\section{Introduction}\label{sec:into}

The discharge of granular materials through an orifice at the bottom of a silo is a phenomenon profusely studied (see for example \cite{kadanoff1999,degennes1999,Jaeger1996,duran2000,tighe2007,beverloo,NeddermanBook}). This is in part due to the technological interest of the problem, but also because the significant challenges the scientific community has encountered in predicting basic quantities such as flow rate and wall pressure during discharge. The flow rate can be calculated by using the empirical Beverloo equation \cite{beverloo}. However, there is not a full theoretical derivation of this equation based on first principles. The main features of the Beverloo equation are explained by heuristic postulates such as the \emph{free fall arch} and the \emph{empty annulus} \cite{tighe2007,NeddermanBook}. Bottom pressure and wall pressure during discharge is also poorly understood. It is often assumed that the Janssen equation \cite{janssen,sperl2006} can predict the internal pressure under flowing conditions, despite a number of experiments demonstrating that this is not the case \cite{aguirre2010,wang2015,perge2012} and a few attempts to expand the Janssen approach to flowing silos \cite{walters1973,walters1973b,walker1966,walker1967}.

A remarkable feature of free granular discharges is the constant flow rate observed despite the drop in the height of the granular column. This is in contrast with viscous fluids, which show a decreasing flow rate as the level of the container falls. It is also surprising that the flow rate seems to be rather insensitive to the details of the grains properties (friction, elastic modulus, poison ration, etc.). However, this is somehow consistent with viscous fluids where the flow rate through an orifice on a plate does not depend on its viscosity \cite{Linford1961}.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.9\linewidth,trim= 0cm 1.2cm 0 1cm, clip]{fig1.eps}
	\caption{(a) Particle flow rate $Q$ as a function of the column height $h$ for various overweights $W$ (see legend). $Q$ is scaled by the Beverloo flow rate ($Q_{\rm B}$).  (b) Bottom pressure during the discharge. The vertical lines correspond to the different column heights $h_{\rm fill}$ along the discharge for which we have measured the various velocity profiles. Inset: Column height $h'$ at which the acceleration phase starts as a function of the overweight $W$ (note the log-lin scale).}
	\label{fig:q-p-vs-t}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[width=0.99\linewidth,trim=0 2cm 0 0,clip]{fig2.eps}
	\caption{Vertical velocity heat map, $v_z(r,z)$, of the silo at different stages of the discharge (different $h_{\rm fill}$) for a free discharge (left) and for a forced discharge with an overweight of $W =114 W_{\rm D}$ (right). Note that only the right half of the silo is shown. The labels $h_{\rm fill}$ on each figure indicate the point at which the heat map has been extracted during the discharge according to the vertical lines indicated in Fig. \ref{fig:q-p-vs-t}(a). The horizontal red lines indicate the position $z=z_{\rm view}$ at which profiles are measured in the rest of the paper.}
	\label{fig:vy-heatmap}
\end{figure}

Recently, it has been shown that the use of an overweight on top of the granular column of a discharging silo induces an increase of the flow at the final stages of the discharge \cite{madrid2018}. This is once again in contrast to viscous fluids since these decrease their flow rate during discharge from a tank even if a constant overpressure is applied to the free surface. More striking, however, is the fact that the increase in granular flow rate does depend on the grains material properties, which has no parallel effect in the case of viscous fluids. The increase in flow rate using an overweight is indeed an unexpected feature. Common wisdom among specialized scholars indicates that ``... it is pointless to try to extrude a granular material through a converging passage by the application of a stress.'' as stated in page 304 of Ref. \cite{NeddermanBook}. However, experimental observations are against this prediction \cite{madrid2018}.

{One may expect that the use of an overweight is equivalent to the addition of extra granular material on top of the original granular column. However, any additional granular material will contribute to the dissipation of energy during the discharge due to the nonconservative particle--particle and particle--wall interactions. A solid overweight operates differently since it lays a constant weight on the free surface of the granular column and does not dissipate energy except for the particle--overweight nonconservative contacts.}

{The basic mechanism to explain the increase in flow rate due to an overweight is based on energy balance \cite{madrid2018}. In a dense granular system, the collision rate is very high, which leads to \emph{inelastic collapse}. Therefore, all the power injected by gravity during the discharge of a silo is continuously dissipated. As a consequence, the discharge proceeds at constant velocity since the energy input cannot be converted into kinetic energy. Since the number of collisions per unit time is so large, all the energy is dissipated even if the material properties are varied. This renders the flow rate material independent. When a piston does work on the system at a high rate, the collisions are unable to fully dissipate the energy input, particularly when the number of particles in the column decreases. As a result, the flow rate increases due to the non-dissipated injected power that is converted into kinetic energy.}

{Due to the increase in flow rate, the mean velocity inside the silo increases in the final phase of the discharge of a forced silo. However, one may expect that the internal} {velocity profile is not simply upscaled but changed in a more complex way. This becomes an important piece of information when it comes to applications. Mass flow and funnel flow patterns are desirable/undesirable depending on the applications.} In this work, we focus on the velocity profiles inside a discharging silo and compare free discharges with forced discharges. We find a previously unnoticed behavior where free discharges present unsteady velocity profiles in some regions of the silo, while forced discharges present a more steady profile throughout the silo during the initial stages before the flow rate increases. During the phase of flow rate increase, the forced flows tend to flatter profiles in comparison with free discharges.

