Kepler’s Laws and the Motion of Planets
While scientists today largely take for granted the framework under which our solar system works, the nature of this framework was the subject of substantial debate for literally thousands of years. The debate touched on not just astronomy and science in general, but also philosophy and religion. This effort to describe the motion of planets in our solar system had long lasting applications for other objects in our solar system and indeed other solar systems as well.
Questions about the framework of our solar system and the motion of the planets can show up in a wide variety of situations. General education introduction to astronomy classes devote typically one or two weeks to these topics, astronomy lab classes will do calculations using these concepts, upper level astrophysics courses will apply these principles to unique and new situations and discoveries, and physics classes might derive where the equations and laws that describe how our solar system works come from using equations for motion and gravity.
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Introduction to Views on the Solar System
While there may have been a few forward thinkers in their day (Aristarchus perhaps most famously), the prevailing view of our solar system from ancient times until the 1500 and 1600s was that the Earth was at the center of the solar system and everything else, including the Sun, orbited around the Earth. This type of theory is called a geocentric (Earth-centered) model. While today we take for granted the absurdity of such a model, it is still worth considering the following question: do you feel that you are flying through space around the Sun at around 67,000 miles each hour or do you feel that you are at rest and the sky, sun, planets, and stars are moving around us?
Astronomy is a science, which means that astronomers are constantly making observations to test and refine existing theories. With the development of better techniques to measure the location of planets compared to the stars in the sky, scientists began to realize that the predicted locations of the planets from the geocentric model did not match up with their actual locations. In response, Ptolemy modified the prevailing geocentric model to include epicycles, small imaginary circles that held the planets. As the epicycles orbited the Earth in circular orbits, the planets themselves moved around their epicycles. At first this model may seem strange, and Ptolemy made no attempt to explain why there should be epicycles or how they physically worked. However, it met one substantial goal: predicting the planet positions accurately – or at least it did in the moment. However, as scientific skill in measuring planet positions increased, astronomers found that the observations did not match the predictions from Ptolemy’s geocentric model. They tried to fix the model by adding even more epicycles (without any explanation as to why they should be there) until the entire model was a complete mess.
Was there a better solution? Yes there was, and this idea is credited to the Polish astronomer, Nicolaus Copernicus, although the basis of the theory dates back to ancient Greece and was also posited by Islamic astronomers in the middle ages. Copernicus, in his 1543 work, De revolutionibus orbium coelestium, outlined what is now known as the heliocentric or “Sun-centered” model of our solar system. In this model, the Sun, not the Earth, is now the center of the solar system and all the planets, including the Earth, are in circular orbit around the Sun. Copernicus’s model, however, was far from perfect. With planets orbiting the Sun in perfect circles, Copernicus could not match observations exactly to his heliocentric theory. In fact, he famously added small epicycles that each planet traveled along as it orbited the Sun to make the numbers work. Despite his inability to develop a model of our solar system that does not rely on an idea with so little physical basis as epicycles, Copernicus nevertheless ushered in the controversial new idea about our heliocentric solar system and the Earth’s role in it that after more than a hundred years of fight and pushback finally became the accepted model of our solar system.
Johannes Kepler (1571-1630)
Almost 100 years after Copernicus’s groundbreaking work was published, a simple working model of our solar system had yet to be developed. That would soon change thanks to the insight of the German astronomer and mathematician, Johannes Kepler. Through a set of circumstances, Kepler gained access to the extensive, highly precise planet position measurements conducted for many years by the eccentric astronomer Tycho Brahe. Kepler first turned his attention to understanding the orbit of Mars and after almost 10 years of research and trial and error, developed a model of the solar system that accurately predicted Mars’s orbit without the reliance on artificial constructs such as epicycles. While he initially matched his model’s predictions only to the actual motion of Mars, Kepler claimed (correctly) that it applies to not only Mars but all the planets of the solar system. He presented his model in a 1609 work, Astronomia nova.
Kepler’s Three Laws of Planetary Motion
Kepler’s model of the solar system was firmly rooted in the heliocentric model of Copernicus. He presented his model in the form of three laws of planetary motion that applied to every planet in our solar system as it orbited the Sun.
Kepler’s First Law: Planets orbit the Sun in ellipses with the Sun at one of the foci of each ellipse.
