Revised dynamics or dark matter in galactic and extra galactic scales?
Y. Sobouti
Allowing the energy of a gravitational field to serve partially as its own source allows gravitating bodies to exhibit stronger fields, as if they were more massive. Depending on degree of compaction of the body, the field could be one to five times larger than the newtonian field. This is a comfortable range of increase in field strength and may prove to be of convenience in the study of velocity curves of spirals, of velocity dispersions in clusters of galaxies and in interpreting the Tully-Fisher or Faber-Jackson relations in galaxies or systems of galaxies. The revised gravitation admits of superposition principle but only approximately in systems whose components are widely separated. The revised dynamics admits of the equivalence principle in that, the effective force acting on a test particle is derived from a potential, and could be eliminated in a freely falling frame of reference. \keywords{ --Dark matter-- Missing mass-- Alternative dynamics-- Spiral galaxies }
revised_dynamics_forarxive.tex
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%   \thesaurus{06     } A\&A Section 6: dark matter, missing mass


   \title{Revised dynamics or dark matter in galactic and extra galactic scales? }


   \author{Y. Sobouti}

          %\fnmsep\thanks{Just to show the usage
          %of the elements in the author field}
      \offprints{Y. Sobouti}
      \institute{Institute for Advanced Studies in Basic Sciences, P. O. B. 45195-1159, Zanjan,
      Iran\\
      email:  sobouti@iasbs.ac.ir}



    \date{ }
\titlerunning{Revised dynamic or dark matter in galactic and extra galactic ...}

\maketitle



   \begin{abstract}
   Allowing the energy of a gravitational field to serve partially
   as its own source allows gravitating bodies to exhibit stronger
   fields, as if they were more massive. Depending on degree of
   compaction of the body, the field could be one to five times
   larger than the newtonian field. This is a comfortable range of
   increase in field strength and may prove to be of convenience
   in the study of velocity curves of spirals, of velocity
   dispersions in clusters of galaxies and in interpreting the
   Tully-Fisher or Faber-Jackson relations in galaxies or systems
   of galaxies. The revised gravitation admits of superposition
   principle but only approximately in systems whose components
   are widely separated. The revised dynamics admits of the
   equivalence principle in that, the effective force acting on
   a test particle is derived from a potential, and could be
   eliminated in a freely falling frame of reference.
\keywords{ --Dark matter-- Missing mass-- Alternative dynamics--
Spiral galaxies }
   \end{abstract}


%
%_
\begin{quote}
{\small The only justification for our concepts and system of
concepts is that they serve to represent the complex of our
experiences. Beyond this they have no legitimacy. I am convinced
that the philosophers have had a harmful effect upon the progress
of scientific thinking in removing certain fundamental concepts
from the domain of empiricism, where they are under our control,
to the intangible heights of the a priori. Albert Einstein, 1922.}
\end{quote}
% --------------------------------------------------------------------------------------------------
\section{Introduction}

{\it The problem of dark matter surrounding spiral galaxies},
wrote Van Albada et al. (1985), {\it is one of the most enigmatic
questions in present day astrophysics. A number of years of
intensive research have brought little or no clarification.}

Earlier suggestions to resolve mass discrepancy were on the
conservative side. Bahcall and Cassertano (1985) spoke of {\it
missing mass} as synonymous to {\it missing light}. Tremaine and
Lee (1987)with {\it dark matter} meant the matter whose existence
was inferred  only through its gravitational effects. Critics were
vocal. Outstanding among them Milgrom (1983 a,b,c), the architect
of {\it MOND}, spoke of the {\it dark side of the dark matter
hypothesis} and {\it of its arbitrariness in that one invokes the
dark matter in the correct amount and spatial distribution needed
to explain the mass discrepancy in each and every case for itself}
(1987).