\begin{figure*}[t]
	\centering
	\includegraphics[width=0.8\linewidth,trim= 2cm 1.cm 2cm 1cm, clip]{fig3.eps}
	\caption{Scaled vertical velocity $v^*_z(r)=v_z(r)/v_z(z=0,r=0)$ at various heights $z_{\rm view}$ (as indicated in Fig. \ref{fig:vy-heatmap}) and for the different column heights $h_{\rm fill}$ during the discharge. Results for different overweights are compared in each plot: free discharge (blue), $W=4W_{\rm D}$ g (orange), $W=20W_{\rm D}$ (purple), $W=35W_{\rm D}$ (red) and $W=114W_{\rm D}$ (green). Each row corresponds to a different observation plane $z_{\rm view}$ in the silo, while each column corresponds to a different column height $h_{\rm fill}$ during the discharge. Inset: Sample of the velocity profile before scaling.}
	\label{fig:vy-vs-r}
\end{figure*}

\section{Simulations}\label{sec:NumSetup}

We use Discrete Element Method (DEM) simulations via the LIGGGHTS \cite{liggghts} implementation with a particle--particle Hertz interaction and Coulomb criterion using a Young modulus $Y=70$ MPa, Poison ratio $\nu=0.25$, restitution coefficient $e=0.95$ and friction coefficient $\mu = 0.5$ \cite{liggghts}. The same interaction applies for the particle--wall contacts.  Particles are spherical with diameter $d=1$ mm and density $\rho=2500$ kg/m$^3$. {The silo is formed by a cylinder of diameter $D_{\rm s}=2R_{\rm s}=30$ mm. The base of the silo is flat and has a circular orifice of diameter $D=15$ mm in its center.}  For the forced flow simulations, we introduce as an overweight a cylindrical piston with the desired weight $W$. Particles are poured to fill a height in the silo above $12 D_{\rm s}$ (which implies up to $3 \times 10^5$ grains). {The bulk volume fraction obtained is $\rho_{\rm b}=1500$ kg/m$^3$}. The orifice is initially blocked by a plug. After the grains come to rest in the silo (we wait until the kinetic energy per particle falls below $5 \times 10^{-12}$ J), we remove the plug and allow its discharge. The acceleration of gravity is $g=9.81$ m/s$^{2}$ and the time step $5\times10^{-6}$ s. {We measure the overweight in terms of the typical weight $W_{\rm D}$ of a column of grains of height $D_{\rm s}$, i.e., $W_{\rm D}=2 \pi R_{\rm s}^3 \rho_{\rm b}$.}

To assess the velocity and pressure profiles at different stages of the discharge we save snapshots with the particle positions, velocities and contact forces every $5\times10^{3}$ time steps (i.e., $40$ fps). We average properties over snapshots grouped by the number of grains that remain in the silo (in blocks of $1\times10^{4}$ particles). This allows us to compare discharges having different flow rates (free and forced discharges) by putting side by side data that correspond to the same average column height. A square grid of side $d$ over the $z-r$ plane is used to create profiles, which are averaged over the azimuth $\theta$ taking advantage of the cylindrical symmetry. The averaged velocity and pressure for each bin in the grid is calculated over the particles in the bin for each snapshot in the desired stage of the discharge. To focus on a particular position in the silo we take slices (usually horizontal slices) of the grid.

\section{Results}\label{sec:Results}

\subsection{Flow rate and bottom pressure}\label{sec:Results-0}

In Fig. \ref{fig:q-p-vs-t}(a), we show the particle flow rate as a function of column height for a free discharging silo along with forced flows using different overweights. As we can see, in the initial stages of a forced discharge, the flow rate is the same as the free flow rate. However, after this initial steady state, an accelerated flow is observed for all forced discharges. The heavier the overweight, the sooner the acceleration phase starts and the faster is the growth of the flow rate. This is consistent with the experimental results shown in Ref. \cite{madrid2018}. {The inset to Fig. \ref{fig:q-p-vs-t} shows that the column height $h'$, at which the accelerated flow starts, grows logarithmically with the overweight $W$. Currently, we can not provide a plausible explanation for this.} As we will see in the next section, despite observing the same flow rate during the initial phase of the discharge, the forced flows display some interesting subtle differences when compared against unforced discharges.

The pressure at the bottom of the silo during free and forced discharges is shown in Fig. \ref{fig:q-p-vs-t}(b). While the pressure drops quickly at the end of the discharge in an unforced flow, it rises for the forced flows. There seems to be an apparent direct relation between bottom pressure and the flow rate. However, this is not the case. The same particle flow rate can be observed for very different bottom pressures by simply using denser materials and/or wider silos \cite{madrid2018,staron2012}. Also, the bottom pressure increases linearly with gravity, yet the flow rate increases with $g^{0.5}$ \cite{dorbolo2013,arevalo2014}. A discussion on the relation between pressure, flow rate and dissipated power can be found in Ref. \cite{madrid2018}. In this paper we focus only on velocity profiles and leave the discussion on pressure profiles for a future work.

