\documentclass[journal]{rmaa}


\newcommand{\casitiny}{\fontsize{6}{7}\selectfont}


\title{A variant of the Titius-Bode Law}


\author{
J. Daniel Flores-Guti\'errez\altaffilmark{1} 
and Carlos Garc\'ia-Guerra\altaffilmark{2}}

\altaffiltext{1}{Instituto de Astronom\'ia, Universidad Nacional Aut\'o\-noma de M\'exico, Mexico.}
\altaffiltext{2}{Facultad de Arquitectura, Universidad Nacional Aut\'o\-noma de M\'exico, Mexico.}


\fulladdresses{
\item J. Daniel Flores-Guti\'errez: Instituto de Astronom\'ia, Universidad Nacional Aut\'onoma de M\'exico,  Apdo. Postal 70-264, 
04510 M\'exico, D. F., Mexico (daniel@astro.unam.mx).
\item Carlos Garc\'ia-Guerra: Facultad de Arquitectura, Universidad Nacional Aut\'onoma de M\'exico, Mexico (carlosgg2000@hotmail.com).}


\shortauthor{Flores-Guti\'errez \& Garc\'ia-Guerra}

\shorttitle{A variant of the Titius-Bode Law}


\listofauthors{J.~D.~Flores-Guti\'errez \& C.~Garc\'ia-Guerra}

\indexauthor{Flores-Guti\'errez, J.~D.}
\indexauthor{Garc\'ia-Guerra, C.} 


\SetVolume{47} 
\SetFirstPage{173} 
\SetYear{2011}
\ReceivedDate{2010 June 28} 
\AcceptedDate{2011 February 15}


\abstract{The study of the geometric distribution of the planets invariably mentions the Titius Bode
law as an attempt to classify their heliocentric distances through a succession of integer numbers.
With the discovery of more planetary objects better support is given to the basic principle of the law
through their orbital parameters. We briefly refer to a classification of orbital parameters using
Kepler's proportionality parameter and discuss the problem of applying the TB Law to extrasolar systems. 
We also review work by various authors who have proposed functions which reproduce with some
degree of certainty the planetary heliocentric distances, and we compare them to our variant of the
Titius-Bode Law.\medskip}

\resumen{En el estudio de la distribuci\'on geom\'etrica de los planetas invariablemente se menciona
la ley de Titius Bode como un intento de clasificar \emph{grosso modo} sus distancias helioc\'etricas 
mediante una sucesi\'on de n\'umeros enteros. Con el descubrimiento de un gran n\'umero de objetos 
planetarios ahora se sustenta, en cierta medida, el principio b\'asico de dicha ley al trav\'es de la 
determinaci\'on de m\'as elementos orbitales. Abordamos brevemente una clasificaci\'on de los par\'ametros 
orbitales utilizando la constante de proporcionalidad de Kepler y discutimos el problema de la aplicaci\'on 
de la ley TB a sistemas extrasolares. Tambi\'en analizamos las funciones propuestas por diversos autores 
las cuales reproducen las distancias helioc\'entricas de los planetas, y efectuamos una comparaci\'on con 
la variante a la ley Titius Bode que proponemos aqu\'i.}


\addkeyword{planets and satellites: formation}
\addkeyword{planets and satellites: general}
\addkeyword{solar system: general}


\begin{document}


\maketitle


\section{INTRODUCTION}
\label{sec:intro}

In the late eighteenth century Johann Daniel Tietz proposed a numerical sequence with which he tried to 
reproduce the heliocentric distances of the planets, and which was included in the compendium \emph{Contemplation 
of the Nature} written by Charles Bonnet in 1764. In 1772 the famous German astronomer Johann Bode included 
Titius' ideas (\emph{Latin conversion of Tietz}) in his texts on astronomy. According to Titius' scheme, 
the mathematical formulation is written in terms of the parameter $N$ given by the integers 0, 3, 6, 12, 24, 
48, 96 and 192 according to the order of the planets Mercury, Venus, Earth, Mars, Jupiter, Saturn and Uranus.


%%%%% TABLA 1

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Empirical series reproducing the heliocentric distances of the planets}
\label{tab:tabla1} 
\setlength{\tabnotewidth}{0.92\textwidth}
\setlength{\tabcolsep}{1.2\tabcolsep}
\tablecols{14}
\begin{tabular}{lcccccccccccccc}
\toprule 
\multicolumn{14}{c}{MODIFIED TITIUS BODE LAW} \\
\cmidrule{1-14}
\multicolumn{2}{c}{This Paper} &  & \multicolumn{2}{c}{This paper} &  &  &  &  &  &  &  &  &  \\
\cmidrule{1-2}
\cmidrule{4-5}
\cmidrule{7-14}
\multicolumn{1}{c}{Body} & TB    &  & \multicolumn{2}{c}{TBmod}  &  &  &  & $a$   & $T$     & $a_p$  & $T_p$   & $\Delta a_n$  &  $\Delta T_n$  \\
         & $N$   &  & $X$  & $X'$ &  & $N'$ &  & AU    & Year    & AU     & Year    &   \%          &   \%  \\
\midrule
Mercury  & ~~0   &  & ~~0  & ~0   &  & ~~0  &  & ~0.4  & ~~0.253 & 0.387  & ~~0.251 &  ~~3.4        &  ~~0.8 \\
Venus    & ~~3   &  & ~~0  & ~3   &  & ~~3  &  & ~0.7  & ~~0.586 & 0.723  & ~~0.615 & $-$3.2        & $-$4.8 \\
Earth    & ~~6   &  & ~~3  & ~3   &  & ~~6  &  & ~1.0  & ~~1.000 & 1.000  & ~~1.000 &  ~~0.0        &  ~~0.0 \\
Mars     & ~12   &  & ~~6  & ~6   &  & ~12  &  & ~1.6  & ~~2.024 & 1.523  & ~~1.881 &  ~~5.0        &  ~~7.6 \\
Ceres    & ~24   &  & ~12  & 12   &  & ~24  &  & ~2.8  & ~~4.685 & 2.765  & ~~4.600 &  ~~1.2        &  ~~1.9 \\
Jupiter  & ~48   &  & ~24  & 24   &  & ~48  &  & ~5.2  & ~11.858 & 5.202  & ~11.862 & $-$0.1        & $-$0.0 \\
Saturn   & ~96   &  & ~48  & 48   &  & ~96  &  & 10.0  & ~31.623 & 9.510  & ~29.458 &  ~~5.2        &  ~~7.3 \\
Uranus   & 192   &  & ~96  & 96   &  & 192  &  & 19.6  & ~86.773 & 19.25  & ~84.013 &  ~~1.8        &  ~~3.3 \\
Neptune  &       &  & 192  & 96   &  & 288  &  & 29.2  & 157.788 & 30.19  & 164.749 & $-$3.3        & $-$4.2 \\
Pluto    &       &  & 288  & 96   &  & 384  &  & 38.8  & 241.684 & 39.50  & 247.492 & $-$1.8        & $-$2.3 \\
\bottomrule
\tabnotetext{}{TB is the original succession N defining the Titius-Bode law. TBmod is the modified 
TB law by means of the empirical succession $X$ and $X$' and $N'=X+X'$. In Columns~$a$ and $T$ we 
give the semi-major axes of orbits of the planets calculated by the TBmod model $(N'+4)/10$ and the 
period computed by the way of the Third Law of Kepler, respectively. In Columns~$a_p$ and  $T_p$ we 
present the astronomical values of the observed orbital parameters. In the last two columns we give 
the determination of the error between the computed heliocentric distances and the astronomical ones 
$\Delta a$ and $\Delta T$, where the values tending to zero indicate a good degree of certainty.}
\end{tabular}
\vspace{0.3cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


