-nasa-9"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-nasa_9-0">↑</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20110815190823/http://erc.ivv.nasa.gov/mission_pages/stereo/news/SECCHI_P2003.html">"SECCHI Makes a Fantastic Recovery!"</a>, Brian Dunbar 2008</li> <li id="cite_note-Holton3-10"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Holton3_10-0">↑</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r6066747"><cite id="CITEREFGerald_James_Holton,_Stephen_G._Brush2001" class="citation book cs1">Gerald James Holton, Stephen G. Brush (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/?id%3DczaGZzR0XOUC%26pg%3DPA45%26dq%3DKepler%2B%2522harmonic%2Blaw%2522">Physics, the Human Adventure</a>. Rutgers University Press. Lk 45. <a href="https://et-m-wikipedia-org.translate.goog/wiki/Rahvusvaheline_raamatu_standardnumber?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rahvusvaheline raamatu standardnumber">ISBN</a> <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Raamatuotsimine/0813529085?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eri:Raamatuotsimine/0813529085"><bdi>0813529085</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics%2C+the+Human+Adventure&rft.pages=45&rft.pub=Rutgers+University+Press&rft.date=2001&rft.isbn=0813529085&rft.au=Gerald+James+Holton%2C+Stephen+G.+Brush&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DczaGZzR0XOUC%26pg%3DPA45%26dq%3DKepler%2B%2522harmonic%2Blaw%2522&rfr_id=info%3Asid%2Fet.wikipedia.org%3AKepleri+seadused" class="Z3988"></li> <li id="cite_note-Burtt-11"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Burtt_11-0">↑</a> <a href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Edwin_Arthur_Burtt&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Edwin Arthur Burtt (pole veel kirjutatud)">Burtt, Edwin</a>. The Metaphysical Foundations of Modern Physical Science. p. 52.</li> <li id="cite_note-astro-12"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-astro_12-0">↑</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r6066747"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20111105015158/http://www.astro.lsa.umich.edu/undergrad/Labs/extrasolar_planets/pn_intro.html">"Arhiivikoopia"</a>. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.astro.lsa.umich.edu/undergrad/Labs/extrasolar_planets/pn_intro.html">Originaali</a> arhiivikoopia seisuga 5. november 2011. Vaadatud 6. novembril 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Arhiivikoopia&rft_id=http%3A%2F%2Fwww.astro.lsa.umich.edu%2Fundergrad%2FLabs%2Fextrasolar_planets%2Fpn_intro.html&rfr_id=info%3Asid%2Fet.wikipedia.org%3AKepleri+seadused" class="Z3988"><code class="cs1-code">{{<a href="https://et-m-wikipedia-org.translate.goog/wiki/Mall:Netiviide?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mall:Netiviide">netiviide</a>}}</code>: CS1 hooldus: arhiivikoopia kasutusel pealkirjana (<a href="https://et-m-wikipedia-org.translate.goog/wiki/Kategooria:CS1_hooldus:_arhiivikoopia_kasutusel_pealkirjana?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kategooria:CS1 hooldus: arhiivikoopia kasutusel pealkirjana">link</a>)</li> <li id="cite_note-Almanac-13"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Almanac_13-0">↑</a> Astronomical Almanac for 2008, lehekülg K7</li> <li id="cite_note-Principia-14"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Kepleri_seadused?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Principia_14-0">↑</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://books.google.com/books?id%3D6EqxPav3vIsC%26pg%3DPA232">'Principia', Book 3, Proposition 12</a></li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="Kirjandus">Kirjandus</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Kirjandus"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul> <li>Kepleri elulugu on võetud kokku lehekülgedel 523–627, samuti viiendas raamatus tema magnum opuses, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Harmonice_Mundi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Harmonice Mundi">Harmonice Mundi</a> (harmonies of the world), uuesti trükitud lehekülgedel 635–732 raamatus On the Shoulders of Giants: The Great Works of Physics and Astronomy (works by Copernicus, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Johannes_Kepler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Johannes Kepler">Kepler</a>, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Galileo_Galilei?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Galileo Galilei">Galileo Galilei</a>, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Isaac_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Isaac Newton">Newton</a>, and <a href="https://et-m-wikipedia-org.translate.goog/wiki/Albert_Einstein?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Albert Einstein">Einstein</a>). <a href="https://et-m-wikipedia-org.translate.goog/wiki/Stephen_Hawking?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stephen Hawking">Stephen Hawking</a>, 2002 <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Raamatuotsimine/0762413484?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 0-7624-1348-4</a></li> <li>J. L. Meriam "Dynamics" leheküljed 161–164, ilmunud 1966 ja 1971, toimetaja John Wiley, New york, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Raamatuotsimine/0471596019?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 0-471-59601-9</a></li> <li>Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Raamatuotsimine/0521575974?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 0-521-57597-4</a></li> <li>V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, <a href="https://et-m-wikipedia-org.translate.goog/wiki/Eri:Raamatuotsimine/0387968903?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 0-387-96890-3</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="Välislingid">Välislingid</h2> <a role="button" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Muuda alaosa "Välislingid"" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> muuda </a> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> <ul> <li>B.Surendranath Reddy; Kepleri seaduste animatsioon: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.surendranath.org/Applets/Dynamics/Kepler/Kepler1Applet.html">applet</a>.</li> <li>Crowell, Benjamin, Conservation Laws, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.lightandmatter.com/area1book2.html">http://www.lightandmatter.com/area1book2.html</a>.</li> <li>David McNamara and Gianfranco Vidali, Kepler's Second Law – Java Interactive Tutorial, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.phy.syr.edu/courses/java/mc_html/kepler.html">http://www.phy.syr.edu/courses/java/mc_html/kepler.html</a>.</li> <li>Audio – Cain/Gay (2010) <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.astronomycast.com/history/ep-189-johannes-kepler-and-his-laws-of-planetary-motion/">Astronomy Cast</a> Johannes Kepler and His Laws of Planetary Motion.</li> <li>University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion <a rel="nofollow" class="external autonumber" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://csep10.phys.utk.edu/astr161/lect/history/kepler.html">[1]</a>.</li> <li>Equant compared to Kepler: interactive model <a rel="nofollow" class="external autonumber" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://people.scs.fsu.edu/~dduke/kepler.html">[2]</a>.</li> <li>Kepler's Third Law:interactive model <a rel="nofollow" class="external autonumber" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://people.scs.fsu.edu/~dduke/kepler3.html">[3]</a>.</li> <li>Solar System Simulator (<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://user.uni-frankfurt.de/~jenders/NPM/NPM.html">Interactive Applet</a>).</li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.phy6.org/stargaze/Skeplaws.htm">Kepler and His Laws</a>.</li> <li><a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.youtube.com/watch?v%3D82p-DYgGFjI">http://www.youtube.com/watch?v=82p-DYgGFjI</a> (ingliskeelne video)</li> </ul>   </section> </div> <noscript> <img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Pärit leheküljelt "<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/w/index.php?title%3DKepleri_seadused%26oldid%3D6270518">https://et.wikipedia.org/w/index.php?title=Kepleri_seadused&oldid=6270518</a>" </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://et-m-wikipedia-org.translate.goog/w/index.php?title=Kepleri_seadused&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"> Lehekülje viimane muutmine: 16. detsember 2022, 11:43 </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Keeled</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://af.wikipedia.org/wiki/Kepler_se_wette" title="Kepler se wette – afrikaani" lang="af" hreflang="af" data-title="Kepler se wette" data-language-autonym="Afrikaans" data-language-local-name="afrikaani" class="interlanguage-link-target">Afrikaans</a></li> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://als.wikipedia.org/wiki/Keplersche_Gesetze" title="Keplersche Gesetze – šveitsisaksa" lang="gsw" hreflang="gsw" data-title="Keplersche Gesetze" data-language-autonym="Alemannisch" data-language-local-name="šveitsisaksa" class="interlanguage-link-target">Alemannisch</a></li> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D9%2582%25D9%2588%25D8%25A7%25D9%2586%25D9%258A%25D9%2586_%25D9%2583%25D8%25A8%25D9%2584%25D8%25B1_%25D9%2584%25D9%2584%25D8%25AD%25D8%25B1%25D9%2583%25D8%25A9_%25D8%25A7%25D9%2584%25D9%2583%25D9%2588%25D9%2583%25D8%25A8%25D9%258A%25D8%25A9" title="قوانين كبلر للحركة الكوكبية – araabia" lang="ar" hreflang="ar" data-title="قوانين كبلر للحركة الكوكبية" data-language-autonym="العربية" data-language-local-name="araabia" class="interlanguage-link-target">العربية</a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Lleis_de_Kepler" title="Lleis de Kepler – astuuria" lang="ast" hreflang="ast" data-title="Lleis de Kepler" data-language-autonym="Asturianu" data-language-local-name="astuuria" class="interlanguage-link-target">Asturianu</a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/Kepler_qanunlar%25C4%25B1" title="Kepler qanunları – aserbaidžaani" lang="az" hreflang="az" data-title="Kepler qanunları" data-language-autonym="Azərbaycanca" data-language-local-name="aserbaidžaani" class="interlanguage-link-target">Azərbaycanca</a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Hukum_Gerakan_Planet_Kepler" title="Hukum Gerakan Planet Kepler – indoneesia" lang="id" hreflang="id" data-title="Hukum