\subsection{Velocity profiles}\label{sec:Results-I}

\begin{figure}[t]
	\centering
	\includegraphics[width=0.9\linewidth]{fig4.eps}
	\caption{Vertical velocity $v_z(r)$ at various heights $z_{\rm view}$ (as indicated in Fig. \ref{fig:vy-heatmap}) for the free discharge (left column) and for the forced discharge with an overweight of $W=4 W_{\rm D}$ (right column). The different curves represent different column heights $h_{\rm fill}$ during the discharge (see legend).}
	\label{fig:vy-vt-t}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[width=0.9\linewidth]{fig5.eps}
	\caption{Radial velocity profiles $v_r(r)$  at various observatin planes $z_{\rm view}$ (as indicated in Fig. \ref{fig:vy-heatmap}) for the free discharge (left column) and for the forced discharge with an overweight of $W=20W_{\rm D}$ (right column). The different curves represent different column height $h_{\rm fill}$  during the discharge (see legend).}
	\label{fig:vr-vs-r}
\end{figure}

In Fig. \ref{fig:q-p-vs-t}, we have indicated four different column heights $h_{\rm fill}$ during the discharge (see vertical lines) where we have focused our attention. We compare velocity profiles for different discharges at these different points by ensuring that the granular column has a similar number of grains inside the silo. The corresponding vertical velocity heat maps, $v_z(r,z)$, are shown in Fig. \ref{fig:vy-heatmap} for a free discharge and for a forced discharge with the heaviest overweight studied.
For the free discharge, we observe the usual overall features described in the literature: (a) there exists a stagnant zone at the base surrounding the orifice ($v_z(r>D/2,z<\alpha r) \approx 0$, with $\alpha$ a constant), (b) there is a focused high velocity region at and above the orifice ($r<D/2, \, z<L\approx D/2$), (c) the free surface of the granular column develops a depression at the center, and (d) the velocity is higher at the central axis of the silo than at the walls ($v_z(r=0)>v_z(r=D_{\rm s}/2)$). Some of these features are hard to see in the scale used in Fig. \ref{fig:vy-heatmap}. The main contrasts shown by the forced discharge are: (a) the stagnant zone is shallower ($\alpha_{\rm forced} < \alpha_{\rm free}$), (b) the high velocity region above the orifice is lower in height ($L_{\rm forced} < L_{\rm free}$), and (c) the overweight forces a flat top surface of the granular column. Of course, we also see the overall increase of the velocities at the final stages where the flow increases as shown in Fig. \ref{fig:q-p-vs-t}(a).

To enhance the contrast between the forced and unforced velocity profiles, we have plotted in Fig. \ref{fig:vy-vs-r} horizontal sections of the profiles (i.e., $v_z(r)$) at different heights $z_{\rm view}=d/2$, $D/2$, $D_{\rm s}/2$ and $D_{\rm s}$ in the silo and for the four column heights $h_{\rm fill}$ considered during the discharge. We have indicated these sections as horizontal lines in Fig. \ref{fig:vy-heatmap}. Moreover, the velocities have been scaled at each point during the discharge (each column) by the velocity at the center of the orifice (notice that for forced flows this velocity increases during discharge).

If we focus on the initial stage of the discharge (left column in Fig. \ref{fig:vy-vs-r}), the scaled velocity profiles at all heights $z_{\rm view}$ and for all the overweights coincide. Actually, in this initial stage of the discharge even the unscaled profiles match. At $z_{\rm view}=D_{\rm s}$ (top row), $v_z(r)$ is rather flat, with a slightly lower velocity next to the walls. However, further down into the granular column the profile becomes steeper. Finally, next to the plane of the base ($z_{\rm view}=d/2$) $v_z$ drops to zero everywhere except at the orifice where the velocity displays a circular profile consistent with Refs. \cite{janda2012,rubio2015}.

Figure \ref{fig:vy-vs-r} shows that, when the flow rate increases for the forced discharges (two rightmost columns), the velocity profile displays some differences with the free discharge. The most evident feature is an increase in the velocity at the orifice (this can be appreciated in the unsacaled profiles of the inset, bottom--right). However, the shape of the profile when scaled is conserved at the aperture. The flat profile at $z_{\rm view}=D_{\rm s}$ also remains flat upon acceleration. Nevertheless, the region $D/2 < z < D_{\rm s}/2$ presents nonlinear changes in the profile. In general, the scaled profile becomes less steep as the overweight is heavier, reducing $v_z$ at $r=0$ and increasing its value at $r=D_{\rm s}/2$.

