Jeez, that was close

Sam Wormley wrote in
news:IuTFi.81306$Xa3.73263@attbi_s22:

John Schutkeker wrote:
Sam Wormley wrote in
news:jp%Ei.76616$Xa3.56894@attbi_s22:

John Schutkeker wrote:
I came this close to sending my letter to the journal with a
mistake in it. But it's all fixed now, and all I have to do is
screw up the nerve to click the 'send' button on arxiv.
Click the 'send' button--You nned the feedback.

It's not gonna see the light of day. It's got an unfixable mistake
in it. ;(

On the brighter side... it was good you caught the mistake. :-)

Unfortunately, that's merely a silver lining to a dark cloud, and not
enough to really compensate for the loss of a publication, even if it
was only a letter. This has been sitting on my desk for a many years
now, which means that I've been operating under a misconception for all
that time.

Since I'm giving up on it, I think I'll throw it open to the whole
physics community, or at least that part of it that resides here in this
BBS.

I was trying to find an equation for the following succession of powers
of '2', (0,1,2,4,8,16, 32,64,128,256). As you can see, the problem is
with the first two items in the list, '0' and '1', because '0' is not
strictly a power of '2'. I believe that anyone who can do that, will be
able to publish at least a letter, because it's relevant to an important
problem in physics and astronomy.

"John Schutkeker" wrote in message
. 17.102...
...
Since I'm giving up on it, I think I'll throw it open to the whole
physics community, or at least that part of it that resides here in this
BBS.

I was trying to find an equation for the following succession of powers
of '2', (0,1,2,4,8,16, 32,64,128,256). As you can see, the problem is
with the first two items in the list, '0' and '1', because '0' is not
strictly a power of '2'. I believe that anyone who can do that, will be
able to publish at least a letter, because it's relevant to an important
problem in physics and astronomy.

'1' is not a problem but '0' is, in reverse order
the log to the base 2 of your series is

... 8, 7, 6, 5, 4, 3, 2, 1, 0, -infinity.

George

In sci.astro.amateur John Schutkeker wrote:


Since I'm giving up on it, I think I'll throw it open to the whole
physics community, or at least that part of it that resides here in this
BBS.

Hi, there, John. As someone on the Usenet newsgroup sci.astro.amateur,
please let me clarify that while you may be using a BBS to access the
groups to which you are posting, you are indeed posting to three Usenet
newsgroups. Since Usenet does in some ways resemble, for example,
FidoNet, I can understand how this distinction isn't always so obvious.

As a layperson, I'm curious about your mathematical quandary even if not
so expert at addressing it, so please let me try a question.


I was trying to find an equation for the following succession of powers
of '2', (0,1,2,4,8,16, 32,64,128,256). As you can see, the problem is
with the first two items in the list, '0' and '1', because '0' is not
strictly a power of '2'. I believe that anyone who can do that, will be
able to publish at least a letter, because it's relevant to an important
problem in physics and astronomy.

Please let me ask if I'm right that these numbers represent integers equal
to powers of two, actually starting with 2^0=1, then 2^1=2, 2^2=4, etc.?
If so, then 0 is a problem for the reason that you state: it could be
more and more closely approximated by very high negative powers of 2
(e.g. 2^-100) with very small sizes, but never reached; I guess it might
be the limit of 2^-x when x approaches an infinitely large size.

Apart from the 0, of course, you have a simple power series, 2^x where
n is a nonnegative integer. I'd need to look up the formal notation for
such a series, but it should be pretty straightforward.

I'm curious about the application, and the main mathematical question,
it would seem to me as a layperson, is to how to define a series that
would start with that initial 0, the rest being a straightforward
series of powers of 0 (2^0, 2^1, 2^2, ..., 2^n-1).

Most appreciatively,

Margo Schulter

Lat. 38.566 Long. -121.430

Dear John Schutkeker:

On Sep 12, 12:50 pm, John Schutkeker
wrote:
....
I was trying to find an equation for the following
succession of powers of '2', (0,1,2,4,8,16, 32,64,
128,256). As you can see, the problem is with
the first two items in the list, '0' and '1', because
'0' is not strictly a power of '2'. I believe that
anyone who can do that, will be able to publish
at least a letter, because it's relevant to an
important problem in physics and astronomy.

