
Planets evenly spaced on log scale - tacon
https://www.johndcook.com/blog/2018/04/05/solar-system-on-log-scale/
======
antognini
As a former dynamicist, the underlying reason for this correlation is that
planetary systems with many planets are always on the edge of dynamical
instability. In other words, these kinds of planetary systems with lots of
planets are always packed with as many planets as they can support over any
given lifetime. It then turns out that the stability criteria imply that the
system cannot be any more packed than having approximately log-equal spacing.
In fact, if you take a star and put planets around it with a log uniform
distribution and evolve it for a while, you find that eventually the planets
that remain roughly fit the Bode-Titus law [1]. (I don't think the reasons for
this are entirely well understood.)

One consequence of this is that our own solar system has been around for about
5 billion years, and is therefore probably unstable on timescales of 5-50
billion years. Indeed, long-term simulations indicate that there is a few
percent chance that Mercury will get kicked out of the solar system or crash
into Venus or the Sun before the Sun dies in ~5 billion years [2] [3].

[1]: [https://arxiv.org/abs/astro-ph/9710116](https://arxiv.org/abs/astro-
ph/9710116)

[2]:
[http://www.scholarpedia.org/article/Stability_of_the_solar_s...](http://www.scholarpedia.org/article/Stability_of_the_solar_system#Marginal_stability_of_the_Solar_System).

[3]:
[http://adsabs.harvard.edu/abs/2009Natur.459..817L](http://adsabs.harvard.edu/abs/2009Natur.459..817L)

~~~
yummybear
I wonder what effects Mercury crashing into the sun would have - large solar
flares? Nothing at all?

~~~
dbasedweeb
Nothing much. The Sun is most of the mass in the Solar system, by over 97%.
Most of the rest of that mass is Jupiter, followed at quite a distance by
Saturn. Jupiter has more than 5700 times the mass of Mercury. In other words,
next to the Sun, Mercury is a teeny little speck that would barely even leave
a “splash” as it roasted to a cinder in its corona. It would be the most
underwhelming event you can imagine, for Sol. For Mercury it would be an
exciting journey of vaporizing outer layers, being torn to little shreds, and
those shreds vaporizing completely (all before it contacted anything like a
“surface”).

~~~
nonbel
What would be the effects of Venus crashing into the sun next?

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dbasedweeb
This may be disappointing, but it would be the same for all of the planets,
even Jupiter. Ignoring the effects on orbital dynamics from the planets
changing positions or falling past other planets, they’d just burn up. Jupiter
probsbly explode, or be drawn into a rig around the sun that would eventually
fall in, and burn up.

All in all, the Sun is just _huge_ and wouldn’t be disturbed by any of it. Its
gravity would tear things apart before an impact, just like the moon would be
shredded if it came too close to Earth. Any sattelite has what’s called a
Roche Limit, and if passes that it breaks apart from tidal stresses. That’s
how you get rings around planets, and I guess a star too. Although in most
cases you wouldn’t have a ring around a star, just burning fragments falling
Sunward.

~~~
MR4D
Dumb question, but would the asteroid belt count as a ring around the sun?

Never thought about it before until I read your comment, so I thought someone
more knowledgeable than me could provide some insight here.

~~~
dbasedweeb
Everything in the Solar system is gravitationally bound by the Sun, which is
what makes it part of the system in the first place. In that sense, yes, and
really the whole Solar system is too. There’s some nuance lost in that
formulation though, because unlike a moon that came too close to its planet,
broke apart and became a ring, the asteroid belt is much more distant and
diffuse. It’s also true that Jupiter sort of “shepherds” the asteroid belt, so
unlike something like planetary rings, it’s a much more complex system. I hope
that goes some way to answering your question, which wasn’t dumb at all.

~~~
dormento
Just wanted to chime in and say thank you. HN needs more comments like this.

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perlgeek
One has to be careful with inspecting log graphs by mere looks. What looks
like a small difference from expected to actual value in log scale can be very
large difference in terms of actual values.

