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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Posts Tagged ‘distance to the sun’
Calculating the Distance to the Sun
Thursday, June 25th, 2015Earth’s Distance to the Sun — A Roadmap
Thursday, June 18th, 2015
We left off with Edmund Halley’s proposed method to solve the riddle of Earth’s distance to the sun. Halley posited that when Venus’ orbit brought it directly between the Earth and sun, then principles of astronomy, trigonometry, and geometry could be combined to calculate that distance. Instrumental to Halley’s theory were a number of elements discussed previously in this blog series, including the work of Johannes Kepler. We’ll mesh those elements today and chart the course for future discoveries. To begin things, Halley knew that Kepler’s Third Law of Planetary Motion set out the distance between Earth and the sun in theoretical terms as, 1AU = r_{Venus} ÷ 0.28 which meant that if the distance from Earth to Venus, r_{Venus}, could be calculated, then the distance from Earth to the sun was easily deduced, a matter of simple division. Crucial to the calculation of r_{Venus} is to find the value for the angle α which forms between observers’ lines of sight while charting Venus’ travel across the face of the sun, something which only happens during a rare astronomical event known as the Transit of Venus. See Figure 1. Figure 1 Figure 1 shows two observers positioned on opposite sides of the Earth, busily surveying Venus’ movement across the sun’s face. Their lines of sight converge at a vertex point, or point of intersection, on Venus, then move beyond it to the sun. Due to the principle of vertical angles, which stipulates that angles which share the same vertex point also share the same angle measurement, we know that the angle α that’s formed between Observer A and B‘s lines of sight is of the same value between Earth and Venus as it is between Venus and the sun. Once a is determined, its numerical value can be plopped into an equation we’ve been working with for some time now in this blog series. It’s similar to the equation previously used to calculate Earth’s distance to the moon, r = d x tan(θ) Follow this link to Optically Measuring Cosmic Distances for a review. And here is that equation with terms modified to reflect our new quest, the distance from Earth to Venus, r_{Venus} = d ÷ tan(α) As for the variable d, the distance between the two observers, we’ve worked with that before, too. Follow this link to Determining Chord Length on Circle Earth for a refresher. Next time we’ll see how Venus’ travel path is key to determining the angle α, shown in green on the illustration, and how this angle is crucial to our discovery of the distance between Earth and the sun.
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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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How Far Away is the Sun?
Wednesday, December 24th, 2014
Astronomers tell us that the sun is approximately 93,000,000 miles from Earth. Since no direct method of measuring this distance exists, how did scientists determine this? They had to resort to indirect methods. In the early 18^{th} Century, British astronomer, mathematician and physicist Edmund Halley, of Halley’s Comet fame, devised a method to measure the distance between Earth and its sun with reasonable accuracy. His method was based in part on an optical effect known as the principle of parallax, which had been discovered more than 2000 years ago by Greek astronomer Hipparchus when he also employed it to try and determine that same distance. Halley also employed the ideas of 17^{th} Century German mathematician, astronomer, and astrologist Johannes Kepler. Kepler’s work on planetary motion, in particular the geometry of planetary orbits in our solar system, were vital to Halley’s distance measuring method. Next time we’ll see how Kepler’s Third Law of Planetary Motion brought Halley one step closer to methodology that would determine the distance of our home planet to its sun. _______________________________________
