Sun is NOT the Center of Solar System: A Lesson in Elliptical Orbits

The Sun is NOT the Center of Solar System: A Lesson in Elliptical Orbits

We are going to cover elliptical orbits, ellipses, creating ellipses, sine and cosines, and Johannes Kepler's Second Law of Planetary Motion with an ultra-simple, easy to understand method using a piece of string! 

Kepler was trying to prove that planets sweep out equal areas in equal time periods...except he kept getting errors as he attempted to calculate the orbit of Mars. He tried fitting his calculations to a circle shaped orbit and an egg shaped orbit: no luck. Then he tried an ellipse, succeeded and also proved the sun wasn't at the center of the planets' orbits!

Our sun is not the center of the universe. It is not the center of our solar system. The sun is at one of the two foci of an ellipse, the other focus is an empty area of space. The foci are like two centers of two circles joined together into a blob. BUT THE orbits are so close to circular looking that diagrams of our solar system are basically circles within circles with our Sun at the center (but technically a smidge off). It's called the "barycenter".

In fact you can create a circle by bringing the two foci together-and we'll do that after learning how to easily create ellipses with a piece of string, a compass and two pins. This piece of string will also easily show us Kepler's law of planetary motion: something that is an absolute mathematical truth...but looks totally untrue to human eyes.


Let's start by marking the length and width of the ellipse we want:

Put a pin at each end of the length and connect with a string:

Set the compass to the distance from the center to a length end point:

With the compass set to the distance from the center to an end point of the length, put the sharp part onto a width point and draw marks where the compass-swing hits the line of the length.

Take your pins and string and stick them into the compass marks.

Take pencil and draw while keeping the string pulled tight. There is your awesome new ellipse! You'll notice that the string forms a triangle, kind of like the sine/cosine triangle in the circle from a few posts ago about Lissajous Figures. We'll come back to this triangle in a moment, but first let's Reverse engineer this ellipse into a circle.

As we move the two foci closer and closer together the pencil and string triangle sweeps out a shape that is more and more circular. Put the pins in the same hole (or just use one pin) you'll get a perfect circle.

Back to the triangle sweeping:

Kepler's Second Law of Planetary Motion says that as a planet sweeps along its orbit, in different parts of the ellipse (even the smaller ends) the area it sweeps out is always equal to any other area it sweeps out given the same time to sweep. Two seconds of sweeping in the little end gives the same area as two seconds of sweeping the bloated middle.

That sounds and looks wrong, even in fancy computer animations...but just remember our string triangle: it moves all about and changes angles as it's dragged by the pencil but its overall length never changes. The string doesn't get shorter or longer. The base of the triangle never changes as long as the pins don't fall out. The length of the two other sides of the triangle always add up to the length of the string.

Sameness.

Couple the sweeping in an ellipse with the sine/cosine triangles...

...and add a few more little trigonometry bits like tan and arctan and you get the field of calculating xy ballistics trajectories for non-powered projectiles (bullets, rocks in a catapult, etc.).

Wait, I can use a piece of string to calculate the path of birds?!?

 They'll be non-powered once I gnaw their wings off.

Making Waves: Oscilloscopes, Lissajous and a Smattering of Cymatics

Making Waves: Oscilloscopes, Lissajous and a Smattering of Cymatics

Lissajous Figures are curves that are complex enough to make a pattern. These patterns are great tests for oscilloscopes because you can learn information from them by looking at how they move and their shape. They can tell you the frequency, phase, and even angle ratio which you can then look up in a trigonometry sine value for the phase angle of two waves: which lets you figure out the power factor of an electrical component.

Sine waves: sin and cosine. Get it?

Above is a circle I drew with a triangle. The bottom of the triangle is cosine. This is a 45° triangle. Trigonometry and Lissajous Figures are ratios. The ratio of cosine and sine is 70.7% and the angle is 45°. So, sin45 = 0.707. The same priciple applies to waves, but showing it on a circle is easier.

Fancy-smanshy, but we're just looking neat waves here. Lissajous waves are beautiful and quite reminiscent of Art Nouveau "whiplash" style curves and lines used in everything from painting and sculpture to wild furniture and architecture.

So who discovered Lissajous Figures? A man named Nathaniel Bowditch of course! Huh? Yeah, Bowditch experimented with them for a while, then like 50 years later a guy named Jules Lissajous attached a tuning fork to a lightbulb to create these patterns. Previously, people would use a pointed pendulum swinging through fine sand to create what much later became known as Lissajous Figures. I used an analog synthesizer and an old oscilloscope.

I also put a second line out from the synthesizer to a speaker so I could hear the waves while viewing them. A brand new 20Mhz probe and BNC female to UHF male adapter round out this setup.

Lissajous Figures are easy to use. You just count the loops running across the top of the pattern and down the side. This gives you your ratio, a 3 to 1 means 180 cycle voltage, 3 to 5 means a 60 cycle signal, etc. You set the signal input and then calibrate the oscilloscope by at adjusting until you get a familiar pattern. In the above video the settings I talk about are on the synthesizer, not the oscilloscope: I was going for beauty and variation, not calibration.

There are tons of ever more complex Lissajous Figures / Bowditch Curves. Above are the most common ones you'll probably encounter.

Another type of waves are Chladni Figures after Ernst Chladni in 1787 (later renamed Cymatics in 1967 by Hans Jenny). So, who first investigated them in recent times? Our poor, old friend Robert Hooke! You remember him right? He was the guy who discovered "Newton" Rings and "Newtonian" telescopes. I guess we should count ourselves lucky these aren't called Newton Figures.

Cymatics is sound waves directed through water.  Chladni used sand on the bodies of acoustic guitars, but the principle is the similar; although the sand forms geometric patterns that are more hard edged. Here are a few variations of Cymatics waves:

At 456 MHz the water spattered out of the plate:

Shocking and messy!


Here are some still photos of various Cymatic waves:

Chladni Figures are usually made in sand on flat plates. They're basically Cymatic waves in sand but there is a huge difference in how they look because the sand bouncing around, while the sound waves push the water into itself, bumping the waves along: which is why sound travels 4 times faster in water than air! Of course sound moves fastest through our good old friend beryllium from my neutron gun experiment. 

Chladni Figures, related to Cymatics (water and sound waves) showing sounds creating patterns in sand. For more geometric results a square plate and direct vibration coupling works much better. I was piping sound to a metal file cabinet, which then went to the small plate. At first I was bummed they weren't geometric, but after extensive image searching I found exact pattern duplicates from someone using a wooden desk with no plate.

This was just a spur of the moment test after looking at Lissajous Figures on my oscilloscope at 2 AM waiting for the cloud to clear so I can drag my telescope outside.


Here's another attempt:

Again, I found this same pattern duplicated by someone else, so I wasn't totally annoyed at the less-than-awesome pattern. I think once I get a perfectly flat metal plate instead of a slightly domed dish (which was levitating off the surface due to the sound pressure) I think I'll be able to get the hard edged geometric lines. As Paul Camp said, "There is a balance between discovering for oneself and being told."

I'll also be able to better acoustically couple the metal plate to the sound source. I'll update here once it's done, but I'm working on some other projects first.

Just make the noise stop-I can't cover my ears! Meow.