In the course of the past two decades acurate observations of
velocity curves of spiral galaxies have become available.See, e.
g., Begeman et al. (1991), Sanders (1996), Sanders and Verheijen
(1998), Mc Gauph and de Blok (1998). Credible velocity dispersions
of galactic systems have emerged. See, for example, Tullly et al.
(1996) for data on Ursa Majoris cluster. On the theoretical side
many ingenious and bold candidates for dark matter have been
offered. A list compiled by Ostriker and Steinhardt (2003)
includes items such as cold collisionless dark matter, strongly
self interacting dark matter, warm dark matter, repulsive dark
matter, fuzzy dark matter, self annihilating dark matter, decaying
dark matter, etc.

In spite of all these developments, however, there is no clue as
how to detect the dark matter and get to the physics that it
obeys. The issue does not seem less enigmatic than what von Albada
and his colleagues described in 1985. Arguments pro and con are
the same as a quarter of century ago. Proponents of dark matter
assume the validity of Newtonian dynamics at all distance scales
and look for one or another form of hypothetical matter to provide
the missing gravity. Skeptics, on the other hand, see no logic in
resorting to a concept that solves the riddle of dynamics but
remains unamenable to further validation by any other known
physical means.

A historical reminder might be timely. In the closing decade of
the 19th. century, physicists had agreed that light was an
electromagnetic wave. And out of the experience with waves in
other contexts had required a medium of propagation for it, the
ether. Yet ether evaded all attempt of detection no matter how
ingeniously the detection devices were designed. On the other hand
the notion of medium of propagation seemed so obvious to everyone,
that some of the brightest minds of the time invoked theories that
would have kept the ether, but in a concealed form. The famous
transformations of Lorentz were developed primarily to explain the
null results of the experiments of Fiseau, Michelson, Michelson \&
Morely, etc., rather than as a foundation for the special
relativity that emerged later. It was up to Einstein to think and
later to speak out that {\it our concepts have no legitimacy
beyond what the complex of our experiences bestows upon them}. If
experiments do not reveal the ether it could be dispensed with. If
experiments show that the speed of light is the same in all
reference frames, that could be a fact to build the new physics
upon it.

In these early years of the 21st. Century, it would not be unwise
to take Einstein's advice seriously. Accept the validity of
newtonian dynamics in solar and similar systems, where it has been
tested, but look for alternatives in larger galactic and
extragalactic scales, where it has not been verified. Many
researchers have actually adopted such a point of view before. See
Sanders and Mc Gauph (2002) for a review of Milgrom's {\it MOND}
and earlier attempts. Here, we adopt a variational approach to the
problem. We amend the classical action by adding a term to it that
eventually makes provision for the field energy to serve partially
as its own source. The procedure is apt to a general relativistic
generalization, in preparation for a future presentation.



%------------------------------------------------------------------------------------------------

\section{Variational formulation}

The system to be considered is a collection of point masses $m_i$
at positions ${\bf x}_i$, $i=1,\cdots,N$; plus the field
$\phi({\bf x})$ at point ${\bf x}$. Both newtonian equation of
motion and equation of gravitational field are derivable from the
following action integral.
\begin{eqnarray}
\al\al I=\int L[\textbf{x}_i,\phi(\textbf{x})]~ dt,\\
\al\al L[\textbf{x}_i,\phi(\textbf{x})]=\sum
m_i\left[\frac{1}{2}\dot{\bf x}_i^2+\phi\left({\bf
x}_i\right)\right]-\frac{1}{8\pi G}
\int\left|\nabla\phi({\textbf{x}}\right|^2d^3x.\nonumber
\end{eqnarray}
At point ${\bf x}$ the gravitational energy density is $(8\pi
G)^{-1}|\nabla\phi(\x)|^2$. By Einstein, this energy has an
equivalent mass. We propose to entertain the possibility of using
this mass or a fraction $\alpha$ of it as the source of extra
gravitation on the particles. The effect on $m_i$ at position
${\bf x}_i$ will be $\alpha Gm_i(8\pi G
c^2)^{-1}|\nabla\phi(\x)|^2|\x-\x_i|^{-1}$. The contribution to
the action integral from all space and all particles will be
\begin{eqnarray}
\al\al I_{int}=\int L_{int}[\textbf{x}_i,\phi(\textbf{x})]~ dt,\nonumber\\
\al\al L_{int}[\x_i,
\phi(\textbf{x})]=\frac{1}{8\pi}\frac{\alpha}{c^2}\sum_i m_i
\int\frac{|\nabla\phi({\bf x}|^2}{|{\bf \textbf{x}}-{\bf
\textbf{x}}_i|}d^3 x.
\end{eqnarray}
The dimensionless constant $\alpha$ is of the order of unity. It
is introduced for possible later adjustments. Note that $Gm_i/c^2$
is of the order of the Schwarschild radius of $m_i$.

Before proceeding further, let us emphasize that the argument
given in composing $I_{int}$ is for mnemonic purposes only. The
formal statement of the assumption is the following: The classical
action integral from which Newton's laws are inferred is
inadequate in galactic and extragalactic scales. We propose to
amend it by adding $I_{int}$ to the classical action. The addition
is a scalar, quadratic in field gradients, and contains the
particle coordinates and masses. It will alter both the field
equation and the equations of motion.

The field equation for $\phi(\x)$ is obtained by requiring the
functional derivative of the total action, $I+I_{int}$, with
respect to $\phi(\x)$to vanish. Thus,
\begin{eqnarray}
\frac{\delta (I+I_{int})}{\delta\phi(\textbf{x})}&=& {\bf\nabla}.
\left[{\bf \nabla}\phi(\textbf{x})\left\{1-\frac{\alpha
G}{c^2}\sum
\frac{m_i}{|\textbf{x}-\textbf{x}_i|}\right\}\right]\nonumber\\&+&
4\pi G\sum m_i\delta(\textbf{x}-\textbf{x}_i)=0.
\end{eqnarray}
By writing
$\delta(\textbf{x}-\textbf{x}_i)=\frac{1}{4\pi}\nabla^2{|\textbf{x}-\textbf{x}_i|^{-1}}$
one immediately integrates Eq. (3) into
\begin{eqnarray}
\nabla\phi\left\{1-\frac{\alpha
G}{c^2}\sum_i\f{m_i}{|\textbf{x}-\textbf{x}_i|}\right\}=-G\nabla\sum
\frac{m_i}{|\textbf{x}-\textbf{x}_i|}.
\end{eqnarray}
Denoting the newtonian potential by
\begin{eqnarray}
\al\al
\phi_N(\textbf{x})=-G\sum\f{m_i}{|\textbf{x}-\textbf{x}_i|},
\end{eqnarray}
Eq.(4) integrates into
\begin{eqnarray}
\phi(\x)&=&\f{c^2}{\alpha}\ln\left[1+\alpha\f{\phi_N (\bf
x)}{c^2}\right]\nonumber\\&=&
\phi_N\left[1-\f{1}{2}\alpha\f{\phi_N}{c^2}+\f{1}{3}\alpha^2\f{\phi^2_N}{C^4}+\cdots\right].
\end{eqnarray}
Equation of motion for $m_i$ is obtained likewise,
\begin{eqnarray}
\al\al\frac{\delta (I+I_{int})}{\delta \textbf{x}_i}=\nonumber\\
\al\al\hspace{0.5cm} m_i\left[\ddot{
\textbf{x}}_i-\nabla_i\phi(\textbf{x}_i)-\f{1}{8\pi}\f{\alpha
}{c^2}\nabla_i\int\f{|\nabla\phi(\textbf{x})|^2}{|\textbf{x}-\textbf{x}_i|}d^3
x\right]=0.
\end{eqnarray}
Newtonian limits in all cases are obtained by letting $\alpha=0$.
To elucidate the significance of the proposed amendment to the
classical action and the way it relates to mass discrepancy,
several simple examples are worked out in section 3.