In Fig. \ref{fig:vy-vt-t}, we show some profiles from Fig. \ref{fig:vy-vs-r} but comparing in each plot different points during the discharge for two forcing condition ($W=0$ and $W=4 W_{\rm D}$). A peculiar feature emerges in these plots. Whereas the velocity profile for the unforced discharge remains unaltered during discharge at the orifice and at a high plane ($z_{\rm view}=D_{\rm s}$), it does evolve in the ``transition region'' $D/2 < z < D_{\rm s}/2$. In this region, $v_z(r=0)$ increases (and $v_z(r=D_{\rm s}/2)$ decreases) as the silo discharges. This is somewhat unexpected since the flow rate is steady and one may assume that the internal velocity profiles are steady too. It is worth mentioning that previous experimental studies have pointed out this velocity drift of the ``plug zone'' \cite{sielamowicz2005,waters2000}. Interestingly, for forced flows, in the transition region $D/2 < z < D_{\rm s}/2$, the profile $v_z(r)$ does remain unaltered during discharge until the start of the accelerated phase of the flow. This is a remarkable feature and implies that one could homogenize the velocity profile in a silo by using an overweight. Of course, in the final stages of the discharge this would cause an increase of the flow rate.

Some profiles for the radial velocity in the silo are shown in Fig. \ref{fig:vr-vs-r}. The values of $v_r$ are at least one order of magnitude lower than those of $v_z$. In all cases, the radial velocity is zero both at the walls ($v_r(r=D_{\rm s}/2)=0$) and at the axis of the silo ($v_r(r=0)=0$). A maximum in $v_r$ is always found at an intermediate value of $r$. In particular, for $z_{\rm view}=d$ the maximum occurs at $r=D/2$ as expected. Once again, a forced flow yields profiles at all heights that are steady and do not change much with time until the accelerated phase develops. However, the unforced discharge displays increasing radial velocities at $z_{\rm view}=D_{\rm s}/2$, while radial velocities drop during discharge at $z_{\rm view}=D/2$ and $z_{\rm view}=d$. One interesting feature is that for the forced flow, at $z_{\rm view}=D_{\rm s}/2$, the radial velocity decreases significantly during he accelerated phase. Beyond this, overall, the radial velocity is not affected by the fact that the forcing induces an acceleration in the flow rate.

\section{Conclusions}\label{Sec:Concl}

We have shown that the velocity profiles during the discharge of a silo can be affected {in a non trivial way} by the forcing of the flow via an overweight. Since the flow rate increases during the discharge due to forcing, the profiles display larger values in the vertical velocities. However, the radial velocities seem to be little affected.

When the vertical velocity profiles are scaled by the maximum velocity at the orifice, one can consider changes in the shape of the profiles. We observed that the profiles far from the orifice (one silo diameter above the base, or four orifice diameters above the base) remain flat as in a free discharge. Likewise, the scaled profile at the plane of the orifice conserves the shape of the one observed for a free discharge. However, at a plane one orifice radius above the base, the vertical velocity profile becomes flatter as the overweight is increased.  {These results suggest that the use of an overweight helps to develop a mass flow rate profile even for these flat-bottomed silos, which may be of particular interest in the industry.}

{A number of open questions remain with respect to forced flows, such as: (i) Is there a simple way to upgrade the Beverloo rule to a height-dependent expression that takes into account the effect of an overweight? (ii) Is the 5/2 power in the Beverloo rule still valid? (iii) Since the acceleration of the flow rate depends on the material, what material properties are more relevant? (iv) Is the pressure profile inside the silo dramatically different from the unforced discharge? (v) What is the undelying mechanism that induce the accelerated phase to start logarithmically sooner as the overweight increases? (vi) Are there connections with related experiments where an overweight has induced drastic changes in the granular dynamics \cite{metcalfe2002}? These, and other questions, need to be addressed. The new knowledge that we earn regarding forced discharges may help develop new models to understand even the apparently simpler case of a free discharge.}

\textbf{Acknowledgments} We thank J. R. Darias for valuable discussions. This work has been supported by ANPCyT (Argentina) through grant PICT 2012-2155 and UTN (Argentina) through grant PID MAUTNLP-2184.


\section{Introduction}

In these lines I intend to show that there exists consistent evidence
pointing to the need of revision and further study of what seem at present a
settled issue, namely the independence of the speed of electromagnetic
radiation on the motion of its source.

The main point in the evidence is the range disagreement during the Earth
flyby of the spacecraft NEAR in 1998. Its range was measured near the point
of closest approach using two radar stations of the Space Surveillance
Network (SSN), and compared with the trajectory obtained from the Deep Space
Network (DSN)\cite{antreasian98}. As for the range, the two measurements
should match within a meter-level accuracy (the resolution is 5 m for Millstone
and 25 m for Altair), but actual data showed a difference that varies
linearly with time (with different slopes for the two radar stations) up to
a maximum difference of about 1 km, i.e. more than 100 times larger than the
accuracy of the equipment used (see figure 10 of \cite{antreasian98}).
Further, when NEAR crossed the orbits of GPS satellites (orbital radius
26,600 Km) the measured range difference was 650 m, that is, a time
difference of 2 $\mu$s. Is it reasonable that any standard GPS receiver
performs better than DSN or SSN?

There has not been a complete explanation for the range discrepancy. It is
very difficult to find any physical reason that may produce this anomaly,
for any physical disturbance of the path of the spacecraft should manifest
equally in SSN and DSN measurements. Guruprasad\cite{chirp} proposed
an explanation that points to a time lag in the DSN signals, but the model
is, at best, within 10\% of the measured data and, more important, it fails
to explain an important feature, that is, the different slope for the two radars.
If we assume that systems are working properly, then the measured range
difference (time lag) could be due to different propagation time of the
employed signals.