In terms of powers of two ($2^n$), the orbital parameters are determined by the equation $R_n=4+3 \cdot 2^n$ where $N=-\infty, 0, 1, 2, 3,...$, 
also in the order of the planets (Nieto 1970) or as proposed most recently $R_n=0.4+0.3 \cdot 2^{n-2}$ with $n=1, 2, 3,...$ (Holton \& 
Brush 2001). An interesting historical analysis of the progression was presented by Jaki (1972). Many discussions have arisen around 
the Titius-Bode model like those of Ovenden (1972, 1975) who analyzed the evolution of the circular and elliptical orbits and found that 
a Titius-Bode planetary distribution is a distribution of least interaction action between the planets, as well as how they evolve numerically 
into a resonant system from arbitrary initial conditions of position and velocity. Nieto (1970, 1975) made a detailed study of the 
Titius-Bode Law, and Horedt, Pop, \& Ruck (1977) described a numerical application of the generalized Titius law for more than nine 
planetary objects. Patton (1988) obtained the TB Law for Lagrange multipliers and applied it with acceptable results to eleven large asteroids. 
Through the approach of Monte Carlo, Lynch (2003) concluded that it is not possible to assure that the Titius-Bode Law explains the 
distribution of planets and satellites and Chang (2008) found that the distribution of the ratio of rotational periods in the 55 Cnc 
system is apparently inconsistent with that derived from the Titius-Bode Law. More recently Kotliarov (2009) has presented an interesting 
discussion of the Butusov model and proposed a generating function for the distribution of the planets.


\section{Modified Titius-Bode Law}
\label{sec:modified}

We propose a modification to the Titius-Bode Law (TBmod) by combining the sequences $X$ (0, 0, 3, 6, 12, 24, 48, 96, 192, 288, 
384, 576, 960, 1728) and $X'$ (0, 3, 3, 6, 12, 24, 48, 96, 96, 96, 192, 384, 768, 1536), which were derived empirically by 
Garc\'ia Guerra (2005). The sum of the sequences {$N'=X+X'$} reproduces one by one the terms of the succession of Titius-Bode 
(TB) from Mercury to Pluto, including the approach to the asteroid Ceres and other distant planetary objects. The semi-major axes 
of the planets are derived from the modified TB formulation {$a=(N'+4)/10$} which shows the curvilinear distribution of distances 
for each one of them (Figure~1). We presented the exponential function fit $r=e^{0.6122x-2.8176}$, $R^2=0.9990$ obtained from the 
interpolation ($r=A e^{x/t}$) and we compared it with the semi-major axes of major planetary bodies and other small objects, among 
them Ceto, Eris, Sedna and the distant object SQ372.


%%%%% FIGURA 1

\begin{figure}[!t]
\includegraphics[width=0.99\columnwidth]{dfloresfig1.eps}
\caption{Distribution of the semi-major axes of the planets and some small objects, given by the
original and modified Titius-Bode Law, TB and TBmod respectively. We also present two interpolations 
that summarize the combination with an exponential function ($r=A e^{x/t}$) and a seventh order 
polynomial (we only display the five first terms of the series), showing interpolated
fits from Neptune to the interior region of the Kuiper Band.} 
\label{fig:Figure 1}
\vspace{0.6cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In Table~1 we display the orbital parameters $a$ and $T$ calculated with the TBmod model, and the data for $a_p$ and $T_p$ 
(Astronomical Almanac 2010); in the columns $\Delta a=(a_n-a_p)/a_p$ (Kotliarov 2008) and $\Delta T=(T_n-T_p)/T_p$ we give 
the proportionality ratios between the calculated and observed values to measure the degree of certainty, which is considered 
acceptable if their values are close to zero. In our case these TBmod ratios have values of less than 5\% and 8\% respectively. One 
of our goals is to determine some kind of mathematical expression that would represent the general TBmod behavior, and to compare 
it with tendencies that show the different models deduced from the TB law summarized in Table~2.

