开普勒第三定律也叫行星运动定律。开普勒第三定律的常见表述是:绕以太阳为焦点的椭圆轨道运行的所有行星,其各自椭圆轨道半长轴的立方与周期的平方之比是一个常量。
德国天文学家约翰尼斯·开普勒根据丹麦天文学家第谷·布拉赫等人的观测资料和星表,通过开普勒本人的观测和分析后,于1609年在他出版的《新天文学》上发表了关于行星运动的前两条定律,又于1618年,在《宇宙谐和论》提出了第三条定律。
定律定义
开普勒在《宇宙谐和论》上的原始表述:绕以太阳为焦点的椭圆轨道运行的所有行星,其各自椭圆轨道半长轴的立方与周期的平方之比是一个常量。推导过程
万有引力定律是用开普勒第三定律导出的,因此不能再用万有引力定律来推导开普勒第三定律,循环论证是不严谨的。开普勒第三定律是开普勒根据第谷的观测数据来计算出来的,没有见过推导,推导过程只能是与万有引力定律的联系,不能叫推导。[3]
观测数据
开普勒整理数据发现,右图下方的坐标中各点大致连成一条直线,因此他认为行星的运行周期
和
成正比(其中
为轨道半径),并计算出该直线的斜率为
,即
。
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D12/sign=078529ddaaec8a13101a53e2f60380d8/e4dde71190ef76c6f746f6199e16fdfaaf51676f.jpg)
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D19/sign=183228ddaaec8a13101a53e9f7038054/e4dde71190ef76c6e8f1f7199e16fdfaaf5167e4.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D12/sign=9f32ed0536fae6cd08b4af630fb34749/d6ca7bcb0a46f21fb6105261f5246b600c33aedf.jpg)
![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D42/sign=9e19c21b8518367aa9897edf2e73bd79/64380cd7912397dd8f0c97365a82b2b7d0a2878e.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D74/sign=ee3824158982b90139adc137728da7fb/b7003af33a87e95014eefc5e13385343fbf2b402.jpg)
常规方法
方法一:
现实中的星体运动的轨道大多数是椭圆,于是便有以下推导:
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D122/sign=9c19c634d833c895a27e9c79e3127397/86d6277f9e2f0708bb059ba3e824b899a901f215.jpg)
面积速度为 ![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D214/sign=f5fe545ba686c9170c035538fd3c70c6/e7cd7b899e510fb36f10c734d833c895d1430c03.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D214/sign=f5fe545ba686c9170c035538fd3c70c6/e7cd7b899e510fb36f10c734d833c895d1430c03.jpg)
设各行星绕太阳运行周期为T,椭圆半长轴为a、半短轴为b、太阳到椭圆中心的距离为c
则行星绕太阳运动的周期
。
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D100/sign=3414548072f082022992953f7bfafb8a/b999a9014c086e0684ca8fe803087bf40ad1cb37.jpg)
选近日点A和远日点B来研究,由ΔS相等可得 ![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D118/sign=357a905d37d3d539c53d0bc20286e927/7aec54e736d12f2e45a81ec34ec2d562843568e0.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D118/sign=357a905d37d3d539c53d0bc20286e927/7aec54e736d12f2e45a81ec34ec2d562843568e0.jpg)
![](https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D227/sign=15b2c63a8326cffc6d2ab8b08e004a7d/63d9f2d3572c11df0b1b0f80622762d0f603c24f.jpg)
得: ![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D129/sign=8b54ce5f0b23dd542573a36ae809b3df/738b4710b912c8fc838944f6fe039245d6882196.jpg)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D129/sign=8b54ce5f0b23dd542573a36ae809b3df/738b4710b912c8fc838944f6fe039245d6882196.jpg)
由几何关系得:
,
, ![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D80/sign=a74fe3f0f7246b607f0ebf74eaf8e284/ca1349540923dd544f081acbd009b3de9c82481d.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D70/sign=01314ecf1138534388cf85219213c8df/a686c9177f3e6709fd26f85c3ac79f3df8dc556a.jpg)
![](https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D69/sign=39f93fd642a98226bcc1282e8b82ab8b/14ce36d3d539b600e8ceb214e850352ac65cb715.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D80/sign=a74fe3f0f7246b607f0ebf74eaf8e284/ca1349540923dd544f081acbd009b3de9c82481d.jpg)
所以 ![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D112/sign=6898275078310a55c024daf585444387/63d9f2d3572c11df36420280622762d0f603c286.jpg)
