[seqfan] Re: Another planetary sequence
Felix Fröhlich
felix.froe at gmail.com
Tue Sep 13 22:23:27 CEST 2016
Rick, your idea of a sequence "counting all orbital situations that
can occur with all orbits elliptical and coplanar" sounds interesting.
Best regards
Felix
2016-09-13 22:21 GMT+02:00 Felix Fröhlich <felix.froe at gmail.com>:
> 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/
>>
>
>
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