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>>MIDDLETON: This week, well, running the seminar and being the speaker, I get the pleasure
of introducing myself. I think most of you know who I am. My name's Chad Middleton, I'm
a professor here at CMU. The talk I'm going to give today is entitled Circular Orbits
on a Warped Spandex Fabric. This talk is based on, orginally, Michael Langston, our graduate
from last year, did his senior research project on exactly this topic, and I kind of took
up where he left off last summer, or last year. I spent the summer then kind of learning
from the mistakes that we had made together. This paper, you can't see down below, but
it has been accepted for publication in the American Journal of Physics. I don't know
when it's going to happen, but it's in the editorial process right now. Alright, so let's
go ahead and get started. So I have to start every talk off somehow connecting it to general
relativity, right? So anyway, let me give a little discussion of general relativity
before we get started talking about spandex fabric. So Einstein's theory of general relativity
describes gravity as the warping of space and time due to the presence of matter and
energy. And this equation on the top is called the Einstein Field Equation, it's actually
in general a set of coupled-- well there's 10 in general-- coupled non-linear partial
differential equations in general. You don't even know the structure, but you should realize
there's-- on the left-hand side, that g mu nu, that is called the Einstein tensor, it
describes the curvature of space-time. On the right-hand side there's some constants
and that's multiplying t mu nu. T mu nu is the stress energy tensor, and that's effectively
telling you about the matter and energy that you have in your space-time itself. So effectively,
John Wheeler once eloquently said, uh, summarized general relativity as this: matter tells space
how to curve and space tells matter how to move. That's basically the essence in general
relativity. In general relativity, gravity is not thought of as a force, but rather the
manifestation of the warping of space and time. This is a picture that comes off of
a typical, well, a graduate-level text on general relativity. What people typically
do is they think then in terms of-- to think about this curvature of space-time, you often
envision a spandex fabric, or some type of fabric pulled taut and place a mass on top.
That mass causes that space to warp and then the analogy is often drawn to the thinking
about particle orbits, or orbits of planets, as due to the fact that these planets are
in fact moving in a warped space-time. And this is kind of effectively the analogy that's
often used in general relativity. So basically the essence of this talk is to kind of probe
this conceptual analogy. We're basically working out the physics behind the analogy that's
often used. So in a typical undergraduate general relativity course, one of the first
things you study is what's called the Schwarzchild Solution. This is an exact solution to the
Einstein Field Equations, and what this does is it describes the geometry outside of a
spherically symmetric, non-rotating massive object. So effectively a ball, or a star that's
not rotating. And again, there's the exact solution's easy to write down, so we can write
down the geometry outside of that space-time. And again, that's the whole game in general
relativity, is to understand the geometry outside of your mass or outside of your object,
here being a spherically symmetric mass. Now what you kind of have to do effectively, or
it's nice to do to wrap your brain around it, you're trying to envision this four dimensional
warped space-time and that's quite hard to do, right. So what we, what you do then in
general relativity, often as an exercise, is you construct what's called an embedding
diagram. And what an embedding diagram effectively is, is you take your four-dimensional geometry,
your four-dimensional space-time, you look at one moment in time, so you effectively
think of a photograph of the sky, and then you look at, in spherical polar coordinates,
you look at your polar angle theta, you look at theta equals pi over two. And what that
does is effectively reduce your four dimensional space-time to a two dimensional effectively
surface. And then what you do is you construct what's called an embedding diagram. And embedding
diagram is you take this surface that exists in the two-dimensional surface of a three-dimensional
sense and you force then the geometry of that two-dimensional surface to be equal to the
two-dimensional spacial curvature of that slice. And you think of then, then you can
wrap your brain about, around this concept in general relativity, this warped space and
time. Now, for the Schwarzchild Solutions to actually work it's very easy to do. You
can take this two-dimensional surface in the three-dimensional space and then this, of
course, this two-dimensional surface has cylindrical symmetry, so those cylindrical coordinates
will correspond to z, r and phi, if you will, and there's symmetry in the phi direction,
and then z is a function of r which describes them as two-dimensional surfaces given by