%------------------------------------------------------------------------------------------------

\section{Applications}
\subsection{Gravitational field of spheres }
Consider a sphere of uniform density $\rho$, radius $R$, and total
mass $M$: From Eqs. (5) and (6) the outside solution is
\begin{eqnarray}
\al\al \phi(r) =
\f{GM}{s}\ln\f{r-s}{r}=-\f{GM}{r}\left(1+\f{1}{2}\f{s}{r}+\cdots\right),
\hspace{0.3cm} r\geq R \cr \al\al \f{d\phi}{dr} = \f{GM}{r(r-s)},
 \hspace{.3cm}r\geq R,
\end{eqnarray}
where $s=\alpha GM/c^2$ is of the order of Schwarschild radius of
$M$. For the inside, noting that $\phi_N(r)=-\f{1}{2}\f{G
M}{R^3}\left(3R^2-r^2\right)$, one obtains
\begin{eqnarray}
\al\al
\phi(r)=\f{c^2}{\alpha}\ln\left[1-\f{3}{2}\f{s}{R}+\f{1}{2}\f{s}{R}\f{r^2}{R^2}\right],
\hspace{0.3cm}r\leq R,\cr \al \al
\f{d\phi}{dr}=\f{GM}{R^2}\f{r}{R}\left[1-\f{3}{2}\f{s}{R}+\f{1}{2}\f{s}{R}\f{r^2}{R^2}\right]^{-1},
\hspace{0.3cm} r\leq R.
\end{eqnarray}
\subsection{Motion of test satellites}
Equation of motion is obtained by substituting Eqs. (8) \& (9) in
Eq. (7) and reducing it. The result is
\begin{eqnarray}
    \al\al m\ddot\mathbf{r}=-\f{G m M}{r^2}\left[1+u\left(\f{s}{R}\right)+\f{s}{r-s}\right]\f{{\bf
r}}{r},
\end{eqnarray}
where $u(s/R)$ is derived, numerically calculated, and plotted in
the Appendix. As the central body contracts and $R$ reduces from
an initially large value to its allowed minimum $\f{3}{2}s;
u(s/R)$ increases from zero to 4. The first term on the right hand
side of Eq. (10) is recognized as the classical expression. The
second term, $u(s/R)$, is the pivotal one and as mentioned it
could be four times as large as the classical term. In newtonian
parlance this amounts to saying that a compact object may,
gravitationally, present itself up to five times more massive than
the same object in a diffuse state. This is what we propose as a
partial answer to mass discrepancy!\\ The third term in Eq. (10)
arises from deviations of the force law from $r^{-2}$. It is
infinitesimal compared with the previous two terms, though it may
observably perturb elliptical orbits in secular time scales.
\subsection{N-body problem}
Consider a collection of spheres each of mass $M_i$, radius $R_i$,
and positions $\x_i$. Assume the system members are well separated
such that $|\x_i-\x_j|=r_{ij}\gg R_i$ for all  $i \neq j$. It is
demonstrated in the Appendix, that on account of this assumption,
the integral appearing in Eq. (7) splits into a sum given by Eq.
(A. 7). Thus, one finds
\begin{eqnarray}
\al\al \left. \ddot \x_i-G\sum M_j^{\rm
eff}\f{(\x_i-\x_j)}{|\x_i-\x_j|^3}\right]=0.
\end{eqnarray}
where $M_j^{\rm eff}=M_j(1+u_j), u_j=u(s_j/R_j)$. Actually a term
of the order $G\sum_jM_js_j|r_{ij}-s_j|^{-1}$, is neglected in Eq
(11), a) because of the extra small factor $s_j$ and b) because of
its steeper, $r^{-3}$, decrease with distance compared with
$r^{-2}$ dependence of the retained terms.