So it bothers me a bit that there is no actual fit and error margins.

~~~
wpietri
Mar's Law: "Everything is linear if plotted on a log-log scale with a fat
magic marker."

It's #6 on the excellent (and surprisingly software-applicable) "Akin's Laws
of Spacecraft Design":
[http://spacecraft.ssl.umd.edu/akins_laws.html](http://spacecraft.ssl.umd.edu/akins_laws.html)

~~~
joveian
Those are good. My favorites:

16\. The previous people who did a similar analysis did not have a direct
pipeline to the wisdom of the ages. There is therefore no reason to believe
their analysis over yours. There is especially no reason to present their
analysis as yours.

21\. (Larrabee's Law) Half of everything you hear in a classroom is crap.
Education is figuring out which half is which.

29\. (von Tiesenhausen's Law of Program Management) To get an accurate
estimate of final program requirements, multiply the initial time estimates by
pi, and slide the decimal point on the cost estimates one place to the right.

30\. (von Tiesenhausen's Law of Engineering Design) If you want to have a
maximum effect on the design of a new engineering system, learn to draw.
Engineers always wind up designing the vehicle to look like the initial
artist's concept.

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wl
Mar's law: Everything is linear when plotted on a log-log scale with a fat
marker.

(I know this is a lin-log plot)

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rcthompson
There's an extra large jump from Mars to Jupiter, but that's because there's
an asteroid belt between them, which might be an aborted planet. If you put
the average distance of the asteroid belt as planet number 5, I bet it would
look even more linear.

~~~
dr_zoidberg
The asteroid belt doesn't have nearly enough mass to be/have been a planet[0].
Largest thing there is Ceres, followed by Vesta, of which only the first
qualifies as a dwarf planet.

There was an hypothesis about it being the remnants of a destroyed planet[1],
but that was mostly an idea to support the Titius-Bode law[2], which was
disproved a few hundred years ago.

[0] Currently about 3x Ceres mass, though it may have about Earth mass early
in its history. See
[https://en.wikipedia.org/wiki/Asteroid_belt#Formation](https://en.wikipedia.org/wiki/Asteroid_belt#Formation)
and
[https://en.wikipedia.org/wiki/Asteroid_belt#Evolution](https://en.wikipedia.org/wiki/Asteroid_belt#Evolution)

[1]
[https://en.wikipedia.org/wiki/Phaeton_(hypothetical_planet)#...](https://en.wikipedia.org/wiki/Phaeton_\(hypothetical_planet\)#Phaeton_hypothesis)

[2]
[https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law](https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law)

~~~
mdpopescu
> ... the Titius-Bode law[2], which was disproved a few hundred years ago

Er... the Wikipedia page you're citing at [2] has this to say:

"Results from simulations of planetary formation support the idea that a
randomly chosen stable planetary system will likely satisfy a Titius–Bode
law."

and

"96% of these exoplanet systems adhere to a generalized Titius–Bode relation
to a similar or greater extent than the Solar System does."

~~~
adrianN
"Disproved" in science often means "superseded by more accurate theory".

~~~
johndcook
True, but the article doesn't mention a more accurate theory. It implies that
the rule works better than we have theoretical reasons to expect. (But as I
commented above, the article contradicts itself to some degree.)

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bdamm
It's an interesting conjecture, but what about the anomalies? There are over a
hundred planet systems discovered by Kepler with various levels of
confirmation. None of them are mentioned here. The essay only lists 9 systems.
How do the rest compare? Are there anomalies in unconfirmed systems? Why were
these systems chosen; was it a random selection?

~~~
olympus
You need a system with more than two planets. A system with only two planets
is going to be linearly spaced on any type of scale.

Once you've filtered out systems with two or fewer planets, you need orbital
measurements with decent precision to tell whether a plot of planet spacing is
actually linear on a log scale or not. Measuring a planet's orbit requires a
decent amount of observation (since we can't measure star/planet mass directly
and don't usually measure period directly), so unconfirmed systems likely
don't have good measurements.

It's likely that most of the unmentioned systems get filtered out by one of
the above two criteria.

~~~
grkvlt
Also, even if we detect a system with N > 3 exo-planets, we often cannot tell
if there are more than N, but below the threshold of detectability, making the
plots incomplete and incorrect.