this equation. I am working in units where c and g are set equal to one, here. So this
is just the square root of this r minus 2m, effectively, and what this is is this two-dimensional
surface that has z as a function of r described by this guy has the exact same curvature as
then this two-dimensional spacial slice of the full Schwarzchild Solution. What then
is often done is one takes a logical leap, or one, what often is done is by this embedding
diagram, people then start to think and use this analogy to then envision some object
rolling on this two-dimensional surface as the analogy itself. You should realize that
these are actually different things. This embedding diagram, that is an exact solution
that describes the geometry of the Schwarzchild Solution in two dimensions. But then the question
is, is there any warped, two-dimensional surface that will, if I get a marble, say, rolling
on this surface, can I get then the exact same motion that one would in fact get for
a planet orbiting around the sun? So that's what we want to answer in the next couple
of slides. Alright. So let me then jump from Einstein's theory of general relativity to
Keppler's third law. The reason why we're going to talk about Kepler's third law is
we're going to, this is the-- at the end of the day, this is the expression we're going
to want to come up with, this Kepler-like expression. So for simplicity, this is the
simplest way to do this, if I look at some massive object, say capital M, maybe that's
the sun or something like that, and I have some other mass that's orbiting, maybe it's
a planet or something like that with small m, and it's moving in a perfect circle, this
is the easiest way to do this, I could then say what is that force acting in this little
mass, and of course it's given by a Newtonian theory, Newton's Universal Law of Gravitation,
G m M over r squared. That's in that force acting on my object, I set that equal to mass
times acceleration and the centripetal coordinates, that's [inaudible] squared over r. Then what
I can do is I can connect the speed of this planet orbiting around the sun to, well, the
speed is a constant, so it's the distance traveled, which is the circumferance two pi
r, divided by the period, capital T corresponds to the period of time it takes this planet
to orbit around the sun. If I take this expression for v, substitute it into Newton's second
law up above and rearrange, this is what I find, and this is called Kepler's Third Law,
this was discovered by Johanes Kepler by looking at Tycho Brahe's data before Newtonian Theory
was understood. We're going to come up with a Kepler-like expression for our treatment
here today, as well, so I want you to realize this is just a constant, so who cares about
this. This is the period, the period squared, the time it takes this planet to orbit around.
This is the radius of the orbit and M is the central mass. So this gives me a relationship.
Kepler's Third Law, this gives me a relationship connecting the time it takes the planet to
orbit around the sun to the radius of the orbit itself and to the central mass, and
this is how astronomers are able to in fact determine the mass of a central object, is
because they can measure then the period it takes some planet to orbit around our sun,
for example, with the radius you can calculate the mass of the planet versus using some big
scale. You should also realize that Kepler's third law has nothing to do with the mass
of the planet that's doing the orbiting itself. It doesn't matter how massive this guy is,
Kepler's Third Law holds for whatever this guy is. I'm bringing Kepler's Third Law up
because we are going to find an expression similar to this guy for our spandex fabric.
Let me give you a basic outline of the paper and the talk today as well. There's basically
five parts. The first two parts, I'm not going to lie, it's like a little bit heavy at the
beginning. It will lighten up later on. These first two parts are the theoretical aspect
of this whole thing. These second two parts are what represent the curve-- sorry, the
experimentation itself and our Kepler-like expressions, and then at the very end I'm
going to come back to general relativity and compare it to circular orbits in general relativity
effectively. So the first thing we're going to do is we're going to consider a marble
rolling on a cylindrically symmetric surface. I'm not going to specify this first part,
what that surface is, just some surface that has cylindrical symmetry. We're going to come
up with the equations of motion describing this marble rolling on this surface, and we're
going to use Lagrangian dynamics in order to do that. The second part, and this is the
hairiest of the lot, which Michael can attest to, this is the shape of the spandex fabric.
So what we're going to do is actually determine an equation which determines the shape of
this spandex fabric by using the calculus of variations. From there, then, I can connect
these to effectively come up with a Kepler-like expression. I can't get an exact expression,
but what I can do is I can find a Kepler-like expression in the small curvature regime.
