text:
\section{Introduction}

 Extended Theories of Gravity (ETGs) are acquiring more and more interest
 in modern astrophysics and cosmology due to several inescapable issues and shortcomings that, in principle, could be addressed by revising or extending General Relativity (GR) \citep{poisson,axially,quadrupolo,BH,ivan,SCF,enzo,CARDO,greci}.
In particular,  observational and theoretical issues related to dark matter (DM) and dark energy (DE) could be encompassed in a new picture where gravitational interaction is depending on the scale of the self-gravitating systems without requiring further material ingredients \citep{annalen} that, up to now, seem extremely elusive (see the recent results of the LUX Collaboration 2013). In particular, one of the most famous observational issues which needs  DM, in addition to the visible one, concerns dynamics of stellar objects in galaxies, a problem arising since the beginning \citep{oort32,Oort}.
The disagreement between the observed and expected mass needed to explain dynamics soon
revealed to be present even at cosmological scales \citep{zwicky33}.
Besides, the observed accelerating behavior of the Hubble flow  \citep{Riess} needs to be addressed at fundamental level either
by revising the Standard Cosmological Model or finding out cosmic fluids capable of giving rise to cosmic acceleration.

It is however worth pointing out that the possibility to consider modifications of the Newtonian
dynamics should be taken into account not only when considering the coupling of matter and
gravity field of  each-other influencing distinct objects, but also in the case of self-gravitating systems.
These can be of different types and sizes: collapsing molecular clouds, instability
strip and main sequence stars, hyper-massive stars, and post-evolved very compact objects such
as magnetars \citep{jeans,ivan2}. In particular, exotic compact objects are not only matter of speculative
theories but are gaining observational evidence of their presence \citep{muno06} and very often escape
interpretation in the framework of standard Newtonian theory.
Specifically, the structure and evolution of very massive stars is a key field of investigation
for interpreting peculiar objects like Gamma Ray Burst (GRB). The nature of progenitors could play a crucial role in determining the energetics and hydrodynamical processes related to the central
engine in order to provide the observed enormous amount of $10^{51\div54}$ erg of  electromagnetic
energies.
Suitable candidates for GRB progenitors are thought to be Wolf-Rayet stars with hydrogen-stripped envelopes
and masses of the helium core $M \sim 15\div30 M_{\odot}$.

Additionally,   numerical simulations
claim for the possibility of pair-instability supernovae with $M \ga 130M_{\odot}$ as possible progenitor,
if not for all, at least for a part of GRBs \citep{cct10}.
It is thus evident that addressing the problem of stellar stability over a wide range of
masses and radii has important implications for several  aspects of their pre- and
post- evolutionary history.
These theoretical issues can be considered valid both at  descriptive and predictive
level:  from one side there is the need to explain the existence of objects which seem to have
masses higher than their  expected value ({\it i.e.} Volkoff magnetars);
from the other side,  providing theoretically stable stellar models (not yet observed) can have important
implications  both for  stellar evolution and for testing gravity.
In particular,  modified gravity theories may lead to extra branch of solutions for
 neutron stars as recently pointed out in \cite {Astashenok}.

In general, the stability of stellar structures is strictly related to  the hydrostatic equilibrium
conditions. Equations governing  such equilibrium contain  the gradient of the gravitational
potentials and are, ultimately, related to the field equations governing the coupling
between matter and gravitational field.
In the Newtonian theory, if the thermodynamical relations of stellar structures are determined by a polytropic equation
of state between pressure and density, the gravitational potential $\Phi$ is determined
through the well known Lane-Emden equation (LEE).
The weak-field approximation in GR  case leads to a generally non-linear second-order differential equation
governing the behaviour of the gravitational potential over the polytropic stellar structure \citep{kippe}.

However, when the Hilbert-Einstein action is modified considering a function  $f(R)$ of the Ricci scalar, in particular, if we expand this function up to $R^2$, the weak field
equations lead to a system of coupled modified Poisson equations between $\Phi$ and the scalar
curvature invariant $R$ \citep{poisson}. This means, equivalently,  that two gravitational potentials have to be considered being the field equations of fourth order \citep{PRnostro}.
This system gives rise to a \emph{modified Lane-Emdem equation} (MLEE)
for $\Phi$ which contains an additional convolutional term related to the Yukawa-like correction in
the gravitational potential \citep{annalen}.
The importance of this term depends on the interaction length-scale over which it acts.

In this paper, we will consider numerical solutions of MLEE coming from $f(R)$-gravity.  We want to show that, beside standard stellar structures, coming from the Newtonian limit of GR, other structures could be addressed. The eventual detection of such structures could constitute a formidable test for alternative theories of gravity, in particular for ETGs.

The paper is organized as follows: in Section \ref{due} we
review briefly how the MLEE  comes out in the context of $f(R)$-gravity.
In Section \ref{sect_algorithm},  a description of the algorithm used to numerically integrate the MLEE is given.
In Section \ref{results} we show the numerical results  for some values of polytropic index  comparing them with GR results. In particular, we discuss the role of the Yukawa-like correction coming out in the Newtonian potential.
Stellar masses and gravitational energy binding are calculated in Section \ref{cinque}. Finally, discussions
and conclusions are drawn in Section \ref{sei}.

\section{The modified Lane-Emden equation}
\label{due}

The modified Lane-Emden equation (MLEE) has been  derived in the context  of  $f(R)$-gravity, for self-gravitating
systems in the linear approximation of weak field \citep{poisson}.
We refer the reader to that paper for the mathematical details and its derivation.
Here,  we briefly report the the most relevant steps. \\
\noindent In the metric approach, the $f(R)$ field equations are
obtained by varying the action with respect to
$g_{\mu\nu}$ \citep{PRnostro,reviewodi,reviewodi1,reviewmauro,reviewvalerio}. We get

\begin{eqnarray}\label{fieldequationHOG}
\left\{\begin{array}{ll}
f'(R)R_{\mu\nu}-\frac{f(R)}{2}g_{\mu\nu}-f(R)_{;\mu\nu}+g_{\mu\nu}\Box f'(R)=\mathcal{X}\,T_{\mu\nu}^{(m)},\\
\\
3\Box
f'(R)+f'(R)R-2f(R)\,=\,\mathcal{X}\,T^{(m)}\,,
\end{array}\right.
\end{eqnarray}
where the second equation is the trace.
Here,
$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_m)}{\delta
g^{\mu\nu}}$ is the the energy-momentum tensor of matter, while
$T=T^{\sigma}_{\,\,\,\,\,\sigma}$ is the trace,
$f'=\frac{df(R)}{dR}$, $\Box={{}_{;\sigma}}^{;\sigma}$ and
$\mathcal{X}\,=\,8\pi G$\footnote{Here we use the convention
$c\,=\,1$.}. The conventions for the Ricci tensor is
$R_{\mu\nu}={R^\sigma}_{\mu\sigma\nu}$ while for the Riemann
tensor is
${R^\alpha}_{\beta\mu\nu}=\Gamma^\alpha_{\beta\nu,\mu}+...$. The
affinities are the standard Christoffel symbols
$\Gamma^\mu_{\alpha\beta}=\frac{1}{2}g^{\mu\sigma}(g_{\alpha\sigma,\beta}+g_{\beta\sigma,\alpha}
-g_{\alpha\beta,\sigma})$. The adopted signature is $(+---)$ \citep[see][]{landau}.

The field equations (\ref{fieldequationHOG}) can be rewritten as

\begin{eqnarray}
\label{ET}
G_{\mu\nu}=\mathcal{X} \left(T_{\mu\nu}^{(m)}+T_{\mu\nu}^{(c)}\right)\,,
\end{eqnarray}
where $G_{\mu\nu}$ is the Einstein tensor and  $T^{ (c)}_{\mu\nu}$ is the curvature energy-momentum tensor that comes out when we introduce curvature higher order terms in the Lagrangian \citep{CS}. It is defined as
\begin{eqnarray}
&&\mathcal{X}T^{(c)}_{\mu\nu}=\left[1-f'(R)\right]R_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\left[f(R)-R\right]+\nonumber\\&&+\nabla_\mu \nabla_\nu f'(R)-g_{\mu\nu}\Box f'(R)\,,
\end{eqnarray}
and  vanishes as soon as $f(R)=R$.
As we can see, in the right side of  equation (\ref{ET}), there are two fluids: a standard matter fluid and a curvature fluid. In this way we can treat even the ETGs as the GR in the presence of two sources \citep[see][for details]{gae}.
Note that these quantities satisfy the Bianchi identities, in fact,
 using the following  properties
 \begin{equation}[\nabla_\gamma, \nabla_\beta]V^\alpha = - R^\alpha_{\,\,\, \rho \beta\gamma} V^\rho\,, \end{equation}
  where $V^\alpha$ is a generic vector, and
 \begin{equation}[\nabla_\mu, \nabla_\nu]f'(R)=0\,, \end{equation}
  it is straightforward to show that

 \begin{equation}\label{divTc}
    \nabla^\mu (\mathcal{X} T^{(c)}_{ \mu\nu})=0\,,
 \end{equation}

\noindent
so that the total energy-momentum tensor can be written as
  \begin{equation}
 T_{\mu\nu}=T^{(m)}_{\mu\nu}+T^{(c)}_{\mu\nu}\,,
  \end{equation}
and then $\nabla^\mu  T_{\mu\nu}=0$.
Let us now  perturb the metric tensor up to $c^{-2}$ so that the Ricci scalar becomes
  \begin{equation}
  R\sim R^{(2)}+{\cal O}(4)\,,
  \end{equation}
 and the $n$-th derivative of the Ricci function can be developed as
\begin{eqnarray}
f^{n}(R)\,&\sim&\,f^{n}(R^{(2)}+\mathcal{O}(4))\sim\,\nonumber\\
&\sim&\,f^{n}(0)+f^{n+1}(0)R^{(2)}+\mathcal{O}(4)\,,\nonumber\\
\end{eqnarray}
where $R^{(n)}$ indicates a quantity of order $\mathcal{O}(n)$.
From lowest order of field equations (\ref{fieldequationHOG}),  we
have $f(0)\,=\,0$ which trivially follows from the assumption that the space-time
is asymptotically Minkowskian. Equations (\ref{fieldequationHOG})  at $\mathcal{O}(2)$ - order (Newtonian level)
become

\begin{eqnarray}\label{PPN-field-equation-general-theory-fR-O2}
\left\{\begin{array}{ll}
R^{(2)}_{tt}-\frac{R^{(2)}}{2}-f''(0)\triangle
R^{(2)}=\mathcal{X}\,T^{(0)}_{tt}\,,\\\\
-3f''(0)\triangle
R^{(2)}-R^{(2)}=\mathcal{X}\,T^{(0)}\,,
\end{array}\right.
\end{eqnarray}
where $\triangle$ is the Laplacian in the flat space, $R^{(2)}_{tt}=-\triangle\Phi$ and for simplicity we set $f'(0)=1$. We recall that the energy momentum for a perfect fluid considered is the following
\begin{equation}
T_{\mu\nu}=(\epsilon+p)u_{\mu} u_{\nu}-pg_{\mu\nu}\,,
\end{equation}
where $p$ is the pressure  and $\epsilon$ is the energy density of perfect fluid. Then we have

\begin{equation}\label{HOEQ}
\left\{\begin{array}{ll}\triangle\Phi+\frac{R^{(2)}}{2}+f''(0)\triangle R^{(2)}\,= \,-{\cal X}\rho\,,\\ \\
3f''(0)\triangle R^{(2)}+R^{(2)}\,=\,-{\cal X}\rho\,,\end{array}\right.
\end{equation}
where $\rho$ is the mass density. We note that $f''(0)\,=\,0$ we have the Newtonian mechanics: $\triangle\Phi\,=\,-4\pi G\rho$.
This equation can be considered the modified Poisson equation for $f(R)$-gravity.

\noindent From the Bianchi identity (satisfied for first line of eq. \ref{fieldequationHOG}) we find

\begin{equation}\label{equidr}
\frac{\partial p}{\partial x^k}=-\frac{1}{2}(p+\epsilon)\frac{\partial \ln g_{tt}}{\partial x^k}\,.
\end{equation}

Let us suppose now that  matter satisfies a polytropic equation $p\,=\,K\,\rho^\gamma$ with $K$ free independent parameter or a constant with fixed value and $\gamma$ the polytropic exponent. Since $\epsilon\,=\,\rho c^2$,  equation (\ref{equidr}) satisfies this relation
\begin{equation}
\frac{\gamma K}{\gamma-1}\rho^{\gamma-1}\,=\,\Phi\,\,\,\,\rightarrow\,\,\,\,\rho=\left[\frac{\gamma-1}{\gamma K}\right]^{\frac{1}{\gamma-1}}\Phi^{\frac{1}{\gamma-1}}= A_n\Phi^n
\label{densita}
\end{equation}
where $n$ is the polytropic index, which is defined by $n=\frac{1}{\gamma-1}$. We remember that in non relativistic limit we have $\gamma=\frac{5}{3}$ and $n=\frac{3}{2}$ while for the relativistic limit $\gamma=\frac{4}{3}$ and $n=3$. Note that for this cases, where the equation of state a polytropic form, the polytropic constant $K$ is fixed and can be calculate from a natural constants.\\
Here, using the above equation (\ref{HOEQ}) and after some calculations, we get the general form of the MLEE which reads as
\begin{eqnarray}
&& \frac{d^2w(z)}{dz^2}+\frac{2}{z}\frac{dw(z)}{dz}+w(z)^n= \label{le_equation} \nonumber\\&&
=\frac{m\xi_0}{8~z}\int_0^{\xi/\xi_0} dz' z' \{ e^{-m\xi_0 | z-z'|}-e^{-m\xi_0 | z+z'|}\} w(z')^n\,,\nonumber\\
\label{LE}
\end{eqnarray}

\noindent
where $z=R/\xi_0$  is a dimensionless radius given in units of the
polytropic radius, which is defined as

\begin{equation}
 \xi_0=\sqrt{\frac{3}{16 \pi G}} K^{1/2} (1+n)^{1/2} {\rho_c}^{\frac{1-n}{2n}}.
\label{poly_radius}
\end{equation}

\noindent
The dimensionless function $w(z)=\Phi/\Phi_c$   gives the gravitational potential normalized
to its central value

\begin{equation}
 \Phi_c={\rho_c}^{1/n} (n+1) K,
\end{equation}
where ${\rho_c}$ is the central density, while $K$ and $n$ are the constant and the index
of the polytropic equation of state

\begin{equation}
 P=K \rho^{\gamma}~~~~~~~~~~~~~~~ \gamma=1+\frac{1}{n}.
\end{equation}

The quantity $m \xi_0$ in the right-hand side of equation (\ref{le_equation}) is
 related to the effective length scale of the Yukawa-like correction term
to the Newtonian potential. Defining $L_{\rm Y} = 1/m$, we may write $m \xi_0= \xi_0/L_{\rm Y}$;
when $L_{\rm Y}/\xi_0 \gg 1$,  $m \xi_0 \rightarrow 0$, the gravitational potential
asymptotically becomes Newtonian, and  the right-hand side
term of equation (\ref{le_equation}) goes to zero, in turn recovering
the GR case.

\noindent It is well know that analytical solutions for the classical Lane-Emden equation
exist only for three values of the polytropic index $n$, equal to 0, 1 and
5, respectively. Of these, only $n=0$ and $n=1$ give finite-radius solutions.

\noindent The situation of course gets even more complicated in the case of  MLEE,
for its full integro-differential nature-- in this case analytical solutions
are available only for $n$=0 \citep[see][]{poisson}.

\noindent A systematic investigation of polytropic stellar structures thus requires the developement
of numerical methods aimed at obtaining general solution of equation (\ref{le_equation}). This is actually
necessary for both $f(R)$ and GR models, given that, {\it e.g.}, degenerate non-relativistic ($n=3/2$)
and relativistic ($n=3$) polytropic equations of state are needed to  build up  stellar
models.
\section{Description of the algorithm}
\label{sect_algorithm}

Let us numerically solve  equation (\ref{le_equation}) using an iterative
procedure described as follows.
If we define $w_i(z)$ the function at the $i^{\rm th}$-iteration and
define the integral term of the right hand side as $\aleph [w(z')]$,
the iteration algorithm can be written as

\begin{equation}
 \frac{d^2w_i}{dz^2}+\frac{2}{z}\frac{dw_i}{dz}+w_i^n=\frac{m\xi_0}{8~z}\aleph [w_{i-1}].
\label{iter_equation}
\end{equation}

In the framework of a numerical procedure, the term on the right-hand side of
 equation (\ref{iter_equation}) can be viewed as a source term of a
non linear second order differential equation.
We can now rewrite the equation as a system of two first-order differential equations in the form

\begin{equation}\label{system}
\large{\left\{\begin{array}{ll}\frac{dw_i}{dz}=k_i,\\ \\
\frac{dk_i}{dz}=-\frac{2}{z}k_i - w_i^n + \frac{m\xi_0}{8~z} \aleph [w_{i-1}],
\end{array}\right.}
\end{equation}

\noindent
with boundary conditions $w_i(0)=1$ and $k_i(0)=0$, respectively.\\
The last term on the right-hand side of the second equation is proportional to
the  quantity  $\aleph [w_{i-1}]$ which is determined by
the function $w_{i-1}$ obtained at the $(i-1)^{\rm th}$-iteration.
At the first iteration $i=1$, an initial guess for $w_0$ must be given, and
we choosed $w_0=1$.
The solution of the system (\ref{system}) is obtained via a coupled  $4^{\rm th}$-order
Runge-Kutta integration method up to the value $z_{\rm max}=\xi/\xi_0$ with
the above mentioned initial conditions

The regular and smooth behaviour of the function $w(z)$ in its domain of definition allows to compute
the integral on right-hand side of equation (\ref{le_equation})  using the Gauss-Legendre
quadrature according to the general rule

\begin{equation}
 \int_0^{z_{\rm max}} f(z) dz \approx  \frac{\xi}{2 \xi_0} \sum\limits_{j=1}^n q(z_j) f( \frac{\xi}{2 \xi_0} z_j + \frac{\xi}{2 \xi_0} ),
\end{equation}

\noindent
where

\begin{equation}
 q(z_j)=\frac{2}{1-{z_j}^2 [{P_n}'(z_j)]^2},
\end{equation}

\noindent
are the weights and $z_j$ is the $i^{\rm th}$ root of the Legendre polynomial $P_n(z)$.
The values of the function $f(z)$ (see eq. \ref{le_equation}) at the nodes $z_j$
are computed by mean of a linear interpolation.

\noindent

\begin{figure}
\hspace{-0.4cm}
\centerline{\includegraphics[width=6.5cm, angle=-90]{wfunction_vs_index0-3_mcsi0-mcsi1.eps}}
\caption{Numerical solutions of the Lane-Emden equation (\ref{le_equation}) for different values of the polytropic index $n$.
For each value of $n$, two solutions are reported: $m \xi_0=1$ (continuum line) and $m \xi_0=0$ (dashed-line).
The first case gives w(z) corresponding to field equations with $f(R)$, the second to $f(R)=R$.}
\label{wfunct_solutions}
\end{figure}

\begin{figure}
\hspace{-0.4cm}
\centerline{\includegraphics[width=6.5cm, angle=-90]{zmax_vs_lyuk_nindex1-3.eps}}
\caption{Stellar radius in units of the polytropic radius $\xi_0$ (see equation \ref{poly_radius})
as a function of the normalized Yukawa length scale, for different values of the polytropic index $n$,
obtained from the numerical solution of equation (\ref{iter_equation}). The quantity along the x-axis actually is
 equal to $1/m\xi_0$, where $m^2=-1/3f''(0)$.}
\label{zmax_vs_lyuk}
\end{figure}

\begin{figure}
\hspace{-0.4cm}
\centerline{\includegraphics[width=6.5cm, angle=-90]{zmax_vs_n1e-5-3.5_lyuk1e-4-1.eps}}
\caption{Stellar radius $z_{\rm max}$ in units of $\xi_0$ as a function of the polytropic index $n$
for different values of the normalized Yukawa length-scale $L_{\rm Y}/\xi_0$.}
\label{zmax_vs_index}
\end{figure}

As we are interested in finding self-consistent solutions, which in turn
correspond to determine the stellar radius of the polytropic star,
it is worth pointing that $z_{\rm max}$  is not given by hand as a free parameter, but at each iteration
it is determined by the value of $z$ such that $w(z)=0$.

In this way, the two input parameters required to derive a full solution
of equaton (\ref{le_equation}) are the  polytropic index $n$ and
the product $m~\xi_0$.
Bearing in mind that $m^2=-1/3f''(0)$ and $1/m=L_{\rm Y}$ defines the characteristic
scale of the Yukawa-like correction to the Newtonian potential, the quantity $1/m\xi_0=L_{\rm Y}/\xi_0$ just
expresses the Yukawa lenght scale in units of the polytropic radius $\xi_0$.
The fastness in the convergence of the iteration procedure depends on the initial guess
of the $w(z)$ function, and we found that the setting $w^0(z)=1$ provides convergence after
just a few iterations.

\section{Results}
\label{results}

The robustness of the solving algorithm described in Section \ref{sect_algorithm} allowed
us the possibility to explore the solutions of the MLEE for any value of
the poliyropic index $n < 5$ and the Yukawa length scale $L_{\rm Y}/\xi_0$ and
compare them with the case of GR.  The latter case simply corresponds to setting $m \xi_0=0$ in equation (\ref{le_equation}).

In Fig. \ref{wfunct_solutions} we report the function $w(z)$ for $ L_{\rm Y}/\xi_0=1$ and
four different peculiar values of the polytropic index $n$.
For each value of $n$ we also reported the solution of the classical LE equation ($m \xi_0=0$);
we remind the reader the when $n=0$ and $n=1$ the classical solution admits the analytical expressions
  $w^0(z)=1-z^2/6$ and  $w^1(z)={\rm sin}(z)/z$, respectively.

The other two function $w(z)$ in Fig. \ref{wfunct_solutions} correspond to the polytropic
equation of state of a degenerate non-relativistic ($n=3/2$) and relativistic ($n=3)$ gas,
and can be of relevant astophysical interest for instance in the study stellar structure of compact objets.
As already known, for any fixed value of $n$, the radius is higher for $f(R)$-models than GR-models,
and the higher the polytropic index, the higher is the stellar radius $z_{\rm max}=R/\xi_0$.

To investigate how more generally $z_{\rm max}$ depends on the Yukawa length-scale value, we subsequently
performed a set of runs with a grid of values of $L_{\rm Y}/\xi_0$ spanning several order
of magnitude, with results reported in Fig. \ref{zmax_vs_lyuk}.
For clarity plotting purposes we show results  for values of $n$ from 1 to 3
with stepsize of 0.5.
From the figure it can be seen that the deviations from the GR-values occur in a range
$10^{-4} \la L_{\rm Y}/\xi_0 \la 1$, which does not approximatively depend
on the politropic index $n$.
The region characterizing the deviations between GR and $f(R)$ solutions has
a sharp increase around $L_{\rm Y}/\xi_0 \sim 10^{-4}$, followed  by a plateau
and a subsequent fast transition to the classical solution at $L_{\rm Y}/\xi_0 \sim 1$.
For $n \geqslant 2$ the right hand part of the plateau also shows a
secondary superposed increase of $z_{\rm max}$ before the steep decay to the classical
solution and specularly in the region around  $ L_{\rm Y}/\xi_0 \sim 10^{-4}$
the value of $z_{\rm max}$ is slightly lower than its GR-solution.

It is also worth noting that in all cases $ L_{\rm Y}/\xi_0 < z_{\rm max} $,
or $L_{\rm Y} < R_{*}$, thus the Yukawa correction term, no matter
which is its true value in the considered region, acts on a characteristic length
scale lower than the stellar radius. We consider this point
of key importance in the framework of the determination
of the stellar structure.

In Fig. \ref{zmax_vs_index} we report the behaviour of the stellar radius as
a function of the politropic index $n$ between 0 and 4 and for $10^{-4} \leq L_{y}/\xi_0  \leq 10$,
which corresponds to the range of Yukawa length-scales where
deviations occur from the classical solutions (see Fig. \ref{zmax_vs_lyuk}).
The data can be well fitted by a function $z_{\rm max}(n)=a ~{\rm exp}(-b~n)+c$
with  $a$ in the range $\sim  0.16\div0.27$, $b\sim 1$ and c in the range
$\sim 2.4\div2.9$.

\section{Stellar mass and gravitational energy binding}
\label{cinque}

The solution of the LEE  allows, for given values of the central
density, the polytropic index $n$ and the constant $K$ of the
polytropic relation $p=K \rho^{1+1/n}$, to find the mass and the gravitational
binding energy of a polytropic star.
In the case of classical solution of the equation, the values are

\begin{equation}
 M=4 \pi \rho_{\rm c} R^3 \left(-\frac{1}{z}\frac{dw}{dz}\right)_{z=z_{\rm max}},
\label{mass_classical}
\end{equation}

\noindent
and
\begin{equation}
 E_{\rm g}=-\frac{3}{5-n}\frac{G M^2}{R}.
\label{egrav_classical}
\end{equation}

In more general case of MLEE, $E_{\rm g}$ and $M$ can be written
as

\begin{equation}
 M=\int^{R_{\rm max}}_0 \rho({\bf x}) d{\bf x},
\label{mass_general}
\end{equation}

\noindent
and

\begin{equation}
 E_{\rm g}=\int^{R_{\rm max}}_0 \Phi({\bf x}) \rho({\bf x}) d{\bf x}.
\label{egrav_general}
\end{equation}

While keeping in mind the definitions $\Phi=w \Phi_{\rm c}$, $\Phi_{\rm c}= \rho_{\rm c}^{1/n} (n+1) K$,
and  $\rho=\rho_{\rm c} w^n$, equations (\ref{mass_general}) and (\ref{egrav_general}) can be rewritten
as
\begin{equation}
 M= 4\pi \xi^3_0  \rho_{\rm c} \int^{z_{\rm max}}_0   w^{n} z^2 dz,
\label{mass_wz}
\end{equation}

and
\begin{equation}
 E_{\rm g}= -4\pi \xi^3_0 (n+1) K \rho^{1/n+1}_{\rm c} \int^{z_{\rm max}}_0   w^{n+1} z^2 dz,
\label{egrav_wz}
\end{equation}

\noindent
where $w(z)$ is a solution of the LE equation (\ref{le_equation}), and depends on
$n$ and $m~\xi_0$.
When $m~\xi_0 \rightarrow 0$, equations (\ref{mass_general}) and (\ref{egrav_general}) can
be solved analytically and reduce to equations (\ref{mass_classical}) and (\ref{egrav_classical}), respectively,
while in the general case of $f(R)$-gravity they need to be solved numerically.

