[seqfan] Re: Another planetary sequence
Paul Barry
pbarry at wit.ie
Wed Sep 14 10:50:02 CEST 2016
Expand sqrt(1/sqrt(1-12x)).
Cheers,
Paul Barry
________________________________
From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Brad Klee <bradklee at gmail.com>
Sent: 13 September 2016 23:07:30
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Another planetary sequence
Hi Felix,
Again, it's a good idea to compute sequences such as this for the purpose
of data analysis, and maybe even for a laugh. This one makes Mars' moons
Phobos and Deimos look pitiful, and another funny observation... Jupiter is
#1 when it comes to moons. Ha!
But think about the "data flood". We probably don't have the manpower to
enter every data sequence into the OEIS; though, this would be a great
"freedom of information act".
To go back to my original criticism, I think the OEIS is usually focused on
infinite series rather than finite sequences.
There are lots of great series out there, even related to planets.
Consider the "Schwarzschild Solution" , as described in:
https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity .
There is an effective potential, for which the radius of minimum energy can
be found by solving a quadratic equation. Expanding the potential around
that r_0 we have something like:
V(r) = V(r_0) + k*r^2 + . . .
then we can calculate the scaled zero-radius
R = (2 * k * m)^(1/4) * r_0
and substitute into
1 - L/R^2
as in the previous email, the energy-zero-order precession angle ( up to a
factor of 2 pi). We have the exact form for this term, but it's nothing
pretty to look at. It's much too long to fit into one tiny column of an
article page. So then, let us expand in powers of x=(G m Sqrt[M (m +
M)])/(c L):
3*x^2 + (45/2)*x^4 + (405/2)*x^6 + (15795/8)*x^8 + (161109/8)*x^10 + ...
To compare with standards, take the first term:
2 Pi* 3*G^2* m^2 *M *(m + M)/(c* L)^2
as in the wikipedia article, under the usual assumption (M+m)~M.
So, how about the numerator / denominator series?
Numerator: 3, 45, 405, 15795, 161109, 3383289
Denominator: 1, 2, 2, 8, 8, 16
Denominators look somewhat boring, but numerators are interesting, and
Not in the OEIS!
I'll see if I can put this in later when I have double checked everything.
Best Regards,
Brad
On Tue, Sep 13, 2016 at 8:21 PM, Felix Fröhlich <felix.froe at gmail.com>
wrote:
> 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
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
--
Seqfan Mailing list - http://list.seqfan.eu/
________________________________
This email is subject to the following disclaimer(s) available at WIT Disclaimer.<https://www.wit.ie/about_wit/documents_and_policies/email-disclaimer>
Tá an ríomhphost seo faoi réir an tséanta/na séanta seo leanas atá le fáil ag WIT Séanadh.<https://www.wit.ie/about_wit/documents_and_policies/email-disclaimer>
[Waterford Institute of Technology]<https://www.wit.ie>
________________________________
More information about the SeqFan
mailing list