Posts Tagged ‘Kepler’

Let’s Read the Internet! Week 3

October 26, 2008

Self Control and the Prefrontal Cortex John Lehrer at The Frontal Cortex

Summarizes some research that indicates people only have a certain amount of willpower to ration out over the day. My first reaction to reading this article was to think, “yeah, but that’s only for weak people, not me.” My next reaction was to resist the temptation to check my email too frequently. My third reaction was to slaughter eight cats in a murderous frenzy, then to sit forlornly surveying the carnage I had wrought and wonder if this cycle would ever end.

Scott Belcastro’s Lonely Searching from Erratic Phenomena

I’ll admit I don’t know much about art, but I can tell when something looks cool. I saw how similar the paintings were, and felt surprised at first that people don’t get bored doing the same sort of thing over and over. But then I realized it must be because they’re refining, focusing down, and trying to work out subtleties and understand their subject more fully. Not that I see all the subtleties, exactly, but maybe if you read the text they actually talk about that stuff.

The Gallery of Fluid Motion
Videos of fluids being fluidy. Don’t get too excited, though. Despite what it sounds like, this is not a potty cam.

Amazing Super Powers

The Incredible Beauty of Hummingbirds in Flight RJ Evans at Webphemera

Small things can be pretty. They aren’t always pretty, which is sad news for your penis.

Is This The Oldest Eye On Earth? Tom Simonite on New Scientist

“It could be the oldest eye, or even human body part, still functioning or to have ever been in use for so long.” There’s a story for the grandkids.

The Laplace-Runge-Lenz Vector Blake Stacey at Science After the Sunclipse

A clever way to prove that orbits in a r^{-2} potential are conic sections, without solving a complicated differential equation. I’m surprised we didn’t do this in ph1a, although I’m kind of glad we didn’t, because it makes me appreciate it much more now.


Answer: The Kepler Problem

October 19, 2008

Here is the question I’m answering.

It’s a trick question! No possible height profile will perfectly reproduce Kepler orbits. The problem is that in the solar system, any given planet moves in two dimensions around the sun. But since the bowl is a curved surface, the balls wobble up and down through three dimensions, and you can’t match these different scenarios up perfectly.

The dynamics of a planet orbiting the sun come out of the Newtonian gravitational potential
\Phi = -\frac{GM_{sun}m}{r}
So you might think that if you just make the height of the bowl inverse proportional to the distance from the center, so that h = -\frac{1}{r}, the balls would follow Kepler orbits. After all, their potential energy would be the same as the potential energy of a planet orbiting the sun, right?

We need to look more closely at the variable r. For the case of a planet around the sun, r is simply the distance from the planet to the sun. But for the balls circling the bowl, there are two possible interpretations of r. One interpretation is to take a string, lay it flat on the bowl, and measure the distance along the bowl to the center. That would be r. The problem with this approach is that the space is curved. If you were to measure the ratio of the circumference of a circle to its radius using r defined this way, you would not get 2 \pi, You would get something that depends on r. How could you then reproduce orbits through flat Newtonian space?

Instead of treating the surface of the bowl as a two-dimensional space, you might try to treat its projection as a two-dimensional space. So get up directly above the exhibit and look straight down at it with one eye closed. Then you’re looking at a flat space, so could you reproduce Kepler orbits there?

No, because the projection treats the radial and angular directions differently. If a ball has a true velocity of 1 \frac{m}{s} and is going around the center of the bowl in a circle, then in projection it still has an apparent velocity of 1 \frac{m}{s}. On the other hand, if the same ball were plunging straight in towards the center, its velocity would appear slower by a factor of the slope of the bowl, because you wouldn’t notice the portion of its velocity that was up/down in real 3D space. The angle at which a ball appeared to be moving would be distorted by this effect.

If you designed the bowl so that the period of circular orbits followed Kepler’s third law, then in general the projections of balls wouldn’t follow conic sections any more. Projected angular momentum would not be conserved because real angular momentum is conserved, and the projection would hide different proportions of that at different times.

So, while the Kepler exhibit is cool to look at, as best I can tell you can’t truly make it mimic the orbits of planets around the sun.

New Problem: The Kepler Exhibit

October 17, 2008

At the Exploratorium in San Francisco, you can play with this exhibit:

What should the height profile of the bowl be so that balls that roll without slipping (or, so that blocks sliding without friction) would reproduce 2-D Kepler orbits when viewed in projection from above?