Albeit the approximations, Eq. (11) is identical with the
newtonian many body equation of motion, except that instead of the
classical masses, the effective masses, $M_j^{\rm
eff}=M_j(1+u_j)$, play the role. Depending on the degree of
compactness of the constituent bodies, the effective masses could,
in the case of uniform densities, be 1 to 5 times larger than the
classical ones. Having noted this, it is easy to write the first
integrals of Eq. (11):
\begin{eqnarray}
\al\al {\rm Momentum}: {\bf P}=\sum M_i^{\rm eff}\dot\x_i= const.\\
\al\al {\rm Angular~ momentum}: {\bf L}=\sum M_i^{\rm
eff}\x_i\times\dot\x_i= const.\\
 \al \al {\rm Energy}: E=T+V= {\rm const.,}~~~\\
\al\al\hspace{1.37cm} T=\f{1}{2}\sum_iM_i^{\rm eff}\dot\x_i^2,~~~
V=-\f{1}{2}G\sum_{i,j}M_i^{\rm eff} M_j^{\rm eff}/r_{ij}\nonumber\\
 \al\al{\rm Virial~~ theorem}: 2\bar T+\bar V=0,
\end{eqnarray}
 where 'bars' on $T$ and $V$ indicate averages over system members
 or over time periods.\\

Increased effective mass in Eq. (11) and the virial theorem may
prove useful in explaining the missing mass or missing gravity
issues, in the velocity curves of galaxies and/or velocity
dispersions in clusters of galaxies.


%------------------------------------------------------------------------------------------------

\section{Concluding remarks}
\begin{quote}
{\small\it ... it is contrary to the mode of thinking in science
to conceive of some thing that can act itself, but which  cannot
be acted upon. Albert Einstein, 1922.}
\end{quote}

The quotation is from Einstein's argument to set the stage for the
rejection of absolute space-time continuum of special relativity
in the presence of matter, and his admiration of Mach's
reservation to ascribe an absolute meaning to {\it inertia}
irrespective of matter elsewhere. In an attempt to amend newtonian
dynamics, we have followed Einstein's advice and tried to alter
the laws of gravity and of motion simultaneously. The interaction
lagrangian of Eq. (2) contains both the particle coordinates,
$\x_i$, and the field gradient, $\nabla\phi$.


Noteworthy in $L_{int}$ is the field energy that eventually
appears as the increased effective mass, $M^{{\rm eff}}=M(1+u)$ in
Eqs. (10-15). We recall that the main issues in the analysis of
the velocity curves of spirals are: a) insufficiency of the
observed and/or estimated stellar masses to provide the required
dynamical effect, and b) the flatness of the velocity curves
beyond what the distribution of observed masses would permit.
Similarly, in clusters of galaxies, again the observed masses are
much smaller than the virial ones, inferred from the velocity
dispersion in the cluster. The increased effective mass discussed
above could provide a partial answer in both cases of luminous
mass deficiency. The flatness of the rotation curves is a matter
of the distribution of effective masses. For example, if there are
enough low luminosity collapsed faint objects in the outskirts of
spiral arms the velocity curves could be flattened enough.

The field in Eq. (6), unlike the Newtonian one, is not
proportional to the mass of the constituent members and does not
admit of superposition principle. Even in the case of approximate
superposition in systems with widely separated components, the
field of each component is a highly nonlinear function of its
mass. This nonlinearity could prove useful in discussions of
Tully-Fisher relation in spirals, an empirical relation between
the brightness of the galaxy and the asymptotic orbital speed at
the flat end of the velocity curve. The Tully-Fisher relation,
with the assumption of a further empirical mass to light ratio for
the galaxy, translates into a power law between the asymptotic
speed and the total mass, $v_{asymp} \propto M^\beta$, $2.5
<\beta<3.5$. In the present dynamics, $\beta$ will sensitivity
depend on the compaction factor, the fraction, and the
distribution of the compact members of the galaxy. Same
considerations and reservations holds for the Faber-Jackson
relation, another empirical relation between the velocity
dispersion in cluster of galaxies and the luminosity of the
cluster.

In section 3.3, conservation laws were derived for the case of
approximate superposition. One can, however, do better. The total
lagrangian, $L+L_{{\rm int}}$, is invariant under time
translations, coordinate translations, and coordinate rotations.
Therefore, seven first integrals of motion, corresponding to
energy, momenta, and angular momenta, should exist. This point of
view will be presented elsewhere.