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alan
We might be able to start using this as a way to specify planets, rather than
the arbitrary number-by-discovery. Try StarName(with catalog
info)#log(approximate radius of orbit)

Earth would we Sol#0 Kepler-90#-1.3, Kepler-90#-1.1, out to Kepler-90#0 for
the outermost one on his graph.

That way, finding a new planet doesn't require either renumbering to specify
where the planet is relative to the star. (BTW, I'm not real happy about using
AU as the base measurement. It might be better to use Megameters since you're
less likely to end up with negative log)

~~~
saagarjha
What would you do for highly elliptical orbits?

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amelius
Something tells me that the mass of the planets should be in the conjecture
somewhere, because small objects have little meaning and perhaps would not
even classify as planets.

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rdiddly
Anyone care to speculate based on physics why this might have worked out this
way?

~~~
yonkshi
Indulge my wild guess (I'm not a physicist):

If the proto solar system's matter distribution was a Gaussian-like
distribution, it would make sense that matter density droped in a logarithmic
scale, and thus planets formed droped logarithmically w.r.t radius.

More intuitively.. further away from the sun has less probability for
rocks/particles to interact with each other.

~~~
Someone
In our solar system, the small planets are closest to the sun.

Jupiter, with 72% of all planetary mass, is more than twice as heavy as all
other planets, combined.

Saturn, Uranus and Neptune take most of the remaining 28%
([https://en.wikipedia.org/wiki/List_of_Solar_System_objects_b...](https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size#Graphical_overview))

(Of course, there also is way more volume that far away)

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kpwagner
Fascinating that this relationship scales from solar systems in which the
planets are close to the star to solar systems that are more spread out. I
wonder if the mass of the star has an effect on the average distance that the
mass of the rest of the solar system is from the star. I also wonder if the
mass of the star has a correlation with the tightness of the logarithmic
distance relationship exhibited in this article.

~~~
hinkley
I suspect some expert will come in and tell us that there is some fundamental
of orbital mechanics which correlated to the observations. That is, stable
planetary systems have orbits that correlate with the area of the orbit (the
square of the distance from the star)

I recall one of the earliest breakthroughs in orbital mechanics was someone
figuring out how the area of a slice of a elliptical orbit was the same
anywhere in the orbit, when the angle of the arc is calculated as a unit of
time and not degrees.

Not to say you’re wrong. I think you’re right, but the quality you’re
attributing to the system is a behavior of the system that has an underlying
link to physics that may already be well explored. Just nobody has bothered to
put a pretty plot in front of us armchair types and students before.

~~~
vkou
> I recall one of the earliest breakthroughs in orbital mechanics was someone
> figuring out how the area of a slice of a elliptical orbit was the same
> anywhere in the orbit, when the angle of the arc is calculated as a unit of
> time and not degrees.

That would be Kepler's second law. It states that a line between the sun and
the planet sweeps equal areas in equal periods of time.

~~~
hinkley
There you go.

And calculus has an explanation for why that’s the case.

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jwiley
Some more discussion on Reddit:

[https://www.reddit.com/r/askscience/comments/58plow/why_are_...](https://www.reddit.com/r/askscience/comments/58plow/why_are_the_planets_logarithmically_spaced/)

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debt
Was kind of hoping to see pictures of the planets themselves, evenly spaced on
a log scale. Those line graphs were a little less exciting.

~~~
rcthompson
Unfortunately, I believe we only have decent photos for one of the mentioned
solar systems.

~~~
EGreg
Would have made for a great illustration, methinks!

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pacaro
I’m working on a solar system chart for my daughter, to my eye it’s more
aesthetically pleasing to scale both distance from the sun and radius by
square root.

Regardless of your approach the sun is too big to usefully represent on the
same scale and have all other objects be visible

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rhelzerm
Anything is evenly spaded on a log scale...

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mikekij
...sort of like what you’d expect inside a simulation.

~~~
Retra
A simulation is very general concept. You may as well have said "sort of like
what you'd expect to hear with your ears."

Obviously you hear things with your ears. And equally obviously, you can
expect things in a simulation.

