{"id":20121,"date":"2021-05-17T07:22:32","date_gmt":"2021-05-16T23:22:32","guid":{"rendered":"http:\/\/www.qiusir.com\/?p=20121"},"modified":"2021-05-17T07:22:32","modified_gmt":"2021-05-16T23:22:32","slug":"the-lagrange-points","status":"publish","type":"post","link":"https:\/\/www.qiusir.com\/?p=20121","title":{"rendered":"the Lagrange points"},"content":{"rendered":"<p><center><img decoding=\"async\" alt=\"2021\" src=\"http:\/\/www.qiusir.com\/wp-content\/gallery\/physics\/20240329Lps.jpg\" width=\"480\" class=\"framed\" \/><\/center><br \/>\n\uff08\u548cMingsongHu\u4e00\u8d77\u91cd\u65b0\u6784\u9020\u4e86\u62c9\u683c\u6717\u65e5\u70b9s\uff09<br \/>\n\u5728\u4e86\u89e3\u62c9\u683c\u6717\u65e5\u70b9\u4e4b\u524d\uff0c\u5e94\u8be5\u5148\u5bf9\u53cc\u661f\u7cfb\u7edf\u6709\u6240\u77e5\u6653\u3002\u4e3a\u4e86\u65b9\u4fbf\uff0c\u628a\u53cc\u661f\u7cfb\u7edf\u4e2d\u5927\u8d28\u91cf\u7684\u79f0\u4e3a\u6052\u661f\uff0c\u8d28\u91cf$$M_1$$\uff0c\u6bd4\u5982\u592a\u9633\uff1b\u5c0f\u8d28\u91cf\u7684\u79f0\u4e3a\u884c\u661f\uff0c\u8d28\u91cf$$M_2$$\uff0c\u6bd4\u5982\u5730\u7403\u3002<br \/>\n\u7531$$\\frac{GM_1M_2}{R^2}=M_1\\omega ^2 R_1$$\uff0c$$\\frac{GM_1M_2}{R^2}=M_2\\omega ^2 R_2$$\uff0c<br \/>\n\u5f97$$\\frac{R_1}{R_2}=\\frac{M_2}{M_1}$$\uff0c$$\\omega^2=\\frac{G(M_1+M_2)}{R^3}$$<br \/>\n\u53cc\u661f\u4ee5\u76f8\u540c\u89d2\u901f\u5ea6\u56f4\u7ed5\u8d28\u5fc3\uff08\u5230\u65cb\u8f6c\u4e2d\u5fc3\u8ddd\u79bb\u4e0e\u8d28\u91cf\u53cd\u6bd4\uff09\u65cb\u8f6c\uff0c\u89d2\u901f\u5ea6\u7531\u8d28\u91cf\u548c\u8ddd\u79bb\u51b3\u5b9a\u3002<\/p>\n<p>\u5728\u53cc\u661f\u7cfb\u7edf\u4e2d\u5f15\u5165\u7b2c\u4e09\u4e2a\u5929\u4f53\uff0c\u4e3a\u540c\u524d\u9762\u7edf\u4e00\uff0c\u8fd9\u91cc\u79f0\u536b\u661f\uff0c\u7531\u4e8e$$m< <M_2$$\uff0c\u5bf9\u53cc\u661f\u7cfb\u7edf\u8fd0\u52a8\u5f71\u54cd\u5ffd\u7565\uff0c\u5c31\u5982\u4eba\u9020\u5730\u7403\u536b\u661f\u5bf9\u5730\u7403\u548c\u6708\u4eae\u7684\u8fd0\u884c\u7684\u5f71\u54cd\u53ef\u4ee5\u5ffd\u7565\u4e00\u6837\u3002\u5728\u53cc\u661f\u7cfb\u7edf\u7684\u4e8c\u4f53\u5f15\u529b\u573a\u4e2d\uff0c\u6570\u5b66\u5bb6\u6b27\u62c9\u548c\u62c9\u683c\u6717\u7ed9\u4e00\u5171\u63a8\u7b97\u51fa\u4e865\u4e2a\u7279\u6b8a\u7684\u4f4d\u7f6e\uff0c\u80fd\u8ba9\u7b2c\u4e09\u4e2a\u7269\u4f53\u4e0e\u53e6\u4e24\u4e2a\u5929\u4f53\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4fdd\u6301\u4e0d\u53d8\uff0c\u5373\u5728\u53cc\u661f\u7cfb\u7edf\u7684\u975e\u60ef\u6027\u7cfb\u4e2d\uff0c\u536b\u661f\u7684\u52a0\u901f\u5ea6\u4e3a\u96f6\u3002\n\n\u5bf9\u4e8e$$L_1$$\uff0c\u539f\u672c\u4e0e\u6052\u661f\u8ddd\u79bb\u8d8a\u8fd1\u89d2\u901f\u5ea6\u8d8a\u5927\uff0c\u4f46\u7531\u4e8e\u6709\u884c\u661f\u7684\u5f15\u529b\u62b5\u6d88\u4e00\u90e8\u5206\uff0c\u4f7f\u536b\u661f\u4ee5\u884c\u661f\u7684\u89d2\u901f\u5ea6\u8fd0\u884c\u6210\u4e3a\u53ef\u80fd\u3002$$r$$\u4e3a\u536b\u661f\u4e0e\u884c\u661f\u8ddd\u79bb\u3002\n$$\\frac{GM_1m}{(R-r)^2}-\\frac{GM_2m}{r^2}=m\\omega^2(\\frac{M_1}{M_1+M_2}R-r)$$\n\u5316\u7b80\u5f97$$\\frac{M_1}{(R-r)^2}=\\frac{M_2}{r^2}+\\frac{M_1}{R^2}-\\frac{M_1+M_2}{R^3}r$$\n\n\u5bf9\u4e8e$$L_2$$\uff0c\u540c\u7406\uff0c\u539f\u672c\u4e0e\u6052\u661f\u8ddd\u79bb\u8d8a\u8fdc\u89d2\u901f\u5ea6\u8d8a\u5c0f\uff0c\u4f46\u7531\u4e8e\u6709\u884c\u661f\u7684\u5f15\u529b\u53c2\u4e0e\u589e\u5f3a\u4e86\u4e00\u90e8\u5206\uff0c\u4f7f\u536b\u661f\u4ee5\u884c\u661f\u7684\u89d2\u901f\u5ea6\u8fd0\u884c\u6210\u4e3a\u53ef\u80fd\u3002$$r$$\u4e3a\u536b\u661f\u4e0e\u884c\u661f\u8ddd\u79bb\u3002\n$$\\frac{GM_1m}{(R+r)^2}+\\frac{GM_2m}{r^2}=m\\omega^2(\\frac{M_1}{M_1+M_2}R+r)$$\n\u5316\u7b80\u5f97$$\\frac{M_1}{(R+r)^2}+\\frac{M_2}{r^2}=\\frac{M_1}{R^2}+\\frac{M_1+M_2}{R^3}r$$\n\n\u536b\u661f\u5728\u53cc\u661f\u7cfb\u7edf\u7684\u79bb\u5fc3\u52a0\u901f\u5ea6\uff08\u975e\u60ef\u6027\u7cfb\u7684\u79bb\u5fc3\u529b\u6240\u5bf9\u5e94\u7684\u52a0\u901f\u5ea6\uff09$$a_c=\\omega ^2(r-\\frac{M_2}{M_1+M_2}R)$$\uff0c$$a_c=\\frac{G(M_1+M_2)r-M_2R}{R^3}$$\n\u5bf9\u4e8e\u5728\u53cc\u661f\u8fde\u7ebf\u4e0a\u7684\u62c9\u683c\u6717\u65e5\u70b9\uff0c\u53ef\u4ee5\u5229\u7528\u76f8\u5bf9\u52a0\u901f\u5ea6\u4e3a\u96f6\u6c42\u89e3\uff08\u56fe\uff09\u3002$$a=-\\frac{GM_1}{r^2}sgn(r)+\\frac{GM_2}{(R-r)^2}sgn(R-r)+a_c$$\n<br \/>\n<img decoding=\"async\" alt=\"2021\" src=\"http:\/\/www.qiusir.com\/wp-content\/gallery\/physics\/L45.jpg\" width=\"480\" class=\"framed\" \/><\/p>\n<p>\u5bf9\u4e8e\u62c9\u683c\u6717\u65e5\u70b9\u7684\u8ba8\u8bba\uff0c\u66f4\u559c\u6b22$$L_{4,5}$$\u7684\u8ba8\u8bba\uff0c\u7b80\u6d01\u3001\u5de7\u5999\u3002<br \/>\n$$\\frac{GM_1M_2}{R^2}=M_1a_1$$\uff0c\u5f97$$a_1=\\frac{GM_2}{R^2}$$\uff0c\u540c\u7406\u5f97$$a_2=\\frac{GM_1}{R^2}$$\uff0c\u5f97$$a_{14}=\\frac{GM_1}{R^2}$$,$$a_{24}=\\frac{GM_2}{R^2}$$\uff0c\u6839\u636e\u56fe\u4e2d\u51e0\u4f55\u5173\u7cfb$$\\frac{R_1}{R_2}=\\frac{M_2}{M_1}$$\uff0c$$a_4$$\u4e0d\u4ec5\u6307\u5411\u53cc\u661f\u7684\u8d28\u5fc3\uff0c\u7531$$a=\\omega^2r$$\uff0c\u4e14\u6ee1\u8db3\u89d2\u901f\u5ea6\u76f8\u540c\u3002<\/p>\n<p>\u5982\u679c\u4ec5\u4ece\u4e2d\u5b66\u751f\u505a\u9898\u7684\u89d2\u5ea6\u770b\uff0c\u53ea\u9700\u8981\u638c\u63e1\u5728\u62c9\u683c\u6717\u65e5\u70b9\u7684\u536b\u661f\u548c\u53cc\u661f\u5177\u6709\u76f8\u540c\u89d2\u901f\u5ea6\u5373\u53ef\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uff08\u548cMingsongHu\u4e00\u8d77\u91cd\u65b0\u6784\u9020\u4e86\u62c9\u683c\u6717\u65e5\u70b9s\uff09 \u5728\u4e86\u89e3\u62c9\u683c\u6717\u65e5\u70b9\u4e4b\u524d\uff0c\u5e94\u8be5\u5148\u5bf9\u53cc\u661f\u7cfb\u7edf\u6709\u6240\u77e5\u6653\u3002\u4e3a\u4e86\u65b9\u4fbf &#8230; <a title=\"the Lagrange points\" class=\"read-more\" href=\"https:\/\/www.qiusir.com\/?p=20121\" aria-label=\"\u9605\u8bfb the Lagrange points\">\u9605\u8bfb\u66f4\u591a<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[6],"tags":[247,242],"class_list":["post-20121","post","type-post","status-publish","format-standard","hentry","category-mpo","tag-physics","tag-qiusir"],"_links":{"self":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/posts\/20121","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=20121"}],"version-history":[{"count":0,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/posts\/20121\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=20121"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=20121"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=20121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}