Tuesday, March 3, 2020

Earth in space 2

So, we know how long it takes Earth to rotate, that it's tilted in respect to the ecliptic, and the time it takes to travel around the sun. So how far is it from the sun?

Uhrmph...uh. Anybody got a meter stick?

Well, with some high powered telescopes and a lot of patience, we could measure the apparent distance between two stars and half a year later, on the opposite side of the sun, measure the same distance and see how much it's changed. 

Stars don't move in the sky in respect to us - not to any measurable extent, that's why they're called "fixed stars" - so any apparent motion would be due to what's called "parallax error". If you want to see parallax in action, get close to the wall on one side of an analog wall clock (the kind with hands) and read the time, then go to the other side against the wall. You'll get two different readings because you're viewing the hands of the clock from two different angles.

You can use the parallax method if you can measure very tiny angles, but the whole idea of this blog is to get by with little expense and portable equipment.

What to do?

Let's use math!

But first, the background. Back in the early 16th century, Nicholas Copernicus realized that astronomy would make much more sense if the sun were at the center of things instead of the Earth. Tycho Brahe didn't hold with Mr. Copernicus' new fangled ideas but he did have an eye for precision and cranked out a massive library of observations of the visible planets.

Johannes Kepler (born 1571), the hero of our story, did hold with Copernicus' new fangled ideas and needed his teacher, Tycho Brahe's data to prove it, but Brahe wouldn't give...until, on his deathbed, he bequeathed his library to Kepler. 

Kepler started working tirelessly on the data. It was hard for him to keep a job, and he moved from place to place. What he wanted was very pointedly something the powerful church of the time did not want, a heliocentric universe. After his benefactor, King Rudolph of Bohemia, abdicated his throne to his brother Matthias, things began to fall apart for the Kepler family, but Johannes continued working.

His way of understanding Earth in space was to, first, work out the motion of Mars from Brahe's observations. With his results, he applied them to the other visible planets to see if they worked the same way. When it was obvious that all the visible planets complied with his three laws (we talked about those in "Orbits"), he applied them to Earth. The third law is telling. It tells, in fact, how far the Earth orbits from the sun (at least, at its furthest point, and since the Earth's orbit is almost (not quite, but almost) circular, we'll go with that).

Kepler's third law says that the square of the orbital period (the time it takes a planet to orbit the sun) is proportional to the cube of the distance we want. The proportionality constant, it turns out, is equal to the sum of mass between the two bodies (the sun and the planet) multiplied by Newton's universal gravitational constant divided by 4 times pi squared (we had to wait for Newton to figure out the details). All together, the formula, 200 years in the making, is…

r is the distance we're looking for. T is the time it takes for us to travel around the sun, 365.25 days or 31,557,600 seconds. I convert to seconds because the almost-constant a, in SI units is 2.97x10-19 s2/m2. That's a rather messy looking value but remember that constants like that are mainly there to make your answer come out in the right units. I said that it's "almost constant" because it's actually a calculated value that includes the sum of the sun's mass and the mass of the object orbiting the sun. But objects in the solar system are so much smaller than the sun that their mass is negligible. The point is that the ratio of a planet's period of orbit squared to the maximum distance from the sun cubed is almost exactly the same for all the visible planets.

When I do the math, it puts us 149,675,423,264 meters or 93,003,996 miles from the sun at our maximum distance. The Wikipedia says an average of about 92 million miles and, since the Earth's elliptical orbit is almost circular, that's pretty close.

It's sorta exciting when calculations like this come out right. I used my statistical spreadsheet, DANSYS, to do the calculations but you could have done it with any spreadsheet, or even a calculator.

So, I'll see you next time and write about Florida (Station, that is) when we've moved a little further around the sun

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