科技三定律 ：&br&1.任何在我出生时已经有的科技都是稀松平常的世界本来秩序的一部分。 &br&2.任何在我15-35岁之间诞生的科技都是将会改变世界的GM性产物。 &br&3.任何在我35岁之后诞生的科技都是违反自然规律要遭天谴的。 &br&
　　——英国科幻作家道格拉斯·亚当斯&br&&br&社会三定律：&br&1.任何比我早出生10年及以上的人都是裹步不前的老顽固。&br&2.任何出生时间和我相差10年以内的人都是这个社会的精英，中流砥柱。&br&3.任何比我晚出生10年及以上的人都是无可救药垮掉的一代。&br&
——纳什·沃夏尔·硕德
科技三定律 ： 1.任何在我出生时已经有的科技都是稀松平常的世界本来秩序的一部分。 2.任何在我15-35岁之间诞生的科技都是将会改变世界的GM性产物。 3.任何在我35岁之后诞生的科技都是违反自然规律要遭天谴的。 ——英国科幻作家道格拉斯·亚当斯 社会三定…
&figure&&img src=&https://pic4.zhimg.com/v2-9dd8b599ec721b268f37ac3f409e4b43_b.jpg& data-rawwidth=&900& data-rawheight=&444& class=&origin_image zh-lightbox-thumb& width=&900& data-original=&https://pic4.zhimg.com/v2-9dd8b599ec721b268f37ac3f409e4b43_r.jpg&&&/figure&&p&&i&本文首发于公众号熊猫茶园&/i&&/p&&p&&i&————————————————————&/i&&/p&&p&&i& 我把三只大泥锅和两三只泥罐一个搭一个地堆起来，四面架上木柴，木柴底下放上一大堆炭火，然后从四面和顶上点起火来，一直烧到里面的罐子红透为止，而且当心不让它们炸裂。我看见它们已经红透之后，又继续让它们保留五六个小时的热度……我慢慢熄灭火，让那些罐子的红色逐渐退下去，而且整夜地守着它，不让火退得太快。到了第二天早晨，我便烧出了三只很好的瓦锅和两只瓦罐，虽然不能说美观，却烧得再硬也没有了，而且其中的一只由于沙土烧熔了，有一层很好的釉。经过这次试验成功之后，不用说，我不缺什么陶器用了。&/i&&/p&&p&也许有的朋友认出来了，这是《鲁滨孙漂流记》中，写烧制陶器的一段。鲁滨孙在荒岛上的十几年，简直就是早期人类历史的缩影。一开始猎山羊采野果为食，对应狩猎和采集时代；然后种植小麦、驯服山羊、烧陶器，进入了农耕时代；再造船，进入航海时代；然后收服星期五，开始与不同文明的交流融合。&figure&&img src=&https://pic1.zhimg.com/v2-a23c345ec58e471ac81d7_b.jpg& data-rawwidth=&585& data-rawheight=&414& class=&origin_image zh-lightbox-thumb& width=&585& data-original=&https://pic1.zhimg.com/v2-a23c345ec58e471ac81d7_r.jpg&&&/figure&&/p&&p&先民发现篝火烧过的地面会变硬，于是把粘土捏成人偶放在火里烧，就成了最早的陶器。陶人偶是用于宗教活动的，最早的有近三万历史，那时人类很唯心，发明音乐舞蹈绘画等等，初衷都并非审美，而是与鬼神交流。又过了一万年，人们才想起陶还能做成瓶瓶罐罐，装水盛酒，现在最早的陶罐是在日本发现的，二万年历史。烧制陶器是人类进入新石器时代的一个标志，古代所有的农耕文明都会做陶。 &/p&&p&最早的烧陶方法，和鲁滨孙一样，把陶坯埋在柴里烧，为保持温度，又用泥把坯和柴都封起来，这叫堆烧，现在云南一些地民族地区还有这种烧法。堆烧的热量损失太快，柴又经常压坏陶坯，古人把坯和柴分开放置，用厚土层保温，窑出现了。 &figure&&img src=&https://pic2.zhimg.com/v2-77f56d6e4b5f95f5a2c99e_b.jpg& data-rawwidth=&1000& data-rawheight=&690& class=&origin_image zh-lightbox-thumb& width=&1000& data-original=&https://pic2.zhimg.com/v2-77f56d6e4b5f95f5a2c99e_r.jpg&&&/figure&&/p&&p&这是仰韶文化的陶窑，距今七千年。是在粘土层里挖出来的，类似于现在的陕北窑洞。柴在火膛里燃烧，热量通过倾斜的火道输送到窑室里，窑室略高于火膛，叫横穴窑，最早的窑。&figure&&img src=&https://pic4.zhimg.com/v2-0dd035a8f2ab_b.jpg& data-rawwidth=&868& data-rawheight=&1000& class=&origin_image zh-lightbox-thumb& width=&868& data-original=&https://pic4.zhimg.com/v2-0dd035a8f2ab_r.jpg&&&/figure&&/p&&p&四千年前的龙山文化里，窑演变成这个结构。火膛在窑室的正下方，火焰直接升入窑室，叫竖穴窑。理论上竖穴窑比横穴窑先进，不如说横窑比竖窑更粗糙，我曾经瞻仰过半坡的窑址，原始人居然能在那种破土坑里烧出漂亮的陶罐，真牛逼。&figure&&img src=&https://pic2.zhimg.com/v2-f13fb238eeb716c12750fe_b.jpg& data-rawwidth=&640& data-rawheight=&346& class=&origin_image zh-lightbox-thumb& width=&640& data-original=&https://pic2.zhimg.com/v2-f13fb238eeb716c12750fe_r.jpg&&&/figure&&i&马家窑文化彩陶公元前3300&/i&&i&到2100&/i&&/p&&p&窑的初心是升温，大家知道，陶的温度再提升，就成瓷了。在我们祖先发明瓷之前，有个变故，一种蓝色的石头掉到窑里，熔化后析出坚硬的晶体，如果加入锌和锡，还能做成器皿，于是中国进入了青铜时代。&figure&&img src=&https://pic1.zhimg.com/v2-fd587b1ddec066e6ce03e23_b.jpg& data-rawwidth=&1000& data-rawheight=&991& class=&origin_image zh-lightbox-thumb& width=&1000& data-original=&https://pic1.zhimg.com/v2-fd587b1ddec066e6ce03e23_r.jpg&&&/figure&&i&商代青铜壶，早期瓷器曾经大量模仿青铜器&/i&&/p&&p&冶炼青铜的叫炉，是另一个体系了，不过铜肯定是在烧陶的过程中发现的。糟糕的是，青铜器和漆器兴起后，陶器被边缘化了，不入富贵人家，现在惟一能撑场面的陶器是明代发展起来的紫砂。 &/p&&p&问题是，世界上那么多伟大的文明，为什么只有中国发明了瓷，并且垄断二千年？最多的解释是，中国人喜欢玉，希望把陶烧出玉的效果，就是青瓷的由来。也许还能解释为，东周时的陶工眼红青铜器的火暴，把窑温烧的比铜炉还高，终于搞成了大新闻，实现了伟大逆袭。&figure&&img src=&https://pic4.zhimg.com/v2-689ca79e2c5e7f091aa0c7_b.jpg& data-rawwidth=&800& data-rawheight=&913& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic4.zhimg.com/v2-689ca79e2c5e7f091aa0c7_r.jpg&&&/figure&&i&晋代青瓷谷仓&/i&&/p&&p&那么怎样才算瓷呢？有的解释是，陶用粘土，瓷用高岭土，这不是本质区别，和釉也没关系。选高岭土做瓷器，因为其耐高温。瓷土里有一大半二氧化硅，就是玻璃的成分，这些硅在一千一百度时，大部分熔融了，冷却后形成类似玻璃的物质，而陶器就没有这样的过程。所以与陶相比，瓷坚硬、透明、不吸水，以指叩之，其音清脆。 &/p&&p&&figure&&img src=&https://pic4.zhimg.com/v2-1f79c60cf83ce4ce765a19f8bf05a9c7_b.jpg& data-rawwidth=&1024& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic4.zhimg.com/v2-1f79c60cf83ce4ce765a19f8bf05a9c7_r.jpg&&&/figure&&figure&&img src=&https://pic3.zhimg.com/v2-aba2dd0f2a1f0e08114b_b.jpg& data-rawwidth=&746& data-rawheight=&395& class=&origin_image zh-lightbox-thumb& width=&746& data-original=&https://pic3.zhimg.com/v2-aba2dd0f2a1f0e08114b_r.jpg&&&/figure&烧瓷器，横穴竖穴窑都不成，最早的瓷器应该诞生在馒头窑里。这种窑半圆形，状如馒头，始于春秋战国，一直持续到现在。馒头窑不再挖土为窑，而是砖石结构，火膛和窑室合为一体，有单门双门两种。馒头窑窑壁厚，面积小，方便控制，可以把温度烧的很高。宋朝北方木柴紧缺，还出现了烧煤的馒头窑，比如定窑。馒头窑还有马蹄窑、葫芦窑等变种，结构大同小异。&figure&&img src=&https://pic2.zhimg.com/v2-75be2d50bcf_b.jpg& data-rawwidth=&675& data-rawheight=&395& class=&origin_image zh-lightbox-thumb& width=&675& data-original=&https://pic2.zhimg.com/v2-75be2d50bcf_r.jpg&&&/figure&&/p&&p&馒头窑主要分布在北方，南方流行的是龙窑。龙窑大，小的长十几米，长的一百多米，修在山坡上，非常壮观，远看像一条巨龙，故名龙窑。龙窑的主火膛在低部，窑室倾斜上升，就像一个大烟囱，把底部的热量向上抽，窑身上还分部着许多投柴孔，可以分段烧成。龙窑的优点是装烧量大，但不像馒头窑那么易于控制，降温太快，适合烧龙泉系的青瓷。&figure&&img src=&https://pic4.zhimg.com/v2-642d5bdedbb04c3b2d5e1bf544b9c6fc_b.jpg& data-rawwidth=&640& data-rawheight=&403& class=&origin_image zh-lightbox-thumb& width=&640& data-original=&https://pic4.zhimg.com/v2-642d5bdedbb04c3b2d5e1bf544b9c6fc_r.jpg&&&/figure&&i&宜兴烧紫砂的龙窑&/i&&/p&&p&唐代陆龟蒙有一句“九州风露越窑开，夺得千峰翠色来”，写的是越窑烧青瓷的情景。龙窑太大了，难以控制窑内氧气的浓度，而这是烧青瓷的关键。所以龙窑要在深秋或初冬天气干燥时点火，诗中“九州风露”，就是点明了窑工的对时机的选择。而“夺得千峰翠色来”的“夺”字，写出瓷工的斗天斗地，处心积虑，可谓神来之笔。&figure&&img src=&https://pic2.zhimg.com/v2-f56abf5340f_b.jpg& data-rawwidth=&650& data-rawheight=&426& class=&origin_image zh-lightbox-thumb& width=&650& data-original=&https://pic2.zhimg.com/v2-f56abf5340f_r.jpg&&&/figure&&/p&&p&明代湖建德化发明了阶级窑，集合了龙窑和馒头窑的优势。阶级窑就像梯田一样，每一级修一座馒头窑，连起来形成龙窑，容量大而且易于控温。插一句广东石湾，也是龙窑，但当地人称之为“南风灶”，很亲切的感觉，是中国唯一把窑称为灶的地方。&figure&&img src=&https://pic4.zhimg.com/v2-db9d2579cb15aab2a08df_b.jpg& data-rawwidth=&407& data-rawheight=&480& class=&content_image& width=&407&&&/figure&&/p&&p&元代开始，景德镇开始成为制瓷中心。景德镇是拿来主义，先后引进了南方的龙窑，北方的馒头窑，经过了葫芦窑的技术改造，明末定型为标准化的卵形窑，就是镇窑，烧松木柴，当地称为柴窑。&figure&&img src=&https://pic1.zhimg.com/v2-5ba5bcbb2eaeeb096477e_b.jpg& data-rawwidth=&600& data-rawheight=&626& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic1.zhimg.com/v2-5ba5bcbb2eaeeb096477e_r.jpg&&&/figure&&/p&&p&镇窑外形像半个鸡蛋，建在窑房里。窑体长十五米，宽五米，前高后低，前宽后窄，窑底从火膛向后部的烟囱逐渐向上倾斜。窑壁较薄，二十厘米厚，窑壁的两侧再砌砖墙加固，墙窑壁之间有三十厘米的缝隙，是为窑体受热膨胀的预留空间，其内灌满保温用的沙土。镇窑每窑可以烧十几吨瓷器，是中国古代结构最先进的瓷窑，就以此为例，讲一下过去烧窑的流程。 &/p&&p&首先要把瓷坯装进窑里，叫满窑，其难度不亚于把大象装进冰箱。过去满窑是一门职业，叫托坯工，重体力活。瓷窑无论烧柴还是煤，都有大量烟尘，会把瓷器熏黑。唐代开始把瓷器装进圆柱形陶罐里，叫匣钵，把许多匣钵摞起来，最上加个盖，就可以隔绝烧窑时的烟尘。瓷坯装在匣钵里的方式，还有覆烧、叠烧、支钉烧等许多方法。 &/p&&p&托坯工把瓷坯装进相应尺寸的匣钵里，然后搬到窑室里摞起来。第一要紧凑，留出火路的前提下，最大限度地利用窑内空间，第二要把不同的瓷，放在不同的窑位。窑的火膛在前，烟囱在后，热量从前向后传递，有一个递减的过程，所以窑里有五个温度位置。