從太陽北極上方觀察，行星以逆時針方向繞著太陽運行，行星的軌道都與天文學家所說的黃道面對齊。 如果沒有一位名叫約翰內斯·普勒的德國數學家的工作，我們就不可能對行星運動有更深入的了解。
Kepler and Tycho
克普勒生活在動盪的 17 世紀初奧地利格拉茨。 由於那個時代常見的宗教和政治困難，開普勒於1600 年8 月2 日被驅逐出格拉茨。幸運的是，一個為著名天文學家第谷·布拉赫擔任助手的機會出現了，年輕的克普勒舉家搬離格拉茨300 英里穿過多瑙河，到達布拉赫在捷克布拉格的家。 第谷·布拉赫被認為是他那個時代最準確的天文觀測者，並且在早些時候的一次會議上對克普勒的研究印象深刻。 然而，布拉赫不信任克普勒，擔心他聰明的年輕實習生可能會讓他作為當時首席天文學家的地位黯然失色。 因此，他讓克普勒只能看到他大量的行星數據的一部分。 他為克普勒設定了了解火星軌道的任務，而火星的運動與亞里斯多德和托勒密所描述的宇宙並不吻合。
火星的軌道
據信，將火星問題交給克普勒的部分動機是布拉赫希望克普勒能夠解決這個問題，而布拉赫則致力於完善自己的太陽系理論，該理論基於地心模型，其中地球是太陽系的模型。太陽系的中心。 根據這個模型，水星、金星、火星、木星和土星都繞著太陽運行，而太陽又繞著地球運行。 事實證明，克普勒與布拉赫不同，他堅信哥白尼太陽系模型“日心說”，該模型正確地將太陽置於其中心。 但火星軌道有問題的原因是哥白尼系統錯誤地假設行星的軌道是圓形的。 經過一番努力，克普勒最終被迫認識到行星的軌道不是圓形，而是幾何學家稱之為橢圓的拉長或扁平的圓形，而布拉赫在火星運動方面遇到的特殊困難是由於以下事實它的軌道是布拉赫擁有大量數據的行星中最橢圓的。 因此，諷刺的是，布拉赫無意中向克普勒提供了他的部分數據，這些數據將使克卜勒能夠制定正確的太陽系理論，從而否定了布拉赫自己的理論。
橢圓的性質
由於行星的軌道是橢圓，讓我們回顧一下橢圓的三個基本性質。 橢圓的第一個屬性：橢圓由兩個點定義，每個點稱為焦點，合在一起稱為焦點。 橢圓任意一點到焦點的距離總和總是一個常數。 橢圓的第二個性質：橢圓變平的程度稱為偏心率。 橢圓越扁平，其離心率越大。 每個橢圓都有一個偏心率，其值介於零、圓和一之間，本質上是一條直線，技術上稱為拋物線。 橢圓的第三個性質：橢圓的最長軸稱為長軸，最短軸稱為短軸。 長軸的一半稱為半長軸。 約翰尼斯·克普勒得知行星的軌道是橢圓形後，制定了行星運動的三個定律，該定律也準確地描述了彗星的運動。
克卜勒第一定律
克卜勒第一定律：每顆行星繞太陽的軌道都是橢圓形。 太陽的中心始終位於軌道橢圓的一個焦點。 太陽位於一個焦點上。
克卜勒第二定律
行星沿著橢圓軌道運行，這意味著行星到太陽的距離會隨著行星繞軌道運行而不斷變化。 克卜勒第二定律：連接行星和太陽的假想線在行星繞軌道運行的相同時間間隔內掃過相同的空間面積。 基本上，行星不會沿著其軌道以恆定速度移動。 相反，它們的速度各不相同，因此連接太陽和行星中心的線在相同的時間內掃過一個區域的相同部分。 行星最接近太陽的點稱為近日點。 最大分離點是遠日點，因此根據克普勒第二定律，行星在近日點時移動最快，在遠日點移動最慢。
克卜勒第三定律
克卜勒第三定律：行星軌道周期的平方與其軌道半長軸的立方成正比。 克普勒第三定律意味著行星繞太陽運行的周期隨著其軌道半徑的增加而迅速增加。 由此我們發現，最內層的行星水星繞太陽公轉一週只需要88天。 地球需要 365 天，而土星則需要 10,759 天。
Kepler --> Newton
儘管克普勒在提出他的三定律時並不知道萬有引力，但它們對艾薩克·牛頓推導出萬有引力理論起了重要作用，該理論解釋了克普勒第三定律背後的未知力量。 克普勒和他的理論對於更好地理解我們的太陽系動力學至關重要，並且可以作為更準確地近似我們行星軌道的新理論的跳板。
Solar System Dynamics: Orbits and Kepler's Laws -- NASA
The planets orbit the Sun in a counterclockwise direction as viewed from above the Sun's north pole, and the planets' orbits all are aligned to what astronomers call the ecliptic plane. The story of our greater understanding of planetary motion could not be told if it were not for the work of a German mathematician named Johannes Kepler. Kepler lived in Graz, Austria during the tumultuous early 17th century. Due to religious and political difficulties common during that era, Kepler was banished from Graz on August 2nd, 1600. Fortunately, an opportunity to work as an assistant for the famous astronomer Tycho Brahe presented itself and the young Kepler moved his family from Graz 300 miles across the Danube River to Brahe's home in Prague. Tycho Brahe is credited with the most accurate astronomical observations of his time and was impressed with the studies of Kepler during an earlier meeting. However, Brahe mistrusted Kepler, fearing that his bright young intern might eclipse him as the premier astronomer of his day.
