Why do the planets' orbital distances fall on an exponential curve?

Question

Background: I was recently reading a book on the planets to my son and I noticed a pattern in the distributions of the planets. The planets' distances roughly follow an exponential distribution.

Below is a plot of scaled, log orbital distances $$ \tilde{d_n} = \frac{\log(d_n/d_1)}{\log(d_2/d_1)} $$ with the line $an+b$:

Where $d_1$ corresponds to Mercury and so on. Ceres is included, Pluto is excluded. By linear regression, $a = 0.90, b = -1.06$.

For the statistically minded, the data has a Pearson's correlation of 0.996. Note that this is a well known phenomenon, see Pletser and references. The code used to generate the plot may be provided on request.

Question: What is the mechanism that leads to this distribution?

Aside: Is there a good introductory text on solar system formation for the mathematically inclined?

Update: This is also known as Titius–Bode law.

Answer 1

This correlation is known as Titius-Bode's law, which is often stated as

\begin{equation} d=0.4 + 0.3 \cdot 2^n \end{equation}

where d represents planet's mean distance from the Sun in Astronomical Units and n = -∞, 0, 1, 2... for Mercury, Venus, Earth, Mars, asteroid belt, Jupiter and so on.

The rule is not satisfied exactly with Neptune's orbit (n=7) constituting a significant departure from it: according to the law Neptune's mean distance ought to be 38.8 AUs, but is in reality just 30 AUs (disagreement of close to 30% with all other planets agreeing to less than 6%). In fact, this departure is what has historically led to diminishing importance of the law. See also the table and chart in wikipedia.

It is currently thought that if the law is not a pure coincidence then it is a consequence of orbital instabilities and the mechanism through which Solar system was formed. It's been shown that rotational and scale invariance of a protoplanetary disk leads to density maxima in the disk appearing periodically in variable

\begin{equation} x = \ln \frac{r_n}{r_0} \end{equation}

which leads to geometric series for planetary distances similar to that expressed in Titius-Bode's law. See this and this paper for details.

Note that the requirements of rotational and scale invariance are very general. As the nebula from which protoplanetary disk is formed collapses under its own gravity, its rotation increases due to the law of the conservation of angular momentum. This eventually leads to the protoplanetary disk's rotational symmetry. Also, gravity does not have intrinsic length scale, so the nebula is highly likely to possess scale invariance. These two requirements are so general that even if the Titius-Bode's law is real it isn't at all useful to select between Solar system's formation models.

I don't know of an advanced book specifically on Solar system formation, but there is a very good book by A.E. Roy on orbital mechanics which certainly would qualify as a book for the mathematically inclined which in addition to chapters on orbital mechanics, rocket dynamics, interplanetary trajectory design includes few solar system formation and many-body stellar systems. So depending on how broad your interests are you may enjoy it.

Answer 2

They don't exactly it's more a (pretty good) statistical correlation than a law.

The usual explanation is that resonances between other planets or existing planets and the sun, disrupted any planets that began to form at any other distance - but it's not theoretically well understood.

Answer 3

I used Eureqa with the data:
distance km time
d t
57.909999847412109 8
108.20999908447266 7
149.60000610351562 6
227.91999816894531 5
778.57000732421875 3
1433.530029296875 2
2872.4599609375 1
4495.06005859375 0

and obtained this simple solution:
$$495.0601*exp(-0.57141924*t)$$
that is a better approximation than other formulations. That can be interpreted as a periodic event in the solar system (a clock), see abstract and full paper here:
A self-similar model of the Universe unveils the nature of dark energy Figure 1 page 9.
The nature of the clock is not in that paper.

EDIT add
In this 2008 pdf (by Poveda et Lara, Revista Mexicana de Astronoma y Astrofsica, I dont have time to find the abstract) it is approximated by $0.1912 * exp (0.5594 * n)$ that is similar to the one I obtained.
other links:
WP Titius-Bode_law
WP Dermott's_Law

where T(n) is the orbital period of the nth satellite, T(0) is of the order 0.46 days and C is a constant of the planetary system. Specific values are:
Jovian system: T(0) = 0.444 d, C = 2.03
Saturnian system: T(0) = 0.462 d, C = 1.59
Uranian system: T(0) = 0.488 d, C = 2.24
Such power-laws may be a consequence of collapsing-cloud models of planetary and satellite systems possessing various symmetries; see Titius-Bode Law. They may also reflect the effect of resonance-driven commensurabilities in the various systems.

Note: the package Eureqa is very good, free to use in Linux and M$:
If you have data then let Eureqa find the equations and, maybe, you find models.