Birkhof's theorem states that in newtonian and relativistic
regimes the gravitational field of a spherically symmetric system
is independent from the internal structure of the system. This is
not the case in the present theory as it is highlighted by the
presence of $u(s/R)$ in Eq. (10). This may open the possibility of
studying the history of contraction or collapse of an object by
logging the external gravitational field of the object as it
evolves.

From Eq.(7), the effective force on a test particle is derived
from a potential. It could be eliminated in a freely falling frame
of reference, meaning that the dynamics developed here admits of
equivalence principle. This is hardly surprising. For, in the
action integrals of Eqs. (1) and (2) one single mass $m_i$ is used
to compose the kinetic and the gravitational potential energies of
the system.


Finally a criticism: a) The mass equivalent of energy is a
relativistic concept. Yet, a gravitational effect was attributed
to it through the law of gravity of Newton. b) The effective mass
attains its full significance when the source object shrinks into
sizes of the order of Schwarzschild radius. Logically, a covariant
general relativistic approach should be adopted. Presently, we are
looking for a field equation and a geodesic one in which some sort
of mutual interactions between distant points, similar to one in
$L_{{\rm int}}$ are taken into account. As in Eq. (2), there will
be a free parameter $\alpha$ in the formulation. We will require
the emerging dynamics to reduce a) to the general relativistic one
in the limit $\alpha \rightarrow 0$; b) to the present dynamics in
the limit of weak fields but $\alpha \neq0$, and c) to the
newtonian one in the limit of zero $\alpha$ and weak field. This
will be presented elsewhere.
%----------------------------------------------------------------
%\clearpage [\vspace{.5cm}]
\begin{appendix}
\section{Reduction of $L_{int}$, Equation(2)
} We have already seen the formal role of the interaction
lagrangian, $L_{int}$ in altering the equations of motion and of
the field. Its numerical values are needed in the study of orbits,
Eq.(7); virial theorem, Eq. (15), etc. Here, we calculate it a)
for a single uniform spherical mass, and b) for a collection of
unform spheres.\\

 a)For a spherically symmetric field the non
newtonian integral appearing in Eq. (7) is
\begin{eqnarray}
l_{int}(r)=\f{1}{8\pi}\f{\alpha}{c^2}\int\f{|d\phi(r')/dr'|^2}{|{\bf
r}-{\bf r}'|}d^3r'.
\end{eqnarray}
In an expansion of $ [\bf r- \bf r' ]^{-1}$ in Legendre
polynomials, only the $ l=0$ term will have non vanishing
contribution; for $d\phi/dr'$ has no directional dependence. We
carry out angular integrations and split the radial range of the
integration into three intervals $0<r'<R$, $R<r'<r$ and
$r<r'<\infty$. Thus,
\begin{eqnarray}
l_{int}&=&\f{\alpha}{2c^2}\left[\f{1}{r}\int_0^R\left|\f{d\phi(r')}{dr'}\right|^2r'^2dr'\right.
+\f{1}{r}\int_R^r\left|\f{d\phi(r')}{dr'}\right|^2r'^2dr'
\nonumber\\&+& \left.\int
_r^\infty\left|\f{d\phi(r')}{dr'}\right|^2r'dr'\right].
\end{eqnarray}
The expression for the field gradient in the first integral is the
interior solution of Eq. (9) and in the other two are the outer
solution of Eq. (8). The remaining mathematical manipulations are
elementary, though lengthy. One eventually arrives at
\begin{eqnarray}
l_{int}=\f{GM}{r}u\left(\f{s}{R}\right), \hspace{.4cm}s=\f{\alpha
GM}{c^2},
\end{eqnarray}
where
\begin{eqnarray}
u(y)=-\f{1}{2}+\f{3}{y}\left[1-\sqrt{\frac{(2-3y)}{y}} \arctan
\sqrt{\frac{y}{(2-3y)}}~\right]
\end{eqnarray}
The parameter $s/R$ indicates the degree of compaction of the
central body. As the body contracts from an initially dispersed
state, $(s/R)\approx 0$, to its maximum allowable compact state,
$(s/R=2/3)$, $u(s/R)$ increases from zero to 4. A plot of $u(s/R)$
is given in figure below.
\begin{figure}
\special{psfile=fig1.eps vscale=50 hscale=50 hoffset=-30
voffset=-300 } \vspace{7cm} \caption[]{$u(s/R)$ versus $s/R$, the
degree of compactness of the central body. }
\end{figure}
In newtonian language, the effect on the motion of an orbiting
satellite is to make the central body look $(1+u)$ times more
massive than what it does in newtonian regime.
\\