Well, it is not continuous, nor differentiable, but

Result = int( 2^n ), for any integer n.
With int( ) returning the "integer portion of" without rounding.
(Might need to add a small bit (like 0.1) inside the int() operation,
since floating point math is always so "goosy".)

But, if you are looking for a computer algorithm, that will work.

What problem would this solve?

David A. Smith

dlzc wrote in news:1189634716.774606.134360@
57g2000hsv.googlegroups.com:

Dear John Schutkeker:

On Sep 12, 12:50 pm, John Schutkeker
wrote:
...
I was trying to find an equation for the following
succession of powers of '2', (0,1,2,4,8,16, 32,64,
128,256). As you can see, the problem is with
the first two items in the list, '0' and '1', because
'0' is not strictly a power of '2'. I believe that
anyone who can do that, will be able to publish
at least a letter, because it's relevant to an
important problem in physics and astronomy.

Well, it is not continuous, nor differentiable, but

Result = int( 2^n ), for any integer n.
With int( ) returning the "integer portion of" without rounding.
(Might need to add a small bit (like 0.1) inside the int() operation,
since floating point math is always so "goosy".)

But, if you are looking for a computer algorithm, that will work.

What problem would this solve?

It's called Bode's Law, the number series which describes the ratios of
the radii of our sun's planetary orbits. You can read about it in
Abel's (old) introductory astronomy text, "Exploration of the
Universe," or in Ivars Peterson's excellent recent book for the general
audience, "Newton's Clock: Chaos in the Solar System." It's clearly a
solution to the dynamical equations, but being unable to express it as
an equation means that you can't put the solution equation into the
starting equations, to see what comes out.

In sci.astro.amateur John Schutkeker wrote:

It's called Bode's Law, the number series which describes the ratios of
the radii of our sun's planetary orbits. You can read about it in
Abel's (old) introductory astronomy text, "Exploration of the
Universe," or in Ivars Peterson's excellent recent book for the general
audience, "Newton's Clock: Chaos in the Solar System." It's clearly a
solution to the dynamical equations, but being unable to express it as
an equation means that you can't put the solution equation into the
starting equations, to see what comes out.

Hi, John. As I recall, Bode's Law (or maybe better Bode's Model, since
it fits most but not all the data now known for the Solar System), goes
about like this.

Let 10 equal the distance between Sun and Earth -- or, as we now say,
one astronomical unit (1 AU).

To derive the distances to what I might term the macroplanets through
Pluto (excepting Neptune!), those with sufficient mass to attain and
maintain hypostatic equilibrium or a near-spherical shape (apart from
rotational oblation and the like), we start with 4, the distance for
Mercury; note that the series of integers for this and other distances
represent tenths of an AU (e.g. 0.4 AU for Mercury).

Our sequence goes like this:

Mercury 4 + 0 = 4
Venus 4 + 3*2^0 = 7
Earth 4 + 3*2^1 = 10
Mars 4 + 3*2^2 = 16
1 Ceres 4 + 3*2^3 = 28
Jupiter 4 + 3*2^4 = 52
Saturn 4 + 3*2^5 = 100
Uranus 4 + 3*2^6 = 196
[Neptune].............................
Pluto 4 + 3*2^7 = 388

Bode's Model thus quite accurately predicts the orbital distance
for seven of the eight dominant planets (all except Neptune), those
which have "cleared the neighborhood of their orbit[s]" and have
masses far exceeding the total mass of bodies in their orbital
zones not under their gravitational influence; and also notably
for the belt or congregate macroplanet 1 Ceres, in fact discovered
by Giuseffe Piazzi in 1801 in an orbit very close to where Bode's
Model predicted it should be (a discovery also of the first body
in what would would be recognized as the main asteroid belt), and
also 134340 Pluto. also now known to be a belt macroplanet, and
the largest of the Kuiper Belt objects. (While 136199 Eris is
larger, it is a Scattered Disk Object with an orbit beyond that
of the Kuiper Belt.) In IAU terms, these categories are called
respectively "planets" and "dwarf planets."