&figure&&img src=&https://pic4.zhimg.com/v2-d33adfbdc557ce43a40ac30_b.jpg& data-rawwidth=&1200& data-rawheight=&800& class=&origin_image zh-lightbox-thumb& width=&1200& data-original=&https://pic4.zhimg.com/v2-d33adfbdc557ce43a40ac30_r.jpg&&&/figure&&/p&&p&离火膛最近的叫大肚，温度最高，能到1320度，这儿的瓷器都是VIP中P，比如祭红、郎红、官窑用的白胎、天青、钧花等等；&/p&&p&大肚之后是小肚，温度略低，1300度，烧精品青花、釉里红、美人醉、各色青釉；&/p&&p&然后是低温区，到1260度，叫想理或者灰可区，属于散座，烧低档瓷和低温颜色釉，比如浇黄、松石绿；&/p&&p&再后面就是烟囱了，烟囱下面是温度最低之处，叫余膛，烧不了瓷，一般用来烧匣钵；&/p&&p&窑背隆起的最高处叫观音膛，也是低温，烧建窑用的砖。 &/p&&p&满窑结束后，用砖砌上窑门，留两个孔，一个点火，一个进柴，烧窑开始。所有工作都是在一个老者的注视下完成的，他坐在窑门外的交椅上喝茶，时不时指挥几句。他就是把桩，烧窑总管，一度是景德镇收入最高的职业。 &figure&&img src=&https://pic1.zhimg.com/v2-8af63b5ef8cce92d88fe6a5a_b.jpg& data-rawwidth=&642& data-rawheight=&423& class=&origin_image zh-lightbox-thumb& width=&642& data-original=&https://pic1.zhimg.com/v2-8af63b5ef8cce92d88fe6a5a_r.jpg&&&/figure&&/p&&p&烧瓷器最讲究温度，有些颜色釉成瓷温度只在上下十度以内，过去又没有温度计，全靠把桩的经验。判断窑温有三种方法，一是观察火焰的颜色，不同温度的火焰会有不同的颜色。每隔一段时间，把桩用铁钎钩出事先放入窑内的圆瓷片，叫火照，通过看其变软的程度判断温度。还可以向窑里吐痰，口水都不行，必须是痰。痰在蒸发时会变成跳动的水珠，把桩根据其高度判断窑温。如果窑温低，加干柴，窑温高，加湿柴，甚至泼水进去。 &/p&&p&烧窑共四天，满窑一天，烧一天，熄火冷却一天，第四天开窑。开窑工人叫收沙帽，负责把匣钵从窑里搬出来，当时窑内的温度将近一百度，所以收沙帽都是极耐烫之人。为什么要趁热开窑呢？因为要利用窑内的余温烘干下一窑的瓷坯。开窑完工后，满窑工马上进场，开始装下一窑。这种循环持续了几百年，直到上个世纪，事情发生了变化。 &/p&&p&五十年代开始，先后出现了轨道煤窑、重油窑、电窑，九二年出现了天然气窑，摧枯拉朽，三年后景德镇最后一座柴窑停烧了。所以刚才写到的各种窑、各种烧法、各种职业，全成了屠龙剑法，没用了。别误会，我一点也不伤感，写到气窑时我有种忆苦思甜的快感，终于解放了。&/p&&p&&i&&figure&&img src=&https://pic1.zhimg.com/v2-b341e50e6edd4aa4df1c_b.jpg& data-rawwidth=&519& data-rawheight=&346& class=&origin_image zh-lightbox-thumb& width=&519& data-original=&https://pic1.zhimg.com/v2-b341e50e6edd4aa4df1c_r.jpg&&&/figure&气窑，金属外壳，其中是很厚的石棉保温层，窑底四周有多个天然气孔，中间有钢轨，可以把满好瓷坯的窑车直接推进去&/i&&/p&&p&镇窑的柴瓷比是2.5，就是制一吨瓷，要烧掉二吨半木柴，所过去景德镇有句老话“一里窑，十里渣”，指一座窑会烧光十里森林，再铺满灰渣。气窑的好处是不再破坏植被，人畜无害，而且烧气没有烟尘，不会熏黑瓷坯，也就用不到匣钵，省了好多成本。还有一点，没有烟尘就没有pm2.5，不产生雾霾。现在有一些怀旧人士在恢复柴窑，我不赞同，这不是复古，是复辟。&/p&&p& ————————————&/p&&p&&b&抄我文章者，虽远必猪&/b&&/p&&p&有泉堂公众号 youquantang&/p&
本文首发于公众号熊猫茶园———————————————————— 我把三只大泥锅和两三只泥罐一个搭一个地堆起来，四面架上木柴，木柴底下放上一大堆炭火，然后从四面和顶上点起火来，一直烧到里面的罐子红透为止，而且当心不让它们炸裂。我看见它们已…
&figure&&img src=&https://pic1.zhimg.com/v2-2a8b4ff9fcdbed65_b.jpg& data-rawwidth=&640& data-rawheight=&360& class=&origin_image zh-lightbox-thumb& width=&640& data-original=&https://pic1.zhimg.com/v2-2a8b4ff9fcdbed65_r.jpg&&&/figure&(本文由本人翻译自&a href=&https://link.zhihu.com/?target=http%3A//io9.gizmodo.com/what-would-the-earth-be-like-if-it-was-the-shape-of-a-d-& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&io9.gizmodo.com/what-wo&/span&&span class=&invisible&&uld-the-earth-be-like-if-it-was-the-shape-of-a-d-&/span&&span class=&ellipsis&&&/span&&/a&,渣翻，因此中英对照，不要在意)&p& According to the laws of physics, a planet the shape of a donut, or toroid, could actually exist — but it's extremely unlikely to ever form naturally. But what if an advanced alien civilization decided to build one? What properties would a toroid-Earth exhibit? And what would life be like?&br&&br&根据物理定律，一个甜甜圈形的，或者环面形的星球是可能实际存在的-但是这几乎不可能自然形成。不过如果一个高级外星文明想要做一个这样的星球呢？它将展现出怎样的特性？它上面的生物又会是怎样的呢？&br&&/p&&p&One question that came up when I published my Double Earth analysis was &What about a toroidal Earth?& This is by no means a new question, and there has been some lengthy discussions online and earlier modelling. But being a do-it-yourself person I decided to try to analyze it on my own.&br&&br&当我发表我的双地球分析的时候，产生了一个问题：“如果有一个环形的地球会怎样？”这绝不是一个新问题。现在网上已经有了一些冗长的分析，也有一些早期模型。但是作为一个喜欢自己动手的人，我决定做一些自己的分析&br&&/p&&h2&&b&Can Toroid Planets Exist?&br&&/b&&b&一个环形的星球能否存在&/b&&/h2&&br&It is &b&not&/b& obvious that a toroid planet is stable.&br&&br&环形星球的稳定性&b&不是&/b&显然的。&br&&br&For all practical purposes planets are liquid blobs with no surface tension: the strength of rock is nothing compared to the weight of a planet. Their surfaces will be equipotential surfaces of gravity plus centrifugal potential. If they were not, there would be some spots that could reduce their energy by flowing to a lower potential. Another obvious fact is that there exists an upper rotation rate beyond which the planet falls apart: the centrifugal force at the equator becomes larger than gravity and material starts to flow into space.&br&&br&星球实际上是一团没有表面张力的液体团：石头的强度比起星球的重量可以忽略不计。他们的表面一定是重力势能和等效离心势能之和的等势面。如果表面离这个等势面有偏离的话，一定会存在某些点，使得把这些点的物质转移到势能更低的地方可以降低总体的能量。另外一个显然的事实是，存在一个角速度上限，如果行星转速高于这个上限，就会崩溃：赤道上的离心力比重力更大，使得物质开始飞向太空。&br&&br&The equilibrium shapes of self-gravitating rotating ellipsoidal planets have been extensively analyzed. Newton started it (leading to some early heroic expeditions to ascertain the true shape of Earth), and Maclaurin refined it. Then Jacobi discovered that high rotation rates ellipsoids with unequal axes were more stable than the oblate ellipsoids of Maclaurin. Chandrasekar has a nice history of the field. Since then computers have become available, and analytical and numerical calculations of more complex or the relativistic case have been performed.&br&&br&只受自身重力的旋转椭园星球的平衡形状已经被大量的分析过了。牛顿开始了这项工作（推动了几次测量地球真正形状的英勇探险（译者注：指1740年法国科学院组织的，分别前往北欧拉普尔和南美厄瓜多尔的两支探险队）），麦克劳林将其完善。其后，雅可比发现对于高转速的椭圆行星，不等轴椭园会比麦克劳林的扁椭园更加稳定。钱德拉塞卡在这个领域做过很好的研究（译者注：钱德拉塞卡是著名星体动力学家，曾在1983年因在星体结构和进化的研究获得诺贝尔奖）。从那之后，计算机变得实用，更加复杂情况的解析与数值解，或者是相对论情形下的问题也都已经被处理过了&br&&br&Similarly, equilibrium states of self-gravitating toroid shapes have been examined by Poincare, Kowalewsky and Dyson (Dyson 1893, Dyson 1893b). Indeed, one can at least in theory spin up an ellipsoidal planet into a ring, although there is plenty of potential for complex wobbles that destabilize the whole system and it looks like there is a &jump& to the ring state. The ring may itself be unstable, in particular to a &bead& instability where more and more mass accumulates at some meridians than others, leading to breakup into two or more orbiting blobs. Dyson analysed this case and found it relevant when the major radius / minor radius & 3 – thin hoops are unstable. There is also a lower rotation rate where the ring become unstable to tidal forces and implodes into a &hamburger& or ellipsoid. So the total mass and angular momentum needs to be in the right region from the start.&br&&br&同样的，在自身重力场下处于平衡状态的甜甜圈形行星也已经被 Poincare,Kowalewsky和Dyson (Dyson 1893, Dyson 1893b)测试过了。确实，你可以至少在理论上把一个椭园行星转成一个圆环，尽管那些复杂的晃动有很大的潜在可能会把整个系统搞得不稳定，而且整个星球就像是突然跳变到了一个环形的状态。这个环可能自己就不稳定了，特别是可能变成某种“念珠”式的不稳定，质量会不停的在某些特定的子午圈聚集，使得星球最后分裂成两个或者更多相互环绕的团块。Dyson分析了这种情况，并且关键在于外环半径/内环半径 & 3这个不等式-大于3时就会导致这个环不稳定，同时还存在一个更低的角速度，使得这个环在潮汐力下不稳定，并且内爆成“汉堡”或者是椭园形。所以初始总质量和角动量必须要在一个合适的区间内&br&&br&It looks like a toroid planet is not forbidden by the laws of physics. It is just darn unlikely to ever form naturally, and likely will go unstable over geological timescales because of outside disturbances. So if we decide to assume it just is there, perhaps due to an advanced civilization with more aesthetics than sanity, what are its properties?