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D112/sign=6898275078310a55c024daf585444387/63d9f2d3572c11df36420280622762d0f603c286.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D298/sign=15080e45267f9e2f74351a012731e962/472309f790529822c68d8560d6ca7bcb0b46d4d3.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D159/sign=f82eb048b7fd5266a32b381192199799/cefc1e178a82b90133757f90728da9773812efbd.jpg)
整理得
。
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D72/sign=9dcd7d90908fa0ec7bc7660f27979be2/8435e5dde71190ef89db2d7ccf1b9d16fdfa6075.jpg)
方法二:
行星绕太阳运动椭圆轨道的面积,根据椭圆的性质则椭圆的面积
(a为长轴,b为短轴)由于单位时间内极径所扫过的面积 ![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D65/sign=1f3113c2d662853596e0d12490ef2317/962bd40735fae6cdffe8c2510cb30f2442a70fa5.jpg)
![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D55/sign=b743dcd6247f9e2f74351d0d1e30b3f8/0df431adcbef76095dfef7af2ddda3cc7cd99e0c.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D65/sign=1f3113c2d662853596e0d12490ef2317/962bd40735fae6cdffe8c2510cb30f2442a70fa5.jpg)
则周期
(1)
![](https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D174/sign=2a3e1a230af79052eb1f433938f2d738/b17eca8065380cd79bb9cbdda244ad345982815e.jpg)
根据椭圆的性质和开普勒第一定律,半长轴
(2)
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D248/sign=ee9086b449ed2e73f8e98128bf00a16d/3c6d55fbb2fb4316b5c8134a23a4462309f7d32f.jpg)
(2)式得 ![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D141/sign=5d0fa0039145d688a702b6a095c37dab/0b55b319ebc4b745dd5bde26ccfc1e178a821536.jpg)
![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D141/sign=5d0fa0039145d688a702b6a095c37dab/0b55b319ebc4b745dd5bde26ccfc1e178a821536.jpg)
(2)式代入(1)式得
(3)
![](https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D260/sign=ce99c0510cb30f24319aeb05f895d192/8601a18b87d6277f2b3f59082b381f30e924fcd5.jpg)
根据椭圆的性质,椭圆的半短轴
,则
(4)
![](https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D87/sign=b0e71910ba12c8fcb0f3fbcafc036122/1c950a7b02087bf4c8bd3cd9f1d3572c11dfcfa9.jpg)
![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D101/sign=254905b5caea15ce45eee40987003a25/9a504fc2d5628535c3afb91193ef76c6a7ef63b6.jpg)
轨道能量推导
由运动总能量
,得
,则运动周期为
![](https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D157/sign=1aea3526a818972ba73a04cfd1cc7b9d/728da9773912b31b7aadc41b8518367adab4e13b.jpg)
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D172/sign=61200e3f61d0f703e2b291db3afa5148/14ce36d3d539b600d1c0579fea50352ac65cb788.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D420/sign=160f3c17daf9d72a1364111fe42a282a/7dd98d1001e939012d7cd51678ec54e736d1969f.jpg)
即 ![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D70/sign=5b49fc3b94dda144de096eb2b3b7dee7/bba1cd11728b4710ee8ca5e4c0cec3fdfc032372.jpg)
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D70/sign=5b49fc3b94dda144de096eb2b3b7dee7/bba1cd11728b4710ee8ca5e4c0cec3fdfc032372.jpg)
适用范围
成立条件
开普勒定律是一个普适定律,适用于一切二体问题。开普勒定律不仅适用于太阳系,他对具有中心天体的引力系统(如行星-卫星系统)和双星系统都成立。 [11] 围绕同一个中心天体运动的几个天体,它们轨道半径三次方与周期的平方的比值(
)都相等,为
,M为中心天体质量。这个比值是一个与行星无关的常量,只与中心体质量有关,那么M相同是这个比值相同。
![](https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D38/sign=5ef5c224ccfc1e17f9bf8a394a90a917/3b87e950352ac65c0d641341f8f2b21193138a9a.jpg)