In Kepler’s first law we immediately see the departure from Copernicus’s heliocentric idea. Copernicus assumed that planets orbit in circles, the perfect shape from classical mathematics and architecture. However, Kepler’s model showed that the orbits are actually ellipses. An ellipse is essentially an oval or flattened circle. An ellipse can take on many different appearances depending on how flattened of a circle it is. The degree of “flattenedness” is a term called eccentricity. Eccentricity goes from 0 to 1, where 0 is a completely unflattened ellipse (i.e. a circle) and an eccentricity of 1 is a completely flattened ellipse (i.e. a line). Most planets have eccentricities that are close to 0, making them circle-like but not actual circles. Comets, which have orbits that take them very close to the Sun but also far away, in contrast have high eccentricity (some as high as 0.999).
What about the 2nd part of Kepler’s first law, the part suggesting that for planetary orbits the Sun is located at one of the foci of the ellipse? Foci are two key points inside an ellipse – mathematically an ellipse is defined as all points where the sum of the distance between the point and the two foci is a constant. The eccentricity of the ellipse also determines the separation between the foci. At its lower limit (eccentricity = 0), then the foci are not separated at all but located together in the very center of the circular ellipse. As the eccentricity increases then the foci are more and more separated. The fact that the Sun is located at one of the foci of the ellipse means that the distance of the planet from the Sun varies (except for eccentricity = 0) and that the greater the eccentricity, the more the distance varies from closest to farthest. This can be seen just by comparing the closest and farthest distances of Earth to that of a higher eccentricity object like a comet. The Earth at its closest is 0.98 AU and at its farthest 1.02 AU. Compare that with Halley’s Comet which has an eccentricity of 0.97. At its closest, Halley’s Comet is 0.59 AU and 35.08 AU at its farthest from the Sun. Finally, an even more extreme comet, Hale-Bopp, which has an eccentricity of 0.995, ranges from 0.91 AU at closest approach to 370.8 AU at its farthest distance. Planets, in summary, have only minor differences in distance from the Sun, compared with higher eccentricity objects like comets that have extreme differences from closest to farthest distance.
Kepler’s Second Law: A line between the Sun and a planet sweeps equal areas in equal times.
While Kepler’s Second Law is phrased in this seemingly strange way, at its core, Kepler’s Second Law relates to the speed of planets in their orbits around the Sun. Planets move fastest when they are located closet to the Sun and slowest when they are farthest from the Sun. For planets, the difference is often quite small. For example, at Earth’s closest approach to the Sun, it is orbiting at 30.3 km/s compared with 29.3 km/s at its farthest distance, a difference of approximately 3%. Compare that with Halley’s Comet which moves at 54.3 km/s at closest approach and less than 1 km/s (0.91 km/s to be precise) at its greatest distance, a difference of approximately 193%. In other words, the greater the eccentricity of a planet, not only the greater the variation in distance, but also the greater the variation in the speed of the planet from closest to farthest approach.
Kepler’s Third Law: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
First, let’s run through a few definitions: orbital period is the time that it takes for a planet to orbit the Sun; the semi-major axis of a planet is the average of its closest and farthest distances from the Sun. What is key about Kepler’s Third Law is that he determined there is a clear and direct relationship between a planet’s time to orbit and its distance from the Sun. Additionally, while this is likely no surprise after a little bit of thought, the third law determines that planets that are farther from the Sun take longer to orbit. Now, for instance, not only can we say that Mars takes longer to orbit the Sun than the Earth but we can directly calculate that based on its semi-major axis distance of 1.52 AU, Mars will take 1.88 Earth years to orbit. These numbers for Mars illustrate another important conclusion that can be derived from Kepler’s Third Law: the period of objects increases more than linearly as the semimajor axis increases. Mars is only 1.5 AU from the Sun (50% larger distance) but takes almost 100% longer to orbit. Neptune, the most distant planet, has a semi-major axis of 30.1 AU (approximately 30x the semi-major axis of Earth) but takes 165 years to orbit (approximately 165x the orbital period of the Earth).