Let us now consider for any given polytropic index $n$, the solutions of equation (\ref{le_equation}) for
the case of $f(R)$ and GR gravity, which we label as  $w_{\rm f(R)}$ and  $w_{\rm GR}$, respectively.
Using equations (\ref{mass_wz}) - (\ref{egrav_wz}) we can now write

\begin{equation}
 \frac{M_{\rm f(R)}}{M_{\rm GR}}= \frac{\int^{z_{\rm max}}_0   w_{f(R)}^{n} z^2 dz}{\int^{z_{\rm max}}_0   w_{GR}^{n}  z^2 dz}\,,
\label{ratio_integral_mass}
\end{equation}
\begin{equation}
 \frac{E^{\rm g}_{\rm f(R)}}{E^{\rm g}_{\rm GR}}= \frac{\int^{z_{\rm max}}_0   w_{f(R)}^{n+1} z^2 dz}{\int^{z_{\rm max}}_0   w_{GR}^{n+1}  z^2 dz}\,.
\label{ratio_integral_energy}
\end{equation}

It is evident that equations (\ref{ratio_integral_mass}) and  (\ref{ratio_integral_energy}) are very useful for
finding relative masses and gravitational  binding energies between $f(R)$ and GR stellar models, for any
desired choice of the parameters $K$ and $\rho_c$, which indeed do not appear.
The results of these ratios are reported in Fig. (\ref{ratio_mass_egrav_vs_lyuk}) for a range of values of $L_{\rm Y}/\xi_0$

Looking at Fig. (\ref{ratio_mass_egrav_vs_lyuk}), it is worth noting
that the value of the ratios  $M_{\rm f(R)}/M_{\rm GR}$ and  $E_{\rm f(R)}/E_{\rm GR}$ are very closed each other.
This is not surprising, given that in equations (\ref{ratio_integral_mass}) and (\ref{ratio_integral_energy})
the only difference is the change in the power of the $w(z)$-function from $n$ to $n+1$.

The most relevant result is that in $f(R)$-gravity masses and associated gravitational energy
binding values can exceed up to 50\%  of their classical value for $10^{-3} \la L_{\rm Y}/\xi_0 \la 10^{-1}$.
Furthermore, we have two  characteristic gravitational radii, the standard Schwarzschild one
\begin{equation}
 R_{\rm s}=\frac{2 G M}{ c^2}\,,
\end{equation}
where $M$ is defined by equation (\ref{mass_wz}) and, as we have seen,  a length related to the Yukawa correction.
Keeping in mind the definition of $\xi_0$, equation  (\ref{poly_radius}),  and
defining $L_{\rm Y}= \alpha \xi_0 $, where $\alpha= 1/m \xi_0$ and
$m^2=-1/3 f''(0)$, we obtain

\begin{equation}
\frac{ L_{\rm Y}}{ R_{\rm s}}= \frac{\alpha 2 c^2 }{ 3 K \rho_c^{1/n} (1+n) F(w,n)}\,,\label{ratio_lyuk_rscw}
\end{equation}
where
\begin{equation}
 F(w,n)=\int^{z_{\rm max}}_0 w^n z^2 dz\,.
\end{equation}

For values of $n$ in the range $1\div4$, the ratio defined in equation (\ref{ratio_lyuk_rscw}) ranges from  $\sim \alpha\,10^4 K^{-1}_{14}\left( \rho_{100}\right)^{-1}$ to  $\sim \alpha\, 10^5 K^{-1}_{14}\left( \rho_{100}\right)^{-\frac{1}{4}}$ where $\rho_{100}=10^{-2}\rho_c$ and $K_{14}=10^{-14} K$. Looking at  Fig. \ref{ratio_mass_egrav_vs_lyuk}, the range of values of $\alpha=L_{\rm Y}/\xi_0$ for which significant deviations from standard Newtonian theory occur is $10^{-4} \la \alpha \la 0.1 $. This means that a second  gravitational radius, acting on scales larger than  the Schwarzschild radius, can be defined. From a dynamical point of view,  in principle,  this fact can be interpreted as the possibility to stabilize more massive self-gravitating systems that cannot be achieved in standard GR. In other words, very massive exotic objects that could not be obtained as solutions of the standard theory of stellar structures could be achieved in the framework of $f(R)$-gravity.

\begin{figure}
\hspace{-0.4cm}
\centerline{\includegraphics[width=6.5cm, angle=-90]{ratio_mass_egrav_vs_lyuk_n3.eps}}
\caption{Ratio of the masses and  gravitational  binding  energies for $f(R)$ and GR polytropic stars
(see eqs. \ref{ratio_integral_mass}-\ref{ratio_integral_energy} as a function of the
normalized Yukawa lenght scale $L_{\rm Y}/\xi_0$.  The results here reported are obtained for
a politropic index value $n=3$ but actually they are the same for any value of $n$ (see text).}
\label{ratio_mass_egrav_vs_lyuk}
\end{figure}

\section{Discussions and conclusions}
\label{sei}

The developement of an efficient algorithm aimed to solve the Lane-Emden equation for stellar structures in $f(R)$-gravity
 allows to open unexplored possibilities  for self-graviting systems and stellar structures.
In particular, we have seen that higher-order gravitational corrections   give the possibility to extend the range of stable stellar structures  by inducing a  further characteristic gravitational radius that, if compared with the Schwarzschild one can rule these new structures.

In general, one can get more massive structures that remain stable. These toy-models have of course to be refined considering also
  values of physical interest  for  the costant $K$ of the polytropic relation, the central density $\rho_c$ and other secondary parameters.
Specifically, a further numerical step and refinement will  consist in building up realistic  stellar models where the structure is
divided into a given number of shells, each  with its own polytropic equation $p=K_i \rho^{\gamma_i}$.
For each $i^{\rm th}$ shell, one  has to  solve equation (\ref{le_equation}) by using,
as Cauchy initial conditions, the values $w(z^{i-1}_{\rm out})$ and  $w'(z^{i-1}_{\rm out})$ where
$z^{i-1}_{\rm out}$ corresponds to the outer border of the bottom $({i-1})^{\rm th}$ shell.
This would be particularly desired for main sequence model with a central nuclear burning core surrounded
by a convective envelope. Besides, the algorithm could work also to describe structures out of the main sequence,  peculiar objects and systems in the instability strip since boundary conditions on the MLEE are less restrictive with respect to those requested for standard LEE.
Finally, as discussed in \cite{Astashenok}, in the case of observation of self-gravitating systems that escape standard description of GR, this could constitute a straightforward testbed  for alternative theories of gravity.