Additional points in the evidence come from anomalies related to the
tracking of spacecrafts, present in both Doppler and ranging data. The
Pioneer anomaly\cite{anderson98} and the flyby anomaly\cite{anderson08}
refer to small residuals of the differences between measured and modeled
Doppler frequencies of the radio signals emitted by the spacecrafts.
Although these residuals are very small (less than 1 Hz on GHz signals) the
problem is that they follow a non-random pattern, indicating failures of the
model. According to the temporal variation of those residuals the Pioneer
anomaly exhibits a main term, an annual term, a diurnal term and a term that
appears during planetary encounters. It should be clarified that a few years
ago an explanation of the Pioneer anomaly was published\cite{turyshev12}.
However, it is a very specific solution that applies only to the main term
of the Pioneer spacecraft anomaly, but left unresolved many other anomalies,
including those of the spaceships Cassini, Ulysses and Galileo; the annual
term; the diurnal term; the increases of the anomaly during planetary
encounters; the flyby anomaly; and the possible link between all them (it is
hard to think that there are so many different causes for the mentioned
anomalies). For all this, I believe that the issue can not be closed as it
stands.

\section{Range disagreement}

As a matter of fact, the range difference between SSN and DSN, $\delta R$,
is perfectly fitted with
\begin{equation}
\delta R\left( t\right) =-\frac{{\bf R}\left( t\right) \cdot {\bf v
\left( t\right) }{c}  \label{deltar}
\end{equation}
where ${\bf R}\left( t\right) $ is a vector range pointing from the spacecraft
to the radar, ${\bf v}\left( t\right) $ the spacecraft velocity relative to
the radar, and $c$ the speed of light. Figure \ref{range} shows this fit and its
comparison with measured data. The orbital and measured data were taken
from\cite{antreasian98}. Although the exact location of the radar stations
are unknown, the fit is statistically significant for both radar stations ($p<10^{-3}$)
including the first outliers points. It reproduces the (almost) linear dependence
with time during the measured interval, and the two different slopes for
Millstone and Altair stations due to their different locations.

Since range measurements are based on time-of-flight techniques, the
validity of (\ref{deltar}) means that the electromagnetic waves (microwave)
of the DSN and SSN travel at different speeds. Specifically, in the radar
frame of reference, if the SSN waves travel at $c$, then the DSN waves
travel at $c$ plus the projection of the spacecraft velocity in the direction
of the beam, in sharp contrast with the Second Postulate of Special
Relativity Theory (SRT).

In view of the above result one may ask what is established, at present,
about the relation of the speed of electromagnetic radiation (light for
short) to the motion of the source. In order to elaborate this point the
following questions are of relevance:

\begin{enumerate}
\item Are there {\it simultaneous} measurements of the speed of light
from different moving macroscopic sources (not moving images) with different
velocities?;

\item Since ballistic (emission) theories are ruled out (see, for example,
DeSitter \cite{desitter13a}\cite{desitter13b}, Brecher \cite{brecher77} and
Alvager et al \cite{alvager64}), how else could the speed of light depend on
the source movement?

\item How is it possible that there is a first order difference in $v/c$ in
spacecraft range measurements, while at the same time there are many
experiments on time dilation that are consistent with SRT to second order in
$v/c$ (see, for example,\cite{boterman14})?;

\item If the velocity of light depend on the velocity of the source, why has
this not been observed in other phenomena in the past?;
\end{enumerate}

In answer to the previous questions, so far as the author is aware, there is
no known experimental work that simultaneously measures the speed of light
from two different sources (not images), or that simultaneously measures the
speed of light and that of its source. For example, in the work by Alvager
et al,\cite{alvager64} the speed of light is measured at a later time (
\approx $ 200 ns) than the emission time, and there is no measurement of the
speed of the source at the time of the {\it detection} of the light.

Note that measurements involving moving images produce different results
from those produced by mobile sources. For example, under SRT, a moving
source is affected by time dilation while a moving image is not. Therefore,
to ensure the independence of the speed of light from its source movement,
it is essential to have two sources with different movements.

Although controversial and beyond the scope of the this note, time
dilatation phenomena may be of different physical origin from first order
terms, as it may be inferred from the work of Schr\"{o}dinger
\cite{schrodinger25}. Thus, measurements of time dilatation phenomena in
accordance with SRT, does not necessarily imply the independence of the
speed of light with the movement of the source.

The experiments mentioned above \cite{desitter13a}\cite{desitter13b}\cite{brecher77}
\cite{alvager64} only rule out ballistic theories in which radiation maintains the
speed of the source at the time of {\it emission}, but do not rule out other ideas,
like Faraday's 1846\cite{faraday46}.

