球面天文学

if we wish to connect three points on the surface of a sphere using the shortest possible route, we would draw arcs of great circles and hence create a spherical triangle. To avoid ambiguities, a triangle drawn on the surface of a sphere is only a spherical triangle if it has all of the following properties:

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大圆
小圆未通过圆心
figure 1: upper panel: examples of great circles. lower panel: examples of small circles. The black dot represents the c源自文库ntre of the sphere.
球面天文基础知识
Spherical Astronomy
教材与参考书目
教材：马文章 《球面天文学》 参考书目： 夏一飞《球面天文学》 L.G.塔夫 《计算球面天文学》 Robin M. Green 《Spherical Astronomy》 天文年历


球面天文学 天体测量学的一个分支，主要内容是运用球 面三角和矢量运算等数学方法研究投影在天 球上的天体位置，以及由于大气折射和地球 和太阳的空间运动等引起的天体位置的变化。 球面天文学是研究天体测量学、天体力学、 恒星天文学和星系动力学等天文分支学科所 必需的基础理论之一。
What to Study??
观测天体位置的归算，天文参考系 转换的理论，天文参考系的应用。

天球，基本点、圈 各种天球坐标系 周日视运动 时间 大气折射

光行差 视差 岁差、章动 自行 综合归算
Important

天文参考系和天文常数系统的精度，直接 影响到天体力学理论的发展，使天文的基 础性工作之一。球面天文学中导出的计算 天体位置的方法在天文学、天文导航学、 测地学、天文地球动力学等学科都有广泛 的应用。 FK5 星表（Catalogue） 天文常数系统
where θ is now measured in radians and s is still given in whatever length units are used for r. Given that spherical triangles are all made up of great circle arcs, and all great circle arcs on a sphere are of the same radius, it is convenient to take the radius of the sphere as unity and then write s=θ
which shows that the length of a side of a spherical triangle is equal to the angle (in radians) it subtends at the centre of the sphere. For example, if s is onequarter of the circumference of a great circle, s is then π/ 2 radians and there is no ambiguity if it is then expressed as 90°.

球面天文学的研究内容：
1. 天球坐标系的建立和天体的视运动 2. 以地球自转和公转周期为基础的时间计量 系统,以及原子时和协调世界时系统。 3. 大气折射、视差、光行差以及自行对天体 位置的影响 4. 岁差和章动对天体位置的影响 5. 天体视位置的归算方法
Why study??

球面天文学不仅是天体测量与天体力学专 业的基础课，它也是所有其他天文专业方 向的基础课。它所研究的天文参考系和天 体位置的转换是天文中都要用到的内容。

The arc lengths (a,b,c) and vertex angles (A,B,C) of the spherical triangle in Figure 4 are related by the following formulae: The sine formula: (sin a / sin A) = (sin b / sin B) = (sin c / sin C)

Now create a new set of axes, keeping the y-axis fixed and moving the "pole" from A to B (i.e. rotating the x,yplane through angle c). The new coordinates of C are x' = sin(a) cos(180-B) = - sin(a) cos(B) y' = sin(a) sin(180-B) = sin(a) sin(B) z' = cos(a)
figure 2: Triangle PCD is a spherical triangle. Triangles PAB and PCED are not spherical triangles.
It is important to realize that the lengths of the sides of a spherical triangle are expressed in angular measure. This follows from the fact that the length of an arc s which subtends an angle at the centre of a circle of radius r is given by s = ( θ / 360) . 2πr where θ is measured in degrees and s is given in whatever length units are used for r. Recalling that 2 πradians = 360°, this gives s=θr
In this course we use only two: the sine rule and the cosine rule. Consider a triangle ABC on the surface of a sphere with radius = 1. We use the capital letters A, B, C to denote the angles at these corners; we use the lower-case letters a, b, c to denote the opposite sides. (Remember that, in spherical geometry, the side of a triangle is the arc of a great circle, so it is also an angle.)
1 The three sides are all arcs of great circles. 2 Any two sides are together greater than the third side. 3 The sum of the three angles is greater than 180°. 4 Each spherical angle is less than 180°.
The relation between the old and new systems is simply a rotation of the x,z-axes through angle c: x' = x cos(c) - z sin(c) y' = y z' = x sin(c) + z cos(c) That is: - sin(a) cos(B) = sin(b) cos(A) cos(c) - cos(b) sin(c) sin(a) sin(B) = sin(b) sin(A) cos(a) = sin(b) cos(A) sin(c) + cos(b) cos(c) These three equations give us the formulae for solving spherical triangles.
A spherical triangle is made up of three arcs of great circles, all less than 180°. The sum of the angles is not fixed, but will always be greater than 180°. If any side of the triangle is exactly 90°, the triangle is called quadrantal. There are many formulae relating the sides and angles of a spherical triangle.

a spherical triangle with arcs of length (a,b,c) and vertex angles of (A,B,C).
figure 4: the vertex angle B is defined as the angle between the tangents to the two great circle arcs.

The cosine formula: cos a = cos b cos c + sin b sin c cos A
球面三角公式
正弦定理： sin a sin b sin c = = sin A sin B sin C 余弦定理： cos a = cos b cos c + sin b sin c cos A cos b = cos a cos c + sin a sin c cos B cos c = cos b cos a + sin b sin a cos C
球面上的圆 （大圆、小圆）
定理： 任何平面和球面的交线都是正圆。 定义： 通过球心的平面与球面的交线，是直径 最大的圆，叫做大圆。 不通过球心的平面与球面的交线，叫小 圆。
The
shortest path between two points on the surface of a sphere is given by the arc of the great circle passing through the two points. A great circle is defined to be the intersection with a sphere of a plane containing the centre of the sphere. If the plane does not contain the centre of the sphere, its intersection with the sphere is known as a small circle.
Turn the sphere so that A is at the "north pole", and let arc AB define the "prime meridian". Set up a system of rectangular axes OXYZ: O is at the centre of the sphere; OZ passes through A; OX passes through arc AB (or the extension of it); OY is perpendicular to both. Find the coordinates of C in this system: x = sin(b) cos(A) y = sin(b) sin(A) z = cos(b)