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\author{
  %Heinz Dehnen and 
  Hartmut Frommert\\
  \rm E-Mail: \tt spider at seds.org\\[1ex]
  \em Dept.\ of Physics, University of Constance\\
  \em P.O.Box 55 60 M 678, D-78434 Konstanz, Germany
}

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\title{\bf
  Relativistic orbits of classical charged bodies in a spherically
  symmetric electrostatic field.
}
%\maketit

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%\title{\bf
%  Classification of
%  relativistic orbits of classical charged bodies in a spherically
%  symmetric electrostatic field.
%}
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%\title{\bf
%  A new look on the
%  relativistic orbits of classical charged bodies in a spherically
%  symmetric electrostatic field.
%}
\maketitle

%\clearpage

\thispagestyle{empty}
\section*{Abstract}
%\begin{abstract}
The orbits of a relativistic charged body in a static, spherically 
symmetric electrical field are calculated and classified in the
classical theory. Contrary to the non-relativistic problem, we find that
there is a limiting minimal value for the angular momentum, $L_c$.
Should the actual angular momentum of a charged test body be lower than
this limit, the test particle will spiral into the central point charge
instead of having (preceding) Keplerian orbits.
%\end{abstract}

\clearpage
\pagenumbering{arabic}

%\tableofcontents

\section*{Introduction}
%======================

Within Dirac's theory, or Sommerfeld's semiclassical theory 
%(Sommerfeld 1960) 
for the fine-structure of the hydrogene atom 
it is well known that the ground state will become unstable for
nuclear charges (of hydrogene-like ions) 
$Z>1/\alpha$, see e.g.\ Greiner 1981.
It is the aim of this paper to show, that this phenomenon is not
a typical quantum mechanical one, but that also in the classical 
theory of a relativistic charged point particle within a static 
Coulomb field there exist non-stable orbits connected with the 
existence of a critical angular momentum $L_c$; should the actual 
angular momentum of the point particle be lower than $L_c$, the
particle will spiral into the central point charge independently from 
its energy (unstable orbit), while
the non-relativistic treatment yields Keplerian orbits in any case 
(ellipses, parabola and hyperbola).
As in the quantum mechanical case mentioned above we neglect the 
radiative reaction force on the charged point particle.


\section{Lagrange function and the equation of motion}
%=====================================================
%         for the relativistic case}
         %==========================

Neglecting the radiative back reaction, the Lagrange function
% underlying the problem under consideration 
of a relativistic point particle (rest mass $m_0$, charge $e$) 
in an underlying electromagnetic potential $A_\mu$ is given by
\beq
\call =
- m_0 c^2 \sqrt{\eta_{\mu\nu} v^\mu v^\nu}
- {e\over c} v^\mu A_\mu\ ,
\label{eq:L}
\eeq
%such that the action is invariant under (arbitrary) re-parametrization
%$\lambda\longrightarrow\lambda'=\lambda'(\lambda)$; see e.g.\ Greiner
%and Rafelski 1984, p.~108.
($v^\mu$: timelike 4-velocity, $\eta_{\mu\nu}=diag(+1,-1,-1,-1)$).
In the following, the magnetic
potential $A_i, i=1\ldots3$ is assumed to vanish, and $A_0=c\Phi$ where
$\Phi$ is the electrostatic potential and a function of the radial
coordinate
%\footnote{
%  We use coordinate designations $t$, $r$, $\theta$, and $\varphi$ 
%here. },
$r$ only. 
%This can be expressed by the initial condition
%$$
%A_\mu = \delta_\mu^0\cdot c \Phi\ .
%$$
Then the Lagrange function (\ref{eq:L}) simplifies to 
\beq
\call = - m_0 c^2 \sqrt{1-{v^2\over c^2}} - e \Phi(r)\ .
\eeq
where $v$ is the absolute value of the 3-velocity.
Because of the spherical symmetry of $\Phi$ there exists angular 
momentum conservation, and consequently the motion of the particle
will take place in a plane orthogonal with respect to the angular 
momentum vector; we {\em choose\/} as this plane the $x$--$y$-plane.

Using plane polar coordinates $r$ and $\varphi$, $v^2$ simplifies to
\beq
v^2 = \dot r^2 + r^2 \dot \varphi^2\ ,
\label{eq:v}
\eeq
and we can derive the Euler-Lagrange equations for $r(t)$ and $\varphi(t)$.
In case of $\varphi$ we have, since
${\partial \call\over\partial\varphi} \equiv 0$,
a conserved angular momentum $L$, i.e.
\beq
L = {m_0 r^2 \dot\varphi \over \sqrt{1-{v^2\over c^2}}} = const\ .
\eeq
Inserting $v^2$ from (\ref{eq:v}) and resolving for $\dot\varphi$ we 
get: 
\beq
\dot\varphi = {c\over r}
  \sqrt{1 - \dot r^2/c^2 \over 1 + \left({m_0 cr\over L}\right)^2}
\label{eq:varphi}
\eeq
Since $\call$ is not explicitely dependent on $t$ we have, in addition,
energy conservation, i.e.\ Hamilton's function is a constant; this 
results in: 
\beq
E = {m_0 c^2\over \sqrt{1-{v^2\over c^2}}} + e \Phi = const
\eeq
Inserting (\ref{eq:v}) and resolving for $\dot r$ yields
\beq
\dot r = c \sqrt{1-\left({m_0 c^2\over E-e\Phi}\right)^2
                 - \left({r\dot\varphi\over c}\right)^2}\ .
\label{eq:r-dot}
\eeq
Eliminating $\dot\varphi$ by eq.\ (\ref{eq:varphi}) one obtains after 
a short calculation:
\beq
\dot r = c \sqrt{1 - \left({m_0 c^2\over E-e\Phi}\right)^2
                     \left(1+\left(L\over m_0 cr\right)^2\right)}\ .
\eeq
Herewith equation (\ref{eq:varphi}) takes the form: 
\beq
\dot\varphi = {L c^2\over\left(E-e\Phi\right) r^2}
\label{eq:phi-dot}
\eeq
By combination of (\ref{eq:r-dot}) and (\ref{eq:phi-dot}) we get the 
differential equation for calculating the orbit, i.e. $r(\varphi)$:
\beq
{dr\over d\varphi} = {\dot r\over\dot\varphi} =
r^2 {m_0 c\over L} \sqrt{\left({E-e\Phi\over m_0 c^2}\right)^2-1-
                      \left({L\over m_0 cr}\right)^2}
\eeq
Substituting $r = 1/s$, this may be rewritten as
\beq
{ds\over d\varphi} =
- {m_0 c\over L} \sqrt{\left({E-e\Phi\over m_0 c^2}\right)^2-1-
                      \left({L\over m_0 c}\right)^2 s^2}\ .
\label{eq:ds/dphi0}
\eeq
For the Coulomb potential $\Phi = Q/r = Qs$ we get finally:
\beq
{ds\over d\varphi} =
- {m_0 c\over L} \sqrt{\left[\left({E\over m_0 c^2}\right)^2-1\right]
  - 2 {E\over m_0 c^2} {eQ\over m_0 c^2} s
  - \left[\left({L\over m_0 c}\right)^2
          - \left({eQ\over m_0 c^2}\right)^2 \right] s^2}
\label{eq:ds/dphi}
\eeq

Looking now for a more suggestive form of the orbital differential equation,
one can take the square of this equation, differentiate with respect to
$\varphi$, and divide by $2 d\varphi/ds$.
Thus one obtains
\beq
{d^2 s\over d\varphi^2} =
  - \left[1 - \left({eQ\over cL}\right)^2 \right] 
    \left( s \pm s_0 \right) 
\label{eq:d2s/dt2}
\eeq
with
\beq
s_0 =
%\left| 
{ {E\over m_0 c^2} {|eQ|\over m_0 c^2} \left({m_0 c\over L}\right)^2
  \over 1 - \left({eQ\over cL}\right)^2 } 
%\right|
\label{eq:s_0}
\eeq
The upper sign in (\ref{eq:d2s/dt2}) applies if $e$ and $Q$ have same
sign and the lower sign otherwise.
Depending on the value of L, equation (\ref{eq:d2s/dt2}) has 3 different
types of solutions, namely
\begin{enumerate}
\item for $L>L_c$: periodic solutions (trigonometric functions) in 
      $\varphi$, 
\item for $L=L_c$: a limiting algebraic case,
\item and for $L<L_c$: nonperiodic (hyperbolic and exponential functions),
\end{enumerate}
where the ``critical'' angular momentum $L_c$ is given by
\beq
L_c = \left|{eQ\over c}\right|\ .
\eeq
For easier processing, it appears appropriate to substitute three new 
constants for $E$, $L$, and $Q$ (or $L_c$):
\beq
u := {E\over m_0 c^2}\ ,\ \ l := {L\over m_0 c}\ ,\ \
l_c := {L_c\over m_0 c} = {\left|eQ\right|\over m_0 c^2}\ ,
\eeq
where $u$ is a dimensionless ``specific energy'', while the ``specific
angular momentum'' $l$ and the ``specific critical angular momentum'' $l_c$
have the dimension of a length.
The length $l_c$ gives the distance at which the absolute value of 
the electric potential energy, $eQ/r$, of the test particle (i.e., 
between the charges $e$ and $Q$) gets equal to te rest energy $m_0 c^2$
of the particle. This demonstrates the special-relativistic nature of 
the effects where $l_c$ or $L_c$ plays a role.
Then eq.\ (\ref{eq:ds/dphi}) reads:
\beq
{ds\over d\varphi} = - {1\over l}
  \sqrt{\left(u^2-1\right) \mp 2 u l_c s - \left(l^2 - {l_c}^2\right) s^2}
\label{eq:sde}
\eeq
The constant $s_0$, eq.\ (\ref{eq:s_0}) takes the value
\beq
s_0 = %\left|
{u l_c\over l^2-{l_c}^2}
%\right|
\ .
\label{eq:s0}
\eeq

\section{Integration of the equation of motion}
%==============================================

%Now the solutions for all three cases of angular momentum ranges are
%calculated.

\subsection{Large angular momentum $L>L_c$: Keplerian orbits}
%------------------------------------------------------------

In case $l>l_c$ or $L>L_c=|eQ|/c$ we can
rewrite equation (\ref{eq:sde}) in the following form
using (\ref{eq:s0}):
\beq
{ds\over d\varphi} = - {1\over l}
\sqrt{\left(u^2-1\right)
      -\left(l^2-{l_c}^2\right)\left(s^2 \pm 2s_0 s\right)}
\eeq
After quadratic completion we obtain
%This may be rewritten as
\bea
{ds\over d\varphi} &=& - {1\over l}
\sqrt{\left[\left(u^2-1\right)+\left(l^2-{l_c}^2\right){s_0}^2\right]
      -\left(l^2-{l_c}^2\right)\left(s \pm s_0\right)^2}
\nonumber
\\
&=& - {1\over l}
\sqrt{\left[ {u^2 l^2\over l^2-{l_c}^2}-1 \right] \cdot
      \left[ 1 - { \left(l^2-{l_c}^2\right)^2\left(s \pm s_0\right)^2
                   \over \left(u^2-1\right) l^2 + {l_c}^2 } \right]}
\label{eq:20}
\eea
%Substituting
With respect to the radicand we can substitute
\beq
s = \mp s_0 + {\sqrt{\left(u^2-1\right) l^2 + {l_c}^2}\over l^2-{l_c}^2}
              \cos\alpha
\label{eq:21}
\eeq
which yields from (\ref{eq:20}):
\beq
{d\alpha\over d\varphi} = \sqrt{1-\left({l_c\over l}\right)^2}\ \
\Longrightarrow\ \
\alpha = \sqrt{1-\left({l_c\over l}\right)^2} \left(\varphi-\varphi_0\right)
\ .
\label{eq:22}
\eeq
%thus
Combining (\ref{eq:21}) and (\ref{eq:22}) we obtain for $r(\varphi)$:
%\beq
%s = s_0 \left\{\pm 1 + \epsilon
%         \cos\left[\sqrt{1-\left({l_c\over l}\right)^2}
%                   \left(\varphi-\varphi_0\right)\right]\right\}
%\eeq
%The explicit form for $r(\varphi)$ reads from this:
\beq
r(\varphi) = {r_0\over \mp 1 + \epsilon
         \cos\left[\sqrt{1-\left({l_c\over l}\right)^2}
                   \left(\varphi-\varphi_0\right)\right]}
\label{eq:r}
\eeq
where
\beq
\epsilon = {\sqrt{\left(u^2-1\right) l^2 + {l_c}^2}\over u l_c}
  = \sqrt{{1\over u^2} + \left({l\over l_c}\right)^2
          - \left(l\over u l_c\right)^2}
\eeq
is the numerical excentricity. The upper sign applies in the repulsive 
and the lower one in the attractive case.

The solution (\ref{eq:r}) represents a {\em preceding\/} Keplerian 
orbit, i.e.\ a preceding ellipse, parabola, or hyperbola,
depending on the value of $\epsilon<1$, $=1$, $>1$ respectively,
where in case of repulsive force $\epsilon>1$ must hold because of 
$r \geq 0$.
In case of bound states it is a periodic orbit
with period $2\pi/\sqrt{1-\left({l_c\over l}\right)^2} > 2\pi$; thus we 
have a progressive periapsis shift of
\beq
\delta\varphi = 2\pi
  \left( {1\over\sqrt{1-\left({l_c\over l}\right)^2}} - 1 \right)
\eeq
per cycle caused by the critical value of $l_c$.
%for the elliptical orbits.
For large values of $L$ or $l$, this expression can be approximated by
\beq
\delta\varphi
\approx \pi \left( {l_c \over l} \right)^2
= \pi \left( {|eQ| \over Lc} \right)^2
\eeq
In case of scattering states $\epsilon > 1$, we have preceding hyperbola,
which means that the asymptotes, calculated from (\ref{eq:r}), are given by
\beq
\varphi-\varphi_0 
% =: \varphi_\infty
= \pm {1\over\sqrt{1-\left({l_c\over l}\right)^2}}
  \arccos\left(\mp{1\over\epsilon}\right)
\eeq
This preceding hyperbola does not coincide with an exact hyperbola,
with another excentricity. Instead, its asymptotes, together with its
apsid line, precede progressively while the particle moves around the
center, at exactly the same rate per passed angle as for the ellipses 
in the bound above,
so that from approach to escape the trajectory has preceded by a total 
of \beq
\delta\varphi =
  \left( {1\over\sqrt{1-\left({l_c\over l}\right)^2}} - 1 \right)
  \cdot 2 \arccos\left(\mp{1\over\epsilon}\right)
\eeq
If the specific angular momentum $l$ comes close to the limit $l_c$,
the angle $\delta\varphi$ will become larger and larger, so that the 
particle may orbit the central charge one or more times, before it
escapes again to infinity (see Fig.~1). 
 
Such types of orbits are also known from 
general relativity, where both massive and massless test bodies with
small angular momentum have similar trajectories within Schwarzschild's
metrical gravitational field 
(see e.g.\ Laue 1965 and Misner, Thorne, Wheeler 1973). 
However, as stated above our results are a purely special
relativistiv effect, while in the Schwarzschild case the non-linear 
structure of the gravitational field has an additional impact.


\subsection{The limiting case $L=L_c$: Quadratic spiral trajectories}
%--------------------------------------------------------------------

For $L=L_c$, i.e.\ $l=l_c$ (corresponding to $s_0=\infty$)
eq.\ (\ref{eq:sde}) takes the form
\beq
{ds\over d\varphi} = \sqrt{{2u\over l_c} \left(s_1 \mp s\right)}
\eeq
with
\beq
s_1 = {u^2-1 \over 2 u l_c}
\label{eq:s1}
\eeq
(the upper sign is valid for repulsive forces, the lower one for the
attractive case).
The solution reads
\beq
s = {u\over 2 l_c} \left(\varphi-\varphi_0\right)^2 \pm s_1\ ;
\eeq
according to (\ref{eq:s1}), $s_1$ is positive, 0, or negative as
$u>1$, $=1$, or $<1$ respectively.
Then the trajectory is given by
\beq
r = \mp { r_0\over
    a \left(\varphi-\varphi_0\right)^2 \mp 1 }
\label{eq:32}
\eeq
where
\beqs
r_0 = {2ul_c\over|u^2-1|}
\ ,\ \
a = {u^2\over|u^2-1|}
\eeqs
Since $r$ must be positive, it is valid for
\begin{itemize}
\item $u<1$ and attractive forces: The trajectory has a maximal distance
  $r_0$ from the origin at $\varphi=\varphi_0$ and spirals quadratically
  into the origin as $|\varphi-\varphi_0| \longrightarrow \infty$.
\item $u>1$ and attractive forces: 
  \beq
  |\varphi-\varphi_0| > 1/\sqrt{a} = {\sqrt{u^2-1}\over u}\ .
  \eeq
  The trajectory comes from infinity at  
  $|\varphi-\varphi_0| = 1/\sqrt{a}$, 
  and spirals quadratically into the origin for
  $|\varphi-\varphi_0| \longrightarrow \infty$.
\item $u>1$ and repulsive forces:
  \beq
  |\varphi-\varphi_0| < 1/\sqrt{a} = {\sqrt{u^2-1}\over u}\ .
  \eeq 
  In this case, $r_0=r(\varphi=\varphi_0)$ represents the minimal
  distance from the central charge and
  for $|\varphi-\varphi_0| \longrightarrow 1/\sqrt{a}$ 
  the trajectory runs out to infinity without and spiralling.   
\end{itemize}
No solution exists in the repulsive case for $u<1$. 
For $u=1$ ($s_1=0$), we have only a solution in the attractive case,
namely 
\beq
s = (u/2l_c) (\varphi-\varphi_0)^2\ , 
\label{eq:e35}
\eeq
which comes from 
infinity at $\varphi=\varphi_0$, and spirals quadratically into the 
origin with $|\varphi-\varphi_0| \longrightarrow \infty$.


\subsection{Small angular momentum $L<L_c$: Exponential spiral orbits}
%---------------------------------------------------------------------

In the last case $L<L_c$, i.e.\ $l<l_c$, we can rewrite
eq.\ (\ref{eq:sde}) as
\beq
\left({ds\over d\varphi}\right)^2 l^2 =
\left[1+{u^2 l^2 \over {l_c}^2-l^2} \right]
\left\{ { \left( l_c^2 - l^2 \right)^2 \over
          \left( u^2 - 1 \right) l^2 + {l_c}^2
        } \left(s \mp s_0\right)^2 -1 \right\}
\label{eq:e48}
\eeq
($s_0$ according to (\ref{eq:s0}) and upper sign for repulsive, the 
lower one for attractive forces).
In the following, it is convenient to introduce the abbrevation
\beq
  b = \sqrt{\left({l_c\over l}\right)^2 - 1}
\eeq
which is a positive real constant.
We discuss the solutions of this equation of motion for the
{\em attractive\/} case ($eQ<0$) first.
Then we have to distinguish three cases, corresponding to the value of the
``specific energy" constant $u$:

%%%%%%%%         Case A

\noindent
{\bf a}. $u<1$, i.e.\ sum of kinetic and potential energy negative:

In this case the solution of (\ref{eq:e48}) reads (see Fig.~2)
\beq
r =
{   r_0
  \over
    1 + a \left\{
            \cosh\left[b\left(\varphi-\varphi_0\right)\right] - 1
          \right\} }
\label{eq:tr1}
\eeq
where
\bea
r_0 &=&
{   {l_c}^2-l^2
  \over
    \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} - u l_c } 
\ \ >\ \ 0
\\
a &=&
{   1
  \over \displaystyle
    1 - {u l_c \over \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} }}
\ \ >\ \ 1\ .
\eea
The trajectory described by equation
(\ref{eq:tr1}) has its greatest distance $r_0$ from the origin at
$\varphi=\varphi_0$, and spirals exponentially into the central charge 
for both 
$\varphi\longrightarrow\infty$ and $\varphi\longrightarrow-\infty$, as
\beq
r % \stackrel{\varphi\longrightarrow\pm\infty}{\longrightarrow}
  \longrightarrow
  {2 r_0\over a} e^{-b|\varphi-\varphi_0|}\ .
\eeq

%%%%%%%%         Case B

\noindent
{\bf b}. $u=1$, i.e.\ sum of kinetic and potential energy zero:

The solution of (\ref{eq:e48}) takes the form (Fig.~3)
\beq
r = {r_1\over\cosh\left[b\left(\varphi-\varphi_0\right)\right]-1}
\label{eq:tr2}
\eeq
with
\beq
r_1 = {{l_c}^2-l^2\over l_c} > 0
\eeq
The trajectory approaches infinity for $\varphi=\varphi_0$ and spirals 
exponentially into the origin for $\varphi\longrightarrow\infty$ as
\beq
r %\stackrel{\varphi\longrightarrow\pm\infty}{\longrightarrow}
  \longrightarrow
  2 r_1 e^{-b|\varphi-\varphi_0|}\ .
\eeq
%Thus $r_1$ here has the role of $r_0/a$ above.


%%%%%%%%         Case C

\noindent
{\bf c}. $u>1$, i.e.\ sum of kinetic and potential energy positive:

Equation (\ref{eq:e48}) has the solution (Fig.~4) 
\beq
r =
{   r_0
  \over
    a \left\{
        \cosh\left[b\left(\varphi-\varphi_0\right)\right] - 1
      \right\} - 1}
\label{eq:tr3}
\eeq
with
\bea
r_0 &=&
{   {l_c}^2-l^2
  \over
    u l_c - \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2}  }
\ \ > \ \ 0
\\
a &=&
{   1
  \over \displaystyle
    {u l_c \over \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} } - 1}
\ \ > \ \ 0
\eea
Because the distance $r$ must be always positive, the range of $\varphi$
is restricted so that the denominator in (\ref{eq:tr3}) remains
positive:
\beq
|\varphi-\varphi_0| > {1\over b} \arcosh\left(1+{1\over a}\right)
= \varphi_\infty
\eeq
The trajectory does not enter the $\varphi$ interval 
$\varphi_0-\varphi_\infty < \varphi < \varphi_0+\varphi_\infty$.
It comes from infinity at 
\beqs
\varphi=\varphi_0 \pm \varphi_\infty
\eeqs
and spirals exponentially into the origin for 
$|\varphi-\varphi_0| \longrightarrow \infty$ 
as
\beq
r % \stackrel{\varphi\longrightarrow\pm\infty}{\longrightarrow}
  \longrightarrow
  {2 r_0\over a} e^{-b|\varphi-\varphi_0|}\ ,
\eeq
i.e.\ exactly as in the first case.

In a second step we discuss the motion for the repulsive case. 
For this, the solution of (\ref{eq:e48}) reads
%Finally, we give the result for the repulsive case.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Because equation (\ref{eq:e48}) is quadratic in the derivative of
%$s$, we have then two types of solutions. The first type is the
%continuation of the above case 1 to values of $\alpha<1$:
%\beq
%r =
%{   r_0
%  \over
%    1 + a \left\{
%            \cosh\left[b\left(\varphi-\varphi_0\right)\right] - 1
%          \right\} }
%\label{eq:tr4}
%\eeq
%with
%\bea
%r_0 &=&
%{   {l_c}^2-l^2
%  \over
%    u l_c + \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} }
%\\
%a &=&
%{   1
%  \over \displaystyle
%    1 + {u l_c \over \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} }}
%\ \ <\ \ 1\ ,
%\eea
%which again describes a trajectory with maximum distance $r_0$ which
%spirals exponentially into the origin for large values of 
%$|\varphi-\varphi_0|$ as above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%The other type of solutions is actually repulsive: 
\beq
r =
{   r_0
  \over
    1 - a \left\{
            \cosh\left[b\left(\varphi-\varphi_0\right)\right] - 1
          \right\} }
\label{eq:tr5}
\eeq
with
\bea
r_0 &=&
{   {l_c}^2-l^2
  \over
    u l_c - \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} }
\\
a &=&
{   1
  \over \displaystyle
    {u l_c \over \sqrt{{l_c}^2 - \left(1 - u^2\right) l^2} } - 1 }\ .
\eea
Because of $r>0$ it follows that $u>1$ must hold.
The trajectory has a {\em minimal\/} distance $r_0=r(\varphi_0)$ and 
goes to infinity without any spiralling for 
\beqs
|\varphi-\varphi_0| \longrightarrow
\varphi_\infty = {1\over b} 
  \arcosh\left( 
     {u l_c\over\sqrt{ {l_c}^2 - \left(1 - u^2\right) l^2} } \right)\ ;
\eeqs
it never leaves the $\varphi$ interval 
$\varphi_0-\varphi_\infty < \varphi < \varphi_0+\varphi_\infty$.
It comes from the infinity at $\varphi=\varphi_0-\varphi_\infty$,
has its closest approach, $r=r_0$, at $\varphi=\varphi_0$, and leaves 
again to infinity at $\varphi=\varphi_0-\varphi_\infty$.


\section{Summary}
%================

The different classes of orbits discussed above for the special 
relativistic Coulomb problem are summarized in the following tables:
%\vspace{3ex}

%\noindent
%{\bf 
\subsection*{Attractive case:}
%}

\noindent
\begin{tabular}{|c|c|c|c|}
\hline %-----------------------------------------------------
        & $E < m_0 c^2$      & $E = m_0 c^2$    & $E > m_0 c^2$    \\
        & bound states       &                  & scattering states\\
\hline %-----------------------------------------------------
$L>L_c$ & preceding ellipse  & preceding parabola
                                                & preceding hyperbola \\
        & eq.\ (\ref{eq:r}), $0\le\epsilon<1$
                             & eq.\ (\ref{eq:r}), $\epsilon=1$
                                    & eq.\ (\ref{eq:r}), $\epsilon>1$
\\ \hline %--------------------------------------------------
$L=L_c$ & quadratical spiral 
                             & quadratic spiral 
                                               & quadratic spiral from\\ 
        & from maximal $r_0$    
                             & from infinity 
                                           & infinity into the center,\\
%        & $\displaystyle{r_0\over1+a\varphi^2}$ 
        & into the center, 
%                       & $\displaystyle{r_1\over\varphi^2}$
                             & into the center,  
%                               & $\displaystyle{r_0\over a\varphi^2-1}$
                                   & eq.\ (\ref{eq:32}), "+" before RHS,
\\ 
        & eq.\ (\ref{eq:32}), lower signs
                             & eq.\ (\ref{eq:e35})& "$-$" in denominator
\\ [2ex] \hline %-------------------------------------------------- 
$L<L_c$ & exponential spiral & exponential spiral & exponential spiral\\
        & from maximal $r_0$ & from infinity    & from infinity \\
        & into the center,   & into the center, & into the center,\\
        & eq.\ (\ref{eq:tr1}), Fig.~2
                             & eq.\ (\ref{eq:tr2}), Fig.~3
                                         & eq.\ (\ref{eq:tr3}), Fig.~4\\
\hline %-----------------------------------------------------
\end{tabular}

%\noindent
%{\bf
\subsection*{Repulsive case:}
%}

\noindent
\begin{tabular}{|c|c|}
\hline %-----------------------------------------------------
        & $E > m_0 c^2$       \\
        & scattering states
\\ \hline %-----------------------------------------------------
$L>L_c$ & preceding repulsive hyperbola  \\       
        & eq.\ (\ref{eq:r}), upper sign, $\epsilon>1$ 
\\ \hline %-----------------------------------------------------
$L=L_c$ & quadratic approach to and escape from \\
        & minimal $r_0$ to infinity \\
%        & $\displaystyle{r_0\over1-a\varphi^2}$
        & eq.\ (\ref{eq:32}), upper signs
\\ [2ex] \hline %-----------------------------------------------------
$L<L_c$ & exponential approach to and escape from \\
        & minimal $r_0$ to infinity \\
        & eq. (\ref{eq:tr5})
\\ \hline %-----------------------------------------------------
\end{tabular}
\vspace{2ex}


Obviously, in classical special-relativistic electrostatics, there 
exists a limiting angular momentum: If a charged body which is attracted
by the electric field has less angular momentum than this critical
value, it will spiral into the source, and this already without taking
into account the radiative energy losses.
%As can be seen from equation
%(\ref{eq:ds/dphi0}), this effect is in principle present not only for 
%the Coulomb problem but in any spherically symmetric field, such as the
%interior of a charge distribution. 
The physical meaning is that low angular momentum test charges get so
close to the central charge, i.e.\ into such a strong field, that they
are accelerated to relativistic velocities and can then no more escape.

Besides that this result is of interest on its own, one may look for 
applications, which can be expected in that part of physics where 
special relativity plays a role, while quantum effects stay weak.
%To some approximation, this applies for heavy ion collisions, which is 
%however still in the quantum theoretical domain. 
%A macroscopic experiment would be the obvious candidate for a test, but
%it is probably impossible to generate such a densely packed charge (and 
%keep it stable for a sufficient amount of time) that a test particle 
%will feel this effect, i.e.\ be accelerated to relativistic velocities
%by the electric field. 
However with respect to applications the radiative back reaction will be
important and must be taken into account. This will be done in a 
subsequent paper. Nevertheless we will give an estimation of the 
critical situation discussed above:
For electrons as test particles, it is necessary to concentrate a
charge of
$$
Q = l_c \cdot {m_0 c^2\over e} 
\approx 4.2 \cdot 10^{12} e \cdot (l_c/cm) 
$$
in a volume of a radius smaller than $l_c$, i.e.\ for a 1-cm radius,
more than $4\cdot10^{12}$ elementary charges had to be stabilized
and localized in this volume, generating a voltage at its surface which
corresponds to the electron's $m_0\>c^2/e$,
i.e.\ more than $5.11\cdot 10^{5}\ V$; ``classical" test bodies would
require even a much higher central charge. 
%and the voltage corresponding to their mass by the same relation
%$m_0\>c^2/e$, i.e.\ proportional to the mass.
%
%The Reissner-Nordstr\o{}m solution of General Relativity, although
%it appears similar to our problem at first glance, is quite different 
%because of the general relativistic, gravitational influence.
It may be difficult to realize such a densely packed charge. However it 
is the hope that the radiative reaction force will improve the 
experimental conditions.

% The help of S.~Rahoutis with the plots is gratefully acknowledged. 

%\bibliographystyle{hplain}
%\bibliography{\jobname}

\begin{thebibliography}{1}

%\bibitem{gr84}
%W.~Greiner and J.~Rafelski.
%\newblock {\em Spezielle Relativit\"atstheorie\/}.
%\newblock Theoretische Physik, Vol.~3A.
%\newblock Verlag Harri Deutsch, Thun, 1984.

\bibitem{gr81}                                      
W.~Greiner.                         
\newblock {\em Relativistische Quantenmechanik, Wellengleichungen\/}.  
\newblock Theoretische Physik, Vol.~6.
\newblock Verlag Harri Deutsch, Thun, 1981.         

\bibitem{laue65}
M.~v.~Laue.
\newblock {\em Die Relativit\"atstheorie\/}, 2.~Band:
  Die allgemeine Relativit\"atstheorie.
\newblock 5th edition.
\newblock F.~Vieweg \& Sohn, Braunschweig, 1965,
\newblock \S 45.

\bibitem{mtw73}
C.W.~Misner, K.S.~Thorne, J.A.~Wheeler.
\newblock {\em Gravitation\/}.
\newblock W.H.~Freeman, San Francisco, 1973,
\newblock chapter 25.6.

\bibitem{som31}
A.~Sommerfeld.
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\newblock F.~Vieweg \& Sohn, Braunschweig, 1960 (1st edition 1919).


\end{thebibliography}

%\clearpage
%\thispagestyle{empty}
%
%\section*{Figure Captions:}
%%--------------------------
%
%\begin{description}
%\item [Fig.~1]
%  The test particle may orbit the central charge one ({\bf a})
%  or more times ({\bf b}), before it escapes again to infinity, if the
%  angular momentum $L$ approaches $L_c$.
%\item [Fig.~2]
%  The orbits in case of small angular momentum.
%\end{description}

\clearpage
\thispagestyle{empty}

\section*{Figure Captions:}
%--------------------------

\begin{description}
\item [Fig.~1]
  The test particle may orbit the central charge one ({\bf a}) 
  or more times ({\bf b}), before it escapes again to infinity, if the 
  angular momentum $L$ approaches $L_c$.    
\item [Fig.~2]
  Orbits for $L<L_c$, $u<1$ (bound states)
\item [Fig.~3]
  Orbits for $L<L_c$, $u=1$ (limiting case)
\item [Fig.~4]
  Orbits for $L<L_c$, $u>1$ (scattering states)
\end{description}


\end{document}