\section{Faraday's ray vibrations}

In order to remove the ether, Faraday introduced the concept of vibrating
rays\cite{faraday46}, in which an electric charge is conceived as a center
of force with attached \textquotedblleft rays\textquotedblright\ that extend
to infinity. The rays move with their center, but without rotating.
According to this view, the phenomenon of electromagnetic radiation
corresponds to the vibration of these \textquotedblleft
rays,\textquotedblright\ that propagates at speed $c$ relative to the rays
(and the center). That is, the radiation remains linked to the source even
after emitted. Today we could describe the interaction as a kind of
entanglement between the charge and the photon. A framework for the
electromagnetic phenomena according to Faraday's ideas was developed. It was
called \textquotedblleft Vibrating Rays Theory\textquotedblright\ (VRT)
\cite{bilbao14} in reference to Faraday's \textquotedblleft vibrating
rays.\textquotedblright\

Under Faraday's idea, the velocity of radiation at a given epoch will be
equal to {\it c} plus the velocity of the source at the {\it same}
epoch, in contrast with ballistic theories in which the emitted light
retains the speed of the source at the {\it emission} epoch. In this
sense the radiation is always linked to the charge at every time after the
emission. Consequently, the measured Doppler Effect corresponds to the speed
of the source at the time of {\it reception}, as well.

Further, a difference between active and passive reflection is expected,
since the latter is still related to the original source according to VRT.
The Deep Space Network (DSN) works with the so called active reflection (the
spacecraft re-emits in real time a signal in phase with the received signal
from Earth), while the Space Surveillance Network (SSN) works with passive
radar reflection. In consequence, the downlink signal from the aproaching
spacecraft will propagate faster that the reflected one. Using the available
orbital data\cite{antreasian98} we found that, under VRT, the theoretical
time-of-flight difference between active and passive reflection gives exactly the same
range disagreement as (\ref{deltar}), see Part 6 of \cite{bilbao14}.

\section{Pioneer anomaly}

The Pioneer anomaly refers to the fact that the received Doppler frequency
differs from the modeled one by a blue shift that varies almost linearly
with time, and whose derivative is
\begin{equation}
\frac{d ( \Delta f) }{dt} \approx -6 \times 10^{-9} \text{ Hz}/\text{s}
\label{ddfdt}
\end{equation}
where $\Delta f$ is the frequency difference between the measured and the
modeled values.

In the case of a source with variable speed, the main difference in Doppler
(to first order) between VRT and SRT, is that SRT relates to the speed of the
source at the time of {\it emission}, while VRT relates to the speed of the
source at the time of {\it reception}. Precisely, this difference seems to
be present in the spacecraft anomalies.

If VRT is valid, it automatically invalidates all calculations and data analysis
of spacecraft tracking which are based on SRT. So, it is not easy to make a
direct comparison between the expected results from SRT and VRT. However, to
see whether or not the main features predicted by VRT are present in the
measurements, we can evaluate the residual by simulating a measured Doppler
signal assuming that light propagates in accordance to VRT but analyzed
according to SRT.

Calling $t_{2}$ the emission time of the downlink signal from the spacecraft
toward Earth and $t_{3}$ the reception time at Earth, the first order
difference of the Doppler shift between VRT and SRT is (see \cite{bilbao14}
Part 4)
\begin{equation}
\Delta f=f_{VRT}-f_{SRT}\approx f_{0}\hat{{\bf r}}\cdot \frac{{\bf
v}_{2}-{\bf v}_{3}}{c}  \label{df}
\end{equation}
where ${\bf v}_{2}$ and ${\bf v}_{3}$ represent the velocities of the
spacecraft at the corresponding epoch, $\hat{{\bf r}}$ is the unit
vector from the spacecraft to the antenna, and $f_{0}$ the proper frequency
of the signal. That is, the velocity used in the SRT formula is that at the
time of {\it emission} while according to VRT is that corresponding at
the time of {\it reception}.

Since the spacecraft slows down as it moves away, then $\hat{{\bf r}
\cdot \left( {\bf v}_{2}-{\bf v}_{3}\right) >0$, therefore the
difference corresponds to a small blue shift mounted over the large red
shift, as it has been observed in the Pioneer anomaly. It should be noted
that this difference appears because of the active reflection produced by
the onboard transmitter. In case of a passive reflection (for example, by
means of a mirror) the above difference vanishes.

\subsection{Main term}

An estimate of the order of magnitude of \ref{df} is obtained by using that
the variation of the velocity of the spacecraft between the time of emission
and reception is approximately
\begin{equation}
{\bf v}_{2}-{\bf v}_{3}\approx {\bf a}\left( t_{2}-t_{3}\right)
\label{v2v3}
\end{equation}
where ${\bf a}$ is a mean acceleration during the downlink interval. An
estimate for the duration of the downlink is simply
\begin{equation}
t_{3}-t_{2}\approx \frac{r}{c}  \label{t3t2}
\end{equation}
where $r$ is a mean position of the spacecraft between $t_{2}$ and $t_{3}$,
therefore
\[
\Delta f\approx -f_{0}\frac{{\bf r}\cdot {\bf a}}{c^{2}}
\]
Since
\[
{\bf a=-}\frac{GM}{r^{2}}\hat{{\bf r}}
\]
where $G$ is the gravitational constant, and $M$ the mass of the Sun, then,
the time derivative becomes
\begin{equation}
\frac{d\left( \Delta f\right) }{dt}\approx f_{0}\frac{{\bf v}\cdot
{\bf a}}{c^{2}}  \label{ddfdtvrt}
\end{equation}

If the difference (\ref{ddfdtvrt}) is interpreted as an anomalous
acceleration we get
\begin{equation}
a_{a}\approx \frac{v}{c}a  \label{aa}
\end{equation}
that is, the so-called anomalous acceleration is $v/c$ times the actual
acceleration of the spacecraft.

Using data from HORIZONS Web-Interface \cite{nasa} for the spacecraft
ephemeris, some characteristic value for $a_{a}$ can be obtained. Consider
the anomalous acceleration detected at the shortest distance of the Cassini
spacecraft during solar conjunction in June, 2002. The spacecraft was at a
distance of $7.42$ AU moving at a speed of $5.76$ km/s. The anomalous
acceleration given by (\ref{aa}) is $a_{a}\approx 2\times 10^{-9}$ m/s$^{2}$
of the same order of the measured one ($\approx 2.7\times 10^{-9}$ m/s$^{2}$
). Also, the closest distance at which the Pioneer anomaly has been detected
was about $20$ AU. the anomalous acceleration predicted by (\ref{aa}) at
that distance is $a_{a}\approx 7.3\times 10^{-10}$ m/s$^{2}$ of the same
order as the measured one.

The \textquotedblleft anomaly\textquotedblright\ given by (\ref{aa})
decreases in time in a way that has not been observed. Note, however, that
according to Markwardt \cite{markwardt02} the expected frequency at the
receiver includes an additional Doppler effect caused by small effective
path length changes, given by
\begin{equation}
\Delta f_{path}=-\frac{2f_{0}}{c}\frac{dl}{dt}  \label{dfpath}
\end{equation}
where $dl/dt$ is the rate of change of effective photon trajectory path
length along the line of sight. This is a first order effect that can
partially hide the difference between SRT and VRT. Therefore, a more careful
analysis should take into account the additional contribution of (\re
{dfpath}) in (\ref{aa}).

Further, other first order effects may appear, for example, by a slight
rotation of the orbital plane. Since orbital parameters are obtained by
periodically fitting the measurements (mainly due to spacecraft maneuvers or
random perturbations) with theoretical orbits, there is no straightforward
way to weight the importance of this effect in (\ref{aa}). In other words,
data acquisition and analysis may hide part of the VRT signature.

\subsection{Annual term}

Apart from the residual referred to in the preceding paragraph there is also
an annual term. According to Anderson et al \cite{anderson02} the problem is
due to modeling errors of the parameters that determine the spacecraft
orientation with respect to the reference system. Anyway, Levy et al
\cite{levy08} claim that errors such as errors in the Earth ephemeris, the
orientation of the Earth spin axis or the stations coordinates are strongly
constrained by other observational methods and it seems difficult to modify
them sufficiently to explain the periodic anomaly.

The advantage of studying the annual term over the main term, is that the
former is less sensitive to the first order correction mentioned above, and,
for the case of Pioneer, also to the thermal propulsion correction\cite{turyshev12}.
Clearly, the Earth orbital position does not modify those terms.

As before, the annual term is explained by the difference between the
velocity of the spacecraft at the time of emission and that at the moment of
detection, which depends on whether the spacecraft is in opposition or in
conjunction relative to the Sun. When the spacecraft is in conjunction,
light takes longer to get back to Earth than in opposition. The time
difference between emission and reception will be increased by the time the
light takes in crossing the Earth orbit. Specifically, taking into account
the delay due to the position of Earth in its orbit, in opposition equation
\ref{t3t2}) should be written as
\begin{equation}
t_{3}-t_{2}\approx \frac{r+R_{orb}}{c}  \label{dtopo}
\end{equation}
while in conjunction it would be
\begin{equation}
t_{3}-t_{2}\approx \frac{r-R_{orb}}{c}  \label{dtconj}
\end{equation}
where $R_{orb}$ is the mean orbital radius of Earth.

Therefore, an estimate of the magnitude of the amplitude of the annual term
is
\begin{equation}
\Delta f\approx f_{0}\frac{aR_{orb}}{c^{2}}  \label{dfanual}
\end{equation}

For the case of Pioneer 10 at 40 AU we get
\begin{equation}
\Delta f\approx 14\text{ mHz}  \label{df40au}
\end{equation}
and at 69 AU
\begin{equation}
\Delta f\approx 4.8\text{ mHz}  \label{df69AU}
\end{equation}
in good agreement with the observed values.

Using data from HORIZONS Web-Interface \cite{nasa} a more complete analysis
of the time variation of $\Delta f$ has be performed. The residual (that is,
simulated Doppler using VRT but interpreted under SRT) during 12 years time
span is plotted in figure \ref{anual}. Also the dumped sine best fit of the
50 days average measured by Turyshev et al \cite{turyshev99} is plotted
showing an excellent agreement between measurements and VRT prediction.
The negative peaks (i.e., maximum anomalous acceleration) occur during
conjunction when the Earth is further apart from the spacecraft, and
positive peaks during opposition. Also, the amplitude is larger at the
beginning of the plotted interval and decreases with time, as it was
observed\cite{anderson08}\cite{turyshev99}.