Given the sequence of $N'$ integer numbers; we carry out a setup of numerical operations in order to construct a function $2^n$, 
which leads us to write this sequence like multiples of the common factor $3 \cdot 2^5$ whose first value $N'=0$ for $n=1$ (Mercury) 
we took into account in the discussion even though it does not exactly reproduce value the 0.387 of Mercury (Table~1). The 
resulting equations are:
\begin{eqnarray}
r_n = 0.4 + 9.6 \cdot 2^{n-7}\,,~~~~~~~~~~~~ n \leq ~8 \\
r_n = 29.2\,,~~~~~~~~~~~~~~~~~~~~~~~~~~  n = ~9 \\
~r_n = 29.2 + 9.6 \cdot \Sigma^{n}_{i=10}~2^{i-10}. \,\,\,\, n \geq 10 
\end{eqnarray}

If we arrange the elements $n=9$ and $n \geq 9$ we realize that they constitute the total sum of the geometric progression 
given by ($2^{n-9}-1$), so that eventually we obtain the modified TBmod model, governed by two generic intervals of arithmetic 
and geometric progressions, respectively
\begin{eqnarray}
r_n = ~0.4 + 9.6 \cdot 2^{n-7}\,, \qquad n \leq 8 \\
r_n = 19.6 + 9.6 \cdot 2^{n-9}\,, \qquad n \geq 9 
\end{eqnarray}


%%%%% FIGURA 2

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=8cm]{dfloresfig2.eps}
\caption{Titius-Bode models of several authors including the TBmod variant proposed in this work. 
It is to be noted that the models are distributed above or below the real TB distribution of 
planets and the TBmod curvilinear path, which coincides with the values of the observed semi-major 
axes. We have chosen the logarithmic scale to better visualize the linear trace of the other 
six models. The color figure can be viewed online.}
\label{fig:figure 2}
\vspace{0.37cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Among the distributions of planetary bodies shown in Table~1 we observed that they present tendencies in the internal region of 
the Solar System including the TBmod distribution (Figure~2), although some of them diverge for the outer planets and even more 
so for distant objects. Note that all the models have difficulty in reproducing the parameters of Mercury, so that historically 
it has been necessary to postulate special conditions for this planet; we shall see how to avoid this problem.