&br&&br&似乎一个环形的星球是不被物理定律禁止的。只是在自然条件下及其难以产生，并且容易在地质学时间尺度下由于外界干扰变得不稳定（译者注：比如被潮汐力改变了角动量）。所以我们决定假设就是有这么一个行星，也许因为某个崇尚美学高于理性的高级文明造了这么一个东西，那么，它的特性是什么？&br&&br&&h2&&b&Directions&br&&/b&&b&方向&/b&&/h2&&br&I will call the two circles along the plane of rotation the equators (the inner and outer). When it does not matter which one I talk about I will just call it &the equator.& As for the poles, they are the circles furthest away from the equatorial plane.&br&&br&我管转动平面上的两个圆叫赤道（一个内部的一个外部的）。如果不管我说的是哪个都没有区别的时候我就直接叫“赤道”。至于两极，它们是离赤道面最远的圆。&br&&br&Hubward is towards the rotation axis, rimward is away from it. Planewards is towards the equatorial plane. North is towards the closest part of the North Pole circle, south towards the closest part of the South Pole circle.&br&&br&向心方向是朝向旋转轴，向边方向是远离旋转轴。平面方向是朝向赤道面。北方是朝向最接近北极环的方向，南极是最接近南极环的方向。&br&&br&&h2&&b&Toroid Gravity&br&&/b&&b&环重力&/b&&/h2&&br&How does gravity work on a toroid planet?&br&&br&在这样一个环形星球上，重力将如何作用&br&&br&The case of a very large main radius torus is essentially a cylindrical planet. In this case the gravitational force falls off as 1/r, where r is the distance from the axis. The total force on any section will be proportional to the total mass (proportional to R, the major radius) and the gravitational force (proportional to 1/R), so the overall force will be constant as we increase R. Adding some rotation will balance it. The surface gravity is 2G rho/r, where rho is the mass per unit length. So as long as the surface gravity is big enough (by having a small r) this will overcome the centrifugal acceleration and stuff will indeed stay down. But things are much harder to guess for small radius tori.&br&&br&一个有着非常大的主半径的圆环实际上就是一个圆柱体的星球，在这种情况下重力将会以1/r的速度减小，这里r是到转轴的距离。作用在任何一个部分的合力将会和总质量成比例（与主半径R成比例）并且重力（与1/R成比例），所有全部的力之和将会在与R无关。增加一些转动将会使它可以平衡它。表面上的重力大小为2G rho/r，其中rho是线密度。所以只要表面重力足够大（r很小的情况），就可以克服离心加速度，物质从而能够保留在表面。但是在这样一个小的圆环面上事物是怎样的就很难想象了。&br&&br&I decided to use a Monte Carlo method to estimate the equilibrium shape. Given the total planetary mass and angular momentum, I start out by distributing a number of massive but infinitely thin rings (with the potential borrowed from this physics exercise — it is a good thing electric and gravitational potentials look the same in classical physics). I calculated their joint potential and added a centrifugal potential. This allowed me to approximate equipotential surfaces and &fill& the potential near the center of the torus with more and more rings until their mass correspond to the planetary mass. I recalculated the angular speed based on the new mass distribution. Then I repeated the process until the planet either flew apart, imploded into a ball, or until enough iterations went by. Admittedly, this wasn't the most elegant way of doing it (the literature uses series expansions in toridal harmonics), but it worked for me.&br&&br&我决定使用蒙特卡罗方法来估计平衡位形。给出行星的总质量和总角动量，我开始分配一些有限质量但是无限细的环（带有了从这样一个物理技巧中计算出的势能-在经典物理中，电力势和重力势形式一样，在计算上带来了方便）。我计算了他们的势能和，并且加上了等效离心势能。这允许我估算出等势面，并且用越来越多的环“填上”靠近环中心的势能，直到这些环的质量等于行星质量。我基于新的质量分布重新计算了角速度。然后我重复这个过程知道行星既不散开，也不会汇聚成一个球，或者知道迭代了足够多的次数。无可否认，这不是求解这个问题最优雅的办法（文献中使用的方法是做环形谐波的级数展开），不过对我来讲能用。&br&&br&The main result is that toroid planets look feasible for sufficiently large enough angular momentum and mass. The cross-section is neither circular nor elliptic but rather egg-shaped, with a slightly sharper inside curvature than on the outside.&br&&br&主要的结果是对于足够大质量和角动量的星球来说，环形星球的想法看上去是行得通的。星球的横截面看起来既不是圆的也不是椭圆的，更接近是一种鸡蛋形的，内部的曲率要比外部稍大些。&br&&br&Why doesn't the planet get squashed into a plane disk? The rotational pull tries to flatten the planet, but it must act against the local gravity field which tries to turn it into a ball (or cylinder).&br&&br&为什么行星不会压瘪成一张平圆盘呢？旋转会尝试把这个星球压平，但是这个力量必须要和它此处的重力场相抗衡，重力场会倾向于把星球变成球形的。&br&&br&While these planets are stable in my simulation, the range of feasible values is not huge: most combinations of mass and angular momentum are unstable. And I have not examined the tricky issue of bead instability.&p&尽管这些行星确实在我的模拟中稳定了，但是稳定值的范围并不大：大多数质量和角动量的组合都是不稳定的。并且我没对难办的念珠状结构的稳定性进行测试（译者：因为他用的模型是柱坐标下关于转角对称的）&br&&br&&br&I will look at a chubby toroid of one Earth mass and a small central hole (&Donut&), and a wider hoop-like toroid with 6 Earth masses but more earth-like gravity (&Hoop&).&br&&br&我将会考虑一个具有一个地球质量，小的中心孔的胖一些的圆环（“甜甜圈”），和一个更大的，具有6个地球质量但是更像地球重力的箍形圆环（“箍”）&br&&/p&&br&&h2&&b&Donut Earth&br&&/b&&b&甜甜圈地球&/b&&/h2&&figure&&img src=&https://pic3.zhimg.com/v2-86c326a04195acba2d434c7ba8d0e6e9_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic3.zhimg.com/v2-86c326a04195acba2d434c7ba8d0e6e9_r.jpg&&&/figure&&p&Figure 1: Local gravitational acceleration (m/s2) around Donut, as experienced by a co-rotating object.&/p&&br&&p&图片1：环上各点的本地重力加速度，在一个正转物体上实验。&/p&&br&&p&Donut has a hubward/interior equator 1,305 km from the center, and a rimward/exterior equator 10,633 km away. The equatorial diameter is 9,328 km.&/p&&br&&p&甜甜圈内向/内部赤道距离中心1,305km，外向/外部赤道距离10,633km。赤道直径（译者：实际上是内外径差）9,328km&/p&&br&&p&The planet extends 1,953 km from the equatorial plane, with a north-south diameter of 3,906 km. The ratio of the diameters is 2.4.&/p&&br&&p&这个星球从赤道面延伸出1,953km，南北直径3,906km,赤道的比例是2.4&/p&&br&&p&The north-south circumference is 21,587 km (0.54 times Earth), while the east-west circumference is 66,809 km (1.7 of Earth).&/p&&br&&p&南北周长21,587km(地球的0.54倍)，但是东西周长是66,809km（地球的1.7倍）&/p&&br&&p&The total area 8.2*108 km2, 1.6 times Earth. The total volume is 1.1*1012 km3, within 1% of Earth (after all, Donut was selected as a roughly one Earth mass world). The Volume/area = 1300, 61% of Earth: there is more surface per unit of volume.&/p&&br&&p&总面积为8.2×10^8km^2.中体积为1.1*10^12km^3，不到地球的1%（毕竟，甜甜圈被设定成差不多一个地球质量）（译者注：感觉不对，这密度是有多高，差不多100倍地球密度了，然而经过验算，体积确实是对的）&/p&&br&&p&One day is 2.84 hours long.&/p&&br&&p&一天有2.84个小时长&/p&&br&&h2&&b&Hoop Earth&/b&&br&&b&环形地球&/b&&/h2&&br&&figure&&img src=&https://pic4.zhimg.com/v2-417b7dd11ca5662dfe66_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic4.zhimg.com/v2-417b7dd11ca5662dfe66_r.jpg&&&/figure&&br&&br&&p&Figure 2: Local gravitational acceleration around Hoop, as experienced by a co-rotating object.&/p&&br&&p&图片2：环上各点的本地重力加速度，在一个正转物体上实验。&/p&&br&&p&Hoop has a hubward/interior equator 8,633 km from the center, and a rimward/exterior equator 19,937 km away. The equatorial diameter is 11,304 km.&/p&&br&&p&环有一个内向/内部赤道，距离中心8,633km，而外向/外部赤道距离了19，937km远。赤道直径为11，304km&/p&&br&&p&The planet extends 4,070 km from the equatorial plane, with a north-south diameter of 8,141 km. The cross-section has roughly the 4:3 ratio of an old monitor.