In other words, with Bode's Model, the series of integers to be
added to 4 to derive the orbital distances of the relevant bodies
starting with Mercury is is (0,3,6,12,24,48,96,192,384).

Most appreciatively,

Margo Schulter

Lat. 38.566 Long. -121.430

Margo Schulter wrote in
:

In sci.astro.amateur John Schutkeker
wrote:

It's called Bode's Law, the number series which describes the ratios
of the radii of our sun's planetary orbits. You can read about it in
Abel's (old) introductory astronomy text, "Exploration of the
Universe," or in Ivars Peterson's excellent recent book for the
general audience, "Newton's Clock: Chaos in the Solar System." It's
clearly a solution to the dynamical equations, but being unable to
express it as an equation means that you can't put the solution
equation into the starting equations, to see what comes out.

Hi, John. As I recall, Bode's Law (or maybe better Bode's Model, since
it fits most but not all the data now known for the Solar System),
goes about like this.

Let 10 equal the distance between Sun and Earth -- or, as we now say,
one astronomical unit (1 AU).

To derive the distances to what I might term the macroplanets through
Pluto (excepting Neptune!), those with sufficient mass to attain and
maintain hypostatic equilibrium or a near-spherical shape (apart from
rotational oblation and the like), we start with 4, the distance for
Mercury; note that the series of integers for this and other distances
represent tenths of an AU (e.g. 0.4 AU for Mercury).

Our sequence goes like this:

Mercury 4 + 0 = 4
Venus 4 + 3*2^0 = 7
Earth 4 + 3*2^1 = 10
Mars 4 + 3*2^2 = 16
1 Ceres 4 + 3*2^3 = 28
Jupiter 4 + 3*2^4 = 52
Saturn 4 + 3*2^5 = 100
Uranus 4 + 3*2^6 = 196
[Neptune].............................
Pluto 4 + 3*2^7 = 388

Bode's Model thus quite accurately predicts the orbital distance
for seven of the eight dominant planets (all except Neptune), those
which have "cleared the neighborhood of their orbit[s]" and have
masses far exceeding the total mass of bodies in their orbital
zones not under their gravitational influence; and also notably
for the belt or congregate macroplanet 1 Ceres, in fact discovered
by Giuseffe Piazzi in 1801 in an orbit very close to where Bode's
Model predicted it should be (a discovery also of the first body
in what would would be recognized as the main asteroid belt), and
also 134340 Pluto. also now known to be a belt macroplanet, and
the largest of the Kuiper Belt objects. (While 136199 Eris is
larger, it is a Scattered Disk Object with an orbit beyond that
of the Kuiper Belt.) In IAU terms, these categories are called
respectively "planets" and "dwarf planets."

In other words, with Bode's Model, the series of integers to be
added to 4 to derive the orbital distances of the relevant bodies
starting with Mercury is is (0,3,6,12,24,48,96,192,384).

Most appreciatively,

Margo Schulter

Lat. 38.566 Long. -121.430

Margo, you're the hottest babe in the solar system. I now have the
solution, and you can come over Saturday night to claim your reward.

John Schutkeker wrote:
I was trying to find an equation for the following succession of powers
of '2', (0,1,2,4,8,16, 32,64,128,256). As you can see, the problem is
with the first two items in the list, '0' and '1', because '0' is not
strictly a power of '2'. I believe that anyone who can do that, will be
able to publish at least a letter, because it's relevant to an important
problem in physics and astronomy.

There are some trivial solutions. Since the real series of powers of
two, for which there is a simple equation, would bt (1/2, 1, 2, 4,
8...), you could have as your equation:

ceil(2^(i-2)-(3/4))

But I'll assume that the use of ceiling and floor functions and the
like is considered "cheating".

You have ten numbers there. That means you can fit a ninth-degree
polynomial to them if you want a real mathematical formula.

If you've found a non-trivial solution, of course, it is indeed a pity
if it has been lost.

John Savard

John Schutkeker wrote:
Margo, you're the hottest babe in the solar system. I now have the
solution, and you can come over Saturday night to claim your reward.

I'm curious: how was she of assistance?

You, after all, already knew that the sequence of numbers was of
interest to you because of Bode's Law, and all she did was show how
the distances of the planets, in tenths of an A.U., were equal to 4
plus 3 times the numbers in your sequence - which is, of course, what
you started from.