He therefore led Kepler see only part of his voluminous planetary data. He set Kepler, the task of understanding the orbit of the planet Mars, the movement of which fit problematically into the universe as described by Aristotle and Ptolemy. It is believed that part of the motivation for giving the Mars problem to Kepler was Brahe's hope that its difficulty would occupy Kepler while Brahe worked to perfect his own theory of the solar system, which was based on a geocentric model, where the earth is the center of the solar system. Based on this model, the planets Mercury, Venus, Mars, Jupiter, and Saturn all orbit the Sun, which in turn orbits the earth. As it turned out, Kepler, unlike Brahe, believed firmly in the Copernican model of the solar system known as heliocentric, which correctly placed the Sun at its center. But the reason Mars' orbit was problematic was because the Copernican system incorrectly assumed the orbits of the planets to be circular.
After much struggling, Kepler was forced to an eventual realization that the orbits of the planets are not circles, but were instead the elongated or flattened circles that geometers call ellipses, and the particular difficulties Brahe hand with the movement of Mars were due to the fact that its orbit was the most elliptical of the planets for which Brahe had extensive data.
Thus, in a twist of irony, Brahe unwittingly gave Kepler the very part of his data that would enable Kepler to formulate the correct theory of the solar system, banishing Brahe's own theory.
Since the orbits of the planets are ellipses, let us review three basic properties of ellipses. The first property of an ellipse: an ellipse is defined by two points, each called a focus, and together called foci. The sum of the distances to the foci from any point on the ellipse is always a constant. The second property of an ellipse: the amount of flattening of the ellipse is called the eccentricity. The flatter the ellipse, the more eccentric it is. Each ellipse has an eccentricity with a value between zero, a circle, and one, essentially a flat line, technically called a parabola. The third property of an ellipse: the longest axis of the ellipse is called the major axis, while the shortest axis is called the minor axis. Half of the major axis is termed a semi major axis. Knowing then that the orbits of the planets are elliptical, johannes Kepler formulated three laws of planetary motion, which accurately described the motion of comets as well. Kepler's First Law: each planet's orbit about the Sun is an ellipse. The Sun's center is always located at one focus of the orbital ellipse. The Sun is at one focus. The planet follows the ellipse in its orbit, meaning that the planet to Sun distance is constantly changing as the planet goes around its orbit. Kepler's Second Law: the imaginary line joining a planet and the Sun's sweeps equal areas of space during equal time intervals as the planet orbits. Basically, that planets do not move with constant speed along their orbits. Rather, their speed varies so that the line joining the centers of the Sun and the planet sweeps out equal parts of an area in equal times. The point of nearest approach of the planet to the Sun is termed perihelion. The point of greatest separation is aphelion, hence by Kepler's Second Law, a planet is moving fastest when it is at perihelion and slowest at aphelion. Kepler's Third Law: the squares of the orbital periods of the planets are directly proportional to the cubes of the semi major axes of their orbits. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun. The earth takes 365 days, while Saturn requires 10,759 days to do the same.
Though Kepler hadn't known about gravitation when he came up with his three laws, they were instrumental in Isaac Newton deriving his theory of universal gravitation, which explains the unknown force behind Kepler's Third Law. Kepler and his theories were crucial in the better understanding of our solar system dynamics and as a springboard to newer theories that more accurately approximate our planetary orbits.
授課教師
陳永忠 ycchen@thu.edu.tw