Ellipses and Elliptic Orbits

An ellipse is defined as the set of points that satisfies the equation In cartesian coordinates with the x-axis horizontal, the ellipse equation is 椭圆被定义为满足方程的点的集合。在直角坐标系中，x轴水平的情况下，椭圆的方程为

The ellipse may be seen to be a conic section, a curve obtained by slicing a circular cone. A slice perpendicular to the axis gives the special case of a circle. 椭圆可以被视为圆锥曲线，这是一种通过切割圆锥所得的曲线。垂直于轴的切割得到圆的特殊情况。

For the description of an elliptic orbit, it is convenient to express the orbital position in polar coordinates, using the angle θ:

Show 显示

Show 显示

Each of the conic sections can be described in terms of a semimajor axis a and an eccentricity e. Representative values for these parameters are shown along with the types of orbits which are associated with them. 每个二次曲线都可以用半长轴a和偏心率e来描述。这些参数的代表性值如图所示，并列出了与之相关的轨道类型。

Polar Form of Ellipse

From the diagram at left, using the Pythagorean theorem to express r' in terms of r: Using the trigonometric identity this reduces to 从左边的图中，利用勾股定理将 r' 用 r 表示出来：利用三角恒等式，这可以简化为

Using the equation for an ellipse, an expression for r can be obtained

This form is useful in the application of Kepler's Law of Orbits for binary orbits under the influence of gravity.

Area of Ellipse

Using the equation for an ellipse 使用椭圆方程 the height y can be expressed as 高度 y 可以表示为

This gives the area integral