%%%%% TABLA 2

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{Some models that fit the\\ Titius-Bode Law} 
\label{tab:models}
\setlength{\tabnotewidth}{0.99\columnwidth}
\setlength{\tabcolsep}{0.78\tabcolsep}
\tablecols{2}
\small
\begin{tabular}{lr}
\toprule 
\multicolumn{1}{c}{Titius-Bode Model} & \multicolumn{1}{c}{Reference} \\
\midrule
$r_n=0.3+0.3\cdot 2^n$     & Wurm (1803)\tabnotemark{a}  \\
~~~~$n=-1,0,1,2,...,6$         &      \\
$r_n=0.2792 \cdot 1.53^n$   & Armellini (1921)\tabnotemark{a} \\
~~~~$n=1,2,3,...,11$           &  \\
$r_n=0.4+0.0075\cdot 2^n$  &  Nicolini (1957)\tabnotemark{a} \\
~~~~$n=-1,0,1,2,...,6$         &   \\
$r_n=0.2792\cdot 1.52^n$   & Munini \& Armellini (1978)\tabnotemark{a}\\\
~~~$n=1,2,3,...,11$          &   \\
$r_n=0.283\cdot 1.52^n$    & Badolati (1982) \\
~~~~$n=1,2,...$               &  \\
$r_n=0.4+0.3\cdot 2^{(n-2)}$ & Holton \& Brush (2001) \\
~~~~$n=-\infty,0,1,2,...,9$    &  \\ 
$r_n=e^{0.53074n-1.51937}$     & Ortiz et al. (2007)\\
~~~~$n=0,1,2,3,4,...$          &  \\ 
$r_n=e^{0.5894n-1.65026}$      & Poveda \& Lara (2008) \\
~~~~$n=-\infty,0,1,2,...,9$     &  \\
$r_n=e^{0.53707n-1.661}$       & Pankovic \& Radakovic (2009) \\
~~~~$n=-1,0,1,2,...,6$          &  \\
\bottomrule
\tabnotetext{a}{As cited by Badolati 1982.}
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The actual distribution of the planets is contained in the TBmod curvilinear path, which coincides with the values of observed 
semi-major axes. We have chosen the logarithmic scale to better visualize the linear trace of the other six models. The Armellini 
and Bodolati model does not reproduce the distribution of planetary objects beyond Jupiter. The Nicolini and Holton \& Brush 
models match the TBmod up to Uranus, as the Nicolini model is only for the planets. The Armellini and Bodolati models show 
values $a_n$ that are very close. The case of the models of Wurm (not shown in Figure~2) falls short of all values $\Delta a_n$. 
In Ortiz et al. (2007), Poveda \& Lara (2008), and Pankovic \& Radakovic (2009) we observe that their traces exhibit a certain 
parallelism and proximity between the distribution of planetary objects. Finally, these cases including TBmod are those that best 
approximate the actual distribution of planetary objects. Of course the TBmod proposal shows the best approximation to the above mentioned
distribution.


%%%%% FIGURA 3

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=8cm]{dfloresfig3.eps}
\caption{The degree of certainty of the TB models proposed by various authors in terms of the ratio 
$\Delta a=(a_n-a_p)/a_p$. The color figure can be viewed online.} 
\label{fig:Figura3}
\vspace{0.37cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 4

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=8cm]{dfloresfig4.eps}
\caption{Regions of accumulation in the Solar System of the small planetary bodies (taken from IAU Minor Planets 
Center: Observable Distant Minor Planets Orbital Elements). We observe that several of the small objects of the 
Solar System, like Eris, 2006HQ or Sedna, cannot be associated to the Titius-Bode Law. The color figure can be 
viewed online.} 
\label{fig:Figura4}
\vspace{0.6cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Degree of certainty of TBmod}
\label{sec:Grade certainty}

We realize that the modified TBmod model acceptably reproduces the orbital parameters $a_n$. We have examined the 
degree of certainty of the model through the differences between the calculated parameters ($a_n,~T_n$) and those 
observed ($a_p,~T_p$) by the ratio $\Delta a=(a_n-a_p)/a_p$, and similarly by calculating the ratio $\Delta T=(T_n-T_p)/T_p$, 
where the periods are the parameters obtained from Kepler's Third Law. If we apply this procedure to each of the 
models in Table~1 we will have an overview of the degree of certainty that the models possess as a whole.

The fit in Figure~3 shows the ratio between the values calculated for each model and the observed 
parameters. We can see that there are certain similarities up to the planet Saturn. However, there are 
divergences when we approach the outer regions of the Solar System. Note that the trace shows the 
smallest difference with the data for planetary objects due to the lower percentages of up to 10\%. In 
this figure it is easy to identify the problem arising in all the models when reproducing the parameters 
of the planet Mercury; therefore it is normally avoided with an \emph{ad hoc} number applied to this planet.

Another aspect that we want to emphasize is the fact that when general models are proposed what is
implicitly done is to change the pattern of the sequence of integers for that of real numbers given by
continuous functions $T=f(r)$. Thus, any Solar System object in principle will also fulfill this function
and also the TB Law.

Given the large number of solar bodies now known we noticed that they fall on the semi-major axis
distribution, although we know small objects like Eris, 2006HQ or Sedna that cannot be associated to
the integers succession of the Titius-Bode Law. In the case of Ceres, which has historically been associated 
with the fifth place in the TB Law, we must say that we now know of the existence of a large number 
of other, smaller, planetary bodies whose orbital parameters are closer to the fifth place in Titius Bode's law.


%%%%% FIGURA 5

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=8cm]{dfloresfig5.eps}
\caption{Classification of planets and satellites. It shows the ranking of the Solar System and 55 Cnc, 
and satellite systems of planets. It is easy to see that 55Cnc and the Solar System are coincident because 
the masses of the central body are similar, but they differ clearly for the satellites. The color figure 
can be viewed online.}
\label{fig:Figura5}
\vspace{1.2cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 3

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Orbital parameters of several planetary objects and the satellite systems of the planets}
\label{tab:TBmodprocess} 
\setlength{\tabnotewidth}{0.96\textwidth}
\tablecols{9}
\casitiny
\begin{tabular}{ccccccccc}
\toprule 
             &                            & Object & $a$ & $T$ & $m$  & $K$           & $a^3/T^2$        & ${a^3/T^2}/K$ \\
             &                            &        & AU   & yr & kg   & m$^3$~s$^{-2}$ & AU$^3$~yr$^{-2}$  &      \\
\midrule
Sun & $M=1.9891 \cdot 10^{30}$~kg &        &     &     &      &               &                  &  \\
  &  & Mercury    & ~~0.387  & ~~~0.251  & $3.302~ \cdot 10^{23}$ & $3.120` \cdot 10^{18}$ & 0.9201 & 0.9273~ \\
  &  & Venus      & ~~0.723  & ~~~0.615  & $4.869~ \cdot 10^{24}$ & $3.390~ \cdot 10^{18}$ & 1.0006 & 1.0084~ \\
  &  & Earth      & ~~1.000  & ~~~1.000  & $5.974~ \cdot 10^{24}$ & $3.390~ \cdot 10^{18}$ & 1.0000 & 1.0078~ \\
  &  & Mars       & ~~1.524  & ~~~1.881  & $6.419~ \cdot 10^{23}$ & $3.390~ \cdot 10^{18}$ & 1.0005 & 1.0083~ \\
  &  & Ceres      & ~~2.765  & ~~~4.599  & $9.500~ \cdot 10^{20}$ & $3.390~ \cdot 10^{18}$ & 1.0000 & 1.0078~ \\
  &  & Jupiter    & ~~5.203  & ~~11.862  & $1.899~ \cdot 10^{27}$ & $3.390~ \cdot 10^{18}$ & 1.0010 & 1.0079~ \\
  &  & Saturn     & ~~9.509  & ~~29.458  & $5.685~ \cdot 10^{26}$ & $3.360~ \cdot 10^{18}$ & 0.9909 & 0.9984~ \\
  &  & Uranus     & ~19.251  & ~~84.013  & $8.685~ \cdot 10^{25}$ & $3.430~ \cdot 10^{18}$ & 1.0109 & 1.0187~ \\
  &  & Neptune    & ~30.188  & ~164.749  & $1.024~ \cdot 10^{26}$ & $3.440~ \cdot 10^{18}$ & 1.0136 & 1.0215~ \\
  &  & Pluto      & ~39.499  & ~248.430  & $1.300~ \cdot 10^{22}$ & $3.380~ \cdot 10^{18}$ & 0.9985 & 1.0063~ \\
  &  & 2002 TC302 & ~55.096  & ~410.620  & $1.800~ \cdot 10^{21}$ & $3.360~ \cdot 10^{18}$ & 0.9919 & 0.9997~ \\
  &  & Eris       & ~67.668  & ~557.000  & $1.700~ \cdot 10^{22}$ & $3.380~ \cdot 10^{18}$ & 0.9987 & 1.0065~ \\
  &  & 2007 UK126 & ~72.814  & ~623.870  & $7.100~ \cdot 10^{20}$ & $3.360~ \cdot 10^{18}$ & 0.9919 & 0.9996~ \\
  &  & 2005 QU182 & 113.278  & 1210.530  & $1.200~ \cdot 10^{21}$ & $3.360~ \cdot 10^{18}$ & 0.9919 & 0.9997~ \\
  &  & Sedna      & 524.453  & 12059.06  & $7.200~ \cdot 10^{21}$ & $3.360~ \cdot 10^{18}$ & 0.9920 & 0.9997~ \\
Jupiter      & $M=1.8990 \cdot 10^{27}$~kg      &        &      &    &      &             &                  &  \\
  &  & Metis      & ~0.0009  & ~~0.0010  & $9.4900 \cdot 10^{16}$ & $3.2102 \cdot 10^{15}$ & $9.210~ \cdot 10^{-4}$ & 0.97240 \\
  &  & Adrasthea  & ~0.0009  & ~~0.0010  & $1.9000 \cdot 10^{16}$ & $3.2102 \cdot 10^{15}$ & $9.428` \cdot 10^{-4}$ & 0.99540 \\
  &  & Amalthea   & ~0.0012  & ~~0.0010  & $7.2200 \cdot 10^{18}$ & $3.2102 \cdot 10^{15}$ & $9.375~ \cdot 10^{-4}$ & 0.98980 \\
  &  & Tebe       & ~0.0015  & ~~0.0020  & $7.6000 \cdot 10^{17}$ & $3.2102 \cdot 10^{15}$ & $8.707~ \cdot 10^{-4}$ & 0.91920 \\
  &  & Io         & ~0.0028  & ~~0.0050  & $8.9400 \cdot 10^{22}$ & $3.2103 \cdot 10^{15}$ & $9.464~ \cdot 10^{-4}$ & 0.99910 \\
  &  & Europa     & ~0.0045  & ~~0.0100  & $4.8000 \cdot 10^{22}$ & $3.2102 \cdot 10^{15}$ & $9.465~ \cdot 10^{-4}$ & 0.99920 \\
  &  & Ganymmede  & ~0.0071  & ~~0.0200  & $1.4820 \cdot 10^{23}$ & $3.2104 \cdot 10^{15}$ & $9.470~ \cdot 10^{-4}$ & 0.99970 \\
  &  & Callisto   & ~0.0126  & ~~0.0460  & $1.0760 \cdot 10^{23}$ & $3.2103 \cdot 10^{15}$ & $9.470~ \cdot 10^{-4}$ & 0.99980 \\
  &  & Leda       & ~0.0740  & ~~0.6540  & $5.7000 \cdot 10^{15}$ & $3.2102 \cdot 10^{15}$ & $9.472~ \cdot 10^{-4}$ & 1.00000 \\
  &  & Himalia    & ~0.0765  & ~~0.6860  & $9.4900 \cdot 10^{18}$ & $3.2102 \cdot 10^{15}$ & $9.523~ \cdot 10^{-3}$ & 1.00530 \\
  &  & Lisisthea  & ~0.0781  & ~~0.7100  & $7.6000 \cdot 10^{16}$ & $3.2102 \cdot 10^{15}$ & $9.471~ \cdot 10^{-4}$ & 0.99990 \\
  &  & Elara      & ~0.0783  & ~~0.7110  & $7.6000 \cdot 10^{17}$ & $3.2102 \cdot 10^{15}$ & $9.476~ \cdot 10^{-4}$ & 1.00040 \\
  &  & Ananque    & ~0.1413  & ~~1.7280  & $3.8000 \cdot 10^{16}$ & $3.2102 \cdot 10^{15}$ & $9.459~ \cdot 10^{-4}$ & 0.99860 \\
  &  & Carme      & ~0.1507  & ~~1.8950  & $9.4900 \cdot 10^{16}$ & $3.2102 \cdot 10^{15}$ & $9.528~ \cdot 10^{-3}$ & 1.00590 \\
  &  & Pasiphae   & ~0.1567  & ~~2.0120  & $1.9000 \cdot 10^{17}$ & $3.2102 \cdot 10^{15}$ & $9.495~ \cdot 10^{-4}$ & 1.00250 \\
  &  & Sinophe    & ~0.1580  & ~~2.0750  & $7.6000 \cdot 10^{16}$ & $3.2102 \cdot 10^{15}$ & $9.158~ \cdot 10^{-4}$ & 0.96680 \\
Saturn       & $M=5.97 \cdot 10^{26}$~kg  &  &  &  &  &  &  &  \\
  &  & Mimas      & ~0.0012  & ~~0.0030  & $3.7500 \cdot 10^{19}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95350 \\
  &  & Enceladus  & ~0.0016  & ~~0.0040  & $1.0800 \cdot 10^{20}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95170 \\
  &  & Tethys     & ~0.0020  & ~~0.0050  & $1.0240 \cdot 10^{20}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95170 \\
  &  & Dione      & ~0.0025  & ~~0.0070  & $1.0950 \cdot 10^{21}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95170 \\
  &  & Rhea       & ~0.0035  & ~~0.0120  & $2.3060 \cdot 10^{21}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95150 \\
  &  & Titan      & ~0.0082  & ~~0.0440  & $1.3450 \cdot 10^{23}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95140 \\
  &  & Hiperion   & ~0.0099  & ~~0.0580  & $1.7100 \cdot 10^{19}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.94960 \\
  &  &  Iapetus   & ~0.0237  & ~~0.2170  & $1.8800 \cdot 10^{21}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.94990 \\
  &  & Febe       & ~0.0864  & ~~1.5070  & $3.9800 \cdot 10^{17}$ & $1.0100 \cdot 10^{15}$ & $3.00~~ \cdot 10^{-4}$ & 0.95090 \\
Uranus       & $M=8.69 \cdot 10^{25}$~kg  &  &  &  &  &  &  &  \\
  &  & Miranda    & ~0.0009  & ~~0.0040  & $6.5900 \cdot 10^{19}$ & $1.4700 \cdot 10^{14}$ & $4.2800 \cdot 10^{-5}$ & 0.99000 \\
  &  & Ariel      & ~0.0013  & ~~0.0070  & $1.3500 \cdot 10^{21}$ & $1.4700 \cdot 10^{14}$ & $4.3300 \cdot 10^{-5}$ & 1.00000 \\
