[meteorite-list] INNER SOLAR SYSTEM

From: Sterling K. Webb <sterling_k_webb_at_meteoritecentral.com>
Date: Tue Sep 26 02:21:26 2006
Message-ID: <003c01c6e133$f640c760$0d7a4b44_at_ATARIENGINE>

Hi, All,

    If you're talking orbits, you have to start with
old Isaac. What would he have given to get his
hands on one of our bigger computers?

    "The orbit of any one planet depends on the
combined motion of all the planets, not to mention
the actions of all these on each other. To consider
simultaneously all these causes of motion and to
define these motions by exact laws allowing
of convenient calculation exceeds, unless I am
mistaken, the forces of the entire human intellect."
                    ---Isaac Newton, 1687

    Isaac was right; it's not a "convenient" calculation!

    This three million year calculation was done for
all NINE (not eight) planets; this movie just shows the
inner system. I would love to see the whole thing animated,
but I don't think it's been done.

    Here's the URL for the original three million year
orbit calculation of the solar system:
http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1991AJ....101.2287Q&amp;data_type=PDF_HIGH&amp;type=PRINTER&amp;filetype=.pdf

    Before this numerical integration was done, the longest
time period calculated for the solar system was only 4400
years! I include the reference on the (very) off chance somebody
wants to check it out; it's a bear of a paper.

    This URL is not the original 3,000,000 year calculation,
but an article by the calculators on the calculating problem
and some of the implications of these calculations:
http://delivery.acm.org/10.1145/320000/314166/p1-lake.pdf?key1=314166&key2=0568329511&coll=portal&dl=ACM&CFID=11111111&CFTOKEN=2222222

    Here's a few chewy quotes:

    "LONGSTOP integrations
were not limited by CPU time but by roundoff
error[35]. New fourth order "mixed variable symplectic"
(MVS) integrators reduce roundoff error such
that a 10 billion year integration of all nine planets is
possible with 128 bit precision."

    So, in theory, we could calculate the entire lifetime
of the solar system (in a decade or two)! I don't think
anybody's tried it yet!

    "These new integrators
were used for an accurate integration of all nine planets
and the Earth's spin axis for 3.05 Myr into the past,
and future[36,27]-roughly the limit of 64 bit precision.
General relativity was included in an extremely elegant
way[40]."

    Among other things, when you calculate "backwards,"
you have to keep changing the mass of the Sun! The solar
wind carries mass away from the Sun so that it becomes
progressively lighter over long timespans. You have to work
out the rate of mass loss and keep adding mass to the Sun as
you go back in time!

    "A comparison with the heroic secular perturbation
calculation shows remarkable agreement over their
common range including the existence of the secular
resonance claimed to be responsible for the chaos[27], but
all planetary orbits appear to be regular over the 6 Myr
interval..."

    Jupiter is the "heavy" in this drama, the big bumper.
Rob asked for the same simulation with Jupiter missing.
My first thought is that things would be more regular
and less eccentric, but my second thought is, what if the
strong influence of Jupiter suppresses influences that,
left to themselves, could get out of hand?
    For example, the Earth and Venus are very close to
being in a 5:8 resonance, but seem to have never achieved
it. I say they've never achieved it because if they had ever
locked into that resonance and stayed undisturbed, the Earth
would have pumped up Venus' orbit in eccentricity until
Venus' orbit at aphelion would have reached all the way
out to the Earth's orbit and big trouble would have ensued!
    Big trouble... Maybe the heavy hand of Jupiter keeps
that resonance from happening?

    "The accuracy of the
integrator at a given timestep is increased by the ratio
of Jupiter's mass (the biggest kicker) to the sun's mass.
Sussman and Wisdom[46] performed a 100 Myr
integration on a somewhat special purpose computer
that used the potential approximation to General
Relativity[SZ], but was otherwise similar to the 6
Myr integration[36]. They found an initial divergence
timescale of 12 Myr, but a 4 Myr divergence dominates
after 60 Myr."

    Note: A "divergence" is when things go whacky
or chaotic. It can show up because the calculation
is not sufficiently accurate, but that's not the case
here. For it to show here suggests a prior time span
in which something unexpected has happened which
we can't "look back" through.
    The "divergence time" is a mathematical measure
of stability, basically, the answer to the question: If
things are stable now, how long would it take for this
stable system to go all to hell? The answer is: not long.

    In the way back when, LaPlace determined that
all the planetary perturbations were "periodic" or
harmonic and could be represented by the sum of
a series of periodic terms. Then, later, Poincare
showed that these series never converged and there
was no proof of stability. Then, in 1989, "Laskar[23]
tested the quasi-periodic hypothesis by numerically
integrating the perturbations calculated to second
order in mass and fifth order in eccentricities and
inclinations, ---150,000 polynomial terms. Fourier
analysis of his 200 million year integration reveals that
the solution is not a sum of periodic terms and implies
an instability that is surprisingly short, just 5 Myr."

    Yeah, that's what I want to do with my spare time,
repeatedly solve a 150,000-term polynomial!

    "The 4 Myr divergence occurs much later
in the outer planets than in the inner planets hinting
at two distinct mechanisms (or three since Pluto has
its own distinct chaotic behavior). They were cautious
about identifying the underlying dynamical mechanism
for the chaos..."

    Things in the solar system must have changed
over time, if this is a true characteristic of the system.
Maddeningly, you can't "calculate" your way back
though a change in a dynamic system caused by
a re-arrangement of bodies, the ejection of a body,
or an outside gravitational influence. It's a bottleneck
that you can't thread in reverse.

    They think resonances did it.

    "Understanding the source of Solar
System chaos awaits an analytical demonstration that
the resonances involved are sufficiently strong and close
for resonance overlap. Nonetheless, the Solar System is
almost certainly chaotic. Laskar[24] looked at the fate of
Mercury and estimates the chance of ejection in the next
few billion years approaches 50%. Our belief in the regularity
of the Solar System would be dashed if the ejection of Mercury
were in the historical record."

    Well, I would take it hard, that's for sure...

    Mercury is a hard body to eject, what with the Sun
hanging on to it so tightly. So, if it's possible to lose
Mercury from the solar system, it's even more possible
(or possible more often) to lose massive bodies further
from the Sun! They conclude that: "There could have
been a dozen or more planets just a few billion years ago."

    Anybody lost any planets lately?


Sterling K. Webb
-------------------------------------------------------------
----- Original Message -----
From: "Matson, Robert" <ROBERT.D.MATSON_at_saic.com>
To: "Pete Pete" <rsvp321_at_hotmail.com>; <meteorite-list@meteoritecentral.com>
Sent: Monday, September 25, 2006 2:52 PM
Subject: RE: [meteorite-list] INNER SOLAR SYSTEM


Hi Pete and List,

With regard to the time-lapse view of the inner planets' orbits,
Pete asked:

> Considering how this solar system is believed to have formed, do you
> think that these dynamics are something that could be expected in
> other systems, too?

Absolutely.

> What could have kick-started it? Only the Sun, somehow, right?

Nope -- read on...

> Or would you think that some past near-catastrophe could have spun
> the plates? (one of them there new-fangled wandering stars)
> It is just a little difficult for me to imagine these wobbles
> were borne without an outside influence.

Ah, but there ~is~ an outside influence -- a big one. Jupiter is
what causes the majority of the harmonic changes in inner planets'
eccentricity and inclination. (Saturn, Uranus and Neptune also
provide smaller perturbations.) What would be interesting to see
for comparison is a second movie in which Jupiter's influence is
turned off.

--Rob
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Received on Tue 26 Sep 2006 02:21:10 AM PDT