b)Multi-component systems: In Eq. (A.1) The main contribution to
the integral comes from the immediate vicinity of the central
body. As one approaches the gravitating body from outside,
$\left|d\phi(r')/dr'\right|^2$ grows roughly as $r'^{-4}$ up to
the surface of the body and fades out to zero at its center.
Assume a system consisting of two well separated spheres $(M_1,
R_1)$ at $\x_1$ and $(M_2, R_2)$ at $\x_2$ with $|\x_1-\x_2|\gg
R_1 \& R_2$. From Eqs. (5) and (6), we read
\begin{eqnarray}
|\nabla\phi(\x')|^2&=&\left[1-\f{\alpha
G}{c^2}\left(\f{M_1}{|\x'-\x_1|}+\f{M_2}{|\x'-\x_2|}\right)\right]^{-2}\nonumber\\
&\times&
G^2\left[\f{M_1^2}{|\x'-\x_1|^4}+\f{M_2^2}{|\x'-\x_2|^4}\right.\nonumber\\
&&\hspace{0.7cm}+\left. 2M_1M_2\f{(\x'-\x_1).
(\x'-\x_2)}{|\x'-\x_1|^3|\x'-\x_2|^3}\right].
\end{eqnarray}
Again, the main contributions come from the vicinities of the two
spheres. In the immediate vicinity of sphere 1, however,
$G^2M_1^2|\x'-\x_1|^{-4}$ is large. While, with the assumption of
large separation of the two spheres, $G^2M_2^2|\x'-\x_2|^{-4}$ is
insignificant. Vice versa for the vicinity of sphere 2. The
expression containing $(\x'-\x_1).(\x'-\x_2)$ practically adds up
to zero upon integration over $\x'$, because of its directional
dependencies. We conclude that \vspace{6.5cm}
\\
\\
\\
\begin{eqnarray}
l_{int}(x)\approx \f{GM_1u_1}{|\x-\x_1|}+\f{GM_2u_2}{|\x-\x_2|},~~
u_i=u\left(\f{s_i}{R_i}\right).
\end{eqnarray}
Similarly, for a system of many, but well separated, components
one obtains
\begin{eqnarray}
l_{int}(x)\approx G\sum_i\f{M_i u_i}{|\x-\x_i|}, \hskip 2em
u_i(s_i/R_i).
\end{eqnarray}
\end{appendix}

%------------------------------------------------------------------------------------------------
%\clearpage

\begin{thebibliography}{}

\bibitem[1985]{von}  Albada, von, T. S., Bahcall, J. N., Begman, K.,
   and Sancisi, R.: 1985, Astrophys. J., {\bf 295}, 305

\bibitem[1985]{Bahcall} Bahcall, J. N.,  Cassertano, S.: 1985,
   Astrophys. J. Lett., 293, L7

\bibitem[1991]{Begeman}Begeman, K. G., Broeils, A. H., and Sanders, R. H.:
   1991, MNRAS, {\bf 249}, 523

\bibitem[1922]{Einstein}Einstein, A., 1922, The Meaning of
   relativity, Princeton Univ. Press,3rd. paperback,1972

\bibitem[1998]{Mc Gaugh}Mc Gaugh, S. S., and de Blok, W. J. G.:
   1998 a,b, Astrophys. J., {\bf 499}, 41 (a), 66 (b)