  &  & Umbriel    & ~0.0018  & ~~0.0110  & $1.2000 \cdot 10^{21}$ & $1.4700 \cdot 10^{14}$ & $4.3300 \cdot 10^{-5}$ & 1.00000 \\
  &  & Titania    & ~0.0029  & ~~0.0240  & $3.5000 \cdot 10^{21}$ & $1.4700 \cdot 10^{14}$ & $4.3300 \cdot 10^{-5}$ & 1.00000 \\
  &  & Oberon     & ~0.0039  & ~~0.0370  & $3.0140 \cdot 10^{21}$ & $1.4700 \cdot 10^{14}$ & $4.3300 \cdot 10^{-5}$ & 1.00000 \\
 Neptune     & $M=1.03 \cdot 10^{26}$~kg  &  &  &  &  &  &  &   \\
  &  & Triton     & ~0.0024  & ~~0.0160  & $1.34~~ \cdot 10^{23}$ & $1.7400 \cdot 10^{14}$ & $5.04~~ \cdot 10^{-5}$ & 0.98000 \\
  &  & Nereida    & ~0.0368  & ~~0.9860  & $2.06~~ \cdot 10^{19}$ & $1.7400 \cdot 10^{14}$ & $5.10~~ \cdot 10^{-5}$ & 1.00000 \\
Earth        & $M=5.97 \cdot 10^{24}$~kg  &  &  &  &  &  &  \\
  &  & Luna       & ~0.0026  & ~~0.0750  & $7.3480 \cdot 10^{22}$ & $1.0200 \cdot 10^{13}$ & $3.0100 \cdot 10^{-6}$ & 1.00000 \\
Mars         & $M=6.42 \cdot 10^{23}$~kg  &  &  &  &  &  &  &   \\
  &  & Fobos      & ~0.0001  & ~~0.0010  & $9.6300 \cdot 10^{15}$ & $1.0900 \cdot 10^{12}$ & $3.2000 \cdot 10^{-7}$ & 1.00000 \\
  &  & Deimos     & ~0.0002  & ~~0.0040  & $1.9300 \cdot 10^{15}$ & $1.0900 \cdot 10^{12}$ & $3.2000 \cdot 10^{-7}$ & 1.01000 \\
Pluto        & $M=1.30 \cdot 10^{22}$~kg  &  &  &  &  &  &  &   \\
  &  & Charon     & ~0.0001  & ~~1.5200  & $0.0170 \cdot 10^{21}$ & $2.4500 \cdot 10^{10}$ & $5.2000 \cdot 10^{-9}$ & 0.72000 \\
\bottomrule
\noalign{\smallskip}
\tabnotetext{}{In the first, second, third, and fourth columns, the names of the bodies, the major 
semi-axes, their periods and their respective masses, are given. In the fifth and the sixth columns 
the values of the proportionality parameter of Kepler for each body, and the quotient are given, and 
finally, in the seventh column we give the quotient of these parameters of proportionality whose 
value is very near to one.}
\end{tabular}
\vspace{0.4cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


With the intention of identifying the regions of the major accumulation of small planetary objects, we
set ourselves the task of assembling the information (Figure~4) of IAU Minor Planets Center
(Observable Distant Minor Planets Orbital Elements). We compare their semi-major axis accumulation
points with the values corresponding to every integer $n$ of the TBmod succession. The result indicates
to us that only seven regions ($m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$, $m_{5}$, $m_{6}$ and $m_{7}$) can be associated 
with some integer 10, 11, 12 (2002TC30), 13 (Ceto), 14, 15 and 16 respectively, which makes us see the generic 
representation of the Titius Bode Law.


%%%%% TABLA 4

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{Orbital parameters of the exoplanetary system Gliese 876, G1581, \\$\upsilon$~And, 55 Cnc, HD160691, HD69830 and 61 Vir}
\label{tab:Exoprocess}
\setlength{\tabnotewidth}{0.9\textwidth}
\tablecols{9}
\scriptsize
\begin{tabular}{ccccccccc}
\toprule 
Object  & $M$  &  & $a$  & $T$  & $m$  & $K$           &  $a^3~T^{-2}$            & $a^3~T^{-2}/K $ \\
        & kg   &  & AU   & yr   & kg   & m$^3$~s$^{-2}$ & $\mathrm{AU^3~yr^{-2}}$  &      \\
\midrule
$\upsilon$ And & $2.525~ \cdot 10^{30}$~kg  &   &   &   &   &   &   &    \\
  &  & 1   & 0.059   & ~0.01  & $~1.3100 \cdot 10^{27}$  & $4.27 \cdot 10^{18}$ & 1.29 & 1.02\\
  &  & 2   & 0.822   & ~0.65  & $22.0036 \cdot 10^{27}$  & $4.31 \cdot 10^{18}$ & 1.31 & 1.03\\
  &  & 3   & 2.550   & ~3.57  & $19.5356 \cdot 10^{27}$  & $4.30 \cdot 10^{18}$ & 1.30 & 1.03\\
55Cnc          & $1.97~~ \cdot 10^{30}$~kg  &   &   &   &   &   &   &    \\
  &  & 1   & 0.038   & ~0.01  & $0.0456  \cdot 10^{27}$  & $3.36 \cdot 10^{18}$ & 0.92 & 0.93\\
  &  & 2   & 0.115   & ~0.04  & $1.5644  \cdot 10^{27}$  & $3.36 \cdot 10^{18}$ & 0.95 & 0.95\\
  &  & 3   & 0.240   & ~0.12  & $0.3209  \cdot 10^{27}$  & $3.36 \cdot 10^{18}$ & 0.94 & 0.95\\
  &  & 4   & 0.781   & ~0.71  & $0.2734  \cdot 10^{27}$  & $3.36 \cdot 10^{18}$ & 0.94 & 0.95\\
  &  & 5   & 5.770   & 14.29  & $7.2808  \cdot 10^{27}$  & $3.37 \cdot 10^{18}$ & 0.94 & 0.95\\
61 Vir         & $1.8890 \cdot 10^{30}$~kg  &   &   &   &   &   &   &     \\
  &  & 1   & 0.050   & ~0.01  & $0.0304  \cdot 10^{27}$  & $3.19 \cdot 10^{18}$ & 0.95 & 1.01\\
  &  & 2   & 0.218   & ~0.10  & $0.1088  \cdot 10^{27}$  & $3.19 \cdot 10^{18}$ & 0.95 & 1.01\\
  &  & 3   & 0.476   & ~0.34  & $0.1367  \cdot 10^{27}$  & $3.19 \cdot 10^{18}$ & 0.95 & 1.01\\
HD69830        & $1.71~~ \cdot 10^{30}$~kg  &   &   &   &   &   &   &     \\
  &  & 1   & 0.079   & ~0.02  & $0.0627  \cdot 10^{27}$  & $2.89 \cdot 10^{18}$ & 0.86 & 1.01\\
  &  & 2   & 0.186   & ~0.09  & $0.0721  \cdot 10^{27}$  & $2.89 \cdot 10^{18}$ & 0.86 & 1.01\\
  &  & 3   & 0.630   & ~0.54  & $0.1101  \cdot 10^{27}$  & $2.89 \cdot 10^{18}$ & 0.86 & 1.01\\
HD181433       & $1.55~~ \cdot 10^{30}$~kg  &   &   &   &   &   &   &     \\
  &  & 1   & 0.080   & ~0.03  & $0.0452  \cdot 10^{27}$  & $2.62 \cdot 10^{18}$ & 0.78 & 1.00\\
  &  & 2   & 1.760   & ~2.63  & $1.2150  \cdot 10^{27}$  & $2.62 \cdot 10^{18}$ & 0.79 & 1.01\\
  &  & 3   & 3.000   & ~5.95  & $1.0252  \cdot 10^{27}$  & $2.62 \cdot 10^{18}$ & 0.76 & 0.99\\
Gliese876       & $1.55~~ \cdot 10^{29}$~kg  &   &   &   &   &   &   &     \\
  &  & 1   & 0.021   & ~0.01  & $0.0399  \cdot 10^{27}$  & $1.12 \cdot 10^{18}$ & 0.32 & 0.97\\
  &  & 2   & 0.130   & ~0.08  & $1.3559  \cdot 10^{27}$  & $1.13 \cdot 10^{18}$ & 0.32 & 0.97\\
  &  & 3   & 0.208   & ~0.17  & $4.3202  \cdot 10^{27}$  & $1.13 \cdot 10^{18}$ & 0.32 & 0.97\\
  &  & 4   & 0.334   & ~0.34  & $0.0873  \cdot 10^{27}$  & $1.12 \cdot 10^{18}$ & 0.32 & 0.97\\
G1581          & $6.16~~ \cdot 10^{29}$~kg  &   &   &   &   &   &   &     \\      
  &  & 1   & 0.030   & ~0.01  & $0.0116  \cdot 10^{27}$  & $1.04 \cdot 10^{18}$ & 0.36 & 1.18\\
  &  & 2   & 0.041   & ~0.01  & $0.0934  \cdot 10^{27}$  & $1.04 \cdot 10^{18}$ & 0.32 & 1.04\\
  &  & 3   & 0.070   & ~0.04  & $0.0320  \cdot 10^{27}$  & $1.04 \cdot 10^{18}$ & 0.27 & 0.89\\