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D36/sign=3bc1b896a7c27d1ea1263dc21ad578b7/908fa0ec08fa513dc96966ec3e6d55fbb3fbd9c5.jpg)
拓广形式
开普勒第三定律也可以表示为:
![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D184/sign=d170a46ea1ec08fa220017af6def3d4d/37d3d539b6003af31cf4dcb1342ac65c1138b690.jpg)
引入天体质量后可表示为:
![](https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D202/sign=44f0c2e2908fa0ec7bc7630d1496594a/37d3d539b6003af31ca7dcb1342ac65c1138b6c5.jpg)
其中
,
为两个相应的行星质量,
,
为两个相应行星围绕同一恒星运动的周期,
,
为两个行星围绕同一恒星运动的平均轨道半径。 [2] 通过拓展形式,可以根据绕同一行星的两星体轨道半径估测星体质量,或根据星体质量估测运行轨道。
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D19/sign=d644837a3801213fcb334ad555e72d9d/b7fd5266d0160924f04c15b8d50735fae6cd3423.jpg)
![](https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D20/sign=a565ace08ad4b31cf43c93bb86d6b849/b3b7d0a20cf431ad2467f49a4a36acaf2fdd98d8.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D15/sign=1c1d7b3ca818972ba73a04cfe6cd4a55/9922720e0cf3d7cab7c0785ef11fbe096b63a9e0.jpg)
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D15/sign=1506392eafaf2eddd0f14dec8d10d359/1ad5ad6eddc451dae2c44bc3b5fd5266d01632e3.jpg)
![](https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D17/sign=aee9f03eccfc1e17f9bf88364a90a90d/342ac65c1038534316b2c0099013b07eca80889d.jpg)
![](https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D18/sign=595a99dcb44543a9f11bfec41e17ac11/3bf33a87e950352a2c1f62205043fbf2b2118b98.jpg)
应用实例
天体
实际星体问题大多数为二体问题,实际应用时,人们把开普勒定律看成是牛顿定律和万有引力定律的表现形式。(M为中心天体质量,m为行星质量)
由此可见,开普勒定律只是一个近似定律。
通过开普勒第三定律,在天体运行中有以下应用:
- 通过测出形体的绕转周期以及半长轴,算出双星的质量及估计中心天体的质量;
- 在星—箭分离问题中,通过星箭椭圆运动周期之比,计算星箭运动轨迹半长轴之比。
二体问题
二体问题是天体力学中的一个基本问题,它是指可视为质点的两个天体在相互间唯一的万有引力作用下的运动规律问题。二体问题可以用牛顿万有引力定律和牛顿运动定律来描述并得到完全解决。开普勒三定律是二体问题的解。
航天
开普勒轨道的定义:
- 符合开普勒三定律的天体或航天器的运行轨道;
- 由二体问题的解的道德天体或航天器的运行轨道。
由定义可知,开普勒的轨道也称为二体问题轨道,符合上述定义的开普勒轨道也称为理想的开普勒轨道。航天器的开普勒轨道可由如下二体问题的基本方程解得:
![](https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D79/sign=0e9d47d79352982201333bcad6caec96/3b292df5e0fe9925f48f124037a85edf8db1711a.jpg)
上述方程描述在惯性坐标系中航天器相对于天体的轨道运动,式中的
是从天体(质量记为
)到航天器(
)的位置矢量,
是二体系统的引力常数,G是万有引力常数。由于
,可以只考虑
对
的引力,这种情况可把航天器开普勒轨道看成是限制性二体问题的解,即看成是在惯性固体天体中心引力场中的运动(有心力运动)轨迹。
![](https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D15/sign=b8cff82039f33a879a6d041fc75ccd4a/c8177f3e6709c93d1d3975e79c3df8dcd00054d2.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D19/sign=6cccb23f4234970a4373142695ca1575/8601a18b87d6277f5a3048282b381f30e924fc84.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D20/sign=34181ffa82cb39dbc5c06056d11670a9/42166d224f4a20a4afff47d793529822720ed039.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D104/sign=5503dd65d588d43ff4a995f2491ed2aa/aec379310a55b31942c4f86540a98226cffc17e4.jpg)
![](https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D64/sign=990062d5e3fe9925cf0c6a5434a83178/4afbfbedab64034fdbd9b816acc379310a551d8d.jpg)
![](https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D19/sign=6cccb23f4234970a4373142695ca1575/8601a18b87d6277f5a3048282b381f30e924fc84.jpg)
![](https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D20/sign=34181ffa82cb39dbc5c06056d11670a9/42166d224f4a20a4afff47d793529822720ed039.jpg)