\bibitem[1983]{Milgrom}Milgrom, M.: 1983 a,b,c, Astrophys. J.,
   {\bf 270}, 365 (a), 375 (b), and 384 (c)


\bibitem[1987]{ Milgrom}Milgrom, M.: 1987, in {\it ark matter in
the universe}, p. 231, eds.: Bahcall,J. Piran,T. and Weinberg, S.,
Wold Scientific Publ. Co., Singapore

\bibitem[200?]{Ostriker}Ostriker, J. M. and Steinhardt, P.,
2003, arXiv:astro-ph/0306402

\bibitem[1996]{Sanders}Sanders, R. H.: 1996, Astrophys. J., {\bf
   473}, 117

\bibitem[1998]{Sanders}Sanders, R. H., and Verheijen, M. A. W.:
   1998, Astrophys. J., {\bf 503}, 97

\bibitem[2002]{Sanders}Sanders, R. H., and Mc Gaugh, S. S.:
   2002, arXiv:astro-ph/0204521 v1

\bibitem[1987]{Tremaine} Tremaine, S., Lee, H. M.: 1987, in {\it Dark matter in the
   universe},  p.,410, eds. Bahcall, J., Piran, T., and Weinberg,
   S., World Scientific Pub. Co., Singapore


\bibitem[1996]{Tully}Tully, R. B., Verheijen, M. A. W., Pierce,
   M. J., Huang, J. S., and Wainscoat, R.: 1996, Astron. J., {\bf
   112}, 2471








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astro-ph0505369.extracted_bib
\begin{thebibliography}{}

\bibitem[1985]{von}  Albada, von, T. S., Bahcall, J. N., Begman, K.,
   and Sancisi, R.: 1985, Astrophys. J., {\bf 295}, 305

\bibitem[1985]{Bahcall} Bahcall, J. N.,  Cassertano, S.: 1985,
   Astrophys. J. Lett., 293, L7

\bibitem[1991]{Begeman}Begeman, K. G., Broeils, A. H., and Sanders, R. H.:
   1991, MNRAS, {\bf 249}, 523

\bibitem[1922]{Einstein}Einstein, A., 1922, The Meaning of
   relativity, Princeton Univ. Press,3rd. paperback,1972

\bibitem[1998]{Mc Gaugh}Mc Gaugh, S. S., and de Blok, W. J. G.:
   1998 a,b, Astrophys. J., {\bf 499}, 41 (a), 66 (b)

\bibitem[1983]{Milgrom}Milgrom, M.: 1983 a,b,c, Astrophys. J.,
   {\bf 270}, 365 (a), 375 (b), and 384 (c)


\bibitem[1987]{ Milgrom}Milgrom, M.: 1987, in {\it ark matter in
the universe}, p. 231, eds.: Bahcall,J. Piran,T. and Weinberg, S.,
Wold Scientific Publ. Co., Singapore

\bibitem[200?]{Ostriker}Ostriker, J. M. and Steinhardt, P.,
2003, arXiv:astro-ph/0306402

\bibitem[1996]{Sanders}Sanders, R. H.: 1996, Astrophys. J., {\bf
   473}, 117

\bibitem[1998]{Sanders}Sanders, R. H., and Verheijen, M. A. W.:
   1998, Astrophys. J., {\bf 503}, 97

\bibitem[2002]{Sanders}Sanders, R. H., and Mc Gaugh, S. S.:
   2002, arXiv:astro-ph/0204521 v1

\bibitem[1987]{Tremaine} Tremaine, S., Lee, H. M.: 1987, in {\it Dark matter in the
   universe},  p.,410, eds. Bahcall, J., Piran, T., and Weinberg,
   S., World Scientific Pub. Co., Singapore


\bibitem[1996]{Tully}Tully, R. B., Verheijen, M. A. W., Pierce,
   M. J., Huang, J. S., and Wainscoat, R.: 1996, Astron. J., {\bf
   112}, 2471








\end{thebibliography}