&/p&&p&The center of mass circle is 14,294 km from the center.&/p&&br&&p&星球向赤道面外延展出4,070km，南北直径8,141km，横截面差不多有一个旧显示器比例的4：3（译者？？？）&/p&&p&质心环距离中心14,294km&/p&&br&&p&The north-south circumference is 30,794 km (0.77 of Earth) while the east-west circumference is 125,270 km (3.1 times Earth). The total area is 2.5*109 km2, 4.9 times Earth, and the total volume 6.5* times Earth. The volume/area = 1500, 70% of Earth.&/p&&br&&p&南北周长是30,794km（0.77倍地球周长），但是东西直径是125,270km(是地球的3.1倍)。总面积2.5*109 km2，是地球的4.9倍，总体积是6.5*1012km3，是地球的6倍，体积/面积比为1500，地球的70%。&/p&&br&&p&The day is 3.53 hours.&/p&&br&&p&一天有3.53小时&/p&&h2&&b&Environment&br&&/b&&b&环境&/b&&/h2&&br&&p&So, what is life on these torus-Earths?&/p&&br&&p&所有环形地球上的生命会是怎样的&/p&&br&&p&&b&Gravity&br&&/b&&b&重力&/b&&/p&&br&&p&The surface gravity depends on location. It is weakest along the interior and exterior equator, while strongest slightly hubward from the &poles.& This can be a fairly major difference.&/p&&br&&p&表面上的重力强度取决于你所处的位置，最弱的地方是在内部和外部赤道上，但是最强的地方在“两极”更靠里一些的地方。这可以是个相当重要的区别&/p&&br&&p&&b&&i&Donut&/i&&/b&&/p&&p&&b&&i&甜甜圈&/i&&/b&&/p&&br&&p&Donut has just below 0.3 G gravitation along the equators and 0.65 G along the poles. The escape velocity is not too different from Earth, 11.4 km/s.&/p&&br&&p&甜甜圈在赤道上的重力差不多只有0.3G，而在两极有差不多有0.65G。逃逸速度和地球没什么区别，差不多11.4km/s。&/p&&br&&br&&figure&&img src=&https://pic4.zhimg.com/v2-e9a735d6bbd74_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic4.zhimg.com/v2-e9a735d6bbd74_r.jpg&&&/figure&&br&&p&Figure 3: Surface gravity (m/s2) of Donut.&/p&&br&&p&图片3：甜甜圈上的表面重力（单位m/s^2）&/p&&br&&p&The geosynchronous orbit of Donut is very close to the outer equator, less than 2,000 km up. A satellite orbiting there will stay over one spot, but unlike on Earth it will not be able to cover a hemisphere with transmissions, just a smaller region.&/p&&br&&p&甜甜圈的同步轨道非常接近于外部赤道，高度低于2,000km。一个在那个轨道上的卫星会停留在一个位置，但是和地球上不一样，它没法覆盖半个星球的通讯，最多就是一小片区域。&/p&&br&&p&On the other hand, the circumferential velocity at the equator is 6.5 km/s, making launches easier. Launching east a rocket needs just 4.9 km/s velocity to escape.&/p&&br&&p&另一方面，赤道上的环绕速度是6.5km/s，这使得入轨更容易，向东发射的火箭只需要4.9km就能入轨。&/p&&br&&p&There is a central unstable Lagrange point at the middle of the hole. A satellite will be attracted to the equatorial plane, but any deviation outwards will be amplified.&/p&&br&&p&在环孔中央有一个不稳定平衡的拉格朗日点。那里的卫星会被吸引到赤道平面，但是微小的偏差都会被增强。&/p&&br&&p&&b&&i&Hoop&/i&&/b&&/p&&p&&b&&i&环&/i&&/b&&/p&&br&&p&Hoop has 1.1 G gravity along the poles but just 0.75 G along the rimward equator. The hubward equator has slightly higher gravity, 0.81 G.&/p&&br&&p&环在两极有1.1G的重力，但是在外赤道只有0.75G，内部赤道重力大些，差不多0.81G。&/p&&br&&br&&figure&&img src=&https://pic1.zhimg.com/v2-e8a36a4dcb06dc58fb3ef2_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic1.zhimg.com/v2-e8a36a4dcb06dc58fb3ef2_r.jpg&&&/figure&&br&&p&Figure 4: Surface gravity (m/s2) of Hoop.&/p&&br&&p&图片4：环的表面重力（单位m/s^2）&/p&&br&&p&Escape velocity is 19 km/s (remember, the planet weighs in at 6 earth masses). Rimward equator velocity is 9.9 km/s – a rocket will need to provide 10 km/s to escape if it launches eastward.&/p&&br&&p&逃逸速度是19km/s（别忘了，这个星球有6倍地球质量）。外部赤道环绕速度是9.9km-如果向东发射，一个火箭得有差不多10km/s速度来入轨&/p&&br&&p&Note again that having a low gravity equator and high gravity poles does not mean stuff will roll or drift towards the poles: as mentioned before, the surface is an equipotential surface, so gravity (plus the centrifugal correction) is always perpendicular to it.&/p&&br&&p&再说明一下，赤道重力低两极重力高不意味着物体会向两极滚动或者漂移：就像前面说过的一样，表面是等势面，所以重力（包含离心力部分）总是垂直于星球表面（译者注：这显然的说明里这里的势能是等效势能，这里的重力不是真正的有势力，不过这也很显然）&/p&&br&&p&But an air mass flowing towards the pole will be squeezed together. In fact, the different gravities will create horizontal pressure differences that are going to interact with temperature differences to set up jet streams in nontrivial ways.&/p&&br&&p&但是流向两极的空气会被压缩。事实上，不同的重力会产生一个水平压力差，这将会与温度差相互作用，以一种非平凡的方式产生喷射气流。&/p&&br&&p&&b&Light&br&&/b&&b&光照&/b&&/p&&br&&p&First, the nights and days of these worlds are very short. There is not much time for the environment to cool down or heat up during the diurnal cycle. What really matters is how much light they get over longer periods like seasons. Assuming these worlds orbit at an Earth-like distance from a Sun-like star, these are long enough to matter.&/p&&br&&p&首先，这些世界的白天和夜晚都非常的短。在昼夜循环中，没有许多时间让环境冷却或者升温。更重要的问题是在一个长周期上，比如季度的尺度上，它们得到的光照强度问题。假设这些世界是以接近地球轨道的距离绕着一个接近太阳的恒星，那么对考虑这些问题就足够长了。&/p&&br&&p&An an aside, if the torus-worlds orbited closer, tidal forces would really start to bite and before long the planets would become unstable. Since luminosity grows roughly as the fourth power of star mass and the life zone radius scales as the square root of luminosity, in the life zone the experienced tidal forces scale as M/(√(M4))3=1/M5. That is, bright stars have far less tidal effect on habitable planets: maybe Donut and Hoop better orbit some blue-white F star rather than a G star like the sun to be really safe.&/p&&br&&p&我个人提一下，如果这些环状世界轨道靠的太近了，潮汐力会开始撕咬这些星球，并且不久这些星球就会变的不稳定。鉴于亮度会恒星质量的四次方增长,而宜居带范围是和亮度的方根成比例,所有在宜居带内受到的潮汐力范围是M/(√(M4))3=1/M5.就是说,亮星会有更小的潮汐力作用在宜居星球:也许一个甜甜圈型的或者环形的星球围绕一颗发出蓝白光的F等亮星运转,比绕像太阳一样的G等星要安全的多了.&/p&&br&&p&Torus-shaped worlds have an outer rim that is not too different from a normal ellipsoidal planet. Days occur with a sunrise at the eastern horizon and a sunset at the western horizon. The sun moves along a great circle that slowly shifts north and south over the year, giving seasons. However, on the interior side things are different. Here other parts of the planet can shadow the sun: to a first approximation we should expect far less solar energy.&/p&&br&&p&环形世界有一个和一般椭球世界（环境）差不多的外部轮廓。太阳从东方升起，白天开始，直到晚上太阳从东方落下。太阳沿着一个大圈运动，并且在一年中缓慢的向南向北偏移，从而产生四季。然而，在内部情况就完全不一样了。这里的阳光会被星球的其他部分挡住：作为第一个估计我们可以预料这里会收到少的多的太阳能。&/p&&br&&p&We can look at three different cases: zero axial tilt, 23 degrees (like Earth) and 45 degrees.&/p&&br&&p&我们可以考虑三个不同的情况：零黄赤交角，23度角（和地球一样）还有45度角。&/p&&br&&p&&b&&i&Zero tilt&/i&&/b&&/p&&p&&b&&i&零度倾角&/i&&/b&&/p&&p&For zero tilt the hubward side will never get any sunlight: the sun is always hidden below the horizon or by the arc of the world. At the poles the sun is moving just along the horizon, and slightly inwards there will be a perennial dawn/dusk. The temperature difference will be big, with the interior at subarctic temperatures: this is not entirely different from a tidally locked world, and we should expect water (and maybe carbon dioxide) to condense permanently here. The end result would be an arid (but perhaps not super-hot) outer equator, possibly habitable twilight polar regions, and an iced-over interior.