  &  & 4   & 0.220   & ~0.18  & $0.0424  \cdot 10^{27}$  & $1.04 \cdot 10^{18}$ & 0.32 & 1.04\\
\bottomrule
\noalign{\vspace{0.06cm}}
\tabnotetext{}{In the Columns~1 to 5, the names of the bodies, the mass of the star, the major semi-axes, 
their periods and the planet masses, are given. In Columns~6 and 7 the values of the proportionality parameter 
of Kepler for each body, and the quotient are given, and finally, in Column~8 we give the quotient of these 
parameters of proportionality, whose value is very close to one.}
\end{tabular}
\vspace{0.3cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Another aspect that we want to review is whether the distribution of planets in our Solar System
could be compared with that of extrasolar systems (often very different from ours), and so we
enquired whether the TB model could be applied to such planetary systems.


\section{Systems of satellites and exoplanets}
\label{sec:Exoplanets}

In the interpretation of the Titius-Bode Law a perennial concern has existed about its application to
satellite systems that orbit the planets of our Solar System and, more recently, to planetary systems
orbiting other stars such as $\upsilon$ And, 55 Cnc, 61 Vir, HD69830, HD181433, Gliese 876 and G1581 
(data from Schneider 2010\footnote{http://exoplanet.eu/catalog-all.php.}). Of course, we must recognize 
the importance of characterizing the orbital period of the planetary objects under study by applying 
Kepler's Third Law {$a^3$/$T^2=K$}, where $K=\pi^2G(M+m)$ is called the Proportionality Kepler Parameter 
(Tables~3 and 4); we must have a clear idea that a self-gravitating system is characterized by means 
of its central mass and that this acts like a main factor of classification. This way,  we observe in 
the fifth column that the values of $K$[m$^3$~s$^{-2}$] are different for every self-gravitating system. 
Certainly this behavior is also estimated in the sixth column; the values of the ratio $a^3/T^2=[\mathrm{AU^3~yr^{-2}}]$ 
are of the order of one for our self-gravitating Solar System and 55Cnc, since both satisfy Kepler's 
proportionality. Finally, we must say that if the dimensionless ratio $(a^2/T^3)/K$ has 
fluctuations close to one, then the parameters $T$, $a$, $M$ and $m$ are well-established.


%%%%% FIGURA 6

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=7.8cm]{dfloresfig6.eps}
\caption{Orbital parameters of the exoplanetary system. Observe that the systems whose central 
masses are similar to that of the Sun are {$\upsilon$ And}, 55 Cnc, 61 Vir, HD 69830 and HD 181433, 
and differ from those of Gliese 876 and Gliese 1581.} 
\label{fig:Figura6}
\vspace{0.3cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 7

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=7.8cm]{dfloresfig7.eps}
\caption{Distribution of exoplanets of 55Cnc and 61Vir. As the central mass of both systems is
similar to that of the Sun, we will assume that they satisfy the equations we have called TBF.
We include the members of the Solar System with the purpose of comparing the extrasolar objects
with those familiar to us. On the abscissa axis we give the heliocentric distances of the planetary 
objects according to the order of the integer succession TB law.} 
\label{fig:Figura7}
\vspace{0.3cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Thus, also the systems of satellites of the planets are characterized in a unique way, although the 
constants of proportionality they obey have very small values (Figure~5); the extrasolar
planets system possesses specific values of the above mentioned constant that differ or not from that
of our Solar System, so much as mass of the star differs from the mass of the sun (Figure~6).

Now assume that the planetary systems 61 Vir and 55 Cnc obey our TBmod model. To reproduce the 
values of their semi-major axes, it is necessary to take the negative part of the integer numbers. We 
chose the values $-2$, $-1$, 0, 1 therefore obtaining TBmod values 0.01, 0.1, 0.2 and 0.3 respectively, 
among which it is easy to find the values of the semi-major axes of the planets of 55 Cnc and 61 Vir 
(Figure~7). Note that the objects 55Cnc(f) and 55Cnc(d) correspond to the planets Venus and Jupiter, 
respectively, and 61Vir(d) to a place between Mercury and Venus. This leads us to reassess the general 
functions proposed here for the Solar System, since the two equations do not allow us to calculate 
semi-major axes of less than 0.4. In order to avoid this problem we note that with the equation~(4) we 
can determine these small values, and the remaining ones by means of the following equations (TBF): 
\begin{eqnarray}
r_n = 0.008 + 9.6 \cdot 2^{n-6}\,, \qquad -4 \leq n < 2 \\ 
r_n = 0.4~~~ + 9.6 \cdot 2^{n-7}\,,~~ \qquad  2 \leq n < 9 \\
r_n = 19.6~ + 9.6 \cdot 2^{n-9}\,.~~~~~~~~ \qquad  n \geq 9 
\end{eqnarray}


%%%%% FIGURA 8

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=9.0cm]{dfloresfig8.eps}
\caption{Distribution of the planetary objects in three regions defined by the intersections of the 
functions The intersection of the first two, assuming them as continuous functions, defines the point 
$(n1,~a1)=(2.5,~0.8~\mathrm{AU})$, and the intersection of the second and the third ones gives the 
points $(n2,~a2)=(8.4,~26.5~\mathrm{AU})$. Through these points the three regions of the Solar System 
are defined and it is possible to apply them to extrasolar systems with central masses similar to those 
of the Sun, 55 Cnc and 61 Vir. We include the accumulation points ($m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$, 
$m_{5}$, $m_{6}$ and $m_{7}$).The color figure can be viewed online.} 
\label{fig:Figura8}
\vspace{0.6cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Now we present the distribution of the objects that induces each of the equations~(6--8) in order to 
discern their tendencies and to display the intervals that are valid in each (the values of $\chi^2$ for 
