2019年USAAAO决赛第5题

英文题目

5. (15 points) An alien spaceship from the planet Kepler 62f is in search of a rocky planet for a remote base. They’re attracted to Earth because of a fortunate coincidence: its axis of rotation points directly at their home planet. That means they can have uninterrupted communication with home by planting fixed transmitters on The North Pole. But first, they need to find out if Earth’s axis will always point in the same direction or if it undergoes precession. They can’t know without years of observation, so they hope that we, its now-extinct intelligence, have left behind the answer. While orbiting Earth, they see a few remarkable structures, including the Hoover Dam in Nevada. Zooming in on the dam, a colorful plaza with peculiar markings on its floor catches their attention. Descending on the plaza, they realize the markings are a map of the sky when the dam was built, left to indicate the date to posterity. Figure 1 is an overhead architectural map of this plaza. The center-point depicts the north ecliptic pole, and the large circle represents the path of the Earth’s axis throughout its counter-clockwise procession. As they interpret the map, they’re dismayed to realize that their star has not been and will not be Earth’s north star for very long.

Figure 1: Overhead architectural plan of the Hoover Dam plaza depicting Polaris as north star

For the purpose of this question, assume that the Earth’s axial tilt is a constant $$i = 23.5^{\circ}$$ and its axis precesses at a constant rate.

a) Using the values on the map, and knowing that the aliens used carbon-aging to determine that the dam is 12,000 years old, find all possible values for the period of Earth’s axial precession.

b) Using the most optimistic answer (longest period) from part (a), calculate how many arcseconds the Earth’s axis precesses each day. Use the period you calculate here in the next two sections.

c) If they hadn’t been lucky enough to come across the star map and decided to build a radio interferometer to observe the movement of the celestial pole over the course of 30 days instead, how many kilometers would the baseline of their telescope array have to be, assuming it operated at a 20cm wavelength?

d) As a last resort, to keep Earth’s axis fixed, the aliens decide to counter the forces that cause the Earth’s precession by building giant nuclear thrusters on the Earth’s surface. Assume Earth’s precession is caused by external forces alone and calculate the average force (in kN) that a strategically positioned thruster would have to exert to counter them.

中文翻译

5. (15 points)来自开普勒62f星的一艘外星飞船正在寻找一个适合建立偏远基地的岩石行星。他们因为一个幸运的巧合被地球吸引：地球的自转轴正好指向他们的母星。这意味着他们可以通过在北极安装固定的发射器与母星保持不间断的通信。但首先，他们需要弄清楚地球的自转轴是否会一直指向同一方向，或者是否会发生岁差。如果没有多年的观测，他们无法得知，所以他们希望我们——现在已经灭绝的智慧生命——留下了答案。在环绕地球飞行时，他们看到了几处引人注目的建筑，包括内华达州的胡佛水坝。放大观看水坝时，一个地面上带有奇特标记的彩色广场引起了他们的注意。当他们降落在广场上时，意识到这些标记是一张水坝建造时天空的地图，用来向后世指示日期。图 1 是这个广场的建筑俯视图。中心点表示北黄极，大圆表示地球自转轴在逆时针的岁差运动中的路径。当他们解读这张地图时，沮丧地意识到他们的星星过去并没有，也不会在很长时间内成为地球的北极星。

为此，假设地球的轴倾角为常数$$i = 23.5^{circ}$$，且其轴以恒定速率进动。

a）使用地图上的数据，已知外星人使用碳年代测定法确定大坝有 12,000 年历史，找出地球轴向岁差周期的所有可能数值。

b）使用第(a)部分中最乐观的答案（最长周期），计算地球轴每天进动多少角秒。在接下来的两个部分中使用你在这里计算的周期。

c）如果他们运气不足，没有遇到星图，而是决定建造一个射电干涉仪来观察天极在30天内的运动，那么假设其工作波长为20厘米，他们的望远镜阵列的基线长度需要多少公里？

d）为了保持地轴固定，作为最后的手段，外星人决定通过在地球表面建造巨大的核动力推进器来抵消地球岁差。假设地球的岁差仅由外力引起，并计算一个战略性放置的推进器必须施加的平均力（以千牛计）。