Gerakan Planet Kepler" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneesia" class="interlanguage-link-target">Bahasa Indonesia</a></li> <li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ms.wikipedia.org/wiki/Hukum_gerakan_planet_Kepler" title="Hukum gerakan planet Kepler – malai" lang="ms" hreflang="ms" data-title="Hukum gerakan planet Kepler" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target">Bahasa Melayu</a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bn.wikipedia.org/wiki/%25E0%25A6%2595%25E0%25A7%2587%25E0%25A6%25AA%25E0%25A6%25B2%25E0%25A6%25BE%25E0%25A6%25B0%25E0%25A7%2587%25E0%25A6%25B0_%25E0%25A6%2597%25E0%25A7%258D%25E0%25A6%25B0%25E0%25A6%25B9%25E0%25A7%2580%25E0%25A6%25AF%25E0%25A6%25BC_%25E0%25A6%2597%25E0%25A6%25A4%25E0%25A6%25BF%25E0%25A6%25B8%25E0%25A7%2582%25E0%25A6%25A4%25E0%25A7%258D%25E0%25A6%25B0" title="কেপলারের গ্রহীয় গতিসূত্র – bengali" lang="bn" hreflang="bn" data-title="কেপলারের গ্রহীয় গতিসূত্র" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target">বাংলা</a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%2597%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D1%258B_%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25B0" title="Законы Кеплера – valgevene" lang="be" hreflang="be" data-title="Законы Кеплера" data-language-autonym="Беларуская" data-language-local-name="valgevene" class="interlanguage-link-target">Беларуская</a></li> <li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be-tarask.wikipedia.org/wiki/%25D0%2597%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D1%258B_%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25B0" title="Законы Кеплера – valgevene (taraškievitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Законы Кеплера" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="valgevene (taraškievitsa)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Keplerovi_zakoni" title="Keplerovi zakoni – bosnia" lang="bs" hreflang="bs" data-title="Keplerovi zakoni" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target">Bosanski</a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%2597%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D0%25B8_%25D0%25BD%25D0%25B0_%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580" title="Закони на Кеплер – bulgaaria" lang="bg" hreflang="bg" data-title="Закони на Кеплер" data-language-autonym="Български" data-language-local-name="bulgaaria" class="interlanguage-link-target">Български</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Lleis_de_Kepler" title="Lleis de Kepler – katalaani" lang="ca" hreflang="ca" data-title="Lleis de Kepler" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580_%25D1%2581%25D0%25B0%25D0%25BA%25D0%25BA%25D1%2583%25D0%25BD%25C4%2595%25D1%2581%25D0%25B5%25D0%25BC" title="Кеплер саккунĕсем – tšuvaši" lang="cv" hreflang="cv" data-title="Кеплер саккунĕсем" data-language-autonym="Чӑвашла" data-language-local-name="tšuvaši" class="interlanguage-link-target">Чӑвашла</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Keplerovy_z%25C3%25A1kony" title="Keplerovy zákony – tšehhi" lang="cs" hreflang="cs" data-title="Keplerovy zákony" data-language-autonym="Čeština" data-language-local-name="tšehhi" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cy.wikipedia.org/wiki/Deddfau_mudiant_planedau_Kepler" title="Deddfau mudiant planedau Kepler – kõmri" lang="cy" hreflang="cy" data-title="Deddfau mudiant planedau Kepler" data-language-autonym="Cymraeg" data-language-local-name="kõmri" class="interlanguage-link-target">Cymraeg</a></li> <li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Keplers_love" title="Keplers love – taani" lang="da" hreflang="da" data-title="Keplers love" data-language-autonym="Dansk" data-language-local-name="taani" class="interlanguage-link-target">Dansk</a></li> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Keplersche_Gesetze" title="Keplersche Gesetze – saksa" lang="de" hreflang="de" data-title="Keplersche Gesetze" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target">Deutsch</a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%259D%25CF%258C%25CE%25BC%25CE%25BF%25CF%2582_%25CE%25B1%25CF%2583%25CF%2584%25CF%2581%25CE%25B9%25CE%25BA%25CF%258E%25CE%25BD_%25CF%2580%25CE%25B5%25CF%2581%25CE%25B9%25CF%2586%25CE%25BF%25CF%2581%25CF%258E%25CE%25BD" title="Νόμος αστρικών περιφορών – kreeka" lang="el" hreflang="el" data-title="Νόμος αστρικών περιφορών" data-language-autonym="Ελληνικά" data-language-local-name="kreeka" class="interlanguage-link-target">Ελληνικά</a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Kepler%2527s_laws_of_planetary_motion" title="Kepler's laws of planetary motion – inglise" lang="en" hreflang="en" data-title="Kepler's laws of planetary motion" data-language-autonym="English" data-language-local-name="inglise" class="interlanguage-link-target">English</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Leyes_de_Kepler" title="Leyes de Kepler – hispaania" lang="es" hreflang="es" data-title="Leyes de Kepler" data-language-autonym="Español" data-language-local-name="hispaania" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Le%25C4%259Doj_de_Kepler" title="Leĝoj de Kepler – esperanto" lang="eo" hreflang="eo" data-title="Leĝoj de