[seqfan] Re: Another planetary sequence

[Felix Fröhlich]

Thanks for all the further replies. What about the following sequence. Its
terms are really integers.

Rank of size of largest natural satellite of n-th planet from the Sun among
size of all natural satellites of the planets, or 0 if the planet has no
natural satellites.

Terms are 0, 0, 5, 67, 1, 2, 8, 7

Of course this sequence may be a bit problematic as well. It is still a
time-dependent sequence, although it probably won't change frequently. Also
I don't know if that new sequence is that more interesting and it's of
course still finite and relatively short.

Like others in this thread, I have also thought about other possible
planetary sequences. What came to my mind was to make a sequence related to
orbital resonances (like the 1:2:4 resonance involving Io, Europa and
Ganymede), although I am not sure at the moment what that sequence could be
exactly.

Best regards
Felix

2016-09-12 4:58 GMT+02:00 Brad Klee <bradklee at gmail.com>:

> 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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>
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