2015IOAA理论短问题第1题-共振轨道
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英文题目
Several exoplanets have been observed in the Gliese 876 system $$\left(M_{G} = 0.33\pm 0.03 M_{\odot}\right)$$ as given in the following table,
| Gliese System | Mass | Semi Major Axis (AU) |
|---|---|---|
| Gliese 876 b | $$2.276M_{J}$$ | 0.2083 |
| Gliese 876 c | $$0.714M_{J}$$ | 0.1296 |
| Gliese 876 d | $$6.8M_{\oplus}$$ | 0.0208 |
| Gliese 876 e | $$15M_{\oplus}$$ | 0.334 |
where $$M_{\odot}$$ is mass of Sun, $$M_{J}$$ is mass of Jupiter, $$\left(M_{J}=1.89813 \times 10^{27} \mathrm{~kg}\right)$$, and $$M_{\oplus}$$ is mass of Earth. Assume that all these planets revolve around Gliese 876 in the same direction. Two planets are said to be in resonant orbits if the synodic period of one planet with respect to the other planet is an integer multiple of the orbital period of the second planet.
Find if any of the exoplanets of Gliese 876 system may have resonant orbits.
中文翻译
下表中给出了一些在Gliese 876星系中被观测到的系外行星，
| Gliese 星系 | 质量 | 轨道半长轴 (AU) |
|---|---|---|
| Gliese 876 b | $$2.276M_{J}$$ | 0.2083 |
| Gliese 876 c | $$0.714M_{J}$$ | 0.1296 |
| Gliese 876 d | $$6.8M_{\oplus}$$ | 0.0208 |
| Gliese 876 e | $$15M_{\oplus}$$ | 0.334 |
其中，$$M_{\odot}$$表示太阳质量，$$M_{J}$$表示木星质量$$\left(M_{J}=1.89813 \times 10^{27} \mathrm{~kg}\right)$$，$$M_{\oplus}$$表示地球质量。假设这些行星绕Gliese 876公转的方向均相同。如果一颗行星相对于另一颗行星的会合周期是后者轨道周期的整数倍，则称这两颗行星处于共振轨道中。
在Gliese 876星系中找出可能存在轨道共振的行星。
解答
官方答案
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官方答案 |
(50%)Determine planet periods
Synodic period of planet 1 and 2 (assume$$ P_1 < P_2$$) is $$1/P_s = 1/P_1 - 1/P_2 \rightarrow P_s = \frac{P_1 P_2}{P_2 - P_1}$$
Resonant orbit happens if
$$P_s = mP_1 \rightarrow \frac{P_1 P_2}{P_2 - P_1} = mP_1 \rightarrow m = \frac{P_2}{P_2 - P_1} > 1$$, integer
Using the data of the Gliesean system we have
| Gliese System | Mass ($$kg$$) | Semi Major Axis ($$m$$) | Period (Earth, days) $$P = 2\pi \sqrt{\frac{a^3}{GM_G}}$$ |
| Gliese 876 star | $$6.64359 \times 10^{29}$$ | ||
| Gliese 876 b | $$4.31938 \times 10^{27}$$ | $$3.1161 \times 10^{10}$$ | $$60.07568$$ |
| Gliese 876 c | $$1.35564 \times 10^{27}$$ | $$1.9388 \times 10^{10}$$ | $$29.48304$$ |
| Gliese 876 d | $$4.079 \times 10^{25}$$ | $$3.1116 \times 10^{9}$$ | $$1.8956$$ |
| Gliese 876 e | $$8.7194 \times 10^{25}$$ | $$5.0011 \times 10^{10}$$ | $$122.1432$$ |
Calculate resonant orbit for each planet
From the results above we see that
(25%)Gliese 876 c and Gliese 876 b have resonant orbit \[m = \frac{P_b}{P_b - P_c} = 1.96 \approx 2 \]
(25%)Gliese 876 b and Gliese 876 e have resonant orbit \[m = \frac{P_e}{P_e - P_b} = 1.9679 \approx 2 \]
翻译
AI翻译
(50%)确定行星周期
行星1和2的会合周期（假设 $$P_1 < P_2$$）为$$ 1/P_s = 1/P_1 - 1/P_2 \rightarrow P_s = \frac{P_1 P_2}{P_2 - P_1}$$
共振轨道发生在满足以下条件时：
$$P_s = mP_1 \rightarrow \frac{P_1 P_2}{P_2 - P_1} = mP_1 \rightarrow m = \frac{P_2}{P_2 - P_1} > 1$$, 整数
使用 Gliese 系统的数据，我们有
| Gliese 星系 | 质量 ($$kg$$) | 半长轴 ($$m$$) | 周期 (地球日) $$P = 2\pi \sqrt{\frac{a^3}{GM_G}}$$ |
| Gliese 876 star | $$6.64359 \times 10^{29}$$ | ||
| Gliese 876 b | $$4.31938 \times 10^{27}$$ | $$3.1161 \times 10^{10}$$ | $$60.07568$$ |
| Gliese 876 c | $$1.35564 \times 10^{27}$$ | $$1.9388 \times 10^{10}$$ | $$29.48304$$ |
| Gliese 876 d | $$4.079 \times 10^{25}$$ | $$3.1116 \times 10^{9}$$ | $$1.8956$$ |
| Gliese 876 e | $$8.7194 \times 10^{25}$$ | $$5.0011 \times 10^{10}$$ | $$122.1432$$ |
2 计算每颗行星的共振轨道
从上述结果中我们可以看到
(25%)Gliese 876 c 和 Gliese 876 b 拥有共振轨道 \[m = \frac{P_b}{P_b - P_c} = 1.96 \approx 2 \]
(25%)Gliese 876 b 和 Gliese 876 e 拥有共振轨道 \[m = \frac{P_e}{P_e - P_b} = 1.9679 \approx 2 \]