Everyone else presumably thought that since the sequence would have
gone on to infinity, Mercury came about in place of all the tiny
planets it would have predicted. I would think that if you were
looking for a "reason" for the distribution of the planets in the
solar system, you would start by using the exact distances of the
planets (including Neptune instead of Pluto).

But we already know that the planets didn't always have their current
distances from the Sun. At one time, while Jupiter's orbital period
was its present 12 years, that of Saturn was 18 years instead of 30.
This resonance led to Saturn's orbit becoming larger, leading to the
Late Heavy Bombardment. (I just learned about this stuff at the last
astronomy club meeting I attended...)

So that means that the spacing of planets in the Solar System is very
much the result of historical causes, which would seem to mitigate
against any simple regular law. Or the law might be really simple -
planets in adjacent orbits repel one another, until the ratio in
distance from the sun is nearly, but not quite, 2 to 1, and the ratio
in orbital period is therefore something like the 13:8 of Earth and
Venus, or the 5:2 of Saturn and Jupiter.

John Savard

Quadibloc wrote in news:1189822323.290526.231200@
50g2000hsm.googlegroups.com:

John Schutkeker wrote:
Margo, you're the hottest babe in the solar system. I now have the
solution, and you can come over Saturday night to claim your reward.


I'm curious: how was she of assistance?

Typing my reply to her helped me organize my thoughts, and I sudden;y
realized that I had had an idea for it that I hadn't tried yet. I tried
it and it didn't work, but a similar function did.

You, after all, already knew that the sequence of numbers was of
interest to you because of Bode's Law, and all she did was show how
the distances of the planets, in tenths of an A.U., were equal to 4
plus 3 times the numbers in your sequence - which is, of course, what
you started from.

Everyone else presumably thought that since the sequence would have
gone on to infinity, Mercury came about in place of all the tiny
planets it would have predicted. I would think that if you were
looking for a "reason" for the distribution of the planets in the
solar system, you would start by using the exact distances of the
planets (including Neptune instead of Pluto).

But we already know that the planets didn't always have their current
distances from the Sun. At one time, while Jupiter's orbital period
was its present 12 years, that of Saturn was 18 years instead of 30.
This resonance led to Saturn's orbit becoming larger, leading to the
Late Heavy Bombardment. (I just learned about this stuff at the last
astronomy club meeting I attended...)

So that means that the spacing of planets in the Solar System is very
much the result of historical causes, which would seem to mitigate
against any simple regular law. Or the law might be really simple -
planets in adjacent orbits repel one another, until the ratio in
distance from the sun is nearly, but not quite, 2 to 1, and the ratio
in orbital period is therefore something like the 13:8 of Earth and
Venus, or the 5:2 of Saturn and Jupiter.

John Savard

As I recall correctly, you're a mathematical whiz, which means that
Danby's book will be useful to you. I'd recommend spending the $35 at
Willmann-Bell for such a well written hardcover, of such mathematical
rigor and such a good price. AFAIK, you can't get current hardcover
texts from any other source, although there are plenty of good used ones
at the usual sources.

Once you've read Danby, you're ready for Laskar's papers, which are the
state of the art. Wisdom's work is useful for everything except
equations, because he's using Hamiltonian rather than Newtonian
mechanics, so his equations are impossible for anybody but a total
genius to decipher. Someday I hope he writes another text on the
Hamiltonian analysis of the problem, because it would be a wonderful
contribution. He had to trade first authorship of his first text to
Jerry Sussman, in exchange for the kick starting his career.

For now, Laskar's work is the only thing us ordinary mathematicians have
to work with. I've looked closely at his equations, and there are a
couple of trivial simplifications possible, but I can't see anything
worth a paper. Since you're good at math, you might be able to spot the
next change of variables that goes somewhere useful.

Or you can be the one to put Bode's Law into the perturbed equations of
n-body motion, and show that it generates a meaningful solution. That's
what I say in the conclusion of this letter I'm working on, although I
haven't yet found out whether the content will be acceptable to whomever
reviews it.

It's not too different from control theory, which is mathematically a
very rich subject.