inverse-square law through geometry

Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law of gravity using
only geometrical methods--without analysis, in other words. If you find
that sort of stuff interesting (and I can't see why you would), I've
made a short PDF of it available at

http://astro.isi.edu/notes/gravity.pdf

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law


Hi Brian:

To each his own, and I'm glad you use your spare time in such a fashion. Me? I
just dug up my DVD of "Girls Gone Wild!" :-)

Serously, that's a very nice presentation--thanks!

Peace,
Rod Mollise
Author of _Choosing and Using a Schmidt Cassegrain Telescope_
Like SCTs and MCTs?
Check-out sct-user, the mailing list for CAT fanciers!
Goto http://members.aol.com/RMOLLISE/index.html

Brian Tung wrote in message ...
Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law of gravity using
only geometrical methods--without analysis, in other words. If you find
that sort of stuff interesting (and I can't see why you would), I've
made a short PDF of it available at

http://astro.isi.edu/notes/gravity.pdf

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

looks more like a tautology . You're basing it on Kepler's Laws (and some
geometry/trig ), which already include the effects of gravity . Entertaining
reading anyway, almost like an Umberto Eco novel ( seriously ) .

best regards,
matt tudor

Brian Tung:
http://astro.isi.edu/notes/gravity.pdf

Rod Mollise:
To each his own, and I'm glad you use your spare time in such a fashion.
Me? I just dug up my DVD of "Girls Gone Wild!" :-)

John Steinberg:
It sure is purty. You can do some beautiful work with LaTeX, but as
someone who continues to struggle with long division, I have no idea
what it all means.

Oy! I am constantly surprised at how much we have in common--though you
are a bit ahead of me in the long-division front. Your post sent me to
the dictionary to find out what long division is.

Are you still in the USA? The Canada Immigration sites have been
overwhelmed, or so I read. I'm thinking summers in Québec and winters
on Virgin Gorda, British Virgin Islands.

The apparent end of 228 years of liberal democracy is profoundly sad;
that it would come eventually might have been expected by persons
smarter than me, but that it came so soon, and in my lifetime, was
coompletely unexpected. So much -- everything, actually -- in the way
of rationalism and critical thinking was completely lost on the Bush
voters. I have been asked if I find it ironic that we are fighting wars
against people just like us. "Not at all," I reply. There are many ways
of viewing history, and the view of recorded history as a series of
struggles between religious extremists is a valid as any other, IMO.

Sadly yours,

Davoud

--
usenet *at* davidillig dawt com

Brian Tung wrote:
Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law of gravity using
only geometrical methods--without analysis, in other words. If you find
that sort of stuff interesting (and I can't see why you would), I've
made a short PDF of it available at

http://astro.isi.edu/notes/gravity.pdf

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt


Time well spent! Thank you Brian.
-Sam

Matt Tudor wrote:
looks more like a tautology . You're basing it on Kepler's Laws (and some
geometry/trig ), which already include the effects of gravity . Entertaining
reading anyway, almost like an Umberto Eco novel ( seriously ) .

Kepler's laws are empirical only. They give the shape and the dynamics
of the orbit, but not the magnitude of the force or acceleration. In a
way, I guess it is a tautology, but then so is finding the derivative of
any function. Nonetheless, doing it without the calculus is interesting
to contemplate.

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

I (Brian Tung) wrote:
Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law of gravity using
only geometrical methods--without analysis, in other words. If you find
that sort of stuff interesting (and I can't see why you would), I've
made a short PDF of it available at

I've had to change the URL; the new URL is

http://astro.isi.edu/notes/newton.pdf

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

nice demonstration.
Jerry




Brian Tung wrote:

I (Brian Tung) wrote:
Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law of gravity using
only geometrical methods--without analysis, in other words. If you find
that sort of stuff interesting (and I can't see why you would), I've
made a short PDF of it available at

I've had to change the URL; the new URL is

http://astro.isi.edu/notes/newton.pdf

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

Hi Brian,

Brian Tung wrote:

Since it's cloudy here and the aurora probably couldn't be seen this
far down anyway even if it were clear, I passed some time demonstrating,
somewhat less than rigorously, the inverse-square law of gravity using
only geometrical methods--without analysis, in other words. If you find
that sort of stuff interesting (and I can't see why you would), I've
made a short PDF of it available at

http://astro.isi.edu/notes/gravity.pdf

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

That looks good, :-)

However, I have a [quite possibly very stupid] question. What I'm
going to write below may be complete nonsense, I don't know...

Isn't it much simpler than that? Let for the beginning think not
about gravity, but light intensity at a given distance from a point
source. With the distance, the light intensity decreases with
the inverse square law, simply because the same number of photons
are distributed on the wider area. The same applies for gravity -
whatever force acts, it is "distributed" over wider and wider
distances.

To check if this is right, one can easily elliminate one of the
dimensions - for example, to put two long rods close enough to
one another and to check the forces acting on them. This is
basically equivalent as eliminating the dimension along the length
of the rod. In this situation, the interaction is governed by not
the inverse square law (1/x^2), but by a simple reciprocial (1/x)
law. Again, pureli geometric considerations - the "action" is
distributed this time along a perimeter of a circle. As far as
I know, the electrical force acting between two close wires is
defined exactly by this law, and I assume the same is for gravity.

The force between two flat plates, positioned very cluse to each
other will be then not dependent on the distance between the
plates (this is when we keep the distance between the plates
small compared to the plate dimensions). This is again from
purely geometrical considerations, and AFAIK the force is constant
indeed.

So, as a generalization, one can say that in an N-dimensional
space, the force of gravity, electromagnetism, or whatever, as function
of the distance will be 1/x^(N-1).

Do you think this makes sense?

Regards,

- Alex

P.S. BTW, it's very interesting to test how planets orbits will look
if space had more than three dimensions. With four-dimensional
space, the orbits look like flowers, not ellipses, :-)

Alexander Avtanski wrote:
Isn't it much simpler than that? Let for the beginning think not
about gravity, but light intensity at a given distance from a point
source. With the distance, the light intensity decreases with
the inverse square law, simply because the same number of photons
are distributed on the wider area. The same applies for gravity -
whatever force acts, it is "distributed" over wider and wider
distances.

That presupposes that gravity relies on some wavefront, and that the
force applied by the wavefront is directly proportional to the intensity
of that wavefront. This is certainly plausible, but it is very far from
a proof of the inverse-square law. Just because it makes sense that it
obeys the same relationship that light does, doesn't mean that it's
necessarily *so*.

The task is not to convince people that the inverse-square law is right,
but to show that it follows directly from Kepler's laws. (Newton also
showed that the inverse-square law implies that the orbit is a conic
section-- of which Kepler's laws are the subset that correspond to bound
orbits.)

One can also do this with analysis, and that's much easier to follow,
although it's still not a trivial problem--one indication that the
equivalence of Kepler's laws and the inverse-square law is a deep one.

To check if this is right, one can easily elliminate one of the
dimensions - for example, to put two long rods close enough to
one another and to check the forces acting on them. This is
basically equivalent as eliminating the dimension along the length
of the rod. In this situation, the interaction is governed by not
the inverse square law (1/x^2), but by a simple reciprocial (1/x)
law. Again, pureli geometric considerations - the "action" is
distributed this time along a perimeter of a circle. As far as
I know, the electrical force acting between two close wires is
defined exactly by this law, and I assume the same is for gravity.

You can't assume that. It may be so, but you have to prove it's so.
(I don't mean you specifically--rather, anyone who is interested in
establishing the inverse-square law from Kepler's laws.)

Brian Tung
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt