\section{Introduction}
\label{sec:0}
White dwarf binaries are thought to be the most common binaries in the
Universe, and in our Galaxy their number is estimated to be as high as 10$^8$.
In addition most stars  are known to be part of binary systems, roughly half
of which have orbital periods short enough that the evolution of the two stars
is strongly influenced by the presence of a companion. Furthermore,  it has
become clear from observed close binaries, that a large fraction of binaries
that interacted in the past must have lost considerable amounts of angular
momentum, thus forming compact binaries, with compact stellar components. The
details of the evolution leading to the loss of
angular momentum are uncertain, but generally this is interpreted in the
framework of the so called ``common-envelope evolution'': the picture that in a
mass-transfer phase between a giant and a more compact companion the
companion quickly ends up inside the giant's envelope, after which
frictional processes slow down the companion and the core of the
giant, causing the ``common envelope'' to be expelled, as well as
the orbital separation to shrink dramatically \cite{Taam and Sandquist (2000)}.

Among the most compact binaries know, often called ultra-compact or ultra-short
binaries, are those hosting two white dwarfs and classified into two
types:
\emph{detached} binaries, in which the two components are relatively widely
separated and \emph{interacting} binaries, in which mass is transferred from one
component to the other. In the latter class a white dwarf is accreting from a
white dwarf like object (we often refer to them as AM CVn systems, after the
prototype of the class, the variable star AM CVn; \cite{warn95,Nelemans
(2005)}).
\begin{figure}
  \includegraphics[height=7.5cm,angle=0]{P_Mtot_SPYnew1}
\caption{Period versus total mass of double white dwarfs. The points and arrows
are observed systems \cite{Nelemans et al. (2005)}, the
grey shade a  model for the
Galactic
population. Systems to the left of the dashed line will merge within a Hubble
time, systems above the dotted line  have a combined mass above the
Chandrasekhar mass. The top left corner shows the region of possible type Ia
supernova progenitors,   where the grey shade has been darkened for better
visibility (adapted from \cite{Nelemans (2007)}). }
\label{fig:P_Mtot}
\end{figure}

In the past many authors have emphasised the importance of studying white dwarfs
in DDBs. In fact, the study of ultra-short
white dwarf binaries  is relevant to some important astrophysical questions
which have been outlined by several author. Recently, \cite{Nelemans
(2007)} listed the
following ones:

 \begin{itemize}

\item {\em Binary evolution} Double white dwarfs are excellent tests of
  binary evolution. In particular the orbital shrinkage during the
  common-envelope phase can be tested using double white dwarfs. The
  reason is that for giants there is a direct relation between
  the mass of the core (which becomes a white dwarf and so its mass is
  still measurable today) and the radius of the giant. The latter
  carries information about the (minimal) separation between the two
  components in the binary before the common envelope, while the
  separation after the common envelope can be estimated from the
  current orbital period. This enables a detailed reconstruction of
  the evolution leading from a binary consisting of two main sequence
  stars to a close double white dwarf \cite{Nelemans et al.(2000)}.
The interesting
  conclusion of this exercise is that the standard schematic
  description of the common envelope -- in which the
  envelope is expelled at the expense of the orbital energy -- cannot
  be correct. An alternative scheme, based on the angular momentum,
  for the moment seems to be able to explain all the observations
  \cite{Nelemans and Tout (2005)}.

 \item {\em Type Ia supernovae} Type Ia supernovae have peak
  brightnesses that are well correlated with the shape of their light
  curve \cite{Phillips (1993)}, making them ideal standard candles to determine
  distances. The measurement of the apparent brightness of far away
  supernovae as a function of redshift has led to the conclusion that
  the expansion of the universe is accelerating
  \cite{Perlmutter et al. (1998),Riess et al.(2004)}. This depends on the
assumption
that these
  far-away (and thus old) supernovae behave the same as their local
  cousins, which is a quite reasonable assumption. However, one of the
  problems is that we do not know what exactly explodes and why, so
  the likelihood of this assumption is difficult to assess
  \cite{Podsiadlowski et al. (2006)}. One of the proposed models for the
  progenitors of type Ia supernovae are massive close double white
  dwarfs that will explode when the two stars merge \cite{Iben and Tutukov
(1984)}.
  In Fig.~\ref{fig:P_Mtot} the observed double
  white dwarfs are compared to a model for the Galactic population of
  double white dwarfs \cite{Nelemans et al. (2001)}, in
which the merger rate of
  massive double white dwarfs is similar to the type Ia supernova
  rate. The grey shade in the relevant corner of the diagram is
  enhanced for visibility. The discovery of at least one system in
  this box confirms the viability of this model (in terms of event
  rates).

\item {\em Accretion physics} The fact that in AM CVn systems the mass losing
star is an evolved, hydrogen deficient star,
  gives rise to a unique astrophysical laboratory, in which accretion
  discs made of almost pure helium \cite{Marsh et al. (1991),Schulz et
al.(2001),Groot et al. (2001),Roelofs et al. (2006),Werner et al. (2006)}.
This opens  the possibility to test the behaviour of accretion discs of
  different chemical composition.

\item {\em Gravitational wave emission}
Untill recently the DDBs with two NSs were considered among the best sources to
look for gravitational wave emission, mainly due to the relatively high chirp
mass expected for these sources, In fact, simply inferring the strength of the
gravitational wave amplitude expected for from  \cite{Evans et al. (1987)}
\begin{equation}
h = \left[ \frac{16 \pi G L_{GW}} {c^3 \omega^2_g 4 \pi d^2} \right] ^{1/2}  =
10^{-21} \left(  \frac{{\cal{M}}}{{\it M}_{\odot}}
\right)^{5/3} \left ( \frac{P_{orb}}{\rm 1
 hr} \right)^{-2/3} \left ( \frac{d}{\rm 1  kpc} \right)^{-1}
\end{equation}
where
\begin{equation}
L_{GW} = \frac{32}{5}\frac{G^4}{c^5}\frac{M^2 m^2 (m+M)}{a^5} ;
\end{equation}
\begin{equation}
{\cal{M}}=\frac{(Mm)^{3/5}}{(M+m)^{1/5}}
\end{equation}

where the frequency of the wave is given by $f = 2/P_{orb}$. It is evident that
the strain signal $h$ from DDBs hosting neutron stars is a factor 5-20 higher
than in the case of DDBs with white dwarfs as far as the orbital period is
larger than approximatively 10-20 minutes.  In recent years, AM CVns have
received great
attention as they represent a large population of guaranteed  sources for the
forthcoming \textit{Laser Interferometer Space Antenna}
\cite{2006astro.ph..5722N,2005ApJ...633L..33S}. Double WD
binaries enter the  \textit{LISA}
observational window (0.1 $\div$ 100 mHz) at an orbital period  $\sim$ 5 hrs
and, as they evolve secularly through GW emission, they cross the  whole
\textit{LISA} band. They are expected to be so numerous ($\sim 10^3  \div 10^4$
expected), close on average, and luminous in GWs as to create a  stochastic
foreground that dominates the \textit{LISA} observational window  up to
$\approx$ 3 mHz \cite{2005ApJ...633L..33S}. Detailed knowledge of the
characteristics of their background signal would thus be needed to model it and
study weaker background GW signals of cosmological origin.
\end{itemize}

 \begin{figure}
  \includegraphics[height=12cm,angle=-90]{Galactic_GWR1}
  \caption{Expected signals of ultra-compact binaries,
  the ones with error bars from (adapted from  \cite{Roelofs et al.
(2006),Nelemans (2007)}.}
\label{fig:fh_HST}
\end{figure}

A relatively small number of ultracompact DDBs systems is presently known.
According to \cite{2006MNRAS.367L..62R} there exist 17
confirmed objects with
orbital periods in the $10 \div 70$ min in which a hydrogen-deficient mass
donor, either a semi-degenerate star or a WD itself, is present. These are
called AM CVn systems and are roughly characterized by optical emission
modulated at the orbital period, X-ray emission showing no evidence for a
significant modulation (from which a moderately magnetic primary is suggested,
\cite{2006astro.ph.10357R}) and, in the few cases where
timing analyses could be
carried out, orbital spin-down consistent with GW emission-driven mass
transfer.

In addition there exist two peculiar objects, sharing a number of observational
properties that partially match those of the ``standard'' AM CVn's. They
are
RX\,J0806.3+1527\  and RX\,J1914.4+2456, whose X-ray emission is $\sim$ 100\% pulsed, with
on-phase and off-phase of approximately equal duration. The single modulations
found in their lightcurves, both in the optical and in X-rays, correspond to
periods of, respectively, 321.5 and 569 s
(\cite{2004MSAIS...5..148I,2002ApJ...581..577S}) and were
first interpreted as orbital periods. If so, these two objects
are the binary systems with the shortest orbital period known and could belong
to the AM CVn class. However, in addition to peculiar emission properties with
respect to other AM CVn's, timing analyses carried out by the above cited
authors demonstrate that, in this interpretation, these two objects have
shrinking orbits. This is contrary to what expected in mass transferring
double white dwarf systems (including AM CVn's systems) and suggests the
possibility that the binary is detached, with the orbit shrinking because of GW
emission. The electromagnetic  emission would have in turn to be caused by some
other kind of interaction.

Nonetheless, there are a number of alternative models to account for the
observed properties, all of them based upon binary systems.  The intermediate
polar (IP) model (\cite{Motch et al.(1996),io99,Norton  et al. (2004)}) is
the only one in
which the pulsation periods are not assumed to be orbital. In this model, the
pulsations are likely due to the spin of a white dwarf accreting from
non-degenerate secondary star. Moreover, due to geometrical constraints the
orbital period is not expected to be  detectable. The other two models assume a
double white dwarf binaries in which the pulsation period is the orbital period.
Each of them invoke a semi-detached, accreting double white dwarfs: one is
magnetic, the double degenerate polar model
(\cite{crop98,ram02a,ram02b,io02a}), while the
other is non-magnetic, the
direct impact model (\cite{Nelemans et al. (2001),Marsh and
Steeghs(2002),ram02a}), in which,
due
to the compact dimensions of these systems, the mass transfer streams is forced
to hit  directly onto the accreting white dwarfs rather than to form an
accretion disk .