Kepler" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target">Esperanto</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Keplerren_legeak" title="Keplerren legeak – baski" lang="eu" hreflang="eu" data-title="Keplerren legeak" data-language-autonym="Euskara" data-language-local-name="baski" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D9%2582%25D9%2588%25D8%25A7%25D9%2586%25DB%258C%25D9%2586_%25DA%25A9%25D9%25BE%25D9%2584%25D8%25B1" title="قوانین کپلر – pärsia" lang="fa" hreflang="fa" data-title="قوانین کپلر" data-language-autonym="فارسی" data-language-local-name="pärsia" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Lois_de_Kepler" title="Lois de Kepler – prantsuse" lang="fr" hreflang="fr" data-title="Lois de Kepler" data-language-autonym="Français" data-language-local-name="prantsuse" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/Dl%25C3%25ADthe_Kepler" title="Dlíthe Kepler – iiri" lang="ga" hreflang="ga" data-title="Dlíthe Kepler" data-language-autonym="Gaeilge" data-language-local-name="iiri" class="interlanguage-link-target">Gaeilge</a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Leis_de_Kepler" title="Leis de Kepler – galeegi" lang="gl" hreflang="gl" data-title="Leis de Kepler" data-language-autonym="Galego" data-language-local-name="galeegi" class="interlanguage-link-target">Galego</a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EC%25BC%2580%25ED%2594%258C%25EB%259F%25AC%25EC%259D%2598_%25ED%2596%2589%25EC%2584%25B1%25EC%259A%25B4%25EB%258F%2599%25EB%25B2%2595%25EC%25B9%2599" title="케플러의 행성운동법칙 – korea" lang="ko" hreflang="ko" data-title="케플러의 행성운동법칙" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target">한국어</a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D4%25BF%25D5%25A5%25D5%25BA%25D5%25AC%25D5%25A5%25D6%2580%25D5%25AB_%25D6%2585%25D6%2580%25D5%25A5%25D5%25B6%25D6%2584%25D5%25B6%25D5%25A5%25D6%2580" title="Կեպլերի օրենքներ – armeenia" lang="hy" hreflang="hy" data-title="Կեպլերի օրենքներ" data-language-autonym="Հայերեն" data-language-local-name="armeenia" class="interlanguage-link-target">Հայերեն</a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%2595%25E0%25A5%2587%25E0%25A4%25AA%25E0%25A5%258D%25E0%25A4%25B2%25E0%25A4%25B0_%25E0%25A4%2595%25E0%25A5%2587_%25E0%25A4%2597%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A4%25B9%25E0%25A5%2580%25E0%25A4%25AF_%25E0%25A4%2597%25E0%25A4%25A4%25E0%25A4%25BF_%25E0%25A4%2595%25E0%25A5%2587_%25E0%25A4%25A8%25E0%25A4%25BF%25E0%25A4%25AF%25E0%25A4%25AE" title="केप्लर के ग्रहीय गति के नियम – hindi" lang="hi" hreflang="hi" data-title="केप्लर के ग्रहीय गति के नियम" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target">हिन्दी</a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Keplerovi_zakoni" title="Keplerovi zakoni – horvaadi" lang="hr" hreflang="hr" data-title="Keplerovi zakoni" data-language-autonym="Hrvatski" data-language-local-name="horvaadi" class="interlanguage-link-target">Hrvatski</a></li> <li class="interlanguage-link interwiki-os mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://os.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D1%258B_%25D0%25B7%25D0%25B0%25D0%25BA%25D1%258A%25C3%25A6%25D1%2582%25D1%2582%25C3%25A6" title="Кеплеры закъæттæ – osseedi" lang="os" hreflang="os" data-title="Кеплеры закъæттæ" data-language-autonym="Ирон" data-language-local-name="osseedi" class="interlanguage-link-target">Ирон</a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/L%25C3%25B6gm%25C3%25A1l_Keplers" title="Lögmál Keplers – islandi" lang="is" hreflang="is" data-title="Lögmál Keplers" data-language-autonym="Íslenska" data-language-local-name="islandi" class="interlanguage-link-target">Íslenska</a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Leggi_di_Keplero" title="Leggi di Keplero – itaalia" lang="it" hreflang="it" data-title="Leggi di Keplero" data-language-autonym="Italiano" data-language-local-name="itaalia" class="interlanguage-link-target">Italiano</a></li> <li class="interlanguage-link interwiki-he badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%2597%25D7%2595%25D7%25A7%25D7%2599_%25D7%25A7%25D7%25A4%25D7%259C%25D7%25A8" title="חוקי קפלר – heebrea" lang="he" hreflang="he" data-title="חוקי קפלר" data-language-autonym="עברית" data-language-local-name="heebrea" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ka.wikipedia.org/wiki/%25E1%2583%2599%25E1%2583%2594%25E1%2583%259E%25E1%2583%259A%25E1%2583%2594%25E1%2583%25A0%25E1%2583%2598%25E1%2583%25A1_%25E1%2583%2599%25E1%2583%2590%25E1%2583%259C%25E1%2583%259D%25E1%2583%259C%25E1%2583%2594%25E1%2583%2591%25E1%2583%2598" title="კეპლერის კანონები – gruusia" lang="ka" hreflang="ka" data-title="კეპლერის კანონები" data-language-autonym="ქართული" data-language-local-name="gruusia" class="interlanguage-link-target">ქართული</a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580_%25D0%25B7%25D0%25B0%25D2%25A3%25D0%25B4%25D0%25B0%25D1%2580%25D1%258B" title="Кеплер заңдары – kasahhi" lang="kk" hreflang="kk" data-title="Кеплер заңдары" data-language-autonym="Қазақша" data-language-local-name="kasahhi" class="interlanguage-link-target">Қазақша</a></li> <li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ky.