题目内容
(12分)如图为宇宙中有一个恒星系的示意图,A为该星系的一颗星,它绕中央恒星O运动轨道近似为圆,天文学家观测得到A行星运动的轨道半径为R,周期为T
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250005492671465.jpg)
(1)中央恒星O的质量是多大?
(2)长期观测发现,A行星实际运动的轨道与圆轨道总存在一些偏离,且周期性地每隔t时间发生一次最大的偏离,天文学家认为形成这种现象的原因可能是A行星外侧还存在着一颗未知的行星B(假设其运行轨道与A在同一平面内,且与A的绕行方向相同),它对A行星的万有引力引起A轨道的偏离.根据上述现象及假设,预测未知行星B绕中央恒星O运行轨道半径有多大?
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250005492671465.jpg)
(1)中央恒星O的质量是多大?
(2)长期观测发现,A行星实际运动的轨道与圆轨道总存在一些偏离,且周期性地每隔t时间发生一次最大的偏离,天文学家认为形成这种现象的原因可能是A行星外侧还存在着一颗未知的行星B(假设其运行轨道与A在同一平面内,且与A的绕行方向相同),它对A行星的万有引力引起A轨道的偏离.根据上述现象及假设,预测未知行星B绕中央恒星O运行轨道半径有多大?
(1)
;(2)R0![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549313803.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549298838.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549313803.png)
试题分析:(1)中央恒星质量为M,A行星质量为m,
则有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549376707.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549516534.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549547242.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549563226.png)
解得:M=
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549298838.png)
(2)由题意可知:A、B相距最近时,B对A的影响最大,且每隔t
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549563226.png)
设B行星周期为T
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549610269.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549641460.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549657501.png)
解得:TB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549672583.png)
设B行星的质量为m′。运动的轨道半径为RB,则有
G
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549688740.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549719543.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549547242.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549750440.png)
由①、④、⑤可得:RB=R0
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000549313803.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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