\begin{table}[!ht]
\caption{Overview of observational properties of AM CVn stars (adapted from
\cite{Nelemans (2005)})}
\label{tab:overview}
\smallskip
\begin{center}
\hspace*{-0.5cm}
{\small
\begin{tabular}{lllllllcc}\hline
Name  & $P_{\rm orb}^a$ & & $P_{\rm sh}^a$ & Spectrum & Phot. var$^b$ & dist &
X-ray$^c$ &
UV$^d$  \\
 & (s) & & (s) & & &  (pc) & &  \\
\hline \hline
ES Cet & 621  &(p/s) &  & Em & orb &  350& C$^3$X &  GI   \\
AM CVn & 1029 &(s/p) & 1051 & Abs & orb & 606$^{+135}_{-95}$ & RX & HI  \\
HP Lib & 1103 &(p) & 1119 & Abs & orb & 197$^{+13}_{-12}$ & X & HI \\
CR Boo & 1471 &(p) & 1487 & Abs/Em? & OB/orb & 337$^{+43}_{-35}$& ARX & I \\
KL Dra & 1500 &(p) & 1530 & Abs/Em? & OB/orb &  &  &  \\
V803 Cen & 1612 &(p) & 1618 & Abs/Em? & OB/orb &  & Rx & FHI   \\
SDSSJ0926+36 & 1698.6& (p) &  & & orb & & & \\
CP Eri & 1701 &(p) & 1716 & Abs/Em & OB/orb & &  & H  \\
2003aw & ? & & 2042 & Em/Abs? & OB/orb &  & &    \\
SDSSJ1240-01 & 2242 &(s) & & Em & n &  & &    \\
GP Com & 2794 &(s) & & Em & n & 75$\pm2$ & ARX & HI \\
CE315  & 3906 &(s) &  & Em & n & 77? & R(?)X & H  \\
 & & & & & & & & \\
Candidates & & & & & & & & \\\hline\hline
RXJ0806+15 & 321 &(X/p) & & He/H?$^{11}$ & ``orb'' &  &
CRX &  \\
V407 Vul & 569 &(X/p) &  & K-star$^{16}$ & ``orb'' &  &
ARCRxX &  \\ \hline
\end{tabular}
}
\end{center}
{\small
$a$ orb = orbital, sh = superhump, periods from {ww03}, see
references therein, (p)/(s)/(X) for photometric, spectroscopic, X-ray period.\\
$b$ orb = orbital, OB = outburst\\
$c$  A = ASCA, C = Chandra, R = ROSAT, Rx = RXTE, X = XMM-Newton {kns+04}\\
$d$  F = FUSE, G = GALEX, H = HST, I = IUE
}
\end{table}

After a brief presentation of the two X--ray selected double degenerate binary
systems, we discuss the main scenario of this type, the
Unipolar Inductor Model (UIM) introduced by \cite{2002MNRAS.331..221W}  and
further developed by \cite{2006A&A...447..785D,2006astro.ph..3795D}, and compare
its predictions with the salient observed
properties of these two sources.
\subsection{RX\,J0806.3+1527}
\label{0:j0806}
RX\,J0806.3+1527\ was discovered in 1990 with the {\em ROSAT}\ satellite during the All-Sky Survey
(RASS; \cite{beu99}). However, it was only in 1999 that a
periodic
signal at 321\,s was detected in its soft X-ray flux with the {\em ROSAT}\ HRI
(\cite{io99,bur01}).
Subsequent deeper optical studies allowed to unambiguously identify the optical
counterpart of RX\,J0806.3+1527, a blue $V=21.1$ ($B=20.7$) star (\cite{io02a,io02b}). $B$,
$V$ and $R$ time-resolved photometry revealed the presence of
a $\sim 15$\% modulation at the $\sim 321$\,s X-ray period (\cite{io02b,ram02a}.
 \begin{figure*}[htb]
   \resizebox{16pc}{!}{\rotatebox[]{-90}{\includegraphics{new_spec_norm.ps}}}
\caption{VLT FORS1 medium (6\AA; 3900--6000\AA) and low (30\AA; above
6000\AA) resolution spectra obtained for the optical counterpart of
RX\,J0806.3+1527. Numerous faint emission lines of HeI and HeII (blended
with H) are labeled (adapted form \cite{io02b}).}
\label{spec}
\end{figure*}
   \begin{figure*}[hbt]
   \centering
   \resizebox{20pc}{!}{\includegraphics{israel_f1.eps}
}
    \caption{Left panel: Results of the phase fitting technique used
to infer the P-\.P coherent solution for RX\,J0806.3+1527: the linear term (P
component) has been corrected, while the quadratic term (the \.P
component) has been kept for clarity.  The best \.P solution inferred
for the optical band is marked by the solid fit line.  Right panel:
2001-2004 optical flux measurements at fdifferent wavelengths.}
        \label{timing}
    \end{figure*}

The VLT spectral study revealed a blue continuum with no intrinsic
absorption lines \cite{io02b} . Broad ($\rm FWHM\sim 1500~\rm
km~s^{-1}$), low equivalent width ($EW\sim -2\div-6$ \AA) emission
lines from the He~II Pickering series (plus additional emission lines
likely associated with He~I, C~III, N~III, etc.; for a different interpretation
see \cite{rei04}) were instead
detected \cite{io02b}. These findings, together with the period stability and
absence of any additional modulation in the 1\,min--5\,hr period
range, were interpreted in terms of a double degenerate He-rich binary
(a subset of the AM CVn class; see \cite{warn95}) with
an orbital period of 321\,s, the shortest ever recorded. Moreover,
RX\,J0806.3+1527\ was noticed to have optical/X-ray properties similar to
those of RX\,J1914.4+2456, a 569\,s modulated soft X-ray source proposed
as a double degenerate system (\cite{crop98,ram00,ram02b}).

In the past years the detection of  spin--up was reported, at a rate of
$\sim$6.2$\times$10$^{-11}$\, s~s$^{-1}$,
for the 321\,s orbital modulation, based on optical data taken from the
Nordic Optical Telescope (NOT) and the VLT archive, and by using
incoherent timing techniques \cite{hak03,hak04}.
Similar results were
reported also for the X-ray data (ROSAT and Chandra; \cite{stro03})
of RX\,J0806.3+1527\
spanning over 10 years of uncoherent observations and based on the NOT
results \cite{hak03}.

A Telescopio Nazionale Galileo (TNG) long-term project (started on 2000)
devoted to the study of the long-term timing properties of RX\,J0806.3+1527\  found a
slightly energy--dependent pulse shape with the pulsed fraction increasing
toward longer wavelengths, from $\sim$12\% in the B-band to
nearly 14\% in the I-band (see lower right panel of Figure~\ref{QU};
\cite{2004MSAIS...5..148I}). An additional variability, at a level of
4\% of the optical pulse
shape as a function of time (see upper right panel of Figure~\ref{QU} right) was
detected. The first coherent
timing solution was also
inferred for this source, firmly assessing that the source was spinning-up:
P=321.53033(2)\,s, and \.P=-3.67(1)$\times$10$^{-11}$\,s~s$^{-1}$ (90\%
uncertainties are reported; \cite{2004MSAIS...5..148I}).
Reference \cite{2005ApJ...627..920S} obtained independently a
phase-coherent timing solutions for
the orbital period of this source over a similar baseline, that is fully
consistent with that of \cite{2004MSAIS...5..148I}.  See
\cite{2007MNRAS.374.1334B} for a similar coherent timing solution
also including the covariance
terms of the fitted parameters.

   \begin{figure}[hbt]
   \centering
   \resizebox{33pc}{!}{\rotatebox[]{0}{\includegraphics{israel_f2.eps}}}
   \caption{Left Panel:  The 1994--2002 phase coherently connected X--ray folded
light curves (filled squares; 100\% pulsed fraction) of RX\,J0806.3+1527, together with the
VLT-TNG 2001-2004 phase connected folded optical light curves (filled circles).
Two orbital cycles are reported for clarity.  A nearly anti-correlation was
found.
Right panels: Analysis of the phase variations induced by pulse shape changes
in the optical band (upper panel), and the pulsed fraction as a function of
optical wavelengths (lower panel). }
              \label{QU}%
    \end{figure}

The relatively high accuracy obtained for the optical phase coherent
P-\.P solution (in the January 2001 - May 2004 interval) was used to
extend its validity backward to the ROSAT observations without loosing the
phase coherency, i.e.  only one possible period cycle consistent with
our P-\.P solution. The best  X--ray phase coherent
solution is P=321.53038(2)\,s, \.P=-3.661(5)$\times$10$^{-11}$\,s~s$^{-1}$ (for
more details see \cite{2004MSAIS...5..148I}). Figure~\ref{QU}
(left panel) shows the
optical (2001-2004) and X--ray (1994-2002) light curves folded by using the
above reported P-\.P coherent solution, confirming the amazing
stability of the X--ray/optical anti-correlation first noted by
(\cite{2003ApJ...598..492I}; see inset of left panel of
Figure\,\ref{QU}).
   \begin{figure}
   \centering
   \resizebox{20pc}{!}{\rotatebox[]{-90}{\includegraphics{israel_f3.eps}}}
   \caption{The results of the {\em XMM--Newton}\ phase-resolved spectroscopy
(PRS)
analysis for the absorbed blackbody spectral parameters: absorption,
blackbody temperature, blackbody radius (assuming a distance of
500\,pc), and absorbed (triangles) and unabsorbed (asterisks)
flux. Superposed is the folded X-ray light curve. }
              \label{xmm}%
    \end{figure}
   \begin{figure}
   \centering
   \resizebox{16pc}{!}{\rotatebox[]{-90}{\includegraphics{israel_f4.eps}}}
   \caption{ Broad-band energy
spectrum of
RX\,J0806.3+1527\ as inferred from the {\em Chandra}, {\em XMM--Newton}, VLT and TNG
measurements and {\it EUVE\/} upper limits. The dotted line represents one of
the possible
fitting blackbody models for the IR/optical/UV bands.}
              \label{xmm}%
    \end{figure}

On 2001, a Chandra observation of RX\,J0806.3+1527\ carried out in simultaneity with time
resolved optical observation at the VLT, allowed for the first time to study the
details of the X-ray emission and the phase-shift between X-rays and optical
band. The X-ray spectrum is consistent with a occulting, as a function of
modulation phase, black body with a temperature of $\sim$60\,eV
\cite{2003ApJ...598..492I}. A 0.5 phase-shift was reported for the X-rays
and the optical band \cite{2003ApJ...598..492I}. More recently, a 0.2
phase-shift was
reported by analysing the whole historical X-ray and optical dataset: this
latter result is considered the correct one \cite{2007MNRAS.374.1334B}.

On 2002 November 1$^{\rm st}$ a second deep X-ray observation was obtained with
the {\em XMM--Newton}\ instrumentations for about 26000\,s, providing an increased
spectral accuracy (see eft panel of
Figure~\ref{xmm}). The {\em XMM--Newton}\ data show a lower value of the absorption column,
a relatively constant black body temperature, a
smaller black body size, and, correspondingly, a slightly lower
flux. All these differences may be ascribed to the pile--up effect in
the Chandra data, even though we can not completely rule out the
presence of real spectral variations as a function of time. In any case
we note that this result is in agreement with the idea of a
self-eclipsing (due only to a geometrical effect) small, hot and
X--ray emitting region on the primary star. Timing analysis did not
show any additional significant signal at periods longer or shorter
than 321.5\,s, (in the 5hr-200ms interval).  By using the {\em XMM--Newton}\ OM a
first look at the source in the UV band (see right panel of Figure~\ref{xmm})
was obtained
confirming the presence of the blackbody component inferred from  IR/optical
bands.

Reference \cite{2003ApJ...598..492I} measured an on-phase
X-ray luminosity (in the range
0.1-2.5 keV) $L_X = 8 \times 10^{31} (d/200~\mbox{pc})^2$ erg s$^{-1}$ for
this source. These authors suggested that the bolometric luminosity might even
be dominated by the (unseen) value of the UV flux, and reach values up to
5-6 times higher.
The optical flux is only $\sim$ 15\% pulsed, indicating that most of it might
not be associated to the same mechanism producing the pulsed X-ray emission
(possibly the cooling luminosity of the WD plays a role). Given these
uncertainties and, mainly, the uncertainty in the distance to the source, a
luminosity $W\simeq 10^{32} (d/200~\mbox{pc})^2$ erg s$^{-1}$ will be assumed
as a reference value.\\

\subsection{RX\,J1914.4+2456\ }
\label{0:j1914}
The luminosity and distance of this source have been subject to much debate
over the last years. Reference \cite{2002MNRAS.331..221W} refer
to earlier ASCA measurements that,
for a distance of 200-500 pc, corresponded to a luminosity in the range
($4\times 10^{33} \div 2.5 \times 10^{34}$) erg s$^{-1}$. Reference
\cite{2005MNRAS.357...49R}, based on more recent XMM-Newton
observations and a standard blackbody
fit to the X-ray spectrum, derived an X-ray luminosity of $\simeq 10^{35}
d^2_{\mbox{\tiny{kpc}}}$ erg s$^{-1}$, where $d_{\mbox{\tiny{kpc}}}$ is the
distance in kpc. The larger distance of $\sim$ 1 kpc was based on a work by
\cite{2006ApJ...649..382S}. Still more recently,
\cite{2006MNRAS.367L..62R} find that an optically thin
thermal emission spectrum,
with an edge at 0.83 keV attributed to O VIII, gives a significantly better
fit to the data than a blackbody model. The optically thin thermal plasma
model implies a much lower bolometric luminosity of L$_{\mbox{\tiny{bol}}}
\simeq 10^{33}$ d$^2_{\mbox{\tiny{kpc}}}$ erg s$^{-1}$.
\\
Reference \cite{2006MNRAS.367L..62R} also note that the
determination of a 1 kpc distance
is not free of uncertainties and that a minimum distance of $\sim 200$ pc
might still be possible: the latter leads to a minimum luminosity of $\sim 3
\times 10^{31}$ erg s$^{-1}$. \\
Given these large discrepancies, interpretation of this source's properties
remains ambiguous and dependent on assumptions. In the following, we refer to
the more recent assessment by \cite{2006MNRAS.367L..62R}
of a luminosity $L =
10^{33}$ erg s$^{-1}$ for a 1 kpc distance.\\
Reference \cite{2006MNRAS.367L..62R} also find possible
evidence, at least in a few
observations, of two secondary peaks in power spectra. These are very close to
($\Delta \nu \simeq 5\times 10^{-5}$ Hz) and symmetrically distributed around
the strongest peak at $\sim 1.76 \times 10^{-3}$ Hz. References
\cite{2006MNRAS.367L..62R} and \cite{2006ApJ...649L..99D}
discuss the implications of this possible finding.
\section{The Unipolar Inductor Model}
\label{sec:1}
The Unipolar Inductor Model (UIM) was originally proposed to
explain the origin of bursts of
decametric radiation received from Jupiter, whose properties appear to be
strongly influenced by the orbital location of Jupiter's satellite Io
\cite{1969ApJ...156...59G,1977Moon...17..373P} . \\
The model relies on Jupiter's spin being different from the system orbital
period (Io spin is tidally locked to the orbit). Jupiter has a surface
magnetic field $\sim$ 10 G so that, given
Io's good electrical conductivity ($\sigma$), the satellite experiences an
e.m.f. as it moves across the planet's field lines along the orbit. The e.m.f.
accelerates free charges in the ambient medium, giving rise to a flow of
current along the sides of the flux tube connecting the bodies. Flowing
charges are accelerated to mildly relativistic energies and emit coherent
cyclotron radiation through a loss cone instability  (cfr. \cite{Willes and
Wu(2004)} and references therein):
this is the basic framework in which Jupiter decametric radiation and its
modulation by Io's position are explained.
Among the several confirmations of the UIM in this system, HST UV observations
revealed the localized emission on Jupiter's surface due
to flowing particles hitting the planet's surface - the so-called Io's
footprint  \cite{Clarke et al. (1996)}.
In recent years, the complex interaction between Io-related free charges
(forming the Io torus) and Jupiter's magnetosphere has been understood in much
greater detail  \cite{Russ1998P&SS...47..133R,Russ2004AdSpR..34.2242R}. Despite
these significant
complications, the above scenario maintains its general validity, particularly
in view of astrophysical applications.\\
Reference \cite{2002MNRAS.331..221W} considered the UIM in the
case of close white dwarf binaries.
They assumed a moderately magnetized primary, whose spin is not synchronous
with the orbit  and a non-magnetic companion, whose spin is tidally locked.
They particularly highlight the role of ohmic dissipation of currents flowing
through the two WDs and show that this occurs essentially in the primary
atmosphere.
A small bundle of field lines leaving the primary surface thread the whole
secondary. The orbital position of the latter is thus ``mapped'' to a small
region onto the primary's surface; it is in this small region that ohmic
dissipation - and the associated heating - mainly takes place. The resulting
geometry, illustrated in Fig. \ref{fig:1}, naturally leads to mainly thermal,
strongly pulsed X-ray emission, as the secondary moves along the orbit.
\begin{figure}
\centering
\includegraphics[height=6.18cm]{Wu2004.eps}
\caption{Electric coupling between the asynchronous, magnetic primary star
and the non-magnetic secondary,, in the UIM (adapted from
\cite{2002MNRAS.331..221W}).}
\label{fig:1}
\end{figure}
The source of the X-ray emission is ultimately represented by the relative
motion between primary spin and orbit, that powers the electric circuit.
Because of resistive dissipation of currents, the relative motion is
eventually expected to be cancelled. This in turn requires angular momentum to
be redistributed between spin and orbit in order to synchronize them. The
necessary torque is provided by the Lorentz force on cross-field currents
within the two stars.\\
Reference \cite{2002MNRAS.331..221W} derived synchronization
timescales $(\tau_{\alpha}) \sim$ few
10$^3$ yrs for both RX\,J1914.4+2456\  and RX\,J0806.3+1527\ , less than 1\% of their
orbital evolutionary timescales. This would imply a much larger Galactic
population of such systems than predicted by population-synthesis models, a
major difficulty of this version of the UIM. However,
\cite{2006A&A...447..785D,2006astro.ph..3795D} have shown that
the electrically active phase is actually
long-lived because perfect synchronism is never reached.
In a perfectly synchronous system the electric circuit would be turned off,
while GWs would still cause orbital spin-up. Orbital motion and primary spin
would thus go out of synchronism, which in turn would switch the circuit on.
The synchronizing (magnetic coupling) and de-synchronizing (GWs) torques are
thus expected to reach an equilibrium state at a sufficiently small degree of
asynchronism.\\
We discuss in detail how the model works and how the major observed properties
of RX\,J0806.3+1527\  and RX\,J1914.4+2456\  can be interpreted in the UIM framework. We
refer to \cite{2005MNRAS.357.1306B} for a possible
criticism of the model based on
the shape of the pulsed profiles of the two sources.
Finally, we refer to \cite{2006ApJ...653.1429D,2006ApJ...649L..99D}, who
have recently proposed alternative mass transfer models that can also account
for long-lasting episodes of spin-up in Double White Dwarf systems.
\section{UIM in Double Degenerate Binaries}
\label{sec:1.1}
According to \cite{2002MNRAS.331..221W}, define the primary's
asynchronism parameter
$\alpha \equiv \omega_1/ \omega_o$, where $\omega_1$ and $\omega_o$ are the
primary's spin and orbital frequencies. In a system with orbital separation
$a$, the secondary star will move with the velocity
$ v = a (\omega_o - \omega_1) = [G M_1 (1+q)]^{1/3}~\omega^{1/3}_o (1-\alpha)$
relative to field lines, where $G$ is the gravitational constant, $M_1$ the
primary mass, $q = M_2/M_1$ the system mass-ratio. The electric field induced
through the secondary is thus {\boldmath$E$} = $\frac{\mbox{{\boldmath$v
\times B_2$}}}{c}$, with an associated e.m.f. $\Phi = 2R_2 E$, $R_2$ being the
secondary's radius and {\boldmath$B$$_2$} the primary magnetic field at the
secondary's location.
The internal (Lorentz) torque redistributes angular momentum between spin and
orbit conserving their sum (see below), while GW-emission causes a net
loss of orbital angular momentum. Therefore, as long as the primary spin is
not efficiently affected by other forces, \textit{i.e.} tidal forces (cfr.
App.A in \cite{2006A&A...447..785D}), it will lag
behind the evolving orbital
frequency, thus keeping electric coupling continuously active.\\
Since most of the power dissipation occurs at the primary atmosphere (cfr.
\cite{2002MNRAS.331..221W}),
we slightly simplify our treatment assuming no dissipation at all at the
secondary. In this case, the binary system is wholly analogous to the
elementary circuit of Fig. \ref{fig:2}.
\begin{figure}
\centering
\includegraphics[width=5.8cm]{CircuitComplete.eps}
\caption{Sketch of the elementary circuit envisaged in the UIM. The secondary
star acts as the battery, the primary star represents a resistance connected
to the battery by conducting ``wires'' (field lines.) Inclusion of the effect
of GWs corresponds to connecting the battery to a plug, so that it is
recharged at some given rate. Once the battery initial energy reservoir is
consumed, the bulb will be powered just by the energy fed through the plug.
This corresponds to the ``steady-state'' solution.}
\label{fig:2}
\end{figure}
Given the e.m.f. ($\Phi$) across the secondary star and the system's effective
resistivity ${\cal{R}} \approx(2\sigma R_2)^{-1}~(a/R_1)^{3/2}$, the
dissipation rate of electric current ($W$) in the primary atmosphere is:
\begin{equation}
\label{dissipation}
W = I^2 {\cal{R}} = I \Phi = k \omega^{17/3}_o (1-\alpha)^2
\end{equation}
where $k = 2 (\mu_1/c)^2 \sigma R^{3/2}_1 R^3_2 / [GM_1(1+q)]^{11/6}$ is a
constant of the system.\\
The Lorentz torque ($N_L$) has the following properties: \textit{i}) it acts
with the same magnitude and opposite signs on the primary star and the orbit,
$N_L = N^{(1)}_L = - N^{(\mbox{\tiny{orb}})}_L$. Therefore; \textit{ii})
$N_L$ conserves the total angular momentum in the system, transferring all
that is extracted from one component to the other one; \textit{iii}) $N_L$ is
simply related to the energy dissipation rate: $W = N_L \omega_o (1-\alpha)$.\\
From the above, the evolution equation for $\omega_1$ is:
\begin{equation}
\label{omega1}
\dot{\omega}_1 = (N_L/I_1) = \frac{W}{I_1 \omega_o (1-\alpha)}
\end{equation}
The orbital angular momentum is $L_o = I_o \omega_o$, so that the orbital
evolution equation is:
\begin{equation}
\label{omegaevolve}
\dot{\omega}_o = - 3 (N_{\mbox{\tiny{gw}}}+ N^{(\mbox{\tiny{orb}})}_L)/I_o =
- 3 (N_{\mbox{\tiny{gw}}} - N_L)/I_o =
-\frac{3}{I_o\omega_o}\left(\dot{E}_{\mbox{\tiny{gw}}} -\frac{W}{1-\alpha}
\right)
\end{equation}
where $I_o = q (1+q)^{-1/3} G^{2/3} M^{5/3}_1 \omega^{-4/3}_o$ is the orbital
moment of inertia and $N_{\mbox{\tiny{gw}}} = \dot{E}_{\mbox{\tiny{gw}}}/
\omega_o$ is the GW torque.
\subsection{Energetics of the electric circuit}
\label{sec:2}
Let us focus on
how energy is transferred and consumed by the electric circuit.
We begin considering the rate of work done by $N_L$ on the orbit
\begin{equation}
\label{Eorb}
\dot{E}^{(orb)}_L = N^{(orb)}_L \omega_o = -N_L  \omega_o = - \frac{W}
{1-\alpha},
\end{equation}
and that done on the primary:
\begin{equation}
\label{espin}
\dot{E}_{spin} = N_L \omega_1 = \frac{\alpha}{1-\alpha} W = -\alpha
\dot{E}^{(orb)}_L.
\end{equation}
The sum $\dot{E}_{spin} + \dot{E}^{(orb)}_L = -W$. Clearly, not all of the
energy extracted from one component is transferred to the other one. The
energy lost to ohmic dissipation represents the energetic cost of spin-orbit
coupling.\\
The above formulae allow to draw some further conclusions concerning the
relation between $\alpha$ and the energetics of the electrical circuit.
When $\alpha>1$, the circuit is powered at the expenses of the primary's
spin energy. A fraction $\alpha^{-1}$ of this energy is transferred to the
orbit, the rest being lost to ohmic dissipation. When $\alpha <1$, the circuit
is powered at the expenses of the orbital energy and a fraction $\alpha$ of
this energy is transferred to the primary spin.
Therefore, \textit{the parameter $\alpha$ represents a measure of the
energy transfer efficiency of spin-orbit coupling}: the more
asynchronous a system is, the less efficiently energy is transferred, most of
it being dissipated as heat.
\subsection{Stationary state: General solution}
\label{sec:2.1}
As long as the asynchronism parameter is sufficiently far from unity, its
evolution will be essentially determined by the strength of the synchronizing
(Lorentz) torque, the GW torque being of minor relevance. The evolution in
this case depends on the initial values of $\alpha$ and $\omega_o$, and on
stellar parameters.
This evolutionary phase drives $\alpha$ towards unity, \textit{i.e.} spin and
orbit are driven towards synchronism. It is in this regime that the GW torque
becomes important in determining the subsequent evolution of the system. \\
Once the condition $\alpha =1$ is reached, indeed, GW emission drives a small
angular momentum disequilibrium. The Lorentz torque is in turn switched on to
transfer to the primary spin the amount of angular momentum required for it to
keep up with the evolving orbital frequency.
This translates to the requirement that $\dot{\omega}_1 = \dot{\omega}_o$. By
use of expressions (\ref{omega1}) and (\ref{omegaevolve}), it is found
that this condition implies the following equilibrium value for $\alpha$
(we call it $\alpha_{\infty})$:
\begin{equation}
\label{alfainf}
1 - \alpha_{\infty} = \frac{I_1}{k} \frac{\dot{\omega}_o/\omega_o}
{\omega^{11/3}}
\end{equation}
This is greater than zero if the orbit is shrinking ($\dot{\omega}_o >0$),
which implies that $\alpha_{\infty} <1$. For a widening orbit, on the other
hand, $\alpha_{\infty} > 1$. However, this latter case does not correspond to
a long-lived configuration. Indeed,
define the electric energy reservoir as $E_{UIM} \equiv (1/2) I_1 (\omega^2_1
-\omega^2_o)$, which is negative when $\alpha <1$ and positive when $\alpha
>1$. Substituting eq. (\ref{alfainf}) into this definition:
\begin{equation}
\label{stationary}
\dot{E}_{UIM} = - W,
\end{equation}
If $\alpha = \alpha_{\infty} >1$, energy is consumed at the
rate W: the circuit will eventually switch off ($\alpha_{\infty}=1$). At later
times, the case $\alpha_{\infty}<1$ applies.\\
If $\alpha = \alpha_{\infty} <1$, condition (\ref{stationary}) means that the
battery recharges at the rate $W$ at which electric currents dissipate
energy: the electric energy reservoir is conserved as the binary evolves.\\
The latter conclusion can be reversed (cfr. Fig. \ref{fig:2}): in steady-state,
 the rate of energy dissipation ($W$) is fixed by the rate at which power is
fed to the circuit by the plug ($\dot{E}_{UIM}$). The latter is determined by
GW emission and the Lorentz torque and, therefore, by component masses,
$\omega_o$ and $\mu_1$.\\
Therefore the steady-state degree of asynchronism of a given binary system is
uniquely determined, given $\omega_o$. Since both $\omega_o$ and
$\dot{\omega}_o$ evolve secularly, the equilibrium state will be
``quasi-steady'', $\alpha_{\infty}$ evolving secularly as well.
\subsection{Model application: equations of practical use}
\label{application}
We have discussed in previous sections the existence of an asymptotic regime
in the evolution of binaries in the UIM  framework.
Given the definition of $\alpha_\infty$ and $W$ (eq.
\ref{alfainf} and \ref{dissipation}, respectively), we have:
\begin{equation}
\label{luminosity}
W = I_1 \omega_o \dot{\omega}_o~\frac{(1-\alpha)^2}{1-\alpha_{\infty}}.
\end{equation}
The quantity $(1-\alpha_{\infty})$ represents the \textit{actual} degree of
asynchronism only for those systems that had enough time to evolve towards
steady-state, \textit{i.e} with sufficiently short orbital period.
In this case, the steady-state source luminosity can thus be written as:
\begin{equation}
\label{luminsteady}
W_{\infty} = I_1 \dot{\omega}_o \omega_o (1-\alpha_{\infty})
\end{equation}
Therefore - under the assumption that a source is in steady-state - the
quantity $\alpha_{\infty}$ can be determined from the
measured values of $W$, $\omega_o$, $\dot{\omega}_o$. Given its definition
(eq. \ref{alfainf}), this gives an estimate of $k$ and, thus, $\mu_1$.\\
The equation for the orbital evolution (\ref{omegaevolve}) provides a further
relation between the three measured quantities, component masses and degree of
asynchronism.
This can be written as:
\begin{equation}
\label{useful}
\dot{E}_{\mbox{\tiny{gr}}} + \frac{1}{3} I_o \omega^2_o (\dot{\omega}_o /
\omega_o) =  \frac{W} {(1-\alpha)} \nonumber
\end{equation}
that becomes, inserting the appropriate expressions for $\dot{E}_
{\mbox{\tiny{gr}}}$ and $I_o$:
\begin{equation}
\label{extended}
\frac{32}{5}\frac{G^{7/3}}{c^5}~\omega^{10/3}_o X^2  -\frac{1}{3}~
G^{2/3} \frac{\dot{\omega}_o}{\omega^{1/3}_o} X + \frac{W}{1-\alpha} = 0~,
\end{equation}
where $X \equiv M^{5/3}_1 q/(1+q)^{1/3} = {\cal{M}}^{5/3}$, $\cal{M}$ being the
system's chirp mass.
\section{RX\,J0806.3+1527\ }
\label{rxj08}
We assume here the values of $\omega_o$, $\dot{\omega}_o$ and of the
bolometric luminosity reported in \ref{sec:0} and refer to
See \cite{2006astro.ph..3795D} for a complete
discussion on how our conclusions
depend on these assumptions.\\
In Fig. \ref{fig3} (see caption for further details), the dashed line
represents the locus of points in the $M_2$ vs. $M_1$ plane, for which the
measured $\omega_o$ and $\dot{\omega}_o$ are consistent with being due to GW
emission only, \textit{i.e.} if spin-orbit coupling was absent ($\alpha =1$).
This corresponds to a chirp mass ${\cal{M}} \simeq$ 0.3 M$_{\odot}$. \\
Inserting the measured quantities in eq. (\ref{luminosity}) and assuming a
reference value of
$I_1 = 3 \times 10^{50}$ g cm$^2$, we obtain:
\begin{equation}
\label{j08}
\frac{(1-\alpha)^2} {1-\alpha_{\infty}} \simeq \frac{10^{32}d^2_{200}}
{1.3 \times 10^{34}} \simeq 8 \times 10^{-3} d^2_{200}.
\end{equation}
In principle, the source may be in any regime, but our aim is to check whether
it can be in steady-state, as to avoid the short timescale problem mentioned
in \ref{sec:1}. Indeed, the short orbital period strongly suggests it may
have reached the asymptotic regime (cfr. \cite{2006astro.ph..3795D}).
If we assume $\alpha =\alpha_{\infty}$, eq. (\ref{j08}) implies
$(1-\alpha_{\infty}) \simeq 8 \times 10^{-3}$.\\
Once UIM and spin-orbit coupling are introduced, the locus of allowed points
in the M$_2$ vs. M$_1$ plane is somewhat sensitive to the exact value of
$\alpha$: the solid curve of Fig. \ref{fig3} was obtained, from eq.
(\ref{extended}), for $\alpha = \alpha_{\infty} = 0.992$. \\
From this we conclude that, if RX\,J0806.3+1527\  is interpreted as being in the UIM
steady-state, $M_1$ must be smaller than 1.1 $M_{\odot}$ in order for the
secondary not to fill its Roche lobe, thus avoiding mass transfer.
\begin{figure}[h]
\centering
\includegraphics[height=10.18cm,angle=-90]{Massej08.ps}
\caption{M$_2$ vs. M$_1$ plot based on the measured timing properties of
RX\,J0806.3+1527\ . The dashed curve is the locus expected if orbital decay is driven
by GW alone, with no spin-orbit
coupling. The solid line describes the locus expected if the
system is in a steady-state, with $(1-\alpha) = (1-\alpha_{\infty}) \simeq 8
\times 10^{-3}$. The horizontal dot-dashed line represents the minimum mass
for a degenerate secondary not to fill its Roche-lobe at an orbital period of
321.5 s. Dotted lines are the loci of fixed mass ratio.}
\label{fig3}
\end{figure}
From $(1-\alpha_{\infty}) = 8\times 10^{-3}$ and from eq. (\ref{dissipation}),
$k \simeq 7.7 \times 10^{45}$ (c.g.s.): from this, component masses and
primary magnetic moment can be constrained.
Indeed, $k = \hat{k}(\mu_1, M_1, q; \overline{\sigma})$ (eq.
\ref{dissipation}) and a further constraint derives from the fact that $M_1$
and $q$ must lie along the solid curve of Fig. \ref{fig3}. Given the value of
$\overline{\sigma}$, $\mu_1$ is obtained for each point along the solid
curve. We assume an electrical conductivity of
$\overline{\sigma} = 3\times 10^{13}$ e.s.u.
\cite{2002MNRAS.331..221W,2006astro.ph..3795D}. \\
The values of $\mu_1$ obtained in this way, and the
corresponding field at the primary's surface, are plotted in Fig. \ref{fig2},
from which $\mu_1 \sim$ a few $\times 10^{30}$ G cm$^3$ results, somewhat
sensitive to the primary mass.\\
We note further that, along the solid curve of Fig. \ref{fig3}, the chirp mass
is slightly variable, being: $X \simeq (3.4 \div 4.5) \times 10^{54}$
g$^{5/3}$, which implies ${\cal{M}} \simeq (0.26 \div 0.31)$ M$_{\odot}$.
More importantly, $\dot{E}_{\mbox{\tiny{gr}}} \simeq (1.1 \div 1.9) \times
10^{35}$ erg s$^{-1}$ and, since $W/(1-\alpha_{\infty}) = \dot{E}^{(orb)}_L
\simeq 1.25 \times 10^{34}$ erg s$^{-1}$, we have
$\dot{E}_{\mbox{\tiny{gr}}} \simeq (9\div 15)~\dot{E}^{(orb)}_L $.
Orbital spin-up is essentially driven by GW alone; indeed, the dashed and solid
 curves are very close in the M$_2$ vs. M$_1$ plane. \\
\begin{figure}[h]
\centering
\includegraphics[height=10.18cm,angle=-90]{Muj08.ps}
\caption{The value of the primary magnetic moment $\mu_1$, and the
corresponding surface B-field, as a function of the primary mass M$_1$, for
$(1-\alpha) = (1-\alpha_{\infty}) = 8\times 10^{-3}$.}
\label{fig2}
\end{figure}
Summarizing, the observational properties of RX\,J0806.3+1527\  can be well
interpreted in the UIM framework, assuming it is in steady-state.
This requires the primary to have $\mu_1 \sim 10^{30}$ G cm$^3$ and a spin
just slightly slower than the orbital motion (the difference being less
than $\sim 1$\%). \\
The expected value of $\mu_1$ can in principle be tested by future
observations, through studies of polarized emission at optical and/or radio
wavelenghts \cite{Willes and Wu(2004)}.
\section{RX\,J1914.4+2456\ }
\label{rxj19}
As for the case of RX\,J0806.3+1527\ , we adopt the values discussed in
\ref{sec:0} and refer to \cite{2006astro.ph..3795D} for
a discussion of all
the uncertainties on these values and their implications for the model.\\
Application of the scheme used for RX\,J0806.3+1527\  to this source is not as
straightforward. The inferred luminosity of this source seems inconsistent
with steady-state.
With the measured values of\footnote{again assuming $I_1= 3 \times 10^{50}$ g
cm$^2$} $\omega_o$ and $\dot{\omega}_o$,
the system steady-state luminosity should be $ < 2 \times 10^{32}$ erg
s$^{-1}$ (eq. \ref{luminsteady}). This is hardly consistent even with
the smallest possible luminosity referred to in  \ref{sec:0}, unless
allowing for a large value of ($1-\alpha_{\infty} \geq 0.15 $). \\
From eq. (\ref{luminosity}) a relatively high ratio between the actual
asynchronism parameter and its steady-state value appears unavoidable:
\begin{equation}
\label{j19}
|1-\alpha| \simeq  2.2 (1-\alpha_{\infty})^{1/2}
\end{equation}
\subsubsection{The case for $\alpha>1$}
\label{thecase}
The low rate of orbital shrinking measured for this source and its relatively
high X-ray luminosity put interesting constraints on the primary spin. Indeed,
a high value of $N_L$ is associated to $W\sim 10^{33}$ erg s$^{-1}$. \\
If $\alpha<1$, this torque sums to the GW torque: the resulting orbital
evolution would thus be faster than if driven by GW alone. In fact, for
$\alpha < 1$, the smallest possible value of $N_L$ obtains with $\alpha = 0$,
from which $N^{(\mbox{\tiny{min}})}_L = 9 \times 10^{34}$ erg. This implies an
absolute minimum to the rate of orbital shrinking (eq. \ref{omegaevolve}),
$3~N^{(\mbox{\tiny{min}})}_L / I_o$, so close to the measured one
that unplausibly small component masses would be required for
$\dot{E}_{\mbox{\tiny{gr}}}$ to be negligible.
We conclude that $\alpha <1$ is essentially ruled out in the UIM discussed
here. \\
If $\alpha > 1$ the primary spin is faster than the orbital motion and the
situation is different.
Spin-orbit coupling has an opposite sign with respect to the GW torque.
The small torque on the orbit implied by the measured $\dot{\omega}_o$ could
result from two larger torques of opposite signs partially cancelling each
other.\\
This point has been overlooked by  \cite{Marsh and Nelemans (2005)} who
estimated the
GW luminosity of the source from its measured timing parameters and, based on
this estimate, claimed the failure of the UIM for RX\,J1914.4+2456\ .
In discussing this and other misinterpretations of the UIM in the literature,
\cite{2006astro.ph..3795D} show that the argument by
\cite{Marsh and Nelemans (2005)}
actually leads to our same conclusion: in the UIM framework, the orbital
evolution of this source must be affected significantly by spin-orbit
coupling, being slowed down by the transfer of angular momentum and energy
from the primary spin to the orbit. The source GW luminosity must accordingly
be larger than indicated by its timing parameters.
\subsection{Constraining the asynchronous system}
\label{lifetime}
Given that the source is not compatible with steady-state, we constrain system
parameters in order to match the measured values of $W, \omega_o$ and
$\dot{\omega}_o$ and meet the requirement that the resulting state has a
sufficiently long lifetime.\\
\begin{figure}[h]
\centering
\includegraphics[height=10.18cm,angle=-90]{solnew.ps}
\caption{M$_2$ vs. M$_1$ plot based on measured timing properties of
RX\,J1914.4+2456\ . The dot-dashed line corresponds to the minimum mass for a
degenerate secondary not to fill its Roche-lobe. The dashed curve represents
the locus expected if orbital decay was driven by GW alone, with no spin-orbit
coupling. This curve is consistent with a detached system only for extremely
low masses. The solid lines describe the loci expected if spin-orbit coupling
is present (the secondary spin being always tidally locked) and gives a
\textit{negative} contribution to $\dot{\omega}_o$. The four curves are
obtained for $W = 10^{33}$ erg s$^{-1}$ and four different values of $\alpha =
1.025, 1.05, 1.075$, $1.1$, respectively, from top to bottom, as reported in
the plot.}
\label{fig5}
\end{figure}
Since system parameters cannot all be determined uniquely, we adopt the
following scheme: given a value of $\alpha$ eq. (\ref{extended}) allows to
determine, for each value of M$_1$, the corresponding value of M$_2$ (or $q$)
that is compatible with
the measured $W, \omega_o$ and $\dot{\omega}_o$. This yields the solid curves
of Fig. \ref{fig5}.\\
As these curves show, the larger is $\alpha$ and the smaller the upward shift
of the corresponding locus. This is not surprising, since these curves are
obtained at fixed luminosity $W$ and $\dot{\omega}_o$. Recalling
that  $(1/\alpha)$ gives the efficiency of energy transfer in systems with
$\alpha >1$ (cfr. \ref{sec:2}), a higher $\alpha$ at a given
luminosity implies that less energy is being transferred to the orbit.
Accordingly, GWs need being emitted at a smaller rate to match the measured
$\dot{\omega}_o$.\\
The values of $\alpha$ in Fig. \ref{fig5} were chosen arbitrarily and are
just illustrative: note that the resulting curves are similar to those
obtained for RX\,J0806.3+1527\ .
Given $\alpha$, one can also estimate $k$ from the definiton of $W$ (eq.
\ref{dissipation}). The information given by the curves of
Fig. \ref{fig5} determines all quantities contained in $k$, apart from
$\mu_1$. Therefore, proceeding as in the previous section,
we can determine the value of $\mu_1$ along each of the four curves of Fig.
\ref{fig5}. As in the case of RX\,J0806.3+1527\ , derived values are in the $\sim
10^{30}$ G cm$^3$ range. Plots and discussion of these results are reported by
\cite{2006astro.ph..3795D}.\\
We finally note that the curves of Fig. \ref{fig5} define the value of $X$ for
each (M$_1$,M$_2$), from which the system GW luminosity $\dot{E}_
{\mbox{\tiny{gr}}}$ can be calculated and its ratio to spin-orbit coupling.
According to the above curves, the expected GW luminosity of this source is
in the range $(4.6 \div 1.4) \times 10^{34}$ erg s$^{-1}$. The corresponding
ratios $\dot{E}_{\mbox{\tiny{gr}}}/ \dot{E}^{(orb)}_L$ are $1.15, 1.21, 1.29$
and $1.4$, respectively, for $\alpha =1.025, 1.05, 1.075$ and $1.1$.\\
Since the system cannot be in steady-state a strong question on the duration
of this transient phase arises.
\begin{figure}[h]
\centering
\includegraphics[height=10.18cm,angle=-90]{tauj19.ps}
\caption{The evolution timescale $\tau_{\alpha}$ as a function of the primary
mass for the same values of $\alpha$ used previously, reported on the curves.
Given the luminosity $W\sim 10^{33}$ erg s$^{-1}$ and a value of $\alpha$,
$\tau_{\alpha}$ is calculated as a function of M$_1$.}
\label{fig8}
\end{figure}
The synchronization timescale $\tau_{\alpha} = \alpha / \dot{\alpha}$ can be
estimated combining eq. (\ref{omega1})  and (\ref{omegaevolve}). With the
measured values of $W$, $\omega_o$ and $\dot{\omega}_o$, $\tau_{\alpha}$ can
be calculated as a function of $I_1$ and, thus, of M$_1$, given a particular
value of $\alpha$. Fig. \ref{fig8} shows results obtained for the same four
values of $\alpha$ assumed previously. The resulting timescales range from a
few $\times 10^4$ yrs to a few $\times 10^5$ yrs, tens to hundreds times
longer than previously obtained and compatible with constraints derived from
the expected population of such objects in the Galaxy.
Reference \cite{2006astro.ph..3795D} discuss this point
and its implications in more
detail.
\section{Conclusions}
\label{conclusions}
The observational properties of the two DDBs with the shortest orbital period
known to date have been discussed in relation with their physical nature. \\
The Unipolar Inductor Model and its coupling to GW emission have been
introduced to explain a number of puzzling features that these two sources
have in common and that are difficult to reconcile with most, if not all,
models of mass transfer in such systems.\\
Emphasis was put on the relevant new physical features that characterize the
model. In particular, the role of spin-orbit coupling through the Lorentz
torque and the role of GW emission in keeping the electric interaction active
at all times has been thoroughly discussed in all their implications.
It has been shown that the model does work over arbitrarily long timescales.\\
Application of the model to both RX\,J0806.3+1527\  and RX\,J1914.4+2456\  accounts in a
natural way for their main observational properties. Constraints on physical
parameters are derived in order for the model to work, and can be verified by
future observations.\\
It is concluded that the components in these two binaries may be much more
similar than it may appear from their timing properties and luminosities.
The significant observational differences could essentially be due to the two
systems being caught in different evolutionary stages. RX\,J1914.4+2456\  would be in
a luminous, transient phase that preceeds its settling into the dimmer
steady-state, a regime already reached by the shorter period RX\,J0806.3+1527\ .
Although the more luminous phase is transient, its lifetime can be as long as
$ \sim 10^5$ yrs, one or two orders of magnitude longer than previously
estimated.\\
The GW luminosity of RX\,J1914.4+2456\  could be much larger than previously expected
since its orbital evolution could be largely slowed down by an additional
torque, apart from GW emission.\\
Finally, we stress that further developements and refinements of the model are
required to address more specific observational issues and to assess the
consequences that this new scenario might have on evolutionary scenarios and
population synthesis models.


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

Toward the sixth generation of wireless networks (6G), a number of exciting applications will benefit from sensing services provided by future perceptive networks, where sensing capabilities are integrated in the communication network. Once the communication network infrastructure is already deployed with multiple interconnected nodes, a multi-static sensory mesh can be enabled and exploited to improve the performance of the network itself~\cite{9296833}. Therefore, the joint communications and sensing (JCAS) concept has emerged as an enabler for an efficient use of  radio resources for both communications and sensing purposes, where high frequency bands, that are expected to be available in 6G, can favor very accurate sensing based on radar-like technology~\cite{art:Wild_Nokia}.

Relying on the coordination of the network and a distributed processing, sensing signals can be transmitted from one node, and  the reflections on the environment can be received at multiple nodes, in a coordinated manner~\cite{art:Wild_Nokia}. Thus, distributed multi-static sensing approaches can improve sensing accuracy while alleviating the need of full-duplex operation at sensing nodes. In this context, the high-gain directional beams provided by performing beamforming in multiple-input multiple-output (MIMO) and massive MIMO systems, which are essential for the operation of communication systems at higher frequencies, will be also exploited for improving sensing by considering distributed implementations~\cite{8764485,art:multistatic_Merlano}. In multi-static MIMO radar settings, the synchronization among sensing nodes is crucial, thus, this issue has motivated the study of feasibility of synchronization. For instance, a synchronization loop using in-band full duplex (IBFD) was demonstrated for a system with two MIMO satellites sensing two ground targets in~\cite{art:multistatic_Merlano}.