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580_%25D0%25BC%25D1%258B%25D0%25B9%25D0%25B7%25D0%25B0%25D0%25BC%25D0%25B4%25D0%25B0%25D1%2580%25D1%258B" title="Кеплер мыйзамдары – kirgiisi" lang="ky" hreflang="ky" data-title="Кеплер мыйзамдары" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" class="interlanguage-link-target">Кыргызча</a></li> <li class="interlanguage-link interwiki-la mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://la.wikipedia.org/wiki/Leges_Keplerianae" title="Leges Keplerianae – ladina" lang="la" hreflang="la" data-title="Leges Keplerianae" data-language-autonym="Latina" data-language-local-name="ladina" class="interlanguage-link-target">Latina</a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Keplera_likumi" title="Keplera likumi – läti" lang="lv" hreflang="lv" data-title="Keplera likumi" data-language-autonym="Latviešu" data-language-local-name="läti" class="interlanguage-link-target">Latviešu</a></li> <li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lb.wikipedia.org/wiki/Gesetzer_vum_Kepler" title="Gesetzer vum Kepler – letseburgi" lang="lb" hreflang="lb" data-title="Gesetzer vum Kepler" data-language-autonym="Lëtzebuergesch" data-language-local-name="letseburgi" class="interlanguage-link-target">Lëtzebuergesch</a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Keplerio_d%25C4%2597sniai" title="Keplerio dėsniai – leedu" lang="lt" hreflang="lt" data-title="Keplerio dėsniai" data-language-autonym="Lietuvių" data-language-local-name="leedu" class="interlanguage-link-target">Lietuvių</a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Kepler-t%25C3%25B6rv%25C3%25A9nyek" title="Kepler-törvények – ungari" lang="hu" hreflang="hu" data-title="Kepler-törvények" data-language-autonym="Magyar" data-language-local-name="ungari" class="interlanguage-link-target">Magyar</a></li> <li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B2%25D0%25B8_%25D0%25B7%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D0%25B8" title="Кеплерови закони – makedoonia" lang="mk" hreflang="mk" data-title="Кеплерови закони" data-language-autonym="Македонски" data-language-local-name="makedoonia" class="interlanguage-link-target">Македонски</a></li> <li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ml.wikipedia.org/wiki/%25E0%25B4%2597%25E0%25B5%258D%25E0%25B4%25B0%25E0%25B4%25B9%25E0%25B4%259A%25E0%25B4%25B2%25E0%25B4%25A8%25E0%25B4%25A8%25E0%25B4%25BF%25E0%25B4%25AF%25E0%25B4%25AE%25E0%25B4%2599%25E0%25B5%258D%25E0%25B4%2599%25E0%25B5%25BE" title="ഗ്രഹചലനനിയമങ്ങൾ – malajalami" lang="ml" hreflang="ml" data-title="ഗ്രഹചലനനിയമങ്ങൾ" data-language-autonym="മലയാളം" data-language-local-name="malajalami" class="interlanguage-link-target">മലയാളം</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Wetten_van_Kepler" title="Wetten van Kepler – hollandi" lang="nl" hreflang="nl" data-title="Wetten van Kepler" data-language-autonym="Nederlands" data-language-local-name="hollandi" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E3%2582%25B1%25E3%2583%2597%25E3%2583%25A9%25E3%2583%25BC%25E3%2581%25AE%25E6%25B3%2595%25E5%2589%2587" title="ケプラーの法則 – jaapani" lang="ja" hreflang="ja" data-title="ケプラーの法則" data-language-autonym="日本語" data-language-local-name="jaapani" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Keplers_lover" title="Keplers lover – norra bokmål" lang="nb" hreflang="nb" data-title="Keplers lover" data-language-autonym="Norsk bokmål" data-language-local-name="norra bokmål" class="interlanguage-link-target">Norsk bokmål</a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Kepler-lovene" title="Kepler-lovene – uusnorra" lang="nn" hreflang="nn" data-title="Kepler-lovene" data-language-autonym="Norsk nynorsk" data-language-local-name="uusnorra" class="interlanguage-link-target">Norsk nynorsk</a></li> <li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oc.wikipedia.org/wiki/Leis_de_Kepler" title="Leis de Kepler – oksitaani" lang="oc" hreflang="oc" data-title="Leis de Kepler" data-language-autonym="Occitan" data-language-local-name="oksitaani" class="interlanguage-link-target">Occitan</a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Kepler_qonunlari" title="Kepler qonunlari – usbeki" lang="uz" hreflang="uz" data-title="Kepler qonunlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeki" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li> <li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pa.wikipedia.org/wiki/%25E0%25A8%2597%25E0%25A9%258D%25E0%25A8%25B0%25E0%25A8%25B9%25E0%25A8%25BF_%25E0%25A8%25AE%25E0%25A9%258B%25E0%25A8%25B8%25E0%25A8%25BC%25E0%25A8%25A8_%25E0%25A8%25A6%25E0%25A9%2587_%25E0%25A8%2595%25E0%25A9%2588%25E0%25A8%25AA%25E0%25A8%25B2%25E0%25A8%25B0_%25E0%25A8%25A6%25E0%25A9%2587_%25E0%25A8%2595%25E0%25A8%25BE%25E0%25A8%25A8%25E0%25A9%2582%25E0%25A9%25B0%25E0%25A8%25A8" title="ਗ੍ਰਹਿ ਮੋਸ਼ਨ ਦੇ ਕੈਪਲਰ ਦੇ ਕਾਨੂੰਨ – pandžabi" lang="pa" hreflang="pa" data-title="ਗ੍ਰਹਿ ਮੋਸ਼ਨ ਦੇ ਕੈਪਲਰ ਦੇ ਕਾਨੂੰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li> <li class="interlanguage-link interwiki-mnw mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mnw.wikipedia.org/wiki/%25E1%2580%259E%25E1%2581%259E%25E1%2580%25B1%25E1%2580%25AC%25E1%2580%259D%25E1%2580%25BA%25E1%2580%2580%25E1%2580%25B1%25E1%2580%2595%25E1%2580%25BA%25E1%2580%259C%25E1%2580%25B1%25E1%2580%259B%25E1%2580%25BA_%25E1%2580%2599%25E1%2580%2586%25E1%2580%25B1%25E1%2580%2584%25E1%2580%25BA%25E1%2580%2580%25E1%2580%25B5%25E1%2580%25AF_%25E1%2580%2582%25E1%2580%25BC%25E1%2580%25AD%25E1%2580%25AF%25E1%2580%259F%25E1%2580%25BA%25E1%2580%2590%25E1%2580%25AC%25E1%2580%259B%25E1%2580%25AC%25E1%2580%259A%25E1%2580%2590%25E1%2580%25B9%25E1%2580%2590" title="သၞောဝ်ကေပ်လေရ် မဆေင်ကဵု ဂြိုဟ်တာရာယတ္တ – moni" lang="mnw" hreflang="mnw" data-title="သၞောဝ်ကေပ်လေရ် မဆေင်ကဵု ဂြိုဟ်တာရာယတ္တ" data-language-autonym="ဘာသာမန်" data-language-local-name="moni" class="interlanguage-link-target">ဘာသာမန်</a></li> <li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pms.wikipedia.org/wiki/Lej_%25C3%25ABd_Kepler" title="Lej ëd Kepler – piemonte" lang="pms" hreflang="pms" data-title="Lej ëd Kepler" data-language-autonym="Piemontèis" data-language-local-name="piemonte" class="interlanguage-link-target">Piemontèis</a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Prawa_Keplera" title="Prawa Keplera – poola" lang="pl" hreflang="pl" data-title="Prawa Keplera" data-language-autonym="Polski" data-language-local-name="poola" class="interlanguage-link-target">Polski</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Leis_de_Kepler" title="Leis de Kepler – portugali" lang="pt" hreflang="pt" data-title="Leis de Kepler" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Legile_lui_Kepler" title="Legile lui Kepler – rumeenia" lang="ro" hreflang="ro" data-title="Legile lui Kepler" data-language-autonym="Română" data-language-local-name="rumeenia" class="interlanguage-link-target">Română</a></li> <li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://rue.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B2%25D1%258B_%25D0%25B7%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D1%258B" title="Кеплеровы законы – russiini" lang="rue" hreflang="rue" data-title="Кеплеровы законы" data-language-autonym="Русиньскый" data-language-local-name="russiini" class="interlanguage-link-target">Русиньскый</a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%2597%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D1%258B_%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25B0" title="Законы Кеплера – vene" lang="ru" hreflang="ru" data-title="Законы Кеплера" data-language-autonym="Русский" data-language-local-name="vene" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-szy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://szy.wikipedia.org/wiki/Ke%25E2%2580%2599pu-le%25E2%2580%2599_tinli" title="Ke’pu-le’ tinli – Sakizaya" lang="szy" hreflang="szy" data-title="Ke’pu-le’ tinli" data-language-autonym="Sakizaya" data-language-local-name="Sakizaya" class="interlanguage-link-target">Sakizaya</a></li> <li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sq.wikipedia.org/wiki/Ligjet_e_Keplerit" title="Ligjet e Keplerit – albaania" lang="sq" hreflang="sq" data-title="Ligjet e Keplerit" data-language-autonym="Shqip" data-language-local-name="albaania" class="interlanguage-link-target">Shqip</a></li> <li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://scn.wikipedia.org/wiki/Liggi_di_Kepleru" title="Liggi di Kepleru – sitsiilia" lang="scn" hreflang="scn" data-title="Liggi di Kepleru" data-language-autonym="Sicilianu" data-language-local-name="sitsiilia" class="interlanguage-link-target">Sicilianu</a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Kepler%2527s_laws" title="Kepler's laws – lihtsustatud inglise" lang="en-simple" hreflang="en-simple" data-title="Kepler's laws" data-language-autonym="Simple English" data-language-local-name="lihtsustatud inglise" class="interlanguage-link-target">Simple English</a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Keplerove_z%25C3%25A1kony" title="Keplerove zákony – slovaki" lang="sk" hreflang="sk" data-title="Keplerove zákony" data-language-autonym="Slovenčina" data-language-local-name="slovaki" class="interlanguage-link-target">Slovenčina</a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Keplerjevi_zakoni" title="Keplerjevi zakoni – sloveeni" lang="sl" hreflang="sl" data-title="Keplerjevi zakoni" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target">Slovenščina</a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25BE%25D0%25B2%25D0%25B8_%25D0%25B7%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D0%25B8" title="Кеплерови закони – serbia" lang="sr" hreflang="sr" data-title="Кеплерови закони" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target">Српски / srpski</a></li> <li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Keplerovi_zakoni_planetarnog_kretanja" title="Keplerovi zakoni planetarnog kretanja – serbia-horvaadi" lang="sh" hreflang="sh" data-title="Keplerovi zakoni planetarnog kretanja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbia-horvaadi" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Keplerin_lait" title="Keplerin lait – soome" lang="fi" hreflang="fi" data-title="Keplerin lait" data-language-autonym="Suomi" data-language-local-name="soome" class="interlanguage-link-target">Suomi</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Keplers_lagar" title="Keplers lagar – rootsi" lang="sv" hreflang="sv" data-title="Keplers