Last time we saw how centripetal force is responsible for keeping a child’s ball moving in a circular path when twirled above his head. Today we’ll see how that same force is responsible for keeping Earth in its rotation around the sun. We’ll use Isaac Newton’s Second Law of Motion as it relates to centripetal force to bear that out today. Our example from last week, Figure 1 According to Newton’s Second Law of Motion, F_{c}, the centripetal force acting upon the ball, is calculated by the equation, F_{c} = [m_{ball} × v^{2}] ÷ r_{string} where, m_{ball} is the ball’s mass, v the ball’s velocity, and r_{string} is the length of the string. The mechanics are obviously different with regard to Earth and its orbit around the sun, where there’s no obvious physical link between the objects in rotation. In this case the sun’s gravitational force, F_{g}, acts to tether Earth to it. And just like the string and ball, the sun’s gravitational tether creates a centripetal force, F_{c, } which prevents Earth from leaving its orbital path. See Figure 2. Figure 2 Newton concluded that for the Earth to remain on its permanent, fixed path of orbit around the sun, F_{c} must be equal to F_{g}, and that the forces in play between the sun and Earth were like a gigantic tug of war where neither side is able to pull any harder than the other. This stalemate is responsible for keeping Earth stably in place in its orbit. According to Newton’s Second Law, the sun’s gravitational force acting upon the Earth is calculated by, F_{g} = F_{c} = [m × v^{2}] ÷ r where, m is Earth’s mass, v its orbital velocity, and r the distance between Earth and the sun. Thanks to early scientists like Edmund Halley and Henry Cavendish, the values for m and r had already been determined. That left v, Earth’s orbital velocity, the only variable remaining to be solved. We’ll see how 19^{th} Century scientists accomplished that next time.
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Posts Tagged ‘Halley’
Centripetal Force Makes the Earth Go Round
Thursday, July 9th, 2015Calculating the Distance to the Sun
Thursday, June 25th, 2015
We’ve been paying a lot of attention to Venus and its orbital patterns, as did scientist Edmund Halley hundreds of years ago. Back then he came up with a plan to determine Earth’s distance to the sun, the AU. Two key components were Kepler’s Astronomical Unit, or AU, and an angle, α, which formed between lines of sight followed by observers on Earth during the Transit of Venus. Halley theorized that α together with Kepler’s Third Law of Planetary Motion would make it possible to calculate the AU. We’ll see how today. Figure 1 Figure 1 depicts what Halley had in mind. He theorized that if observers positioned on opposite sides of Earth could determine the precise times it took Venus to travel across the sun’s face from each of their perspectives, they could use this information together with previously gathered information on the time it takes Earth and Venus to make a complete orbit around the sun. This would allow the angle α to be calculated, and from that Earth’s distance to Venus, r_{Venus}. Halley’s calculations for α are beyond the scope of this series, but if you’re interested in reading more about them, you can follow this link. Earth’s distance to Venus, r_{Venus,} is computed in a manner similar to the method we used previously to determine Earth’s distance to the moon, by using this equation, r = d × tan(θ) For a refresher on the subject, follow this link to my past blog on Optically Measuring Cosmic Distances. And here’s the same equation modified to solve for the distance between Earth and Venus, r_{Venus,} r_{Venus} = d ÷ tan(α) (1) Once Earth’s distance to Venus was determined, its value was incorporated into Kepler’s equation for 1 AU, and the distance between Earth and the sun became known. Here again is the equation from Kepler’s Third Law of Planetary Motion, 1 AU = r_{Venus} ÷ 0.28 (2) And here it is with the function for r_{Venus} from equation (1) inserted into equation (2) to solve for 1 AU, 1 AU = [d ÷ tan(α)] ÷ 0.28 1 AU = d ÷ [0.28 × tan(α)] (3) From equation (3) the distance between Earth and the sun, 1 Astronomical Unit, was calculated to be between 92,000,000 and 96,000,000 miles. Unfortunately, a Transit of Venus did not occur within Halley’s lifetime, but scientists that followed him applied his methodology after the next Transit occurred in 1761. Since that time modern technology and the radar have improved measuring accuracy so that we now know the sun is located 92,935,700 miles from Earth. Next time we’ll return full circle to our opening topic in this long blog series when we reopen our discussion on gravity, specifically, how the concept of centripetal force is instrumental in determining the gravitational force exerted upon Earth by the sun.