&/p&&br&&p&零度倾角的情况下环内部将永远不会有阳光照射：太阳总是躲在地平线下或者是星球的弧中。在两极，太阳最沿着地平线移动，而稍稍向里的地方会永远是黎明/黄昏。温度差异会非常大，内部会是极圈的温度：这和被潮汐锁定的世界不是完全不一样，我们可以预料这里会有永久冻结的水（也许还有二氧化碳）。最后产生一个干旱的外部赤道（但也许不会太热），也许黄昏极圈和被冰覆盖的内陆是宜居的。&/p&&br&&p&&b&&i&23 degree tilt&/i&&/b&&/p&&p&&b&&i&23度倾角&/i&&/b&&/p&&br&&p&For a terrestrial 23 degree tilt spring and autumn will be like the zero tilt case: light along the equator, dark inside the hole. But during summer and winter the sun has a chance to shine past the rim and onto the opposite side of the hole. Also, there will be large regions with midnight sun or perpetual night in summer and winter, respectively. On Earth the Polar Regions are small, but here they are at the very least long contiguous circles.&/p&&br&&p&对于一个黄赤交角23度的星球来说，春天和秋天就和零度倾角的情况是一样的：光直射赤道，内孔永远黑暗。但是在夏天和冬天阳光有机会绕过星球的外轮廓照射到孔的对面。同样的，会有一大片区域会分别在夏天和冬天产生极昼和永夜。地球上两极圈是很小的，但是这里他们是非常长的连续圆。&/p&&br&&figure&&img src=&https://pic1.zhimg.com/v2-9dd833b63bc86c50ac25f820337aed63_b.jpg& data-rawwidth=&633& data-rawheight=&410& class=&origin_image zh-lightbox-thumb& width=&633& data-original=&https://pic1.zhimg.com/v2-9dd833b63bc86c50ac25f820337aed63_r.jpg&&&/figure&&br&&br&&p&Figure 5: Seasons on Donut during spring, summer, autumn and winter.&/p&&p&图片5：在甜甜圈世界上的一年四季&br&&br&&/p&&p&The spring dawns and autumn twilights on the hubward side would have some amazing deep colors, since the sun would be rising past the atmosphere of the other side (already pre-dawned or pre-twilighted, you could say). This would be added to the local atmospheric optics, producing some very deep reds and color gradients. Just before or after sunrise/sunset parts of the corona would also be visible.&/p&&br&&p&春天的黎明和秋天黄昏的时候，在环内部的世界能看见一些令人惊异的深色，因为太阳升起的时候会通过星球对侧的大气层（你可以说已经经过了预黎明或者预黄昏）。这会被增加到当地的大气光学中去，产生非常深的红色和颜色梯度。就在太阳将要/已经升起/落下的时候，能够看见一部分日冕&/p&&br&&p&These sights would be more impressive if they weren't so brief. On Earth, the sun moves close to 15° per hour: at its fastest, the sun moves one diameter in 2.1 minutes. On Donut solar motion is 127° and on Hoop 102°: a sunrise takes 15 or 19 seconds, respectively. Coming in at a slanted angle and the delaying effects of atmospheric refraction would prolong things a bit, but to an Earthling it would still look very brief.&/p&&br&&p&这些景观如果不是那么短暂的话会更加令人映象深刻。在地球上，太阳每小时移动15°：最快的时候，太阳2.1分钟移动一个视直径。在甜甜圈上，太阳移动127°每小时，而环形世界是102°：日出的差不多分别持续15和19秒。在一个有倾角的世界上，大气折射效应会稍稍把现象延长一点，但是地球人还是会觉得太短了。&/p&&br&&p&Standing on the hubward surface looking up, the other side will be about 20 degrees across on Hoop and 30 degrees on Donut – an enormous arc across the sky.&/p&&br&&p&站在内侧表面然后抬头看，环的对侧会有差不多20度宽；如果是甜甜圈就是30度-一个横跨天空的巨大弧形&/p&&br&&p&[Why is Donut not much wider? Donut is very flat, so the world is seen very foreshortened in the sky. Incidentally, this means that when sunlight refracts through the atmosphere on the other side to hit the hubward side during a dawn or twilight it will be far deeper red than on Hoop.]&/p&&br&&p&[为什么甜甜圈的对侧宽度没有宽的多的多？甜甜圈非常的平，所有这个世界在天空中就看起来会小的多。顺便一提，这说明黎明或黄昏的时候，太阳光在大气中折射到另外一侧，打到内侧环面的时候，那些光会看起来比在环世界上看的时候红的多]&/p&&br&&p&On the inside, having lit parts of the other side would light things up like moonlight. But the total area could potentially be much larger, making for some very bright (if still nightly) nights. For Hoop, this is potentially 16,000 times stronger than Earth moonlight (8000 lux) when the entire opposite side is lit (assuming an Earthlike albedo), making a night as bright as an overcast day. On Donut this reaches low daylight levels (12000 lux). However, this is the &full opposing side& situation: near the equinoxes only a thin sliver is visible.&/p&&br&&p&在内部，被（阳光）照亮的部分会把对侧照亮，就像月光一样。但是总面积可能会非常的巨大，产生一些非常明亮的（如果还能称之为夜晚）夜晚。在环的情况下，当整个对侧都被照亮，这有可能会比地球上的月夜亮度高16,000倍（8000lux）（假设这个世界的反射率和地球一样），使得晚上像阴天一样亮。在甜甜圈上，这可能达到暗一些的白天的亮度（12000lux）。但是，这是“对面完全照亮”的情况：在春分，秋分的时候，可能只能看到一条很窄的亮带&/p&&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-cbc6a2f38ce195a9d3dd8ca4_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic3.zhimg.com/v2-cbc6a2f38ce195a9d3dd8ca4_r.jpg&&&/figure&&br&&p&Figure 6: Averaged insolation over a day on Donut during spring, summer, autumn and winter for the 23 degree case.&/p&&br&&p&图片6：倾角23度下一天的平均日光量，在春天，夏天，秋天，冬天。&/p&&br&&p&In the case of Donut, the rather flat surface means that the northern or southern hemisphere will also catch a lot of sunlight: the total heating on the planet is larger during these seasons than in spring and autumn, unlike on Earth where it is constant since the receiving area stays constant. There are also slightly nontrivial effects due to the angle between the surface and the sunlight, making the temperate zones get slightly less energy than the Polar Regions and tropics.&/p&&br&&p&在甜甜圈的情况下，表面更平意味着南半球或者北半球会获得更多阳光：夏天和冬天星球收到的总热量要比春天和冬天的时候多。这和地球上不同，因为地球对太阳的投影面保持不变，所以收到的热功率是常数。因为表面的和日光的夹角，还会稍稍产生一些特殊的的效应，使得温带收到的能量比极地和回归带都少一点点&/p&&br&&p&The rimward tropics have a fairly constant inflow of solar energy. As we go towards the poles seasonality becomes stronger: at the tropics there is more energy coming in during summer than ever happens at the equator. But the winters are of course equally darker. At the poles and beyond on the peak-gravity hubward side there is sun for half a year followed by polar night. Here the climate truly swings: the rimward tropics at least have brief 1.5 hour nights, but here they last 6 months. Finally, close to the hubward equator in the hole day and night return even in winter (plus extra light reflected from the other side), making it a bit more temperate&/p&&br&&p&外部的回归带有基本上稳定的能量输入。我们移向两极的时候季节就会体现的更加强烈：在回归线上，夏天的时候收到的能量会比热带还多，但是在冬天的时候当然就会更暗了。在两极并且超过重力最高点的内侧，有半年时间都是极夜。这里的气候真的是变化巨大：外部的回归线上最少也有短暂的1.5小时的夜晚，而这里夜晚会持续六个月。最后，靠近内部赤道的地方即使在冬天也会有正常的昼夜变化（加上对侧的反射），使得这里接近于温带&/p&&br&&figure&&img src=&https://pic4.zhimg.com/v2-e85d501d68ee56fdcc1ecc75d898fca4_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic4.zhimg.com/v2-e85d501d68ee56fdcc1ecc75d898fca4_r.jpg&&&/figure&&br&&br&&p&Figure 7: Averaged insolation during different seasons on Donut, as a function of latitude in the 23 degree case. 0 denotes the rimwards equator, 90 the north pole, 180 the hubward equator in the hole, 270 the south pole.&/p&&br&&p&图片7：不同季节甜甜圈星球上的平均日光，自变量为纬度，倾角23度。0表示外部赤道，90度是北极，180度是内部赤道，270度是南极。 &/p&&br&The rather big difference in energy deposited at the sunlit summer side of the hole and the dark winter side of the hole will tend to drive some strong weather – but as we will see, due to the other peculiarities of these worlds evening out the energy differences is harder than on Earth.