What does that correspond to? The small curvature regime corresponds to where the warping of
that fabric is small, so where the slope of the fabric is small. And then we're going
to also come up with a Kepler-like expression of the large curvature regime.The large curvature
regime is, if I have a huge central mass in the center, the walls of that fabric is going
to have quite a large slope. So when that slope is in fact very large, I can also come
up with a Kepler-like expression in that regime as well. Alright, so first off let's talk
about our marble rolling on a cyllindrically symmetric surface to begin with. This is of
course the Lagrangian that describes an object of mass and moving on the cylindrically symmetric
surface, there's effectively theory terms. This is just v squared. This is the translational
kinetic energy of the marble. This is the rotational kinetic energy of the marble, the
marble's rolling on the surface, minus the potential energy. This is just gravitational
potential energy of the marble itself. Now a couple things to notice: first off, I am
talking about a marble which is a uniform marble. For a uniform spherically symmetric
object, the moment of inertia is 2/5 m r squared, which you know. M being the mass of the marble
and r being the radius of the marble. And what we do in this work is we assume that
the marble does not slip. If the marble doesn't slip, then the angular speed is related to
the translational speed of the center of mass. Now if I multiply these two terms together,
multiply by 1/2, what I find is 1/2 i omega squared is 1/5 m v squared. That allows me
to effectively combine these two terms and I merely change that 1/2 to a 7/2, ok? So
that's the first bit. The second bit to realize if you've had advanced dynamics, again, we
talk about Lagrangian dynamics a lot in that class, it appears that there's three coordinates
here. Three generalized coordinates: r, phi and z in cylindrical coordinates. However,
this marble is going to be constrained to lie on the surface itself, which means that
we have an equation of contraint, effectively you tell me what r is and I'll tell you what
the height of that fabric itself. So what I can then do is in this first part, just
generically say that z is some function of r, and I can then put this into the Lagrangian
and express in the Lagrangian terms of r and phi. So I do in fact have two generalized
coordinates, alright? Now if I have two generalized coordinates, that means I'm going to have
two Lagrangian equations of motion. One Lagrangian equation of motion corresponds with effectively
just tells me that I have conservation of angular momentum. The second one then is the
Langrange equation of motion for the radial coordinate. That's the one that we're interested
in here. If I take the Lagrangian from the previous slide and I recognize the fact that
z is a function of r and stick it into the equation, this is the equation of motion that
emerges. I should have mentioned on the previous slide, which I forgot to, the dots correspond
with the time derivatives, and the slashes correspond to r derivatives. So all we did
was we effectively used the chain rule to say that z dot was zdrdrdt or z slash r dot.
So this is in fact the equation of motion for a marble rolling on a cylindrically symmetric
surface without specifying anything about that surface whatsoever. Now, what I can do,
I'm going to try to answer now the question that I addressed on the slide number two,
I'm going to compare it to the equation of motion for planetary orbits. This is a Newtonian
theory. If I look at g r I get an additional term, that doesn't matter. The point is the
same either way. If I compare these two equations I can say "Is there some surface where z is
a function of r, which will allow me to then get exactly planetary-like motion?" And the
answer is no, there's none. If you compare closely, look, if I show z to be proportional
to one over r, a one over r type of a curve, proportional to, I can get this term and this
term to match up. This term matches up. This term and this term-- I have an r double dot
term here and here, but notice I have an additional r dot squared term. There's nothing I can
do to make these two the same, so the takeaway is this, and this, you can't see any of these,
this thing's too low. I have to go down a little bit with this thing. Nope, that's as
low as it goes, so all of my references down below-- there was a paper published in the
American Journal of Physics last year, in 2012, by English and Moreno that actually
showed this result. That there is no cylindrically symmetric surface which will allow me then
to get exactly the planetary motions that I see in Newton's Second Law. Alright, now
I've got to be a little careful. When r dot is equal to zero, however, there is, right,
but that's not very interesting because most orbits are not circular, they're elliptical.
Alright, so let me go back to that equation of motion. This is my generic equation of
motion, and what we're interested in, we tried to simplify the procedure. We were looking
at circular orbits. For circular orbits, r dot is zero and r double dot is zero, which
means that this term and this term go away. When this term and this term go away, this
expression reduces to this very simple equation of this form. Notice I use the relation 5
dots, or r 5 dot, is 2 pi r over the period, T. Here I have an expression involving r,
the radius of my orbit, T, the period, the time it takes this marble to orbit around
the central mass, and what you can find-- what you see from this expression, is that
this is proportional to z prime. Z prime is the slope of the surface. So if you tell me
the slope of the surface, I'll tell you a Kepler-like expression. That's effectively
the takeaway from this slide. This expression is linearly dependent on the slope of the
spandex fabric. So all I need to know is z prime and I can tell you what the Kepler-like
expressions are. This brings us to part two of the talk, the shape of the spandex fabric
itself. In order to find the shape of the spandex fabric, the way we went about doing
this is we considered the total potential energy of the central mass spandex surface.
