2018年IOAA理论第10题-牛郎和织女
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英文题目
(T10) Vega and Altair(75 points)
As per a very famous Chinese folklore about love, Vega and Altair are two lovers. It is said that they can meet each other once every year on a bridge made up of birds over the Milky Way. The parameters of two stars are given in the table below. For the purpose of this question, assume that the coordinate frame is fixed (i.e. not affected by precession or motion of the Sun).
Based on this data, answer the following questions：
(a) (9 points) What is the angular separation of the two stars?
(b) (6 points) Calculate the distance (in parsecs) between Vega and Altair.
(c) (3 points) Calculate position angles of the proper motion vectors of each of these two stars.
For parts d-g, assume that the angular velocity of the stars on the celestial sphere remains constant. This is not a physical situation but this is an assumption to simplify the problem.
(d) (2 points) How many common points on the celestial sphere are there which can be reached by both these stars?
(e) (20 points) Find the coordinates of the closest such point. (Note: Drawing the situation on a celestial sphere will help you in visualising the situation) (f) (8 points) Find when (which year) each of these stars were / will be at that point.
(g) (5 points) When Altair was / will be at that point, what would be its angular separation from Vega?
(h) (22 points) Find coordinates of any point (if it exists) in 3-D space which was /will be visited by both these stars. Do not ignore radial velocities for this part of the question.
| Star/恒星 | Right Ascension/赤经
(J2000.0) |
Declination/赤纬
(J2000.0) |
Parallax/视差
(mas) |
Proper Motion/自行 | Radial Volocity/视向速度
(km/s) | |
|---|---|---|---|---|---|---|
| μαcosδ(mas/year) | μδ(mas/year) | |||||
| Vega/织女星 | 18h36m56.49s | +38°47'07.7'' | 130.23 | +200.94 | +286.23 | -13.9 |
| Altair/牛郎星 | 19h50m47.70s | +8°52'13.3'' | 194.85 | +536.23 | +385.29 | -26.1 |
中文翻译
(T10)牛郎和织女(75分）
根据中国一段有名的神话传说，牛郎星和织女星是一对恋人。据说他们每年又一次机会在银河上空的鹊桥上相会。两颗星的参数在下表中列出。为了解题方便，假设参考架是固定的（也即，不会受岁差和太阳自身的运动影响。）
根据这些数据，回答下列问题：
(a)(9分)这两颗星的角距离是多少？
(b)(6分)计算这每颗星在空间中的距离，单位为秒差距。
(c)(3分)计算每颗星自行的方向角。
对于d-g小问，假设两颗星在天球上的自行角速度保持恒定。这在实际情况是不可能的，但为了简化问题做出这个假设。
(d)(2分)这两颗星在天球上经过的轨迹有多少个公共点？
(e)(20分)找到离他们最近的公共点。（提示：画出示意图有助于你理解情景）
(f)(8分)计算什么时候（具体到年）每颗星位于这个点。
(g)(5分)当牛郎星位于这个点时，它与织女星的角距离是多少？
(h)(22分)找出它们在空间中轨迹的公共点（如果存在）。这个小问中注意不要忽略视向速度。
[分类:球面三角]