&br&&br&&p&另外一个很大的区别就是在夏季，内部被照亮的部分受到的能量，&/p&&br&&p&Overall, the total energy deposited is 2.5 times higher in the rimward equatorial area than in the temperate and polar areas, and the inside of the hole has about a fourth less energy than the surroundings.&/p&&br&&p&总计一下，外部赤道受到的能量总和会比温带和极地高2.5倍，并且内孔会比周围地区收到的能量低4倍。&/p&&br&&br&&figure&&img src=&https://pic4.zhimg.com/v2-18fb796fef75ca2bd6e6f144dcfd0ab9_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic4.zhimg.com/v2-18fb796fef75ca2bd6e6f144dcfd0ab9_r.jpg&&&/figure&&br&&p&Figure 8: Energy received across a year for different latitudes on Donut.&/p&&br&&p&图片8：甜甜圈上不同维度情况下每年收到的能量&/p&&br&&p&Hoop has less self-shadowing. More importantly, it is not as flattened as Donut.&/p&&br&&p&环有更少的自影现象（自己被自己遮挡）。更重要的是它比甜甜圈世界更平。&/p&&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-bc896c5ee71d5e0d9cb444a331faf3ea_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic3.zhimg.com/v2-bc896c5ee71d5e0d9cb444a331faf3ea_r.jpg&&&/figure&&br&&p&Figure 9: Average insolation during a day on Hoop, 23 degree case.&/p&&br&&p&图片9：环上每天的平均日照，倾角23度&/p&&br&&figure&&img src=&https://pic1.zhimg.com/v2-d0d5ac14f89a410c8ace7_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic1.zhimg.com/v2-d0d5ac14f89a410c8ace7_r.jpg&&&/figure&&br&&p&Figure 10: Averaged insolation during different seasons on Hoop, as a function of latitude in the 23 degree case. 0 denotes the rimwards equator, 90 the north pole, 180 the hubward equator in the hole, 270 the south pole.&/p&&br&&p&图片10：不同季节的时候，环上不同的平均日照量，作为纬度的函数，黄赤交角为23度。0表示外部赤道，90表示北极，180表示内部环，270表示南极。&/p&&br&&p&The seasons at first look like what one would expect. A spring and autumn where the hubward regions are in shadow, summers and winters where one polar circle gets a lot of sunlight and the other far less while the hubward regions get light. Note that this produces a seasonal cycle in the hubward area that is at double frequency of the rimward regions (this is true for Donut too): the warm weather happens in &July& and &January&.&/p&&br&&p&乍一看，这些季节变化就像任何一个人会想到的一样。春天和秋天的时候，环内区域处于星球的影子中，夏天和冬天一个极圈会收到大量阳光而另一个就收到少的多的阳光。&/p&&br&&br&&figure&&img src=&https://pic1.zhimg.com/v2-9a9f4f3bfc48cae897b24c1d232a4f40_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic1.zhimg.com/v2-9a9f4f3bfc48cae897b24c1d232a4f40_r.jpg&&&/figure&&br&&p&Figure 11: Energy received across a year for different latitudes on Hoop.&/p&&br&&p&图片11：环上不同维度下，一年中收到的能量。&/p&&br&&p&Somewhat non-intuitively compared to Donut, here the hubward equator does get more sunlight across the year than the Polar Regions. We can hence expect the climate to be a bit like on Earth, with colder Polar Regions and warmer equatorial regions. The rimward equator still gets 60% more energy, though.&/p&&br&&p&然而对比甜甜圈就不是很直观了，这里环内每年确实收到了比极地更多的能量。我们可以因此预计那里的气候会有点点像地球，有一个寒冷的极地而有一个更加温暖的温带地区。不过外部赤道仍然会多收到60%的能量。&/p&&br&&p&&b&&i&45 degree tilt&/i&&/b&&/p&&p&&b&&i&45度角&/i&&/b&&/p&&br&&p&Perhaps the most surprising thing is that for high enough axial tilt we get four cold zones and four warm!&/p&&br&&p&也许最令人惊奇的事情是，如果转轴倾角够大的话，我们会得到4个寒冷地区和4个温带&/p&&br&&p&The easiest way of understanding this is to consider a planet with 90 degree axial tilt like Uranus. For half of the year the North Pole is turned towards the sun and most of the hemisphere has constant daylight. As equinox approaches the axis points sideways, so the planet gets evenly irradiated. The end result is that the poles get more energy than the equator. On a torus world the same dynamics holds true, but now the Polar Regions are circular too.&/p&&br&&p&理解这个事情最简单的方法，是想象一个有90度倾角的球形星球，比如天王星。每半年，北极就会朝向太阳，并且几乎整个北半球就会变成极昼。春分到来的时候，转轴就会朝向另外一边，所已这个星球会变得更加明亮。最后的结果就是两极会比赤道更热。在一个环形世界，相同的机理还是管用的，但是极圈也是圆形的。&/p&&br&&figure&&img src=&https://pic3.zhimg.com/v2-a670f8b1b7eac770bd0557ec_b.jpg& data-rawwidth=&789& data-rawheight=&524& class=&origin_image zh-lightbox-thumb& width=&789& data-original=&https://pic3.zhimg.com/v2-a670f8b1b7eac770bd0557ec_r.jpg&&&/figure&&br&&p&Figure 12: Energy received across a year for different latitudes on Hoop in the 45 degree case.&/p&&br&&p&图片12：在45度倾角上的环上，不同纬度地区每年收到的能量。&/p&&br&&p&For Hoop the difference is not enormous, about 10% in total insolation. The rimward equator is mildly hotter than the Polar Regions and the hubward equator.&/p&&br&&p&在环上，差别没那么大，大概10%的总光照。外部赤道的热度在两极地区和内部赤道之间。&/p&&br&&p&Donut slightly larger differences but in practice most of the surface is dominated by the mildly warm polar regions. The rimward equator is only slightly warmer than the cooler rimward temperate areas.&/p&&br&&p&在甜甜圈上差别就要稍稍大些但是这个世界上大多数表面都是中间的温暖极地。外部极地仅仅只是比中间的温带稍稍温暖一点。&/p&&br&&p&&b&Geosphere&/b&&/p&&p&&b&地质&/b&&/p&&br&&p&The surface area is larger than on Earth, and the volume/area ratio is smaller (for Donut the ratio is 1,300 km, for Hoop 1,500 km, for Earth 2,124 km). One might hence suspect that more thermal energy is leaking out, reducing volcanism and plate tectonics. However, even a small amount of tidal heating due to influences from the sun might release plenty of energy stored in angular momentum. In the case of Hoop there are also 6 times more radioisotopes inside the planet than on Earth but only 5 times more surface area.&/p&&br&&p&（这些世界的）表面积比地球大，但是体积/面积比要更小（比如甜甜圈世界的比例是1,300km，环是1,500km，地球是2,214km）。有人也许因此推测这些星球会散失更多能量，减少火山和板块运动。但是，即使是很少量的太阳引力产生的潮汐加热效应就能从星球的角动量中释放出很多能量。在环的情况下，内部的同时会有6倍多的放射性同位素量，但是表面积只大了5倍。&/p&&br&&p&Continental drift would be affected by the different inner and outer radii. A circle r km inwards from a circle of radius R will be just 2*pi*r km shorter, and the relative change will be r/R. So for Hoop a continental plate drifting from the outer equator across a pole to the inner equator will have to shrink to 43% of its original width to fit. On Donut the effect is much bigger: it becomes 12% of its original width! Hence continental plates moving hubwards on the inside will tend to experience folding, while plates moving rimwards on the inside will experience rifting. Expect some rugged landscape and archipelagoes near the hubward equator.&/p&&br&&p&大陆迁移会受到内部和外部半径不同的影响。一个半径R的圆向内收缩r的圆会只比原来的周长小2*pi*r千米，并且相对变化为r/R。所以对于环的情况，从外部赤道通过一个极漂移内部赤道的的大陆板块会被压缩到它本来宽度的43%来适应星球的形状。在甜甜圈上，这个效应会大大加强：它会被压缩到原本宽度的12%！可以推测向内运动的大陆板块会倾向于被折叠，而向外运动的板块会经历一个撕裂的过程。可以预期在内部赤道附近会出现一些崎岖的地形和群岛。