We construct the total potential energy as an inigral. We're going to use the calculus
of variations. So I need to write the total potential energy I'll call PE as an inigral
function. Those of you who had me in Advanced Dynamics are probably remembering the language,
I try to keep it the same. We're going to construct a total inigral describing the total
potential energy of my elastic-- of my spandex fabric central mass surface. And we have three
contributions. Number one, as I put a mass on this spandex surface, there's going to
be elastic potential energy in that spandex surface itself. So there's a contribution
to the elastic potential energy of the spandex surface. Number three, I'm going to slap or
put a large central mass on the center, there's a gravitational potential energry to that
large central mass as well. Now, if you saw Michael Langston's talk last year, number
two is a new addition to that. So this summer I realized that this term is actually quite
important. The spandex fabric itself has mass. Just as the spandex itself has mass, it has
potential energy. So in this work I generalized this summer what I did last year with Michael--
I should say what Michael did-- to include the gravitational potential energy of the
spandex. And this actually is quite an interesting contribution. I should note that number one
and three were considered back in 2000-- you can't see it-- but back in 2002 as a separate
HAP paper that actually, that arrived at Gary White's relation-- we'll talk about this later--
but considering exactly the same thing. So what we do, this is a technique we'll going
to do in the next several slides, we're going to be going through the details of this effectively.
We construct a total potential energy from these three terms and then I'm going to have
an inigral functional. I could take the functional and then, well I want to apply or use the
calculus of variations. The calculus of variations effectively will say, I'll take this function,
I'll stick it into the Euler Legrange equation, that will give me an extreme solution which
corresponds either to the maximum or minimum, and in this case it corresponds to the minimum
total potential energy. And it turns out that this spandex fabric would in fact choose a
shape that minimizes the total potential energy of this total system. Alright, so the next
three slides I'm going to go through number one, two and three. Not in too much detail,
but to kind of give you the gist of this thing. First think we're going to consider is the
elastic potential energy of the spandex fabric. Imagine, and I forgot to even talk about it
on the first slide, that spandex fabric was a four-foot diameter trampoline, or we draped
a spandex surface over this guy, or a spandex fabric over this guy. It's effectively a circular
surface with some radius capital R, right? So what I'm going to do is I'm going to consider
a concentric ring, a very thin thickness, of differential thickness. So I can write
down E stands for Elastic Potential Energy. This is an infinitesimal contribution to the
total-- to the elastic potential energy of this differential concentric ring. And all
this effectively is is 1/2 K times the stretched square, which each and every one of you know
from intro to physics. Right. K is the spring constant of this differential concentric ring.
What about this guy here? Well-- pushed the wrong button-- what about this guy here? Let
me look at a little bitty-- again, we're looking at a concentric ring. The width of that guy
has a width of DR. If I stretch the fabric, this guy will then be stretched to this amount
here. Where DR was the originial length, again the fabric doesn't, the fabric is constrained
to its radius. So this is my old DRand now I have a stretch in the z direction, so this
length becomes the square by Pythagorean Theorum, the square root of DR squared plus DZ squared.
The difference between this term and this term corresponds to the stretch. The stretch
squared times 1/2 K, that is in fact the potential energy of this differential concentric ring.
Now, what we need to do then is connect this spring constant, K, to what's called the modulous
of elasticity. The spring constant, K, in regards to the modulous of elasticity, kind
of a nice way to think about it, I was thinking about this the last couple days, is kind of
resistence of a wire to the resistivity of a wire, right? The resistivity of a wire is
a property of the wire itself. The resistence of a wire depends not only on the resistivity,
but also on the geometry, the length and the cross-sectionalarity and things like that.
The same thing goes for the spring constant. A spring constant is constant for a given
spring, but if I take a spring and I cut it in half, I actually change the spring constant
itself. If I take, say for a given radius out, if I look at half the-- half the DR that
I had before, my spring constant is going to double. So for a given R, K times DR is
a constant. And then also, if I look at different radii-- if I look at twice the radius, the
spring constant also changes as well. How so? Well the way to think about this is if
I think of this concentric ring, I can think of a series of little bitty springs wrapped
around. If I double my radius, I double my circumference, and if I double my circumference,
I double the number of little springs that I effectively have for a given circle. Which
means what? If I have twice the number of springs that are in parallel with each other,
I am going to double the spring constant, K. So for a truly elastic fabric, this E,
the modulous of elasticity is a constant and it's related to to the spring constant K by
this form here. So what do I do? I take, I take-- I solve this expression for K, I plug
it into this expression here, I pull out a DR and then what I want to do is I want the
total elastic potential energy, the fabric, which requires me to integrate over the radius
of the fabric. So if I integrate this expression from 0 to R over the radius of the fabric
itself, this is the total elastic potential energy of the spandex itself, alright? Now
we've got to talk about gravitational potential energy. This is the new contribution that
came in this summer. This spandex fabric has a mass and therefore it has a potential energy
as well. If I again think of the same differential concentric ring, I can talk, g is for gravitational,
s is for the spandex. An infintesimal contribution to the potential energy of this concentric
ring is what? Well this is just mg times my height, right, or my mass. Here is a differential
mass, it's the mass of that ring. And z is the height of this guy itself. Now, what I
can do is connect the mass to the unstretched arial mass density. So this sigma sub zero,
you'll see this later on in the talk, sigma sub zero is the unstretched arial mass density.
