% format lplain % (c) 1994,1995 Hartmut Frommert. All rights reserved. %\documentstyle[12pt,titlepage,maketit]{article} \documentstyle[12pt,titlepage]{article} %\documentstyle[12pt,titlepage,epsf]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Second title page %\makeatother %\def\maketit{\begin{titlepage}% % \let\footnotesize\small % \let\footnoterule\relax % \setcounter{page}{0}% % \null % \vfil % \vskip 60\p@ % \begin{center}% % {\LARGE \@title \par}% % \vskip 3em % {\large % \lineskip .75em % \begin{tabular}[t]{c}\@author % \end{tabular}\par}% % \vskip 1.5em % {\large \@date \par}% %\end{center}\par %\@thanks %\vfil %\null %\end{titlepage}\relax} %\makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % HA4.STY <14-Jan-1991> \topmargin 0 pt % Nominal distance from top of paper to top of page \oddsidemargin 0.25 in % Note that \oddsidemargin = \evensidemargin \evensidemargin 0.25 in \marginparwidth 0.75 in \textwidth 5.875 in % Width of text line. \textheight 40\baselineskip \advance\textheight by \topskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % From gencmd1.tex: % Equation, ... %% Advantage: possible redefinition (e.g., for slides) % ------------- \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\beas}{\begin{eqnarray*}} \newcommand{\eeas}{\end{eqnarray*}} \newcommand{\beqs}{\begin{displaymath}} %% <-- HF 24 Apr 1992 \newcommand{\eeqs}{\end{displaymath}} %% <-- HF 24 Apr 1992 % Emty Box % -------- \newcommand{\me}{\mbox{}} % Calligraphic letters % -------------------- \newcommand{\calh}{{\cal H}} \newcommand{\call}{{\cal L}} % "\boldmath" in formula % ---------------------- \newcommand{\bm}[1]{{\mbox{\boldmath$#1$}}} % arcosh \newcommand{\arcosh}{\mbox{\rm arcosh}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\baselinestretch{1.6} %---------------------------------------------------------------------------- \begin{document} %---------------------------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{\bf Relativistic orbits of classical charged bodies in a spherically symmetric electrostatic field. } %\maketit %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\title{\bf % Classification of % relativistic orbits of classical charged bodies in a spherically % symmetric electrostatic field. %} %\maketit %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\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 $LL_c$: 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\ \ 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 %-------------------------------------------------- $L m_0 c^2$ \\ & 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 %----------------------------------------------------- $Lc^2/e$, 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. \newblock {\em Atombau und Spektrallinien\/}, I.~Band. \newblock 8th edition. \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 $L1$ (scattering states) \end{description} \end{document}