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How did a transit of Venus provide astronomers with the first measurement of the Earth-Sun distance?

Astronomers calculated the distance between the earth and sun in about 1769 when Venus passed across the sun. Please could you explain this calculation?

It goes a little like this: By 1769, Newton's laws of gravity had been laid out and shown to work. The period of each planetary orbit had been measured, but not the absolute distances. Kepler's 3rd Law (which, really, is Newton's Gravity written in a special form) relates the relative period of each planet to its absolute distance. In other words, Kepler's 3 Law tells us that if the orbit of Venus is 0.60 years, then its distance from the sun is 71% the distance of the Earth. So, only knowing a single distance tells us the distance to each planet. Even more, if we knew the distance to the Sun, then we'd know the size of Venus' orbit too, and the speed at which it moves. So all of these details can be related to one number: the earth-sun distance.

The rest was determined by what astronomers call parallax.

Imaging you and a friend are standing on one side of a street, but separated by a sizable distance. You friend is to your right, for concreteness. And both of you are staring at a single lamppost in front of you on the other side. A car approaches from your left. As you're staring at the lamppost, the car cuts through your line of sight first, then a short time later, it cuts through your friend's line of sight, right? Because your friend is looking at the lamppost from a different angle.

If you knew how far away you and your friend were standing, and the velocity of the car, and the time difference between you and your friend's crossing, you could use geometry to find your distance to the lamppost.

Now, move that analogy to the transit of Venus. You and your friend are standing at different edges of Europe in two separate observatories staring at the sun, waiting for the transit. You will each see the transit happen at slightly different times. Knowing that and the velocity of Venus, you can measure the absolute distance to the sun. How do you know the velocity of Venus? Well, Kepler's 3rd law relates the velocity (which you want to know) to the absolute distance (which you want to find anyway). So everything can be transformed into a single formula which depends only on the time between your observations and the distance to the sun. And from that, astronomers first measured the distance to the sun.

June 2011, David Bernat (more by David Bernat) (Like this Answer)

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