Mass per area.Two pi R, that's the circumference times DR, that's the area of this concentric
ring. So the product of these two is in fact the mass. It took me a little while to think
about this in reference to the sources and things like that. I started thinking, "Well,
if I put something on top, it's going to stretch the fabric.That's going to change the the
mass density. It will become variable mass density. But that doesn't matter, because
the mass of this differential segment is going to remain the same. So what I can then do
is then use this first equality, plug this expression into this guy here, that tells
me the gravitational potential energy of that ring. If I then integrate over the ring, it
tells me the total gravitational potential energy of the spandex fabric, written in terms
of an integral. I need to write all of these guys in terms of an integral. Here is is.
2 pi sigma zero, this is just some constant, g is 9.8 meters per second squared, times
r z d r. I'm going to integrate over that guy. That's two of the three. The last one
is very simple. How about the gravitational potential energy of the central mass? We put
some big massive ball in the center of this guy, what's the potential energy? It's Mg
times the height. So it's z as a function of r, here I'm looking at the center of the
fabric, so r equals 0 here. Captial M corresponds to the mass of the central object, times Mg
times C, that's the potential-- gravitational potential energy of the central mass. This
is a trick, and we pulled this right out of that previous paper I talked about before.
What you can do is write this in terms of an integral. How so? Well z prime is dzdr,
zdr times dr is just dz. If I integrate this from 0 to r, at r it's 0. So then I need the
minus sign because when I evaluate the second limit, that gives me this term here, ok? So
there it is. Those are the three terms in a nutshell. This is an approximation. Notice
we approximate the central mass as being point-like. In all reality, this guy is a finite size,
too. Anyway, let's put it all together. I put it all together and this gives me the
total potential energy of the central mass spandex system. There's three terms, the elastic
potential energy, the gravitational potential energy of the spandex surface and the gravitational
potential energy of the total mass, which I write as a total integral where this f is
called a functional. It's a function of z, the height, the slope, and r itself. Here's
the functional. Here's my three terms. This first term here is the e you recognize as
the modulous of elasticity. This guy involves r and z prime. This term is the func-- this
is the [inaudible] that's to be integrated to give me the elastic potential energy. This
second term, a bunch of constants up front, sigma 0 being the constant arial mass density
which we can measure. R times z here, and then lastly, this term is just mg times z
prime. This guy depends only on z prime. This is the term due to the central mass. So what
do I do? I've got my functional, I want to extremize this guy. What I want is the extrememum
or the minimum of the total potential energy, because the total of the potential energy
is going to tell me the shape of the spandex fabric. Very simple. If you've taken calculus
of variations, you know this. You take this functional, this is called the Euler-Legrange,
or the Euler equation, you stick it in, take these derivatives, and you generate yourself
a differential equation. Here it is. Last semester when Michael considered this guy,
or last year, I should say, when Michael considered this guy, he got exactly this without this
term on the right-hand side. Luckily, by including this gravitational potential energy of the
spandex surface, I can still integrate it the same way Michael did. So this guy is a
derivative respect of r. This whole thing is equal to this right-hand side, which is
just a function of r. I can multiply by DR and integrate. When I do so, I set the constant
of integration equal to 0 and this is the equation that I arrive at. The left-hand side,
notice it just depends on r and z prime. Z prime's the slope. It's a nasty differential
equation. It's non-linear. It's ordinary, that's the nice thing, but it's non-linear.
If you square this thing out, it's like fourth order I believe, it's disgusting. You tell
me the inputs, effectively the masses I start off with, and I'll tell you what z prime is,
if I could solve it exactly. Let's explore the right-hand side. M is the mass of the
central mass. What about this guy? Well this guy, sigma zero, that's the mass per area
times pi r squared. That's the area of a circle. So what I see then is this is how the spandex
fabric self-contributes, as far as the mass does. The shape of the spandex fabric is determined
not just by the mass of the central mass, but also the mass of the spandex fabric interior
to the orbit of the marble itself. These two terms effectively are competing for dominance.