Additionally, multicarrier signals  such as orthogonal frequency-division multiplexing (OFDM) waveforms have proven to provide several advantages for the use on JSAC systems, including independence from the transmitted user data, high dynamic range, possibility to estimate the relative velocity, and efficient implementation based on fast Fourier transforms~\cite{5776640}.
For instance, uplink OFDM 5G New Radio (NR) waveforms have been effectively used for indoor environment mapping in \cite{art:Baquero_mmWaveAlg}. Therein, a prototype full-duplex transceiver was used to perform range-angle chart estimation and dynamic tracking via extended Kalman Filter.

Moreover, the capabilities of distributed sensing system can be further extended by relying on the advantages of flexible nodes as unmanned aerial vehicles (UAVs), which have already raised significant attention for their applicability in numerous scenarios and even in harsh environments~\cite{8877114}. Therefore, UAVs have been already considered for sensing purposes~\cite{art:Wei_UAV_Safe,art:UAVs_Guerra,Chen_JSACUAVSystem}.
For instance, in~\cite{art:Wei_UAV_Safe}, UAVs are explored to  perform simultaneous jamming and sensing of UAVs acting as eavesdroppers by exploiting the jamming signals for sensing purposes. Therein, sensing information is used to perform optimal online resource allocation to maximise the amount of securely served users constrained by the requirements on the information leakage to the eavesdropper and the data rate to the legitimate users. Besides in~\cite{art:UAVs_Guerra}, a UAV-based distributed radar is proposed to perform distributed sensing to locate and track malicious UAVs using frequency modulated continuous wave (FMCW) waveforms. It was shown that the mobility and distributed nature of the UAV-based radar benefits the accuracy for tracking mobile nodes if compared with a fixed radar. However, it does not make complete use of its distributed nature, as each UAV performs local sensing accounting for only the sensing information of its neighbouring UAVs, and there is no consideration on communication tasks.
In the context of JSAC, in~\cite{Chen_JSACUAVSystem}, a general framework for a full-duplex JSAC UAV network is proposed, where area-based metrics are developed considering sensing and communication parameters of the system and sensing requirements. This work uses a full-duplex operation for local sensing at each UAV while considering reflections from other UAVs as interference.

Different from previous works and considering the complexity of full-duplex systems, this work focuses on half-duplex operation and proposes a framework for performing a grid-based distributed sensing relying on the coordination of multiple UAVs to sense a ground target located on an area of interest. It is considered that MIMO UAVs employ OFDM waveforms and digital beamforming is implemented at the receiver side. A periodogram is used for the estimation of the radar cross-section (RCS) of each cell in the grid, leveraging the knowledge of the geometry of the system. The RCS estimation is performed by all of the non-transmitting UAVs, simultaneously, while one UAV is illuminating a certain sub-area of the grid. This process is performed until all UAVs have illuminated their respective sub-areas, then all UAVs inform the measured RCS per cell on the grid to a UAV acting as a fusion center (FC), which performs information-level fusion. This process allows a half-duplex operation in a distributed sensing setting.

\section{System Model}\label{sec:SysModel}
\begin{figure}[bt]
    \centering
    \includegraphics[width=0.73\linewidth]{sysmod4.pdf}
    \caption{System model.\vspace{-1em}}
    \label{fig:sysModel}
\end{figure}
Consider the system depicted in Fig.~\ref{fig:sysModel}, where a single point-like target of RCS $\sigma_{\mathrm{T}}$ is positioned on a square area $S$ of $\ell$ meters of side length. $U$ UAVs are deployed (for simplicity and insighfulness) at a common altitude $h$ and are coordinated to perform distributed sensing to locate the ground target. Each UAV $u$ in the set of all UAVs $\mathcal{U}$, with $u\in\mathcal{U}$, is positioned at coordinates $\mathbf{r}_u = (x_u,y_u,h)$, with $|\mathcal{U}|=U$. Also, the RCS of a ground cell is denoted by $\sigma_{G}$.

Similar to~\cite{Chen_JSACUAVSystem}, it is assumed that each UAV has two arrays of antennas namely a square uniform planar array (UPA)  (mounted facing downward) for sensing  and a uniform linear array (ULA) (mounted horizontally) to communicate with the FC for information fusion and coordination tasks. The square UPA consists of $n\times n$ isotropic antenna elements spaced $\lambda/2$ from each-other, where $\lambda=f_0/c_0$ is the wavelength of the signal, $f_0$ is the frequency of the signal, and $c_0$ is the speed of light.

\if\mycmd1

To perform sensing, the UAV $u \in \mathcal{U}$ estimates the RCS of a certain point on the ground, denoted as $p$,  located at the coordinates $\mathbf{r}_p=(x_p,y_p,0)$. For this purpose, $u$ utilizes a digital receive beamformer $\mathbf{w}_p\in\mathbb{C}^{n\times 1}$. The reflection from point $p$ arriving at UAV $u$ has an angle-of-arrival (AoA) of $\varphi_{p,u} = (\theta_{p,u},\phi_{p,u})$, where $\theta_{p,u}$ corresponds to the elevation angle and $\phi_{p,u}$ to the azimuth. The corresponding beam-steering vector $\mathbf{g}(\varphi_{p,u})$ has its elements $g_{ij}(\varphi_{p,u})$ for all $i,j = 0,...,n-1$, where $i$ is the index corresponding to the antenna element in the $x$ axis and $j$ in the $y$ axis of the UPA defined as~\cite{book:BalanisAntennas}
\begin{align}\nonumber
    g_{ij}(\varphi_{p,u}) = &e^{-j\pi (i-1)\sin(\theta_{p,u})\sin(\phi_{p,u})} \times \\
    &e^{-j\pi (j-1)\sin(\theta_{p,u})\cos(\phi_{p,u})}.
\end{align}
The steering matrix $\mathbf{G}_u \in \mathbb{C}^{n^2\times H}$ contains the steering vectors corresponding to the $H$ reflections captured at UAV $u$ as
\begin{equation}
    \mathbf{G}_u = [\mathbf{g}(\varphi_{1,u}),..., \mathbf{g}(\varphi_{H,u})]_{n^2\times H},
\end{equation}
where $n^2$ is the total number of antennas at UAV $u$.

After applying the receive beamformer $\mathbf{w}_{p}$ at reception, the beam pattern from the reflections captured at $u$ is given by
\begin{align}
    \pmb{\chi} =  \mathbf{G}_u^T\mathbf{w}_{p} = [\chi(\varphi_{1,u}),...,\chi(\varphi_{H,u})]^T,
\end{align}
where $\chi(\varphi_{p,u})$ is the gain of the reflection coming from $p$ and $\pmb{\chi}$ is the beam pattern vector of size $H\times 1$ at every AoA  by applying beamformer $\mathbf{w}_{p}$ at UAV $u$.

\fi

\section{Distributed Sensing Protocol}\label{sec:Protocol}

For the sensing process, it is considered that the total area $S$ is sectioned into a grid composed of $L\times L$ square cells with dimensions $ d \times d $ with $d = \ell/L$. Each cell is characterised by its middle point $p$ of position $\mathbf{r}_p =(x_p,y_p,0)$ such that $p\in\mathcal{P}$, where $\mathcal{P}$ is the set of all cells. For notational simplicity we will refer a certain cell by its middle point $p$. The point $p^*$ represents the target, which is located in the position $\mathbf{r}_{p^*} =(x_{p^*},y_{p^*},0)$.
It is also considered that, at a certain time, a UAV $u\in\mathcal{U}$ illuminates straight down with its UPA working as a phased array,
thus the half power beam width (HPBW) projection forms a circle on the ground. In this sense, it is assumed that the cells that are completely inside the largest inscribed square of the HPBW projection are the intended ones to be sensed by the reflections produced from the illumination of UAV $u$, and are characterised as the cell set $\mathcal{P}_{u}$, while the set of non-intended illuminated cells $\mathcal{P}_{u}'$ contains the cells that are not inside the largest inscribed squared, which are treated as clutter, as illustrated in Fig.~\ref{fig:gridCells}.
\begin{figure}[bt]
    \centering
    \includegraphics[width=0.95\linewidth]{gridFig_Cellsu.pdf}
    \caption{Illumination grid.}
    \label{fig:gridCells}
    \vspace{-1em}
\end{figure}
In total, the set of illuminated cells
is given as $\mathcal{Q}_{u} = \mathcal{P}_{u}\cup\mathcal{P}_{u}'$.

The distributed sensing framework is summarized as follows
\begin{itemize}[label={},leftmargin=*]
    \item \textbf{Step 1:} The UAVs coordinate and assume their positions to cover the whole area of interest $S$, such that every cell in $\mathcal{P}$ is contained in a single $\mathcal{P}_u$, $u\in\mathcal{U}$.
    \item \textbf{Step 2:} UAV $u\in\mathcal{U}$ illuminates the ground directly below acting as a phased array, illuminating the elements of $\mathcal{Q}_u$, and potentially, the target $p^*$.
    \item \textbf{Step 3:} Every other UAV  $u'\in\mathcal{U}\setminus\{u\}$ processes the incoming reflections by choosing a cell $p\in\mathcal{P}_u$ and for that cell
    \begin{itemize}
        \item computes and applies a digital receive beamformer as described in Section~\ref{sec:BF}, and
        \item computes the periodogram corresponding to $p$, and estimates its RCS as described in Section~\ref{sec:Periodogram}.

    \end{itemize}

    \item \textbf{Step 4:} Repeat Step 3 for all cells $p\in\mathcal{P}_u$.
    \item \textbf{Step 5:} Repeat Steps 2-4 for all UAVs $u\in\mathcal{U}$. After this, each UAV $u$ has an estimated RCS map of the grid, $\mathbf{\Hat{\Gamma}}_u$, which is a matrix of RCS estimates of all cells in $\mathcal{P} \setminus \mathcal{P}_u$. This occurs because the UAV $u$ does not estimate the RCS of the cells in $\mathcal{P}_u$, thus avoiding the need for a full-duplex system at the UAVs.

    \item \textbf{Step 6:} All UAVs $u\in\mathcal{U}$ send their RCS estimation maps $\mathbf{\Hat{\Gamma}}_u$ to the FC for information-level fusion.
    \item \textbf{Step 7:} The FC fuses the estimates together into the fused RCS map $\mathbf{\Hat{\Gamma}}$, and, by assuming a non-reflective ground such that the RCS of the ground is smaller than that of the target ($\sigma_{\mathrm{G}} < \sigma_{\mathrm{T}}$), the target is estimated to be located in the cell of highest estimated RCS, i.e. in $\argmax \mathbf{\Hat{\Gamma}}$, as described in Section~\ref{sec:Fuse}.
\end{itemize}

\section{Beamformer Design}\label{sec:BF}
The receive beamformer is designed to have the main lobe of the resulting beam pattern steered towards the intended cell $p$ in order to estimate its RCS. To this end, two different approaches are considered for the design of the receive beamformer, namely least-squares (LS) heuristic formulation and the minimum variance beam pattern based on Capon method. These approaches are described in the following.

