[seqfan] Re: Another planetary sequence

Charles Greathouse charles.greathouse at case.edu
Thu Sep 15 19:39:27 CEST 2016

```> For an example of "not interesting/contrived" seemingly pertinent to this
discussion:
may I ask what ( https://oeis.org/draft/A276707 ) is supposed to convey?
"Number of planet from the Sun that is orbited by the n-th most massive
natural satellite.
5, 6, 5, 5, 3, 5, 8, 7, 7, 6, 6, 7, 7, 6, 6"

This is a sequence I discussed with Felix after this email chain caught my
interest. It's an attempt to do a moon sequence in the right way: no units,
no rounding, and stable over hundreds of millions or billions of years. How
would you do a sequence like this? What do you think of the other
astronomical sequences in the OEIS?
https://oeis.org/wiki/The_multi-faceted_reach_of_the_OEIS#Astronomy

Charles Greathouse
Case Western Reserve University

On Thu, Sep 15, 2016 at 1:14 PM, Joerg Arndt <arndt at jjj.de> wrote:

> Donning my "grumpy old man" hat ...
>
>
> The OEIS is about INTEGER sequences, finite or infinite.
> One can turn real sequences into integer sequences using
> some funny hammer like floor() or round(), this usually
> results in something of little interest.
>
>
> But integer sequences should still have some merit
> (most often/importantly of mathematical nature).
>
> For an example of "not interesting/contrived" seemingly
> pertinent to this discussion:
> may I ask what ( https://oeis.org/draft/A276707 )
> is supposed to convey?
> "Number of planet from the Sun that is orbited by the n-th most massive
>  natural satellite.
>  5, 6, 5, 5, 3, 5, 8, 7, 7, 6, 6, 7, 7, 6, 6
> "
> This is just depressing.
> And a maintenance problem.
>
>
> It appears to me that good sequences come up when one works on
> something that does NOT have creating sequences for the OEIS in mind
> in the first place.
>
>
> Best regards,   jj
>
> * Alonso Del Arte <alonso.delarte at gmail.com> [Sep 15. 2016 18:05]:
> > There exist infinitely many sequences, both finite and infinite. The
> > problem is to decide which ones are useful and/or interesting enough to
> be
> > worth including in this reference work. And then you also have to think
> > about how people are going to look it up to find it.
> >
> > Try the following Wolfram Alpha query: radius of moons of planets of
> Solar
> > System.
> >
> > Today, it didn't quite understand that query, and instead gave me for an
> > answer that the planets have an average radius of 15,111 miles, and this
> > sequence of number of moons: 0, 0, 1, 2, 63 (known), 61 (known), 27
> > (known), 14 (known). If you put in a search for the first four terms, the
> > ones without the disclaimer, you get a couple thousand results.
> >
> > Number of moons certainly looks like a more straightforward sequence than
> > comparing a planet's radius to the radius of its largest moon. But even
> > with this one we have a bunch of asterisks. And could it actually be the
> > case that one of these planets' largest moon is an unknown moon? Sounds
> > unlikely, but it's still enough to cast doubt on both of these sequences.
> >
> > Al
> >
> > On Wed, Sep 14, 2016 at 4:50 AM, Paul Barry <pbarry at wit.ie> wrote:
> >
> > > 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,
> > >
> > >
> > >
> > >
> > > 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
> > > > > 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/
> > > > >
> > > > > --
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> > > > >
> > > >
> > > > --
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> > >
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> > >
> > >
> > >
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> >
> >
> >
> > --
> > Alonso del Arte
> > Author at SmashWords.com
> > <https://www.smashwords.com/profile/view/AlonsoDelarte>
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> >
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