\section*{Acknowledgments}

MDL is supported by MIUR (PRIN 2009). MDL and SC acknowledge the
support of INFN Sez. di Napoli (Iniziativa Specifica TEONGRAV).

redpajama_set_name: RedPajamaArXiv


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I saw this information from the SAE MOBILUS Knowledge Hubs and I think you will find it very useful.
A U.S. Air Force C-130H Hercules aircraft assigned to the 130th Airlift Wing, McLaughlin Air National Guard Base, Charleston, West Virginia, awaits takeoff to deliver supplies to Puerto Rico on October 5, 2017, at Savannah Air National Guard Base, Savannah, Georgia after Hurricane Maria. (Image courtesy: U.S. Air National Guard/Technical Sergeant De-Juan Haley)
Collins Aerospace helps to update the C-130H with modern avionics
Collins is helping L3Harris Technologies bring National Guard and Reserve C-130Hs into the digital era.
William Kucinski | September 3, 2019
"These aircraft are important to national security, and by working with L3Harris, our integrated avionics system will support the extension of the life of the planes for another 20 years," says Dave Schreck, vice president and general manager for Military Avionics and Helicopters at Collins Aerospace.
The 176-aircraft Air National Guard and U.S. Air Force Reserve Lockheed C-130H Hercules fleet is receiving new Collins Aerospace Systems avionics that will help extend the life of the legacy aircraft by 20 years. Collins and its commercial off-the-shelf Flight2 integrated avionics system was chosen by L3Harris Technologies, Inc. to play a key role in the U.S. Air Force's $499-million C-130H Avionics Modernization Program Increment 2 (AMP INC 2) program.
The H-variant of the C-130 – differentiated by its use of Allison T56-A-15 turboprop engines – was first introduced in 1974. Collins' contribution to the workhorse C-130H platform will be seven multifunctional displays (MFDs), three control display units (CDUs), and a new digital autopilot that will replace over 100 analog instruments in each of the fleet's aircraft.
Read the full article on the SAE International website
Impossible Aerospace US-1 makes long-distance flight over Nevada salt flats
ULA's Vulcan Centaur will launch Sierra Nevada Corporation's Dream Chaser spaceplane
Dolphin Technologies introduces cryptocurrency aimed at reducing mobile phone-related accidents
Bookmark http://www.sae.org/news to keep pace with the latest aerospace technology news and information.
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William Kucinski is content editor at SAE International, Aerospace Products Group in Warrendale, Pa. Previously, he worked as a writer at the NASA Safety Center in Cleveland, Ohio and was responsible for writing the agency's System Failure Case Studies. His interests include literally anything that has to do with space, past and present military aircraft, and propulsion technology.
Contact him regarding any article or collaboration ideas by e-mail at william.kucinski@sae.org.
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text:
Get Your EDM Here!
Most people think of Electronic Document Management Software as a software program for managing documents, something like a big document database. In fact, EDM software is part of an EDM System (EDMS). And, EDMS is a whole collection of technologies that work together to provide a comprehensive solutions for managing the creation, capture, indexing, storage, retrieval, and disposition of records and information assets of an organization. For most businesses, this means archived and current paper and electronic documents. It may also include video, audio and website files.
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Going from Paper to Digital
Many paper-based processes, while at their core essential to a business's operation, can be improved when transformed into digital processes. That's why it's important to go beyond just document management when you go digital, and look at your business processes to see how they can be better managed in the digital realm. The differences are critical.
Document management refers to the organization and storage of documents, usually in a secure system that enables users to find, share and edit documents. Document management systems commonly provide storage, versioning, metadata, security, indexing and retrieval capabilities.
Business processes management (BPM) is a systematic approach to making an organization's workflow more effective, more efficient and adaptable. The goal of BPM is to reduce human error and miscommunication and focus stakeholders on the requirements of their roles.
A document-centric BPM system is one that focuses on the management of documents within the context of the business processes which the documents fulfill. For instance, a legal document may require editing by an administrator, review by a legal assistant, case law research by a legal intern and final signature by a partner. The document needs to be managed as it goes through the processes to its completion.
The paper-based way of doing this is by moving a paper document from one desk to another. A more advanced way is with an electronic document that is emailed from person to person. The problem with either of these methods is the human element.
A more effective way is to remove the human element from each transaction by creating business process rules that "move" the document seamlessly from one person to another. This rules-based workflow allows an administrator to create a rule that dictates the flow of the document through an organization: for instance, an invoice passes through an approval process and then is routed to the accounts-payable department. Dynamic rules allow for branches to be created in a workflow process. A simple example would be to enter an invoice amount and if the amount is lower than a certain set amount, it follows different routes through the organization. Advanced workflow mechanisms can manipulate content or signal external processes while these rules are in effect.
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Helpful Articles:
How Long Does It Take to Find a File?
5 Secrets to Efficient Electronic File Management
Is Electronic Document Management a Good Investment?
What's YOUR Excuse for Not Using EDM?
Nine Criteria to Help You Find the Best ECM Solution
How ECM Saves Your Company $7.50 for Every Dollar Spent.


Petland, Inc. is a franchise operation with quality, full service retail pet centers across the United States, Canada, Japan, China, Mexico, South Africa, Brazil and El Salvador. For more than 49 years, Petland Pet Counselors have been dedicated to matching the right pet with the right person and meeting the needs of both. To its customers who already have pets, Petland is dedicated to enhancing their knowledge and enjoyment of the human-animal bond. Petland was founded in 1967 and is headquartered in south central Ohio.
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While Petland is not currently accepting new applicants on Franchising.com, these other similar opportunities are actively looking for new franchisees.


As proud sponsor of the Thames Valley Venus Awards 2018, we were delighted to announce our five semi-finalists for the Influential Women category last night.
Dubbed by Channel 4 as "The Working Women's Oscars", The Venus Awards celebrates the vital contribution that women in business make to the local, regional and national economy.
Louise Fetigan, founder of Little Troopers, a registered charity supporting all children with parents serving in the British Armed Forces, regular or reserve. Louise and her team provide fundamental resources, initiatives and events to ease and aid repeated separation periods aiming to keep parent and child connected and bonded even when miles apart.
Amanda Ayres, founder of The Business Girls Network, an online and offline business community designed to educate, empower and enable female business ownership.
Katherine Knight, managing director of Intelligent Health Intelligent Health's vision is to create active communities where people connect with the natural spaces within and around the place they live. Katherine's award-winning Beat the Street behaviour change programme has engaged over 800,000 participants in the past five years and helps people make activity part of their everyday life.
Emma-Jane Taylor, founder of a portfolio of businesses which all aim to improve lives. StageWorks Performing Arts School is for children aged 4-16 years old, NutritiousWorks is a fitness company for adults and SHEnetWORKS is a networking company for women to share business ideas, collaborate and connect.
Congratulations to our semi-finalists who richly deserve the recognition. We look forward to the final!