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The Transit of Venus from Different Perspectives
Wednesday, June 3rd, 2015
We’ve been working on calculating Earth’s distance to the sun and discussing how past scientists, including Johannes Kepler and Edmund Halley, contributed to the discussion. Today we’ll see how Halley used his theory on the transit of Venus together with Kepler’s Third Law of Planetary Motion and combined them with known mathematical principles to solve the riddle of this enormous distance, known as one astronomical unit, or AU. Halley’s solution began with stationing two observers with telescopes aimed at Venus on opposite sides of the Earth. Their different lines of sight would cause the principle of parallax to come into play, resulting in them seeing Venus from different perspectives. Their sight would converge at Venus’ center and an angle, α, would form between them. Halley posited that if this angle could be measured, it would be an important first step in calculating the distance between Earth and the sun. See Figure 1. Due to their differing perspectives, Observer A would see Venus traveling a path lower on the sun’s face, while Observer B would see it following a path higher up. See Figure 2. The net result was that the length, L_{A}, of Venus’ path as seen by Observer A was significantly shorter than length, L_{B}, of Venus’ path as seen by Observer B. Because of this, Observer B would have seen Venus pass in front of the sun before Observer A. These differing observations meant that even if both observers were to set their telescopic crosshairs on Venus at the exact same moment it became visible to each of them, it would serve no purpose, because they lacked a common point of reference at which to aim in order to take measurements. This fact made measuring the angle α with a physical device such as a protractor impossible. So Halley gave up on the idea of physically measuring α. Instead, he proposed calculating it based on the time it took for Observers A and B to watch Venus traverse the sun’s face from one side to the other along each of their observational paths. Next time we’ll see how Halley put his idea to work to calculate α and used it in conjunction with Kepler’s Third Law to calculate the AU.
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The Transit of Venus
Monday, May 18th, 2015
Last time we learned of Johannes Kepler’s Third Law of Planetary Motion and his development of the astronomical unit (AU) and how these contributed to bringing ancient scientists a step closer to calculating Earth’s distance to the sun. Today we’ll see why Kepler’s focus on Venus, specifically its travel through space in relation to Earth and the sun — the socalled transit of Venus — would become the crucial element to solving the puzzle. Astronomers had previously used the Earth itself as an optical rangefinder to calculate distance to the moon. But unlike the moon which is relatively close to Earth, the sun is many tens of millions of miles away, too distant to be used in that manner. When it came to finding the distance from Earth to its sun, they were stumped. Then in 1716 Edmund Halley had the insight to combine Kepler’s Third Law and the parallax principle with Venus’ orbital journeys to devise an ingenious solution to the problem. The transit of Venus, first predicted by Kepler in 1627, is a rare astronomical phenomenon which only occurs every 243 years. At this time Venus becomes clearly visible from Earth and appears as a black dot traveling a straight path across the fiery backdrop of the sun’s surface. Figure 1 shows this phenomenon as it would look from Earth. According to Halley’s plan, two observers with telescopes would be positioned on opposite sides of the Earth. Due to the principle of parallax their lines of sight would provide different perspectives of Venus’ path. See Figure 2. From the perspective of Observer B, Venus’s path would appear higher on the sun’s face than the path seen from the perspective of Observer A. As their lines of sight converge on Venus’ center, an angle forms between them, which we’ll name α. The same angle forms as they look past Venus to the sun in its backdrop. Halley theorized that if the angle α could be measured, Kepler’s Third Law could be used together with trigonometry to calculate the distance between Earth and the sun, Kepler’s socalled AU. We’ll review Halley’s methodology next time.
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Gravity and the Mass of the Sun
Friday, December 12th, 2014
As a young school boy I found it hard to believe that scientists were able to compute the mass of our sun. After all, a galacticsized measuring device does not exist. But where there’s a will, there’s a way, and by the 18th Century scientists had it all figured out, thanks to the work of others before them. Newton’s two formulas concerning gravity were key to later scientific discoveries, and we’ll be working with them again today to derive a third formula, bringing us a step closer to determining our sun’s mass. Newton’s Second Law of Motion allows us to compute the force of gravity, F_{g,} acting upon the Earth, which has a mass of m. It is, F_{g} = m × g (1) Newton’s Universal Law of Gravitation allows us to solve for g, the sun’s acceleration of gravity value, g = (G × M) ÷ r^{2} (2) where, M is the mass of the sun, r is the distance between the sun and Earth, and G is the universal gravitational constant. You will note that g is a common factor between the two equations, and we’ll use that fact to combine them. We’ll do so by substituting the right side of equation (2) for the g in equation (1) to get, F_{g} = m × [(G × M) ÷ r^{2}] then, using algebra to rearrange terms, we’ll set up the combined equation to solve for M, the sun’s mass: M = (F_{g }× r^{2}) ÷ (m × G)^{ } (3) At this point in the process we know some values for factors in equation (3), but not others. Thanks to Henry Cavendish’s work we know the value of m, the Earth’s mass, and G, the universal gravitational constant. What we don’t yet know is Earth’s distance to the sun, r, and the gravitational attractive force, F_{g}, that exists between them. Next time we’ll introduce some key scientists whose work contributed to a method for computing the distance of our planet Earth to its sun. _______________________________________