&/p&&br&&p&Gravity affects the height of mountains. On Hoop the difference is not enormous compared to Earth, but on Donut mountains at the poles can be 1.5 times higher (maximum around 12 km) and near the equators 3 times higher (24 km). Combined with the ruggedness near the hole this might make for some dramatic landscapes.&/p&&br&&p&重力影响了 山峰的高度。在环上，区别和地球上不是很大，但是在甜甜圈上，两极的山峰能够比地球上高1.5倍（差不多最大12km），而在赤道上差不多要高3倍（24km）。结合孔附近的崎岖地形，这可能会产生一些巨大的景观。&/p&&br&&p&The fast rotation will likely produce a s unlike on Earth the polar regions will not have auroras since the field lines will not intersect the surface… I think — figuring out dynamo currents in a toroid iron core sounds fun but is beyond me.&/p&&br&&p&高速的旋转会很容易导致一个强的磁场；与地球上不同的是，两极地区不会有极光，因为场线不会与表面相交...我认为 - 求解环形铁核中的感生电流听上去挺有趣，但是那超出了我的能力。&/p&&br&&p&&b&Atmosphere&/b&&/p&&br&&p&&b&大气&/b&&/p&&br&&p&We have seen that the light levels change a lot, and that would make us suspect plenty of wind transporting heat from hot sunlit areas to cool shadowed areas. However, the high rate of rotation means that the Coriolis Effect will influence air and water flows to a large degree.&/p&&br&&p&我们已经知道光照强度改变的非常剧烈，并且我们会因此预计会产生大量的风，将热从被照亮的高温地区输运到被遮挡的低温地区。然而，星球的高旋转速率意味着科里奥利效应会大大影响风和水流的运动，使得他们以大转角流动&/p&&br&&p&The Coriolis Effect makes air moving towards or away from the rotational axis bend away, since it has more or less velocity than the ground. A parcel of air &at rest& near the equator has a lot of actual momentum since the equator is moving fast around the rotation axis: if that air were to flow pole-wards it would now have a noticeable velocity eastwards or westwards. This is why the global airflow is not just simple convection cells from the equator towards the poles: as heat is transferred using air polewards the air flow gets twisted around, producing trade winds.&/p&&br&&p&科里奥利效应使得朝向或者远离转轴的空气轨迹弯曲，因为它同地面有速度差异。一小团空气在赤道附近“静止”会拥有大量的角动量因为赤道本身在绕轴高速旋转：如果空气向两极方向运动，那么它就会得到一个显著的东向或者西向的运动。这就是为什么全球气流不是简单的从赤道向两极的运流单位：热量通过极向的气流传输的同时，气流会被偏转，产生了信风。&/p&&br&&p&On torus worlds the rotation rate is 8 times faster than on Earth and the velocity differences are larger. Air hence tends to be twisted around far more, producing a more banded zonal climate than on Earth. Exactly how banded is hard to tell without detailed atmospheric calculations, but it is likely more like on Jupiter than on Earth. This in turn means that heat transfer is less effective: the temperature differences between the hot and cold regions will be bigger.&/p&&br&&p&在一个环形世界上，转速比地球要快8倍，并且速度差异更高。可以推测，空气会倾向于弯折的厉害的多，产生比地球更强烈的带状气候。准确的弯曲方式是很难讲的，除非做细致的大气运动计算，不过比起地球，那大概会很像是木星上的情况。这就意味着，热量传输会不太有效：炎热地区和寒冷地区的温度差异会变得更大&/p&&br&&p&It is likely that there will be inter-tropical convergence zone (ITCZ, alias the doldrums or equatorial lows) around the rimward equator, where winds approaching from north and south will blow westwards (trade winds) while warm air rises, moves away from the equator, cools and descends at a higher or lower latitude (where we should expect major deserts). The big seasonality changes especially on Donut will make the ITCZ shift north and south, triggering monsoons in some regions. However, the rapid rotation will make the Hadley cell thinner than Earth's 30 degree size (exactly how much thinner is slightly tricky to estimate, since it also depends on the latitude-varying gravity).&/p&&br&&p&那里很有可能会有热带辐合带（ITCZ，别名叫赤道无风带或赤道低点）产生在外部赤道周围，那里从南北来的风都会向西吹（信风），这时温暖空气上升，从赤道离开，然后冷却下降在更高和更低的纬度（我们可以预料到那里会有大型沙漠生成）。在甜甜圈世界上特别的巨大季节变化会使得热带辐合带向北向南移动，在一些地区产生季风，然而告诉的旋转会使哈得莱环流比地球上30度的宽度要细（具体细多少就有点难以估算了，因为这也取决于随纬度变化的重力）&/p&&br&&p&Big temperature differences over short distances are going to power plenty of weather, even if it is hard to predict exactly how it is going to look. Especially near the hole on Donut seasonal weather will be wild: warm air from the sunlit side will flow through it in a big vortex, balanced by cool winds from the dark side circulating in the opposite direction.&/p&&br&&p&短距离上的巨大的温差将会给天气带来极大的动力，即使很难预测到底会产生什么样的效果。特别是在甜甜圈的两极，季节性的天气会变得非常强烈：从被照亮地区来的暖气流会从一个巨大的旋风中通过这个地区，同反方向来的，由寒冷黑暗地球产生的冷空气平衡。&/p&&br&&p&The scale height, how quickly pressure drops off with altitude, is proportional to gravity. Hence clouds will be 3 to 1.5 times taller on Donut, while Hoop clouds will be more Earth-like.&/p&&br&&p&高度范围，气压随高度下降的速度，正比与重力的大小。可以知道在甜甜圈上的云会差不多有3-1.5倍高，而环上的云就和地球上的差不多&/p&&br&&p&Like on Earth cyclones can form at the mid-latitudes. Stronger Coriolis forces would make tighter hurricanes, about four times smaller. However, they would tend to last longer on Donut (since the high scale height gives them far more air to play with). Wind speeds depend on the temperature difference between the top of the atmosphere and the ocean, which could vary a great deal across the year.&/p&&br&&p&就像地球上一样，在中纬度地区可以产生旋风。更强的科里奥利力会把飓风收紧，使得它们差不多小4倍。然而，它们在甜甜圈上会持续更长的时间（因为高度范围更大给了它们更多的空气去维持）。风速决定于大气层顶到海平面的温差，而这个温差在一年中可能会变化的非常大&/p&&br&&p&&b&Hydrosphere&/b&&/p&&p&&b&水圈&/b&&/p&&br&&p&The amount of water on either world is not vastly different from Earth, although Hoop's 6 times greater mass with merely 5 times greater area would provide it with 20% more water volume from the initial accretion (so for the same coverage the oceans would be 20% deeper). The higher mass might also accumulate more cometary infall, but it is hard to judge how much this would be.&/p&&br&&p&两个世界的水量都不会同地球有什么太大的差别，尽管环的质量是地球的6倍，而面积差不多是地球的5倍，使得它能够在初始吸积层多获得20%的水（所有在相同的水面积下会深20%）。更大的质量也会使得星球积累更多来自彗星的水，但是很难讲到底有多少&/p&&br&&p&The big seasonal temperature swings will be more pronounced far from the moderating influence of oceans: continents near the poles will be more extreme than equatorial ones. Whether they can maintain ice caps throughout polar summer depends on their layout and the ba since ice reflects away sunlight effectively and the Coriolis Effect can keep air from warming them it is likely. The same for sea ice, although here there is potential for warming sea currents from hubwards or rimwards. Since the flow of water in oceans is constrained by the shape of the basins, the Coriolis Effect will merely drive gyres rather than prevent north- large oceans like the Pacific will be more east-westerly than narrow north-south oceans like the Atlantic.&/p&&br&&p&巨大的季节温度变化将会非常显著的超过海洋能够调节的水平：两极附近的大陆会比赤道附近的更加极端。两极的大陆冰盖能不能撑过夏天取决于它们的布局，已经背景温度；因为冰盖可以有效的反射阳光，并且科里奥利力可以避免空气像一般情况下一样将该地区加热。对于海冰来说也是相同的情况，尽管他们有潜在可能被来自内部和外部的温暖洋流加热。因为海洋的水流被海盆的形状限制住了，科里奥利力将仅仅只是驱动环流，而不能阻止南北方向的流动；像太平洋一样的巨大的海洋会比像大西洋一样的狭长的南北向海洋有更多的东西向流动。&/p&&br&&p&The low gravity near the equator will make some tall waves on Donut: they can be expected to be three times taller than on Earth. Waves at Donut's poles are still 150% of the ones on Earth. Hoop is closer to normal (133% taller at the equator, 90% height at the poles). The wild hubward summer-winter weather on Donut will likely drive some amazing storm waves.