[seqfan] Re: Another planetary sequence

[Brad Klee]

Hi Felix,

Yes, the planets are of great interest to everyone, glad you are
thinking about it!

This sequence is somewhat interesting, but one shortcomming for OEIS
is that it doesn't go on forever, terminates at n=8.

Around noon it's a good time to go outside and look up to the sun.
Unless it happens to be hidden, Mercury is somewhere in the sky, and
then how can you help but to wonder exactly what is going on,
physically? With the gravity wave signals coming out of LIGO, everyone
now believes in Einstein's theory, and maybe they even talk about it
positively in the beer halls. Time changes everything. But some things
even time has difficulty changing. How many people can do the
calculations? For example, perihelion precession? It seems, not so
many. Maybe we can use OEIS to help with all of that.

We can start with something along the lines of:

http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node115.html

but it's still difficult even if you know squares of the y-intercepts
of the Legendre polynomials. This presentation is confounding, at
least to me. Luckily I did figure out another _Big Equation_ that
works just as well __on all planetary data__.

If you read through the following and referenced articles therein:

http://oeis.org/search?q=pendulum+klee&language=english&go=Search

you should get a decent idea as to how to find the exact solutions of
1-D oscillation problems in Classical Mechanics as expansions in an
energy parameter, which leads along one branch to the famous Jacobian
Elliptic integrals.

In a pseudo-potential formulation that Gauss and later Fitzpatrick
suggest for a first analysis of planetary perihelion precession, you
can essentially reduce the two-dimensional orbital motion to one
dimension ( by conservation of angular momentum ), and do more exact
solving along similar lines. This time not for the period, but for the
precession rate.

Using Mathematica to expedite multiplication and addition, I wrote the
following equation two days ago:

1-L/r^2
+a*( (-3/2)*L/r^4 + (-3)*L*v[3]/r^3 + (-15/4)*L*v[3]^2/r^2 + (3/2)*L*v[4]/r^2 )
+ a^2 *((-15/8 )*L / r^6 + (-15/2)* L * v[3]/r^5 + (-315/16)* L*
v[3]^2/r^4 + (-315/8)* L *v[3]^3/ r^3 + (-3465/64)* L * v[3]^4/ r^2 +
( 45/8 ) * L *v[4]/r^4 + (105/4)* L *v[3] *v[4]/ r^3 + (945/16)*
L*v[3]^2 *v[4]/ r^2 + (-105/16)* L *v[4]^2/r^2 +(- 15/4)* L *v[5]/r^3
+ (-105/8)* L *v[3]* v[5]/ r^2 + (15/8)* L *v[6]/r^2 )
+a^3 *( ...) + ... higher order in a ...

which describes perihelion precession in a general isotropic central
potential in terms of potential expansion coefficients v[i], angular
momentum "L", radius "r", and energy "a". As a check, for values of (
v[i], r ) determined from coulomb potential the whole expansion goes
to zero for any pair of (L,a). The coulomb solutions are Kepler
Ellipses fixed in space, i.e. no precession.

It may seem silly to have such a big equation that gets even bigger,
but lets just look at the results. We have to use potential theory to
calculate values for

{a,r,v[1],v[2],v[3],v[4],v[5], ... }

by transforming the data at

http://nssdc.gsfc.nasa.gov/planetary/factsheet/

This is the only hard part, but doesn't take too long. Plugging in the
numbers for Mercury, and scaling to arcseconds per year, we get, term
by term:

{4.67068, 0.747725, 0.10366, 0.0149433, 0.0022804}

with folded sum

{4.67068, 5.41841, 5.52207, 5.53701, 5.53929}

which converges to a value near the 5.5 reported by Fitzpatrick in his
table 2. For other planets the convergence is much faster, and the
second order equation here should do just fine. The pseudo-potential
for Mercury is the most significantly different from quadratic
harmonic, so higher terms are noticeable.

The interesting part of this analysis is that energy parameter "a"
takes into account the time-changing radius of planetary orbits,
whereas I cannot see any time-changing radius of the orbits in
Fitzpatrick's equations! In our approach there is a contribution
independent of "a", but notice that for Mercury values

4.67068 =/= 5.5 ( Fitzpatrick value )

and

4.6708 =/= 5.32 ( https://en.wikipedia.org/wiki/Tests_of_general_relativity )

so we need to include the part that depends on how high in the
effective potential the system is to get nearer to the correct answer.
How does Fitzpatrick get 5.5 with his method? Still wondering about
that, especially if his expansion does indeed ignore terms depending
on the energy of the orbit. ( Maybe a closed form summation along "a"
? Even then, does his expansion have cross terms such as  L *v[3]
*v[4]/ r^3 ? ).

You may already notice that there is a fractional sequence of
coefficients in the big precession equation. We can ( and will ) put
this into the OEIS, as an irregular triangular. Although there is an
infinite number of v[i] variables, there is a natural grading in the
ring generated by all possible products of the variables, which can
already be seen in the first few terms. If you have term

v[i]^p_i * v[j]^p_j *..../r^p_r

Then the exponents must satisfy

( i - 2 )*p_i + ( j - 2 )*p_j + ... p_r = 2(N+1)

so every row is finite. We'll just need to do a Lexicographic ordering
of the variables and exponents, and accept that the triangle will
probably have lots of zeros. While we're on the subject:

** Does anyone know of a canonical form in OEIS for such a sequence?
Or a sequence along similar lines? **

Anyways it's a good place to leave off. This idea may yet be
confusing, but will be much easier to understand once its been
published through a few different venues including the OEIS.

Thanks for writing, happy planet gazing!


<~~~♀~~~~~~

              ⊙
~~~~~~~~☿~~~~~~~~~~~>

          ~~♁~~~>

Watch out for troublesome syzygies!

                         ~♂~>


Best regards,

Brad






> On Sep 11, 2016, at 11:49 AM, Felix Fröhlich <felix.froe at gmail.com> wrote:
>
> Dear sequence fans
>
> I thought about the following sequence:
>
> Ratio of radius of n-th planet (under the current IAU-definition of planet)
> from the Sun to mean radius of its largest natural satellite, rounded to
> the nearest integer, or 0 if the planet has no natural satellite.
>
> a(1)-a(8) are 0, 0, 4, 308, 27, 23, 32, 18
>
> The true ratios are of course not integer values, but the terms give an
> idea of how large the largest moons of the planets are compared to the
> planets themselves (i.e. these values are still useful for comparison, even
> when rounded to integers in my opinion). The closer the value is to 1, the
> larger the largest moon is relative to its planet. The value for Earth's
> moon is relatively small, meaning the Moon is large relative to Earth.
>
> There are already a number of sequences related to the planets in the OEIS,
> but I would like to hear the opinion of other contributors and/or some of
> the editors before submitting this, mainly because I think the sequence
> will likely be rejected.
>
> It is probably a "dumb" sequence, not really mathematically significant,
> but sometimes such sequences are still enjoyable.
>
> Best regards
> Felix
>
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