Proving Kepler’s Laws
Johannes Kepler’s ground-breaking work lacked one critical component. Astronomia nova described in great detail his three laws of planetary motion. However, he presented these laws without proof, instead focusing on how these laws match the data but without explaining why. In essence, he was saying, “Here are some rules. They work but I’m not exactly sure why”. The “why” they work took almost a century more until the publication of Principia by Sir Isaac Newton. Among other things in this groundbreaking book were his derivation of Kepler’s three laws of planetary motion. Newton took his own general laws of motion and his law of universal gravitation and used them to show why Kepler’s laws work. For example, planets are constantly accelerating as they orbit the Sun because of the small changes in velocity and the constant change in direction of the planet in its orbit. Therefore, a force must be creating this acceleration, and Newton showed that if you assume the law of gravity is causing this acceleration, you can derive Kepler’s three laws of planetary motion.
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The word "astronomy" has Greek origin, and it was created from the Greek words “astro” + “nomos”, which together mean “the law of the stars”. Astronomy is a science that studies the origin and evolution, as well as the physical and chemical properties of all objects outside of Earth's atmosphere.
Part of the galaxy generated by the action of the gravitational force of the Sun is called the Solar planetary system.
General characteristics of the Solar Planetary System
-The Sun is the central Star
-In addition to the Sun, the solar system consists of planets, dwarf planets, satellites, asteroids, meteoroids, comets, objects in the Kuiper belt and interplanetary material
-Dominant gravitational influence comes from the Sun
- All objects in the Solar System move around the Sun along elliptical paths (orbits)
-The planets are the largest objects and many have their followers - satellites. The largest are Ganimed (Jupiter), Titan (Saturn) and Kalisto (Jupiter)
Тhе Sun is 149.6 mil. km from the Earth. It consists of six zones: the core, the radiation zone, the convective zone, the photosphere, the chromosphere and the corona. The energy is generated in the core and transmitted to the surface by radiation and convection.
Mercury is the smallest planet in the Sun's system and closest to the Sun. One day on Mercury is equal to 59 days on Earth, and a year lasts the same as 88 Earth days. It has a solid, rocky surface covered with craters.
Venus is a bit smaller than Earth. One Venus day is longer than one Venus year. One Venus day is comparable to 243 earth days and the period of the revolution of 225 earth days. It rotates in the opposite direction as Earth. Venus’ mass, density and gravity is similar to Earth’s and it’s atmosphere is mostly comprised of carbon dioxide.
Earth is the the only known planet (so far) where life exists. The surface of the planet is rocky and 70% of the planet's surface is covered by water. There is a core which is made of iron and nickel. Around the core, there is a rocky cover. The earth also has an atmosphere, which contains 78% nitrogen, 21% oxygen, and 1% of other elements. Earth has one satellite – our moon.
Mars’ rotation period is similar to the Earth (one day on Mars lasts just over 24 hours). The year on Mars lasts 687 earth days. Its surface is rocky and dry. Mars has visible seasonal changes and has two known satellites, Fobos (Phobos) and Deimos.
Jupiter is the largest planet in the solar system. It is a gaseous planet mostly composed of ammonia. In 1979, rings around Jupiter were discovered. Of Jupiter's 67 known satellites, the largest are Europe (Europe), Ganimed (Ganymede), and Kalisto (Callisto).
Saturn is built mostly of hydrogen and helium. By volume it is 755 times larger than Earth. In the upper layers of the atmosphere, winds on Saturn reach 500 miles/second, which is five times faster than the fastest winds on Earth. These winds cause the formation of yellow and golden clouds around the planet. Due to strong pressures, the inner core of the planet is in solid state. The overheated core is a mixture of hydrogen in liquid and molten metals. The outer layer of the planet is made of hydrogen in a liquid state. Saturn's magnetic field is 578 times stronger than the Earth's magnetic field. Saturn has 62 satellites. With a diameter of 5,150 km, the satellite called Titan is second largest satellite in the solar system.
Uranus rotates retrograde, from the east to the west, and the axis around which it rotates is at an angle of 980 to the orbital plane, so it appears to roll on its side during the revolution. The atmosphere is mostly made of hydrogen and helium. It has 27 satellites, of which Oberon and Titania are the largest.
Neptune is made of ice. One year of Neptune lasts for 165 Earth years and the day last for 16 hours. The winds on Neptune are nine times stronger than the winds on Earth and its magnetic field is 27 times stronger than Earth’s. Neptune has 14 satellites. The greatest is Triton.
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