\subsection{Least-Squares heuristic approach}

For this approach, the beamformer is obtained by solving the following constrained LS optimisation problem \cite{art:Shi_ILS,art:Zhang_DBR}
\begin{subequations}
\begin{alignat}{3}
    \mathrm{\textbf{P1:}}\;\;\;\;&\mathrm{minimise}&\qquad&|| \mathbf{G}^T\mathbf{w}_{p} - \mathbf{v} ||_2^2\\
    &\mathrm{subject~to}&\qquad&|| \mathbf{w}_{p} ||_2^2 = 1,
\end{alignat}
\end{subequations}
where $\mathbf{v}$ is the desired response vector over the $H'$ AoAs in the beam-steering matrix $\mathbf{G}\in\mathbb{C}^{n\times H'}$.
In this approach, a heuristic is employed, in which the AoAs in $\mathbf{A}$ are chosen such that they span evenly on the elevation and azimuth ranges, centred around the intended AoA $\varphi_{p,u}$. The AoAs are taken as a mesh of $n$ elevation angles and $4n$ azimuth angles respectively given by
\begin{alignat}{3}
    \theta_i =&\!\mod{\left( \theta_{p,u} + \frac{i\pi}{2(n-1)}, \frac{\pi}{2} \right)}, \; & i=0,...,n-1\\
    \phi_j =&\!\mod{\left( \phi_{p,u} + \frac{j2\pi}{4n-1}, 2\pi \right)}, \; & j=0,...,4n-1,
\end{alignat}
such that $H' = 4n^2$.

The solution of \textbf{P1} is well known to be $\mathbf{w}_{p} = (\mathbf{A}^T)^\dagger \mathbf{v}$ \cite{art:Zhang_DBR} where $(\cdot)^\dagger$ is the pseudo-inverse, but, since $\mathbf{A}^T$ is a matrix with more rows than columns, it can be efficiently solved by applying Cholesky factorization. Therefore, the iterative LS algorithm proposed in \cite{art:Shi_ILS} can be employed to solve \textbf{P1}.

\subsection{Capon method}

The Capon method provides minimum-variance distortionless
response beamforming and can be formulated as a quadratic program (QP) convex optimisation problem \cite{art:Stoica_Capon}
\begin{subequations}
\begin{alignat}{3}
\mathrm{\textbf{P2:}}\;\;\;\;&\mathrm{minimise}&\qquad&\mathbf{w}_{p}^H\mathbf{R}\mathbf{w}_{p}\\
    &\mathrm{subject~to}&\qquad& \mathbf{w}_{p}^H\mathbf{g}(\varphi_{p,u'})  = 1,
\end{alignat}
\end{subequations}
where $\mathbf{R}\in\mathbb{C}^{n\times n}$ is the covariance matrix of the received signal over the desired direction, which can be defined as $\mathbf{R} = \mathbf{g}(\varphi_{p,u'})\mathbf{g}(\varphi_{p,u'})^H + \alpha \mathbf{I}$~\cite{art:Shi_ILS} ,
where $\mathbf{I}\in\mathbb{R}^{n\times n}$ is the identity matrix and $\alpha$ is a small real number. The solution for \textbf{P2} is obtained as in~\cite{art:Stoica_Capon}, and given by
\begin{equation}
    \mathbf{w_{p}} = \frac{\mathbf{R}^{-1}\mathbf{g}(\varphi_{p,u'})}{\mathbf{g}(\varphi_{p,u'})^H\mathbf{R}^{-1}\mathbf{g}(\varphi_{p,u'})}.
\end{equation}

\section{Periodogram}\label{sec:Periodogram}
\if\mycmdPeriodogram0

\fi
\if\mycmdPeriodogram1
For performing sensing, the UAVs illuminating ground transmit frames consisting of $N$ OFDM symbols, each consisting of $M$ orthogonal subcarriers. The transmitted OFDM frame can be expressed as a matrix denoted by $\mathbf{F_{TX}}=[c^{TX}_{k,l}]\in\mathcal{A}^{N\times M}$ with $k=0,...,N-1$, $l=0,...,M-1$ and $\mathcal{A}$ is the modulated symbol alphabet.  At the side of sensing UAVs, the received frame matrix is denoted by $\mathbf{F_{RX}}=[c^{RX}_{k,l}]$ and is composed by the received baseband symbols corresponding to all reflections from $\mathcal{Q}_{\mathrm{u}}$ at UAV $u'$. The elements of the received frame matrix have the form~\cite{art:Baquero_OFDM,art:OFDM_Samir
\begin{align}\label{eq:bPointTarg_Mult} \nonumber
    &c^{RX}_{k,l} =   ~~b_{p}c^{TX}_{k,l}\chi(\varphi_{p,u'}) e^{j2\pi f_{D,p}T_o l}e^{-j2\pi \tau_{p} \Delta f k}e^{-j\zeta_{p}}\\ \nonumber
            &+\sum_{p'\in\mathcal{Q}_{\mathrm{u}}\setminus\{p\}} b_{p'}c^{TX}_{k,l}\chi(\varphi_{p',u'}) e^{j2\pi f_{D,p'}T_o l}e^{-j2\pi \tau_{p'} \Delta f k}e^{-j\zeta_{p'}}\\

            &+ \delta_{u}b_{p^*}c^{TX}_{k,l}\chi(\varphi_{p^*,u'}) e^{j2\pi f_{D,p^*}T_o l}e^{-j2\pi \tau_{p^*} \Delta f k}e^{-j\zeta_{p^*}}+ z_{k,l},

\end{align}
where $f_{D,p}$ is the Doppler experienced by the reflection from $p$ (assumed constant through the frame), $T_o$ is the OFDM symbol duration (including the cyclic prefix time $T_{CP}$), $\tau_p$ is the delay of the reflection from $p$, $\Delta f$ is the inter-carrier spacing, $\zeta_p$ is a random phase shift of the reflection from $p$, $z_{k,l}$ is additive white Gaussian noise (AWGN) of spectral density $N_0$, and $b_p$ is a term embedding the propagation of the wave and the reflecting characteristics of the reflector in $p$. In this expression, the first term corresponds to the intended cell $p$; the second term corresponds to the interference from the other cells, $p'$, in $\mathcal{Q}_u$; and the third term corresponds to the target reflection, with $\delta_{u}=1$ if the target has been illuminated by UAV $u$, and $\delta_{u}=0$ otherwise. Considering a point-source model, $b_p$ is the amplitude attenuation of the signal, given by~\cite{art:OFDM_Samir}
\begin{equation}\label{eq:b_val}
    b_p = \sqrt{\frac{P_T G_T \sigma_p \lambda^2}{(4\pi)^3 d_{p,1}^2d_{p,2}^2}},
\end{equation}
where $\sigma_p\in\{\sigma_{\mathrm{G}},\sigma_{\mathrm{T}}\}$, $P_T$ is the transmit power, $G_T$ is the transmit antenna gain, $d_{p,1}$ is the distance from $u$ to $p$ and $d_{p,2}$ is the distance from $p$ to $u'$.

The received complex symbols $c^{RX}_{k,l}$ contain the transmitted symbols $c^{TX}_{k,l}$, thus, are data-dependent. In order to process $\mathbf{F_{RX}}$, this data-dependency is removed by performing element-wise division, $\mathbf{F}$$=$$\mathbf{F_{RX}}\oslash \mathbf{F_{TX}}$,  to obtain the processed samples consisting of elements $c_{k,l}$$ = $$c^{RX}_{k,l}/c^{TX}_{k,l}$.

To estimate the delay and Doppler from $\mathbf{F}$, a common approach for OFDM signals is to use the periodogram, which provides the maximum likelihood (ML) estimator~\cite{art:Baquero_OFDM}. The periodogram takes the fast Fourier transform (FFT) of $\mathbf{F}$ over OFDM symbols, followed by the inverse FFT (IFFT) over subcarriers at a given delay-Doppler pair $(n,m)$ as~\cite{art:Baquero_OFDM}
\begin{align}\label{eq:periodogram}
    \nonumber P&_{\mathbf{F}}(n,m) = \\
    &\frac{1}{NM} \left| \sum_{k=0}^{N'-1}\left( \sum_{l=0}^{M'-1} c_{k,l} e^{-j2\pi l\frac{m}{M'}} \right) e^{-j2\pi k\frac{n}{N'}} \right|^2,
\end{align}
where $M'\geq M$ and $N'\geq N$ are the lengths of the FFT and IFFT operations respectively, $n=0,...,N'-1$ and $m=0,...,M'-1$\footnote{If $M' > M$ or $N' > N$ is needed in order to have more $n$ or $m$ values, padding is used by setting the added padded symbols to zero.}.
It has been proven that the ML estimator of the delay and Doppler for a single target coincides with the maximum point in the periodogram as $(\hat{n}, \hat{m}) = \argmax_{n,m} P_{\mathbf{F}}(n,m)$~\cite{art:Baquero_OFDM}, which is maximised when
\begin{align}\label{eq:periodogramMax}
    \frac{f_D}{\Delta f} -\frac{\hat{m}}{M} = 0 \;\;\;\land\;\;\;    \frac{\tau}{T_o} -\frac{\hat{n}}{N} =0.
\end{align}

Then, from \eqref{eq:bPointTarg_Mult}, \eqref{eq:b_val} and \eqref{eq:periodogram}, $\sigma_p$ can be estimated a
\begin{equation}\label{eq:rcsestcalc}
    \Hat{\sigma}_p = \left(\frac{1}{NM}\right) \frac{P_{\mathbf{F}}(\hat{n}, \hat{m})(4\pi)^3d_{p,1}^2d_{p,2}^2}{P_TG_T\lambda^2}.
\end{equation}
Then, considering the the geometry and protocol of the system, each UAV can set $\hat{m}$, $M$, $\hat{n}$ and $N$ so that \eqref{eq:periodogramMax} is met exactly for each cell $p$ to be sensed, and the corresponding RCS estimate is obtained by computing \eqref{eq:rcsestcalc}.
\fi

\section{Information-Level Fusion}\label{sec:Fuse}

After all UAVs $u\in\mathcal{U}$ have sent their local RCS estimates for each cell on the grid $\mathbf{\Hat{\Gamma}}_{u}$ to the FC, it performs information-level fusion of the local estimates to obtain a global estimate $\mathbf{\Hat{\Gamma}}$. Then, the following hypothesis test is performed for all the elements of the grid, $p\in\mathcal{P}$
\begin{subequations}\label{eq:hypothesis}
\begin{alignat}{3}
\mathcal{H}_0:\;\;\;\;&|| \mathbf{r}_{p^*} - \mathbf{r}_{p} ||_\infty > \frac{d}{2}  \\
\mathcal{H}_1:\;\;\;\;&|| \mathbf{r}_{p^*} - \mathbf{r}_{p} ||_\infty \leq \frac{d}{2},
\end{alignat}
\end{subequations}
where $||\cdot||_\infty$ is the $L^\infty$ norm. Hypothesis $\mathcal{H}_1$ states the cases when the target $p^* $ is considered to be located in the corresponding cell $p$, and is considered to be met at the cell $p$ that has the maximum value estimate $\Hat{\sigma}=\max \mathbf{\Hat{\Gamma}}$. On the other hand, $\mathcal{H}_0$ states the cases when the target is not located at $p$, and considered to be met at every other cell $p$ that has an estimate $\Hat{\sigma}$ such that $\Hat{\sigma} < \max \mathbf{\Hat{\Gamma}}$.