&/p&&br&&p&赤道附近的低重力会在甜甜圈上产生巨浪：可以期望它们也许会比地球上高3倍。在甜甜圈两极的浪还是会比地球上的高出50%。环上就比较正常了（在赤道上是地球上的133%，两极的是地球上的90%）。内环上残暴的夏-冬天气现象比较容易产生一些令人惊叹的暴风巨浪&/p&&br&&p&&b&Biosphere&/b&&/p&&p&&b&生物圈&/b&&/p&&br&&p&From these considerations, it seems likely that one could have a fairly Earth-like biosphere on Donut and Hoop. Storms, severe weather and long winters are things species on Earth have adapted to just fine. There might be interesting differences in ecosystems based on latitude, since there are more variations between different bands than on Earth (gravity, seasonality, temperatures etc.). Also, at least on Hoop each band has a much larger surface area: there is more room for species diversity within each eco-zone.&/p&&br&&p&由上面的讨论，看上去在甜甜圈和环上还是很有可能有一个类似地球的生物圈。风暴，及其恶劣的天气，还有漫长的冬天之类的，地球上的生物已经适应的很好的。可能会存在在不同纬度的生态系统有趣的基础差异，因为同地球上不同环带的变化要大（重力，季节，温度之类的）而且，至少在环上每条环带都有更大的面积：有足够的空间在每个生态区产生生物多样性。&/p&&br&&p&&b&Would these worlds be able to keep moons?&/b&&/p&&p&&b&这些世界也能拥有卫星吗？&/b&&/p&&br&&p&A moon orbiting exactly in the equatorial plane in a circular orbit it would just feel a potential looking like it came from a spherical planet of some intermediate density. However, if it orbited in slightly eccentric orbit things would change. The potential field close to the planet falls of more slowly than 1/r (the answer for normal spherical planets): the Kepler ellipse is no longer the right solution. And as soon as the orbit becomes slightly tilted things turn even more complicated – now the moon will feel the flatness.&/p&&br&&p&一个严格的在赤道面上以圆轨道运行的卫星会感觉到一个看起来像是从一个中等密度均匀球形星球发出的引力势。然而，如果它轨道稍稍的有点偏心，情况就完全不同了。靠近星球时，势能会比1/r下降的要慢（1/r是一般球形星球的势能函数）：开普勒的椭圆轨道就在于不对了。如果这个轨道还有一点点倾斜，情况就更复杂了 - 现在这个卫星感受到了星球平坦产生的影响&/p&&p&In many ways this is the problem facing satellite designers already: Earth is oblate enough that orbits are affected. This problem was dealt with in the earliest days of spaceflight (see Wikipedia,(Tremaine & Yavetz 2013) or (Nielsen, Goodwin,& Mersman 1958)).&/p&&br&&p&从很多方面来说，卫星设计者已经面对过这种问题了：地球的椭球性已经足够可以影响卫星轨道了。这些问题已经在最早的太空飞行时期就已经被解决了（查看&a href=&https://link.zhihu.com/?target=https%3A//en.wikipedia.org/wiki/Orbital_perturbation_analysis_%28spacecraft%29& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&维基百科&/a&，(&a href=&https://link.zhihu.com/?target=http%3A//arxiv.org/abs/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Tremaine & Yavetz 2013&/a&)或者(&a href=&https://link.zhihu.com/?target=https%3A//archive.org/details/nasa_techdoc_& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Nielsen, Goodwin,& Mersman 1958&/a&)）&/p&&br&&p&Basically, the main effect is that an elliptic orbit precesses – it slowly changes direction in space, for Earth largely depending on the inclination. Eccentricity can also drift, which is a bigger deal. In any case, for a toroid world these effects are far larger: the multipole moments (measures of just how non-spherical the field is) are of course enormous. In fact, they are so big that the standard methods no longer work and we need to do computer simulations.&/p&&br&&p&基本上，椭圆轨道产生的最大问题是会发生进动 - 轨道的方向在空间中缓慢的改变，对于地球来说这极大的取决于倾斜度。偏心率也会改变，而且是个更大的问题。不管怎样，在环形世界上，这些效应只会变的非常大：星球引力场的多级矩（反映引力场有多非球对称的量）当然会非常大。事实上，这些效应是如此之大以至于常规的模式已经完全无法使用了，我们必须要做电脑模拟了。&/p&&br&&p&However, I feel confident that moons in sufficiently remote and circular orbits will be pretty stable. Most likely they will precess so that their orbit is more of a rosette than an ellipse, but they will not go crazy. Of course, moons in close orbits are another matter…&/p&&br&&p&然而，我很肯定远距离的圆轨道卫星是稳定的。它们可能会旋进，产生像花圈一样的轨道，而不是椭圆轨道，但是这些轨道不会乱跑。当然，低轨道的卫星就是另外一回事了。&/p&&br&&p&Running a simulation (where I did not use the full torus potential, but rather a ring of 30 masses) demonstrate some of the possibilities. Indeed, an equatorial elliptic orbit looks nice and stable but precesses into a rosette.&/p&&br&&p&运行了一个模拟程序（我没有用完整的环形世界势能曲线，而是用来一个30倍质量的圆环）演示出一些可能的情况。确实，一个在赤道面上运行的椭圆形轨道看起来很漂亮，很稳定，但是旋进成了花圈的形状&/p&&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-dfa4e4aa1_b.jpg& data-rawwidth=&800& data-rawheight=&453& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic3.zhimg.com/v2-dfa4e4aa1_r.jpg&&&/figure&&p&A nearly polar orbit has even more precession, not just making it rosette around in a plane but also slowly precessing the plane. The moon could appear in the sky in any constellation.&/p&&br&&p&一个靠近极地的轨道进动就变得非常明显了，不仅产生了花圈一样的轨道，并且轨道平面自身也在旋进。这个卫星可以在天空中任何一个星群中出现。&/p&&figure&&img src=&https://pic3.zhimg.com/v2-a15b69ab287a11e04fdf1e37e467f236_b.jpg& data-rawwidth=&640& data-rawheight=&251& class=&origin_image zh-lightbox-thumb& width=&640& data-original=&https://pic3.zhimg.com/v2-a15b69ab287a11e04fdf1e37e467f236_r.jpg&&&/figure&&br&&p&What about orbits through the hole? As mentioned earlier, the exact center is an unstable Lagrange point. Place a moon there, and any kick will make it fall out. But there are orbits through the center that look stable (or rather, give them a kick and they turn into another similar-looking orbit rather than fall down). The simplest is just a moon bobbing up and down through the hole:&/p&&br&&br&&p&一个穿过孔的轨道会是怎样的？就像之前提到的一样，严格的中心点是一个不稳定的拉格朗日点。在这里放置一颗卫星，然后随便给个扰动就会使它坠落。但是，是有看起来稳定的通过孔的轨道的（说的准确点，给它们一个扰动，这些它们会变换到一个相似的轨道而不是坠落）。最简单的情形是一个卫星在孔里上下摆动。&/p&&br&&figure&&img src=&https://pic2.zhimg.com/v2-1bbfbf7f747de21a9b8edab_b.jpg& data-rawwidth=&640& data-rawheight=&619& class=&origin_image zh-lightbox-thumb& width=&640& data-original=&https://pic2.zhimg.com/v2-1bbfbf7f747de21a9b8edab_r.jpg&&&/figure&&br&&p&In fact, one can have a moon bobbing up and down over a particular longitude in a bent rectangular region.&/p&&br&&p&实际上，你可以制造一个卫星，在一个特定的经度上在一个弯曲的矩形内上下摆动&/p&&br&&figure&&img src=&https://pic4.zhimg.com/v2-f2d0a5d1aabbae_b.jpg& data-rawwidth=&721& data-rawheight=&598& class=&origin_image zh-lightbox-thumb& width=&721& data-original=&https://pic4.zhimg.com/v2-f2d0a5d1aabbae_r.jpg&&&/figure&&br&&p&And given some longitudinal velocity, it will move around the hole, filling out a wobbly hyperboloid of one sheet (a &vase orbit&?).&/p&&br&&p&而且，给予一些纵向（垂直经线）速度，卫星可以在绕孔转动，填满一个摇摆双曲面（一个“花瓶轨道”）&/p&&br&&figure&&img src=&https://pic1.zhimg.com/v2-fceeb1152ae_b.jpg& data-rawwidth=&690& data-rawheight=&734& class=&origin_image zh-lightbox-thumb& width=&690& data-original=&https://pic1.zhimg.com/v2-fceeb1152ae_r.jpg&&&/figure&&br&&p&What about orbits that actually go through the hole in just one direct}