We describe a method that could have been used by Kepler to determine the shape and dimensions of Mars’ orbit around the Sun, starting from observations of its path across the night sky. Though we use Kepler’s work as a motivation, our approach does not intend to suggest that this method was actually used by him. Rather, it is only intended to show beginning undergraduate students what may be discovered about the planets using basic observational data together with simple geometric methods. We employ publicly available Harvard sky patrol (HSP) photographs of Mars at different moments in the Earth’s year to get its ecliptic coordinates, locate them on a Sun-centered coordinate system, and finally plot them on a sheet of paper. From this orbital graph, we obtain the shape and dimensions of Mars’ orbit around the Sun, using the radius of the Earth’s orbit to set the scale. By generalizing this orbital shape to the other planets in the Solar System, students may be led to conclude Kepler’s first and second laws. Other features of the scaled-down Martian orbit may be directly measured from the paper orbit. To obtain the actual values for such features, we only need to scale them up. The quantities thus determined agree well (within a surprising 5%) with those reported by NASA.

我们描述了一种开普勒可以用来确定火星绕太阳轨道的形状和尺寸的方法，该方法从观察其穿过夜空的路径开始。尽管我们以开普勒的工作为动机，但我们的方法并不意味着他实际上使用了这种方法。相反，它仅旨在向入门的本科生展示使用基本观测数据以及简单的几何方法可以发现的有关行星的信息。我们使用地球年度不同时刻公开提供的哈佛火星的哈佛天空巡逻（HSP）照片来获取其黄道坐标，将其定位在以太阳为中心的坐标系上，最后将它们绘制在一张纸上。从这张轨道图，我们可以得出火星绕太阳轨道的形状和尺寸，使用地球轨道的半径设置比例。通过将此轨道形状推广到太阳系中的其他行星，可以引导学生得出开普勒的第一定律和第二定律。缩小的火星轨道的其他特征可以直接从纸轨道上测量。要获得此类功能的实际值，我们只需要按比例放大即可。这样确定的数量与NASA报告的数量非常吻合（令人惊讶的是5％）。