The information-level fusion will be carried out using two techniques, namely averaging and pre-normalising the local estimates before averaging.

\textbf{Average: }
The FC averages the values of the cells over the local maps from all UAVs in $\mathcal{U}$ such that $\mathbf{\Hat{\Gamma}} = \frac{1}{U}\sum_{u\in\mathcal{U}} \mathbf{\Hat{\Gamma}}_{u}$.

\textbf{Pre-normalised average: } An average of the pre-normalised local maps is obtained, in which each local map $\mathbf{\Hat{\Gamma}}_{u}$ is normalised between 0 and 1 as
\begin{equation}
    \Bar{\sigma} = \frac{\Hat{\sigma} - \min{(\mathbf{\Hat{\Gamma}}_{u})}}{\max{(\mathbf{\Hat{\Gamma}}_{u})} - \min{(\mathbf{\Hat{\Gamma}}_{u})}}\enskip,\enskip \forall \Hat{\sigma}\in\mathbf{\Hat{\Gamma}}_{u} \enskip,\enskip \forall u \in\mathcal{U}.
\end{equation}

The resulting normalised local maps $\Bar{\Gamma}_{u}$ are then averaged as in the previous approach.

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

In this section, the performance of the proposed sensing protocol will be evaluated in terms of the probability of detection, where detection is considering to occur whenever $\mathcal{H}_1$ is achieved for the cell that contains the target. For this purpose, Monte Carlo simulations were performed, where the target is randomly located at each iteration, and the simulation parameters are presented in Table~\ref{tab:commonPar}, unless stated otherwise. The value for $\sigma_\mathrm{T}$ is assumed as 10 dBsm\footnote{$\sigma$[dBsm] = 10$\log_{10}(\sigma$[$m^2$]$/1 m^2)$}, which is a reasonable value for large vehicles~\cite{art:RCSvals01}, and -30 dBsm for the ground, which is reasonable for grass or dirt~\cite{art:RCSvalsGND}. The OFDM parameters are taken from \cite{Chen_JSACUAVSystem}. The UAVs are set uniformly distributed across $S$ in a way to cover the whole grid, and each one illuminates $L/\sqrt{U}\times L/\sqrt{U}$ cells and avoiding intersections between the $\mathcal{P}_{u}$ cells, unless stated otherwise.
\begin{table}[h!]\vspace{-1em}\centering
\caption{Common simulation parameters}\label{tab:commonPar}
  \begin{tabular}{|c|c||c|c|}
    \hline
    \textbf{Parameter}  & \textbf{Value}    &   \textbf{Parameter}  & \textbf{Value}       \\ \hline
    $P_T$                   & 1 [W]         &   $M$                   & 64        \\ \hline
    $G_T$                   & 1             &   $n$                     & 8         \\ \hline
    $\ell$                     & 100 [m]       &   $f_0$                   & 24 [GHz]  \\ \hline
    $U$                     & 16            &   $BW$                    & 200 [MHz] \\ \hline
    $N_0$                   & -174 [dBm/Hz] &   $T_{CP}$                & 2.3 [$\mu$s] \\ \hline
    $\sigma_{\mathrm{G}}$   & -30 [dBsm] &   $L$                     & 20            \\ \hline
    $\sigma_{\mathrm{T}}$   & 10 [dBsm]    &   $f_D$                   & 0 [Hz]        \\ \hline
    $N$                   & 16            &                      &           \\ \hline
  \end{tabular}
   \vspace{-1em}
\end{table}

\begin{figure}
    \centering
    \includegraphics[width=0.73\linewidth]{result_d_ALL_N_1_PDF.pdf}
    \caption{Detection probability of the target for different cell length $d$ for different beamforming techniques, fusion techniques and $\sigma_{\mathrm{G}}$ values. The number of UAVs and number of cells illuminated per UAV is kept constant, so larger $d$ values imply total area and higher UAV altitude.\vspace{-1em}}
    \label{fig:resultNew_Pd_d_BF}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=0.73\linewidth]{result_Lx_ALL_N_0_PDF.pdf}
    \caption{Detection probability of the target for different cell length $d$ for different beamforming and fusion techniques. Total area and UAV altitude is kept constant, so larger $d$ values imply less cells illuminated per UAV.\vspace{-2em}}
    \label{fig:resultNew_Pd_d_VAR}
\end{figure}

In Fig.~\ref{fig:resultNew_Pd_d_BF} the detection probability is shown as a function of the cell side length $d$, for different $\sigma_{\mathrm{G}}$ values. The number of intended cells per UAV is maintained constant, thus the size of the cells determine the total size of the area $S$ and the altitude of the UAVs $h$, which increases as $d$ increases to accommodate the same number of larger cells in its HPBW. Note that, $d$ increases as $h$ increases, and the $b_p$ value from \eqref{eq:b_val} is closer to the noise level, then the probability of detection decreases.
There exists a maximum point around $d=2$m, where the best probability of detection is achieved.
As expected, as $\sigma_{\mathrm{G}}$ increases, the difference between the RCS of the ground and the target decreases, so that the probability of detection also decreases. By comparing beamforming techniques, both show a similar behaviour. However, when comparing fusion techniques, pre-normalising the local estimates performs better only for larger $d$ and $\sigma_{\mathrm{G}}$ values.

Conversely, in Fig.~\ref{fig:resultNew_Pd_d_VAR} the total size of the area $S$ and the altitude of the UAVs $h$ is kept constant, while varying $d$. This is accomplished by adjusting the number of cells in the grid $L$.
In this case, note that higher $d$ values lead to better probability of detection, as there is a higher area per cell. However, a local optimum can be appreciated around $d=4$m, which shows the presence of a local optimum that offers more precise detection.

\begin{figure}
    \centering
    \includegraphics[width=0.73\linewidth]{result_Deltad2_ALL_N_0_PDF.pdf}
    \caption{Detection probability of the target at a $\Delta$ cells distance for different $\sigma_{\mathrm{G}}$ values and different cell size $d$.\vspace{-1em}}
    \label{fig:resultNew_Pd_Dd}
\end{figure}

Furthermore, in Fig.~\ref{fig:resultNew_Pd_Dd} the detection probability is plotted for different values of $\sigma_{\mathrm{G}}$, different values of a modified threshold $d(\frac{1}{2} + \Delta$) in the hypothesis test \eqref{eq:hypothesis}, and different values of $d$. The curves show the probability of detection of the target at $\Delta$ cells away from the cell with the maximum value in $\Bar{\Gamma}$. Note that values of $\sigma_{\mathrm{G}}\leq -10$dBsm present probability of detection close to 100\% for $\Delta \geq 1$, which is within a distance of one cell. This suggests a probability of detection above 99.89\%  with high accuracy within $5$cm ($d=0.01$m, $\Delta=2$, $\sigma_{\mathrm{G}}\leq -10$dBsm), which is more accurate than other state-of-the-art works utilising MIMO OFDM bistatic-radar such as in~\cite{art:Lyu_MIMOOFDMEurasip}, where they achieve an accuracy of $3$m which uses passive radar in a multi-user MIMO OFDM setting. The results show that for small $\sigma_{\mathrm{G}}$ values, most misdetections occur in an adjacent cell

\begin{figure}
    \centering
    \includegraphics[width=0.73\linewidth]{result_n_ALL_N_0_PDF.pdf}
    \caption{Detection probability of the target for different total number of antennas for the UAV UPA $n^2$ for different beamforming techniques, fusion techniques and $\sigma_{\mathrm{G}}$ values.\vspace{-2em}}
    \label{fig:resultNew_Pd_n_BF}
\end{figure}

Fig.~\ref{fig:resultNew_Pd_n_BF} illustrates the detection probability as a function of the number of antennas in the UPAs of the UAVs, for different $\sigma_{\mathrm{G}}$ values. Therein, the number of UAVs and the number of illuminated cells per UAV is maintained constant, so that narrower beams imply that the UAVs increase their altitudes to accommodate the same intended cells. It is worth noting that the increase of the number of antennas derives into a narrower main beam, and as the beam becomes narrower (higher $n$ values), it is observed an improvement on the probability of detection due to the increased directionality and precision towards the intended sensed cells. However when the beam becomes too narrow, asmall beam misalignment have a bigger impact on the detection of the target, and the increases in the UAV altitudes causes a stronger pathloss, making the received signal closer to the noise level, thus the probability of detection decreases.
For larger  $\sigma_{\mathrm{G}}$ values, the probability of detection decreases even further, as expected. It is also noticed that both fusion techniques show a similar detection probability results, similar to the case with both beamforming techniques. However, the Capon method shows a slightly better performance for a high number of antennas and a small $\sigma_{\mathrm{G}}$ value. Moreover, for smaller $\sigma_{\mathrm{G}}$ values, the fusion by averaging slightly outperforms the pre-normalised averaging approach, while for higher $\sigma_{\mathrm{G}}$ values the opposite is true.

\begin{figure}
    \centering
    \includegraphics[width=0.73\linewidth]{result_z_ALL_N_0_PDF.pdf}
    \caption{Detection probability of the target at a $\Delta$ cells distance for different $\sigma_{\mathrm{G}}$ values for different UAV altitude $h$ values.\vspace{-2em}}
    \label{fig:result_Pd_zUAV}
\end{figure}

Fig.~\ref{fig:result_Pd_zUAV} illustrates the detection probability as a function of the common UAV altitude $h$ for varying $\sigma_{\mathrm{G}}$ and $\Delta$ values. The UAVs are positioned in a similar configuration to the previous figure, thus, less cells are covered by the main beam of the transmitting UAVs at smaller altitude, thus resulting in cells not being illuminated by any UAV. The maximum altitude is considered to be the one where all cells are illuminated once (no overlapping). As the $h$ value increases, each $\mathcal{P}_{u}$ set goes from allocating $1\times 1$ cell, to $3\times 3$ cells and finally to $5\times 5$ cells, such that all cells are illuminated once. This behaviour can be seen in the $\Delta=0$ curve, where a sudden increasing in the probability of detection is observed at altitudes where more cells are allocated in $\mathcal{P}_{u}$, whereas this tendency is also observed for higher $\sigma_{\mathrm{G}}$ values, with worse performance. For higher $\Delta$ values, the probability of detection is higher and increases smoothly with $h$ as higher $\Delta$ implies that the detection can be considered as successful on non-illuminated cells that are adjacent to illuminated cells. This is particularly observed for $\Delta=2$, where every cell in the grid is considered for detection.

\vspace{-1em}
\section{Conclusions}\label{sec:Conclusions}
\vspace{-0.2em}
In this paper, a half-duplex distributed sensing framework for UAV-assisted networks was proposed
in which the area of interest is sectioned into a grid, and the RCS of each cell is estimated by employing receive digital beamforming and a periodogram-based approach, and later sent to a FC for information-level fusion. Results show that the detection probability of the system increases for ground cells of smaller RCS values and that higher accuracy can be achieved within a one-cell distance. Increasing the number of antennas in the UAVs improves the detection probability of the target, however the increase of the altitude of the UAVs can deteriorate it. Moreover, it was found that detection probability is higher for larger cell size $d$ if the UAV altitude is kept constant, however there is a small $d$ value local maximum. Future works can consider the effect of the Doppler and position control of the UAVs to increase the sensing performance of the framework.

\iffalse

\fi

\section*{Acknowledgement}
This research has been supported by the Academy of Finland, 6G Flagship program under Grant 346208 and project FAITH under Grant 334280
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