Surface of spandex

Gary D. White National Science Foundation and American Institute of Physics gwhite@nsf.gov Spandex models for gravity wells: folklore, facts, and fun Surface of spandex N q mg Gulf Coast Gravity Meeting, Oxford

The Spandex, for demonstrating celestial phenomena: • The Solar System • Orbits, precession • Escape velocity • Planetary Rings • Roche Limit • Density differentiation • Early solar system agglomeration models • Earth and moon • Binary Systems • Tidal effects • See ‘Modelling Tidal Effects’, AJP, April 1993, GDW and students NOTE: “Gravity wells” rather than “curved space-time” or “embedding diagrams” Gulf Coast Gravity Meeting, Oxford

Video fun Gulf Coast Gravity Meeting, Oxford

From XKCD (A webcomic of romance,sarcasm, math, and language, http://xkcd.com/681/) …but are spandex gravity wells really like 3-D space? Gulf Coast Gravity Meeting, Oxford

Wrong things that I thought I knew about the shape of the Spandex • “It is like a soap bubble between rings.” Pull middle ring down---this has long been known to produce a catenary curve rotate Better known as thecurve of a hanging chain …or perhaps in St. Louis, the curve of the Arch …except data can’t be fit to the appropriate hyperbolic cosine… Gulf Coast Gravity Meeting, Oxford

M Wrong things that I thought I knew about the shape of the Spandex • “Oh, right, it is like a weighted drum head.” So, it solves Laplace’s equation with cylindrical symmetry, h=A + B*ln(R) This would make it like 2-D gravity, like orbits around a long stick of mass M …except our original data couldn’t be fit to any logarithmic form… Gulf Coast Gravity Meeting, Oxford

…until we learned to stretch it as we attached itthanks to Don Lemons and TJ Lipscombe, AJP 70, 2002 Gulf Coast Gravity Meeting, Oxford

Connection to general relativity • Wilson (1920!- Phil. Mag. 40, 703) gives the metric for an infinite wire of mass to be (to leading order in m) where ; …incredibly small for any reasonable linear density…in the slow speed, small mass density limit this means that the the Newtonian effective potential predicted by Einstein’s equations of a wire (or long stick or bar galaxy or other prolate distribution, perhaps) is given by …In other words, logarithmic, as perhaps expected. Gulf Coast Gravity Meeting, Oxford

…so “pre-stretched” Spandex potential well is like the well around a skinny stick of mass m…but what about rolling marbles on Spandex? Is that really like planets moving in a logarithmic potential? To relay that story, let’s recall some of the coolest early science So, in natural units, T2 = R3 for planets. (In unnatural units, T2 is proportional to R3) Gulf Coast Gravity Meeting, Oxford 9

We determined Kepler’s law analog for unstretched Spandex for circular orbits by doing some experiments… • For fixed M, unstretched Spandex has ln(T)=(1/3)ln(R2) +b • So, Spandex is T3/R2 = k… • Kepler Law for real planets about sun is T2/R3 = c. • Curiously close, but no cigar; • What is pre-stretched spandex Kepler’s law analog for circular orbits? Let’s come back to that…for now notice how noisy the data is… Gulf Coast Gravity Meeting, Oxford

mgh(x) About rolling on the Spandex…let’s first consider the lower dimensional case---modelling one dimensional oscillations with motion in a vertical plane • One-D motion Diff. wrt time to get Assume , then Rolling in a vertical plane in a valley given by h(x): but and no-slip rolling means so x • Conclusion: • You can model motion of a mass at the end of a spring (1D motion) with a ball rolling in a vertical plane if • the shape of the hill matches the potential, and • 2) if you “adjust” the mass and • 3) if the derivatives of the hill are “small” q So for small d we get SHM with

Likewise for 2D(that is, when we want to model near circular 2D motion in a plane we can use near-circular motion on a Spandex)…WHY? First, let’s look at the planar case Diff. wrt time, assume Again, SHM, constant terms give orbital frequency, coefficient of d gives frequency of small oscillations about orbit, Gulf Coast Gravity Meeting, Oxford

Rolling adds more complications, but when rolling in a horizontal circle we have something similar, but with a few new terms due to the rolling constraint leading to, instead of Kepler’s Law Gulf Coast Gravity Meeting, Oxford

For have or • So if h(R) is power law, yielding Kepler’s law analog Gulf Coast Gravity Meeting, Oxford

Effect of rolling on orbits Gulf Coast Gravity Meeting, Oxford

Returning to the question of what is the pre-stretched Spandex vesion of Kepler’s Law • For have or • So if h(R) is logarithmic, yielding Conclusion: A wire of mass M (or any cigar-shaped matter distribution, from far away, but not too far) has a constant velocity profile…hmm… Gulf Coast Gravity Meeting, Oxford

What if not going in a circle? • For cones, the oscillations about near circular motion satisfy • Can also derive neat analytical expression for “scattering angle” for cones… • Spandex is more complicated… Gulf Coast Gravity Meeting, Oxford

Another video, ball in cone Gulf Coast Gravity Meeting, Oxford

Moving in a cone---exp. vs theory for near circular orbits Gulf Coast Gravity Meeting, Oxford

Two comments 2) Recall that to get logarithmic potential that is like real gravity for wire-shaped mass distributions, we have to stretch the Spandex taut and then add a heavy mass. Why is that? …Why do you have to stretch the Spandex for it to model the real gravity? 1) Should imperfect models, like Spandex and cones be used to convey ideas about gravity, general relativity? • Yup, (Imperfect models are better than “perfect” ones (consider “full-scale” maps!)) Was real gravity pre-stretched? Gulf Coast Gravity Meeting, Oxford

Thanks to • My students, especially Michael Walker, Tony Mondragon, Dorothy Coates, Darren Slaughter, Brad Boyd, Kristen Russell, Matt Creighton, Michael Williams, Chris Gresham, Randall Gauthier. • Society of Physics Students (SPS) interns Melissa Hoffmann and Meredith Woy • Aaron Schuetz, Susan White, • SPS staff, AIP, APS, AAPT, NSF, NASA, and • You! Gulf Coast Gravity Meeting, Oxford

Surface of spandex