\section{Flyby anomaly}

Like the Pioneer anomaly, the Earth flyby anomaly can be associated to a
modeling problem, in the sense that relativistic Doppler includes terms that
are absent in the measured signals. The empirical equation of the flyby
anomaly is given by Anderson et al\cite{anderson08}, which, notably, can be
derived using VRT, as is done in Part 6 of \cite{bilbao14}.

Consider the case of NEAR tracked by 3 antennas located in USA, Spain, and
Australia (a full description of the tracking system is found in a series of
monographs of the Jet Propulsion Laboratory\cite{descanso}). The receiving
antenna was chosen as that having a minimum angle between the spacecraft and
the local zenith.

Using available orbital data, a simulated Doppler signal has been calculated
using VRT. Thus, the simulated residual is obtained by subtracting the
theoretical SRT Doppler, from the VRT calculation. We observed, however,
that the term that contains the velocity of the antennas, that is
\begin{equation}
d=\frac{\gamma _{u_{3}}}{\gamma _{u_{1}}}\frac{1-\hat{{\bf r}
_{23}\cdot {\bf u}_{3}/c}{1-\hat{{\bf r}}_{12}\cdot {\bf u
_{1}/c}  \label{dopplerantena}
\end{equation}
is not enough to completely remove the first order (in $u/c$) Earth
signature (${\bf u}$ is the velocity of the antenna, 1 refers to the
emission epoch and 3 to the reception epoch, as in \cite{bilbao14} Part 4).

This is so because the velocity of the antennas is not uniform and the
evaluation of the emission time is different for VRT and SRT. Then, a small,
first order related term remains. Anyway, since orbital parameters are
obtained by periodically fitting the measurements to theoretical orbits,
thus a similar procedure is needed for VRT. By doing so the first order term
is removed. The only difference between orbits adjusted by SRT and VRT
is a slight rotation of the orbit plane, as mentioned above. Note that, in the
case of range disagreement, two different orbital ajustment are needed by
DSN and SSN due to the different propagation speed. In consequence,
it will be impossible to fit a simultaneous measurement, as it seems to
happen with the range disagreement.

The final result shows that each antenna produces a sinusoidal residual with
a phase shift at the moment of maximum approach. Therefore, if we fit the
data with the pre-encounter sinusoid a post-encounter residual remains and
vice versa.

In figure \ref{flyby} are simultaneously plotted the result of fitting the
residual by pre-encounter data (right half in red, corresponding to figure
2a of \cite{anderson08}) and by post-encounter data (left half in blue,
corresponding to figure 2b of \cite{anderson08}).

Note that the simulated plots are remarkably similar to the reported ones,
including the amplitude and phase (i.e., minima and maxima) of the
corresponding antenna. The fitting of post-encounter data (blue) can be
improved by appropriately setting the exact switching times of the antennas
(which are unknown to the author). The flyby Doppler residual exhibits a
clean signature of the VRT theory.

\section{Conclusions}

In this work I have presented observational evidence favoring a dependence
of the speed of light on that of the source, in the manner implied in
Faraday's ideas of \textquotedblleft vibrating rays.\textquotedblright\

It is remarkable and very suggestive that, as derived from Faraday's
thoughts, simply by relating the velocity of light and the corresponding
Doppler effect with the velocity of the source at the time of detection, is
enough to quantitatively and qualitatively explain a variety of spacecraft
anomalies.

Also, it is worth mentioning that a formulation of electromagnetism
compatible with Faraday's conception is possible, as shown in \cite{bilbao14
, which is also compatible with the known electromagnetic phenomena. This
new formalism introduces a striking concept: that both instantaneous
interaction (static terms) and retarded interaction (radiative terms) are
simultaneously present.

Finally, under VRT the manifestation of the movement of the source in the
speed of light is more subtle than the naive $c+kv$ hypothesis ($k$ is a
constant, $0\le k\le 1 $) usually used to test their dependence\cite{brecher77}.
Thus, it is also of fundamental importance the fact that, from
the experimental point of view, it is very difficult to detect differences
between VRT and SRT, as discussed in \cite{bilbao14}, which is also manifest
in the smallness of the measured anomalies, and in the non clear
manifestation of the effect in usual experiments and observations. For
example, it produces a negligible effect on satellite positioning systems,
see Part 7 of\cite{bilbao14}.

I am aware of how counterintuitive these conceptions are to the modern
scientist, but also believe that, given the above evidence, a conscientious
experimental research is needed to settle the question of the dependence of
the speed of light on that of its source as predicted by VRT, and that has been
observed during the 1998 NEAR flyby. As a closure, I recall Fox's words
regarding the possibility of conducting an experiment on the propagation of
light relative to the motion of the source: ``{\it Nevertheless if one
balances the overwhelming odds against such an experiment yielding anything
new against the overwhelming importance of the point to be tested, he may
conclude that the experiment should be performed.}''\ \cite{fox62}

\section*{Acknowledgements}

I am thankful to Fernando Minotti who read this paper and improved the
manuscript significantly, although he may not agree with all of the
interpretations provided in this paper.