text:
George Jetson
Posts tagged with "thrilling 30" - 1
"Truly you have a dizzying intellect." "WAIT 'TIL I GET GOING!"
Posted September 10, 2013 at 10:30 pm
Hold onto your butts. We're about to get painfully nerdy in here. We kind of have to, because I don't know if I have enough words otherwise to bury two photos of a simple and predictable Seeker redeco in. It's Thundercracker. He's a blue and silver Starscream again. Let's move on.
War For Cybertron Starscream is technically a toy of Aligned Starscream, right? Sure, the War for Cybertron aesthetic is blocky and having-a-nose-y enough that he can pass for other version of Starscream, and sure enough a bunch of the WFC designs were borrowed for the ongoing Robots in Disguise comic which is a Generation 1 book. Some folks don't cotton to the idea of stylistic choices varying so widely within a continuity, and so they're happy to ignore the whole Aligned angle and just consider him G1 since he looks G1y enough to them.
There was also a Thundercracker in War for Cybertron, and now there's a toy basically in those colors as Thundercracker. He's being released in the same line as WFC Starscream, but only after the focus of the line shifted from WFC to a G1/Classics approach. And since these toys are being packaged with comics, specifically comics which were written to these toys, thus placing the WFC Thundercracker design in this G1 world as G1 Thundercracker, he isn't even ambiguously a WFC toy, even though he's still sort of repurposed.
OR IS HE???
Here's the wrinkle that I've been mentally picking at. While the Deluxe Class toys in this portion of the Generations line come with IDW comics featuring those characters, the packaging bios mostly ignore the IDW comics' specific interpration of these characters, content instead to rehash the original tech spec bios from the eighties. Swerve is a guy who doesn't drive so good instead of being a insecure bar owner, for example. The packaging presents the "original" version of the character rather than the version currently-appearing-in-fiction. We're getting a new Armada Starscream toy and the Generation 1 comic books are gong to use that design for its G1 Starscream character. However, if the pattern holds, the bio on the package will still talk about Armada Starscream rather than G1 Starscream. The packaging will claim the toy is a different iteration of the character versus the comic inside.
What I wonder is, does this hold true for Thundercracker? And even if this is a question to ask, is it possible to know? WFC Thundercracker's characterization is likely indistinguishable from G1 Thundercracker's. So of course this Thundercracker's bio is just every Thundercracker bio you've ever read before. And it'd be written this way whether it were WFC Thundercracker or G1 Thundercracker being described.
(I wouldn't expect the writer of this bio to know that Takara's Prime Thundercracker toy's bio takes the character in a different direction, nor would I expect him to feel bound by it.)
What I'm getting at is that this is likely a toy that comes in packaging which claims it's two different versions of the same character, much like everything else in the line, but with Thundercracker there's no outward way for us to know that, because of things.
And that fascinates me.
Because I'm a weirdo.
Tags: thundercracker, generations, war for cybertron, thrilling 30
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Fifty Shades of Gray But Mostly Black And Some Purple
I tell you what, I did not care a lot about Megatron's new stealth bomber body when it was introduced in IDW's first ongoing Transformers title. I mean, I didn't hate it. It just kind of existed. It was undoubtedly a thing, just not a thing I gave a lot of thought about. And it had a big M on his forehead, which I'm not sure if I love or hate for its goofiness.
But then Hasbro decided not only to make a toy of it, but to also commission of comics about the toys they were making to include in the packaging. And so we got this amazing comic book both written and illustrated by Nick Roche. I wish Nick Roche would write more. Hell, I wish he would draw more. .... while he writes. He also both wrote and drew Spotlight: Kup, which is another fantastic Transformers story, easily one of the best. The connect between what the story wants to do and what it actually does is strong. Not an inch is wasted.
You might roll your eyes at a "Spotlight: Megatron" issue because, yeah, oh boy, FINALLY, there's gonna be a focus on Megatron, leader of the Decepticons, ABOUT TIME, but the comic book lives up to and exceeds your expectations. We see Megatron returning to life in a new body amidst his crumbling army, and we see how we begins to build that army back up again. He has a way of things, a formula, and center to that formula is Starscream. However, Starscream's as much in shambles as the rest of the Decepticons, and so Megatron literally spends the issue beating Starscream back into his usual self again. And Jesus God, is it slashy, and not in a kind way. By issue's end, you have a perfect idea of how Megatron's brain works. It's brutal, but amazingly executed.
ANYWAY NOW I CARE A BUNCH ABOUT THIS PARTICULAR MEGATRON BODY I GUESS. Thanks, comic. Stupid excellent storytelling.
Stealth Megatron is a Deluxe. This means he's on the small size for towering over much of your collection as he should, but there's a Starscream who's arriving on pegs at the same time who is just about the right size relative to him. Legends Starscream is also an IDW comics design, but a discarded one that was never used for Starscream himself (just Thundercracker). Also, this Starscream comes with a tiny Waspinator partner/weapon. I'm just piling on the reasons to own these things, aren't I. And so I've been having my Deluxe Megatron smack my Legends Starscream around my desk since Megatron arrived in the mail. They're a good pair.
Despite Stealth Megatron's Deluxeness, he's pretty meaty. His arms have a great mass to them, and he just looks like this intimidating chunk of dude who could mess you up... so long as you don't put him next to anybody else in his size class. He transforms by bunching up into this pentagon-shaped thing, and then you tear his arm cannon in half and plug them into the ends for wings. It's a little complicated and messy the first time you try it, but on the second tries and beyond it gets pretty simple. The learning curve is fairly short.
And of course the comic book comes with it. If you don't own the comic book, pick up the toy just to read it, dammit. (Or here it is on Comixology.)
Tags: generations, megatron, starscream, waspinator, thrilling 30
Why can't you be cool like me?
Posted May 14, 2013 at 12:41 am
("He ends most of his speeches that way.")
Hey, everyone, let me introduce you to The Perfect Springer. And, no, actually, it's not because he's a Triple Changer again for the first time in forever. I mean, that's nice. I like that. It's cool. But what makes him the super best is that he takes the bulk of his visuals from Nick Roche's Springer redesign for Last Stand of the Wreckers -- which, you know, is an awesome story, because I keep telling you that it is.
I would have been happyish with a based-on-the-old-cartoon Triple Changer Springer. But Last Stand of the Wreckers Springer? That gets me instantly to climax. It is Boner City over here. In four hours I'm gonna have to call a doctor.
Admittedly, this particular rendition of LSOTW Springer is taller and thinner than how he was rendered in the story itself. He was a wide mass of dude in that thing. This toy is thinner than that, though still, you know, not being a stick. It's probably one of those sacrifices you gotta make because of physics, as this guy has two altmodes to transform into, and if you bulk up his legs you kind of end up with a fat car and helicopter. His proportions do resemble more Roche's original control art, versus Springer's actual appearances.
There are some other changes to the design for this toy, but they're ones I'm satisfied with. Mostly, it's that his altmodes look less made-up vehicle-y by a smidge. Springer's new car mode looks a lot more like a car versus a four-wheeled vehicle of some undetermined sort.
Let me tangent here: Man, what the hell was with original 1986 Springer's toy, anyway? I'm not sure that thing is a real Triple Changer. It has a robot mode and it has an okayish helicopter mode, but his car mode is just the midtransformation step with the propeller removed. It barely pretends to have rear wheels. What an awful thing. And how awesome is it that this new toy has three distinct modes that are all great? I mean, even the shape of the windshield changes. Looking at either vehicle mode, you couldn't tell at a glance that it has a second one.
Springer doesn't have any engineering hurdles like Blitzwing's shoulders. He's, as I've been saying, pretty damn perfect. The only part of the transformation that gave me pause was how to attach the gun under the cockpit in helicopter mode. The instructions just give you an arrow and the words "ATTACH WEAPON." Well, yeah, great, but HOW? Apparently there are some... notches? I just kind of use excessive force and some shapes that look like they complement each other eventually snap together.
The propeller spins very satisfactorily. And it transforms into a believable sword seamlessly. It's not that giant plank of wood on a peg the original toy had. I'm impressed at the amount of articulation he has, too, despite the engineering demands. He has a turning waist and even though his hands bury inside his own forearms while transformed, he still has articulated wrists. Voyager Class is also the perfect size for a Springer -- dude was larger than everyone else in the cartoon (he was Optimus Prime/Rodimus Prime's height), and in the comic he wasn't no shrimp. So he'll look right next to a Deluxe Hot Rod, Kup, or Blurr.
you will buy this
you will buy this now
Tags: generations, springer, thrilling 30