each interval are 0.002, 0.241 and 0.379 respectively). It is obvious that there are three regions 
bound by the intersection of the equations~(6) and (7), and equations~(7) and (8) (Figure~8) 
and whose approximate values are found to be $(n_1,r_1)=(2.5,~0.8~\mathrm{AU}$) and $(n_2,~r_2)=(8.4,~26.5~\mathrm{AU})$, 
respectively. This suggests that these three intervals delimit the regions between the Sun and Venus, 
inside the region of the hypothetical vulcanoid objects that contains Mercury, between Major Solar 
Bodies but without Mercury, and finally from Neptune to the regions of the outer Solar System, into 
which we have included the accumulation points ($m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$, $m_{5}$, $m_{6}$ and $m_{7}$).

Taking equations~(6--8) we have determined the values of the semi-major axes and have fitted exponential 
functions (values of $R^2$ as 0.9999, 0.9971 and 0.9999 respectively):
\begin{eqnarray}
r = e^{0.63413x-1.81942}\,,~~~~~~ \qquad x < 2 \\ 
r = e^{0.67115x-2.38121}\,, \qquad 2 \leq x < 9 \\
r = e^{0.68879x-3.89173}\,.~~~~~~ \qquad x \geq 9 
\end{eqnarray}


%%%%% FIGURA 9

\begin{figure*}[!t]
\centering
\includegraphics[width=15cm,height=8cm]{dfloresfig9.eps}
\caption{Exoplanets with semi-major axes less 0.39~AU. The color figure can be viewed online.}
\label{fig:Figura9}
\vspace{0.5cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Since equation~(9) represents the vulcanoid objects we wish to compare it with the semi-major axes of the exoplanets 
(Schneider 2010)\footnote{http://exoplanet.eu/catalog-all.php.} known until now. We have found that some accumulation 
points correspond to certain elements of TBF (Figure~9). Nevertheless, it is not possible to represent the totality 
of them. Another interesting aspect is the concentration of vulcanoid objects around $n=-2$. Perhaps it is due to 
the existence of the protoplanetary wall near a central mass recently analyzed (Calvet, D'Alessio, \& Woolum 2005), 
whose radius of sublimation we identify with the red color line between 0.07 and 0.15~AU.

Perhaps the origin of these regions can be found in the early formative stages of the protoplanetary
disk dominated by turbulent forces that led to the formation of ring band regions (annular regions). 
With time, these regions evolved, and were confined into bands by the presolar material. The planets 
would have been formed as a result of the agglomeration of small particles into larger ones. A 
hydrodynamic vision has been included in the recent model of Adame (2010), who describes a turbulent 
protostellar disk (Figure~10) with central small masses. Here we can see certain criteria for the 
formation of annular regions; for example, it is possible to consider that the origin of the TBF 
distribution arises from a stationary wave system, according to the initial conditions, the mass of 
the central body being the main parameter.


%%%%% FIGURA 10

\begin{figure}[!t]
\centering
\includegraphics[width=0.99\columnwidth]{dfloresfig10.eps}
\caption{A simplified distribution of a protostellar turbulent disk model to give an idea of how 
annular regions can form in such a disk.} 
\label{fig:Figura10}
\vspace{0.4cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Final comments and conclusions}
\label{sec:conclusions}
 
We have studied the general structure of the Solar System through the Titius-Bode Law, which we have 
applied to the exoplanets system. It has lead to us to consider that the TBF distribution of planetary
objects (equations~(6--11) occurs in three great groups: Inner Solar System bodies, Major Solar System 
bodies and Outer Solar System bodies. For the case of Mercury ($n=1$) we improved the determination 
of the semi-major axis. The order of the major objects agrees well with the integer TBF sequence, 
including the accumulation region ($n=5$) represented by Ceres or the asteroid belt. In the outer Solar 
System, where objects with very eccentric orbits are located, we observed that the TBF integers are 
associated mainly to regions of accumulation of small agglomerations with values $n=15$ or 16, and to 
the abundant concentrations of small objects for $n=10$, 11 or 12.

If we consider solely those systems with central masses similar to the Sun, i.e., 55 Cnc, 61 Vir and HD69830, 
we can assign to certain objects values $n$ of TBF (vulcanoids and major objects) according to equations~(6--8) 
and Table~4. For example, 55Cnc(1) with $n=-2$, 61Vir(1) with $n=-2$, HD69830(1) with $n=-1$, and HD69830(2) 
with $n=0$.

With the set of equations (equations~6--11) we can see the general orbital behavior of Solar System
bodies, which appears to contain the final status of the presolar cloud processes from the stage of
protoplanetary disk to planet formation. Of course, any theory would have to include other phenomena, 
such as migration of planets (Masset 2008), or magnetohydrodynamic phenomena that may have occurred 
in our Solar System. In this context, when we think about the distribution of planets in the Solar 
System, we are actually considering the primitive physical conditions which dominated the presolar 
cloud which created regions of the protoplanetary ring from which the planets were created after a 
process of accretion of the particles that formed these protoplanetary rings.  

We have speculated that the TBF distribution arose from a stationary wave system in the presolar
cloud where it induced the formation of certain annular regions, simultaneously generating turbulent
accretion phenomena. On the other hand, we imagine that these conditions differ from other
protoplanetary environments where the distribution and size of such rings would be subject to
different chemical composition, shape, distribution and laws of accretion of the solid bodies.

\vspace{0.1cm}

\acknowledgments 

We thank the referee for useful comments which improved our presentation.

\vspace{0.3cm}

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\end{document}
