Posts Tagged ‘Kepler’s Third Law of Planetary Motion’

Earth’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 = rVenus ÷ 0.28

which meant that if the distance from Earth to Venus, rVenus, could be calculated, then the distance from Earth to the sun was easily deduced, a matter of simple division.

      Crucial to the calculation of rVenus 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.

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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,

rVenus = 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.