lagar" data-language-autonym="Svenska" data-language-local-name="rootsi" class="interlanguage-link-target">Svenska</a></li> <li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tl.wikipedia.org/wiki/Mga_batas_mosyon_ng_mga_planeta_ni_Kepler" title="Mga batas mosyon ng mga planeta ni Kepler – tagalogi" lang="tl" hreflang="tl" data-title="Mga batas mosyon ng mga planeta ni Kepler" data-language-autonym="Tagalog" data-language-local-name="tagalogi" class="interlanguage-link-target">Tagalog</a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%2595%25E0%25AF%2586%25E0%25AE%25AA%25E0%25AF%258D%25E0%25AE%25B2%25E0%25AE%25B0%25E0%25AE%25BF%25E0%25AE%25A9%25E0%25AF%258D_%25E0%25AE%2595%25E0%25AF%258B%25E0%25AE%25B3%25E0%25AF%258D_%25E0%25AE%2587%25E0%25AE%25AF%25E0%25AE%2595%25E0%25AF%258D%25E0%25AE%2595_%25E0%25AE%25B5%25E0%25AE%25BF%25E0%25AE%25A4%25E0%25AE%25BF%25E0%25AE%2595%25E0%25AE%25B3%25E0%25AF%258D" title="கெப்லரின் கோள் இயக்க விதிகள் – tamili" lang="ta" hreflang="ta" data-title="கெப்லரின் கோள் இயக்க விதிகள்" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target">தமிழ்</a></li> <li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tt.wikipedia.org/wiki/%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580_%25D0%25BA%25D0%25B0%25D0%25BD%25D1%2583%25D0%25BD%25D0%25BD%25D0%25B0%25D1%2580%25D1%258B" title="Кеплер кануннары – tatari" lang="tt" hreflang="tt" data-title="Кеплер кануннары" data-language-autonym="Татарча / tatarça" data-language-local-name="tatari" class="interlanguage-link-target">Татарча / tatarça</a></li> <li class="interlanguage-link interwiki-te mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://te.wikipedia.org/wiki/%25E0%25B0%2595%25E0%25B1%2586%25E0%25B0%25AA%25E0%25B1%258D%25E0%25B0%25B2%25E0%25B0%25B0%25E0%25B1%258D_%25E0%25B0%2597%25E0%25B1%258D%25E0%25B0%25B0%25E0%25B0%25B9_%25E0%25B0%2597%25E0%25B0%25AE%25E0%25B0%25A8_%25E0%25B0%25A8%25E0%25B0%25BF%25E0%25B0%25AF%25E0%25B0%25AE%25E0%25B0%25BE%25E0%25B0%25B2%25E0%25B1%2581" title="కెప్లర్ గ్రహ గమన నియమాలు – telugu" lang="te" hreflang="te" data-title="కెప్లర్ గ్రహ గమన నియమాలు" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target">తెలుగు</a></li> <li class="interlanguage-link interwiki-th mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B8%2581%25E0%25B8%258E%25E0%25B8%2581%25E0%25B8%25B2%25E0%25B8%25A3%25E0%25B9%2580%25E0%25B8%2584%25E0%25B8%25A5%25E0%25B8%25B7%25E0%25B9%2588%25E0%25B8%25AD%25E0%25B8%2599%25E0%25B8%2597%25E0%25B8%25B5%25E0%25B9%2588%25E0%25B8%2582%25E0%25B8%25AD%25E0%25B8%2587%25E0%25B8%2594%25E0%25B8%25B2%25E0%25B8%25A7%25E0%25B9%2580%25E0%25B8%2584%25E0%25B8%25A3%25E0%25B8%25B2%25E0%25B8%25B0%25E0%25B8%25AB%25E0%25B9%258C%25E0%25B8%2582%25E0%25B8%25AD%25E0%25B8%2587%25E0%25B9%2580%25E0%25B8%2584%25E0%25B9%2587%25E0%25B8%259E%25E0%25B9%2580%25E0%25B8%259E%25E0%25B8%25A5%25E0%25B8%25AD%25E0%25B8%25A3%25E0%25B9%258C" title="กฎการเคลื่อนที่ของดาวเคราะห์ของเค็พเพลอร์ – tai" lang="th" hreflang="th" data-title="กฎการเคลื่อนที่ของดาวเคราะห์ของเค็พเพลอร์" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target">ไทย</a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/C%25C3%25A1c_%25C4%2591%25E1%25BB%258Bnh_lu%25E1%25BA%25ADt_Kepler_v%25E1%25BB%2581_chuy%25E1%25BB%2583n_%25C4%2591%25E1%25BB%2599ng_thi%25C3%25AAn_th%25E1%25BB%2583" title="Các định luật Kepler về chuyển động thiên thể – vietnami" lang="vi" hreflang="vi" data-title="Các định luật Kepler về chuyển động thiên thể" data-language-autonym="Tiếng Việt" data-language-local-name="vietnami" class="interlanguage-link-target">Tiếng Việt</a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Kepler%2527in_gezegensel_hareket_yasalar%25C4%25B1" title="Kepler'in gezegensel hareket yasaları – türgi" lang="tr" hreflang="tr" data-title="Kepler'in gezegensel hareket yasaları" data-language-autonym="Türkçe" data-language-local-name="türgi" class="interlanguage-link-target">Türkçe</a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%2597%25D0%25B0%25D0%25BA%25D0%25BE%25D0%25BD%25D0%25B8_%25D0%259A%25D0%25B5%25D0%25BF%25D0%25BB%25D0%25B5%25D1%2580%25D0%25B0" title="Закони Кеплера – ukraina" lang="uk" hreflang="uk" data-title="Закони Кеплера" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target">Українська</a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25DA%25A9%25DB%258C%25D9%25BE%25D9%2584%25D8%25B1_%25D9%2582%25D9%2588%25D8%25A7%25D9%2586%25DB%258C%25D9%2586_%25D8%25A8%25D8%25B1%25D8%25A7%25D8%25A6%25DB%2592_%25D8%25B3%25DB%258C%25D8%25A7%25D8%25B1%25D9%2588%25DB%258C_%25D8%25AD%25D8%25B1%25DA%25A9%25D8%25AA" title="کیپلر قوانین برائے سیاروی حرکت – urdu" lang="ur" hreflang="ur" data-title="کیپلر قوانین برائے سیاروی حرکت" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target">اردو</a></li> <li class="interlanguage-link interwiki-war mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://war.wikipedia.org/wiki/Balaod_nga_mosyon_mga_planeta_ha_Kepler" title="Balaod nga mosyon mga planeta ha Kepler – varai" lang="war" hreflang="war" data-title="Balaod nga mosyon mga planeta ha Kepler" data-language-autonym="Winaray" data-language-local-name="varai" class="interlanguage-link-target">Winaray</a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E5%25BC%2580%25E6%2599%25AE%25E5%258B%2592%25E5%25AE%259A%25E5%25BE%258B" title="开普勒定律 – uu" lang="wuu" hreflang="wuu" data-title="开普勒定律" data-language-autonym="吴语" data-language-local-name="uu" class="interlanguage-link-target">吴语</a></li> <li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://yi.wikipedia.org/wiki/%25D7%25A7%25D7%25A2%25D7%25A4%25D7%259C%25D7%25A2%25D7%25A8%2527%25D7%25A1_%25D7%2592%25D7%25A2%25D7%2596%25D7%25A2%25D7%25A6%25D7%259F" title="קעפלער'ס געזעצן – jidiši" lang="yi" hreflang="yi" data-title="קעפלער'ס געזעצן" data-language-autonym="ייִדיש" data-language-local-name="jidiši" class="interlanguage-link-target">ייִדיש</a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E9%2596%258B%25E6%2599%25AE%25E5%258B%2592%25E5%25AE%259A%25E5%25BE%258B" title="開普勒定律 – kantoni" lang="yue" hreflang="yue" data-title="開普勒定律" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target">粵語</a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E5%25BC%2580%25E6%2599%25AE%25E5%258B%2592%25E5%25AE%259A%25E5%25BE%258B" title="开普勒定律 – hiina" lang="zh" hreflang="zh" data-title="开普勒定律" data-language-autonym="中文" data-language-local-name="hiina" class="interlanguage-link-target">中文</a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-et.svg" alt="Vikipeedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">Selle lehekülje viimane muutmine: 11:43, 16. detsember 2022.</li> <li id="footer-info-copyright">Sisu on kasutatav litsentsi <a class="external" rel="nofollow" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://creativecommons.org/licenses/by-sa/4.0/deed.et">CC BY-SA 4.0</a> tingimustel, kui pole öeldud teisiti.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Andmekaitsepõhimõtted</a></li> <li id="footer-places-about"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Vikipeedia:Tiitelandmed?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB">Tiitelandmed</a></li> <li id="footer-places-disclaimers"><a href="https://et-m-wikipedia-org.translate.goog/wiki/Vikipeedia:%C3%9Cldine_lahti%C3%BCtlus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB">Lahtiütlused</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Käitumisjuhis</a></li> <li id="footer-places-developers"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://developer.wikimedia.org">Arendajad</a></li> <li id="footer-places-statslink"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://stats.wikimedia.org/%23/et.wikipedia.org">Arvandmed</a></li> <li id="footer-places-cookiestatement"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Küpsiste avaldus</a></li> <li id="footer-places-terms-use"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://foundation.m.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">Kasutustingimused</a></li> <li id="footer-places-desktop-toggle"><a id="mw-mf-display-toggle" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/w/index.php?title%3DKepleri_seadused%26mobileaction%3Dtoggle_view_desktop" data-event-name="switch_to_desktop">Lauaarvuti</a></li> </ul> </div> </footer> </div> </div> <div class="mw-notification-area" data-mw="interface"></div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-694cf4987f-tb9qw","wgBackendResponseTime":207,"wgPageParseReport":{"limitreport":{"cputime":"0.268","walltime":"0.405","ppvisitednodes":{"value":974,"limit":1000000},"postexpandincludesize":{"value":19517,"limit":2097152},"templateargumentsize":{"value":3571,"limit":2097152},"expansiondepth":{"value":7,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":29396,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 224.779 1 -total"," 57.12% 128.394 1 Mall:Viited"," 37.14% 83.492 2 Mall:Cite_book"," 35.78% 80.436 1 Mall:Keeletoimeta"," 28.64% 64.383 1 Mall:Ambox"," 6.97% 15.663 3 Mall:Netiviide"," 3.85% 8.648 1 Mall:Cite_journal"," 2.05% 4.599 3 Mall:Peidetud"]},"scribunto":{"limitreport-timeusage":{"value":"0.123","limit":"10.000"},"limitreport-memusage":{"value":4274148,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-67456955cd-6plqx","timestamp":"20241121110323","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Kepleri seadused","url":"https:\/\/et.wikipedia.org\/wiki\/Kepleri_seadused","sameAs":"http:\/\/www.wikidata.org\/entity\/Q83219","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q83219","author":{"@type":"Organization","name":"Wikimedia projektide kaast\u00f6\u00f6lised"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-07-04T09:12:43Z","dateModified":"2022-12-16T09:43:39Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/98\/Kepler_laws_diagram.svg"}</script> <script>(window.NORLQ=window.NORLQ||[]).push(function(){var ns,i,p,img;ns=document.getElementsByTagName('noscript');for(i=0;i<ns.length;i++){p=ns[i].nextSibling;if(p&&p.className&&p.className.indexOf('lazy-image-placeholder')>-1){img=document.createElement('img');img.setAttribute('src',p.getAttribute('data-src'));img.setAttribute('width',p.getAttribute('data-width'));img.setAttribute('height',p.getAttribute('data-height'));img.setAttribute('alt',p.getAttribute('data-alt'));p.parentNode.replaceChild(img,p);}}});</script> <script>function gtElInit() {var lib = new google.translate.TranslateService();lib.translatePage('et', 'en', function () {});}</script> <script src="https://translate.google.com/translate_a/element.js?cb=gtElInit&hl=en-GB&client=wt" type="text/javascript"></script> </body> </html>

CINXE.COM

Kepleri seadused – Vikipeedia