The image below links to an animation that demonstrates that when a planet is near aphelion the point furthest from the Sun, labeled with a B on the screen grab below the line drawn between the Sun and the planet traces out a long, skinny sector between points A and B. When the planet is close to perihelion the point closest to the Sun, labeled with a C on the screen grab below , the line drawn between the Sun and the planet traces out a shorter, fatter sector between points C and D.
These slices that alternate gray and blue were drawn in such a way that the area inside each sector is the same. That is, the sector between C and D on the right contains the same amount of area as the sector between A and B on the left. Since the areas of these two sectors are identical, then Kepler's second law says that the time it takes the planet to travel between A and B and also between C and D must be the same.
If you look at the distance along the ellipse between A and B, it is shorter than the distance between C and D. Since velocity is distance divided by time, and since the distance between A and B is shorter than the distance between C and D, when you divide those distances by the same amount of time you find that:.
The orbits of most planets are almost circular, with eccentricities near 0. In this case, the changes in their speed are not too large over the course of their orbit. For those of you who teach physics, you might note that really, Kepler's second law is just another way of stating that angular momentum is conserved. That is, when the planet is near perihelion, the distance between the Sun and the planet is smaller, so it must increase its tangential velocity to conserve angular momentum, and similarly, when it is near aphelion when their separation is larger, its tangential velocity must decrease so that the total orbital angular momentum is the same as it was at perihelion.
This is usually referred to as the period of an orbit. Kepler noted that the closer a planet was to the Sun, the faster it orbited the Sun. He was the first scientist to study the planets from the perspective that the Sun influenced their orbits.
What this means mathematically is that if the square of the period of an object doubles, then the cube of its semimajor axis must also double.
The proportionality sign in the above equation means that:. Backstage Pass to Iapetus. With fingers crossed and eyes wide open, sci-fi fans and scientists savor the joys of discovery Backstage Pass to Iapetus Blanket Protection. Spacecraft at JPL. This animated movie shows the simulated solar wind velocity from March 21 to May 10, Animation of Spaceweather Predictors 2. Mars Helicopter Prepares for Takeoff. A mission concept to explore Jupiter's intriguing moon system.
The Europa Jupiter System Mission. Amateur video: Follow the Cassini spacecraft during its journey to and near Saturn. The Cassini Journey. Shallow Lightning on Jupiter. In this movie, strung together from a series of images provided by the framing camera on NASA's Dawn spacecraft, we see a full rotation of Vesta, which occurs over the course of roughly five hours. Vesta Full Rotation Movie.
From Mars, the rover was in position to see the opposite side of Tracking Sunspots from Mars, Summer Io Color Eclipse. Jets of icy particles burst from Saturn's moon Enceladus in this brief movie sequence of four images taken on Nov. Enceladus Plume Movie. The tumbling and irregularly shaped moon Hyperion rotates away from the Cassini spacecraft in this movie taken during a distant encounter in Dec.
Rough and Tumble Hyperion. Moons in Motion. Filters, pixels and math star in this narrated cartoon. How Does a Spacecraft Take a Picture? The next full Moon is the Beaver Moon, and there will be a near-total lunar eclipse. Full Moon Guide: November - December These cosmic ribbons of gas have been left behind by a titanic stellar explosion called a supernova.
Establishing gravity as the cause of the moon's orbit does not necessarily establish that gravity is the cause of the planet's orbits.
How then did Newton provide credible evidence that the force of gravity is meets the centripetal force requirement for the elliptical motion of planets?
Recall from earlier in Lesson 3 that Johannes Kepler proposed three laws of planetary motion. His Law of Harmonies suggested that the ratio of the period of orbit squared T 2 to the mean radius of orbit cubed R 3 is the same value k for all the planets that orbit the sun. Known data for the orbiting planets suggested the following average ratio:.
Newton was able to combine the law of universal gravitation with circular motion principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, then a value of 2. Here is the reasoning employed by Newton:. Consider a planet with mass M planet to orbit in nearly circular motion about the sun of mass M Sun. The net centripetal force acting upon this orbiting planet is given by the relationship.
This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as. Substitution of the expression for v 2 into the equation above yields,. By cross-multiplication and simplification, the equation can be transformed into. The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding.
The right side of the above equation will be the same value for every planet regardless of the planet's mass. Newton's universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler's Law of Harmonies.
Our understanding of the elliptical motion of planets about the Sun spanned several years and included contributions from many scientists. Which scientist is credited with the collection of the data necessary to support the planet's elliptical motion?
See Answer Tycho Brahe gathered the data. Johannes Kepler analyzed the data. Isaac Newton explained the data - and that's what the next part of Lesson 4 is all about. Galileo is often credited with the early discovery of four of Jupiter's many moons.
The moons orbiting Jupiter follow the same laws of motion as the planets orbiting the sun. One of the moons is called Io - its distance from Jupiter's center is 4. Another moon is called Ganymede; it is Make a prediction of the period of Ganymede using Kepler's law of harmonies. Suppose a small planet is discovered that is 14 times as far from the sun as the Earth's distance is from the sun 1.
Use Kepler's law of harmonies to predict the orbital period of such a planet.
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