Now what is this alpha? Alpha we're just going to call it a parameter. It's effectively proportional
to one over the modulous of elasticity. Bigger modulous of elasticity, smaller alpha. Alright,
so let me recap if you missed the last five slides or you were not paying attention or
whatever. Let me just recap. Here's where we stand, right, this is where we needed to
get. I have two equations. This equation is my circular equation of motion. You tell me
z prime, I'll tell you Kepler's Law. This second equation, this guy, determines z prime,
effectively solve this guy for z prime, stick it into here and you have a Kepler-like expression.
Again, I cannot solve this guy exactly. But what I can do, is I can explore the small
curvature regime and the large curvature regime. Again, z prime is just the slope. If I look
in the small slope regime, I could perform everybody's favorite Taylor series expansion
and keep the first two terms. If you look closely, if I stick this guy into here, I'm
going to have one minus that one that goes away. I get a z prime squared times a z prime,
that's z prime cubed. I can solve for z prime then, stick it into here and that gives me
a Kepler-like expression. Yeah? >>AUDIENCE MEMBER: [Inaudible]
>>MIDDLETON: Different Kepler's law for different radii. That's right. So in essence, there
would be one general expression if I could solve this analytically, but I can't, right?
So what I'm going to do is find an approximate solution in each regime. A Kepler-like expression
we're going to talk about. And you'll see that soon. That's where we're going. So I
perform a Taylor series expansion, I can solve this guy for z prime, I can then analyze this
guy in a small curvature regime. We'll do that first. I can also look at the large curvature
regime. If I have a huge central mass, the walls on this guy is sloped nice and large.
So what I can do then is I can perform another Taylor series expansion. Z prime's big. I
can, if I pull it out of the square root then I have 1 plus 1 over something huge. I can
expand that in a Taylor series expansion like so. If I put this in here, I'm going to have
1 minus 1 over z prime. I'm going to keep only that first term. When I multiply it by
z prime, I'm going to have z prime minues 1. I can again solve for z prime, stick it
into here and get another Kepler-like expression. Alright, so now at this point, effectively
the theory is over with. Let's now talk about the experimentation. I'm proud to say this
is the first time I've ever given a talk with data that I've actually taken myself. Experimental
physics is pretty cool. Anyway, yeah, I learned a lot from using, doing this stuff. But anyway,
and I should say Michael and I, we did all this stuff together last year. So anyway,
I do this. I do exactly like I said. I'm looking at the small curvature regime, I do the Taylor
series expansion, I plug it in and here I obtain an expression. What do you see? This
is a big ugly constant. We're going to actually determine this experimentally in a moment.
On the left-hand side I have my period cubed. Over here I have my radius. I can measure
the radius and I can measure the period. I can measure the arial mass density, I can
measure the r. I can measure my central mass, I could plot this guy, and that's what we're
going to do. We're going to plot this versus this, and I should find a straight line. Now
first off, let me show a couple things. What I can do, I effectively have two competing
terms tied up under the square root and the denominator. If I look when this central mass
is much, much bigger, then the mass of the spandex fabric interior to the marble, if
I neglect this term altogether, my Kepler-like expression reduces to this guy here. This
was first found by Gary White and Michael Walker back in 2002. They actually did this
same analysis, but they looked at Newton's Second Law instead of considering energy concepts,
and they first arrived at this and did some experimentation as well. If you remember Kepler's
Third Law, effectively the powers are switched. Kepler's Third Law of planetary motion says
that T squared is proportional to r cubed. For a spandex fabric, I-- or, for a marble
on a spandex fabric in a large curvature regime, it's the opposite. It's T cubed is proportional
to r squared, and also the mass difference is different. It's more of a square root of
them versus them. The other regime, though. What about if my central mass is small? If
my central mass is small, then this term's going to dominate. If this term's going to
dominate, I'm going to have an r squared to the one f power that's r, r squared over r
is r. So I find a different Kepler-like, even in this regime here, I find two different
effectivley asymptotic behaviors depending on the masses. Here T cubed is proportional
to r. So then you can ask the question, when are these guys on equal footing? And you simply
plug the numbers in. We measured this last year, the sigma zero I believe was .19 kilograms
per square meter, if I remember correctly.