In the past few days a large number of links were posted on Twitter in the form of shortend URLs, mostly from the Google's shortening service goo.gl. The text contained words like "cool, "very nice" and Google's search page has done it again".
When clicked on the link, it took the user to malicious site which would pretend to inform the user that it had found many viruses and would advise the user to install a security software called SecurityShieldFraud. This is a malware app which contacted other servers. No further information is available about the behavior of the app.
While all the links on Twitter leading to the malicious sites have been blocked, it is important for users to use Twitter with care. While Twitter can display the original link address if we hover the cursor over any shortened link on Twitter, third party services like Untiny can be used to see the original link of any shortened URL.


text:
Category: Martin Edwards
Gallows Court by Martin Edwards – London in 1930 is not a safe place to be…
London in 1930 is a strange place. Puzzles, death, suspicions, and one man is sure that he has the key. Rachel Savernake is always there – powerful, wealthy but at heart an enigma. Martin Edwards pulls out all the stops to create not only characters that fill the complex and many layered novel, but to create a world in which seemingly anything can happen, expectations can be overturned, and no one is safe. Edwards has used his unique knowledge of the period and the Golden Age Mystery novel to produce a powerful story which does not hinge on one murder, however complex. This novel describes a series of deaths which are perhaps suicide, perhaps extraordinary accidents, perhaps unconnected, but all seem to hang together, as the wisps of a story come together into a strong conclusion. I was delighted to receive a copy of this brilliant book to read and review.
Jacob Flint is a determined young man. He has moved from Yorkshire to make his name as a journalist in a London where self made men were common. For reasons that are perhaps unclear, he becomes convinced that a mysterious young woman, Rachel Savernake, knows something about not only the notorious crime that she has been rumoured to have solved, but also has special knowledge of various unexplained deaths among rich and powerful men. After all, she is extraordinarily well informed about seemingly unconnected things, she is in the area when many of the deaths occur, she has few but absolutely devoted servants, and she has a great deal of money. Jacob begins to make links by determined efforts to be at the right place at the right, or wrong, time. He does begin to wonder, however, as he experiences first hand not only near misses, but the actual death of those around him. While he could walk away, be sensible, be safe, he becomes enthralled by the desperate hunt for the truth. Rachel meanwhile seems not only an observer of these desperate deaths, but also have some strange involvement. Also, throughout the novel there are inserts of a journal of a young woman, concerned with her life on an island where Rachel and her cruel father are frequently mentioned. Who is Rachel, and how does she feature in this account of collective destruction?
This is a colourful book, where no death, no murder, takes place in an isolated or colourless manner. This is a London of seedy boarding houses, new art, gaudy theatre and new ways of death. Not that this is a gloomy novel; there is great strength of purpose and some flashes of humour as Jacob, Rachel and others act, observe and discover various truths. This is a mature, effortlessly confident novel in which Edwards uses all his skills and experience of writing contemporary murder mysteries to create mysteries within a mystery which absorbs and compels the reader to become involved. I was really bewildered at times by this book at times, but in a really good way, as just as I thought that I could see what would happen, it would suddenly plunge off in another direction. This is far from the common woman detective solving murder mysteries in the 1930s novel which is surprisingly common at the moment; this is a novel of unexpected heroes and villains. I can whole heartedly recommend this unusual and unique book to anyone with an interest in the Golden Age of detective writing, the social life of London in the 1930s, and how the motives of a few affect the lives of many in unique ways.
We have gone north over the last few days to view our son's new house and to allow the Vicar to go on a five church crawl around York today. All has gone well, as much effort has gone into the new (actually quite old) house and its period features. http://www.northernvicar.co.uk has got much material for blog posts in the near future, and daughter and I found many wonderful books in Waterstones and the Oxfam bookshop in York. I even found a folio set of Jane Austen in the latter, which is very beautiful…..
Leave a comment Gallows Court by Martin Edwards – London in 1930 is not a safe place to be…
The Christmas Card Crime – A list of the Stories and Authors from the Editor
One of the comments on my previous post about this book asked for a list of Stories and Authors contained in the anthology. Well, I asked the Editor, Martin Edwards, and he has provided me with the list. So, here it is:
Baroness Orczy – A Christmas Tragedy
Selwyn Jepson – By the Sword
Donald Stuart (Gerald Verner) – The Christmas Card Crime
Ronald Knox – The Motive
Carter Dickson – Blind Man's Hood
Francis Durbridge – Paul Temple's White Christmas
Cyril Hare – Sister Bessie
E C R Lorac – A Bit of Wire-Pulling
John Bude – Pattern of Revenge
John Bingham – Crime at Lark Cottage
Julian Symons – Twixt the Cup and the Lip
As I said in the review, quite a range of authors represented. Keen followers of the British Library Crime Classics series will recognise some names here, and some that I have reviewed on this blog. Why not look at a few on my list of authors in the right hand panel? Thank you so much, Martin, not only for providing this list but also discovering and editing so many really great books. If you are a fan of Classic Crime, or in need of a present for one, why not get your hands on a copy of Martin's book of Classic Crime novels? Here is my review https://northernreader.wordpress.com/2017/07/28/the-story-of-classic-crime-in-100-books-by-martin-edwards-the-review/
It is a lovely read, packed with information in a very readable style. A great present for yourself or someone else!
Leave a comment The Christmas Card Crime – A list of the Stories and Authors from the Editor
The Christmas Card Crime and other Stories Edited by Martin Edwards – A British Library collection
Eleven stories, eleven classics, and surely something for everyone in this book of crime classics brought out in time for Christmas by the British Library. As Martin Edwards points out in his excellent introduction to this volume in the Crime Classics series, the short story is a format well suited to crime tales. There is certainly no room for lengthy descriptions, character examinations and guessing at motive when the story is only a few pages long; as Edwards points out, every word must count. In these stories every crime is set up, executed and solved (or the solution is presented) brilliantly. There are no loose ends, empty speculation or confusion, but that is not to say that there is any lack of impetus in these tales of death and double dealing. They are all seriously enjoyable tales, often more thematically winter than actual Christmas events, and I was glad to receive a copy of this, the twelfth collection of short stories in this series.
Another fascinating element of this book is the brief introduction to each author written by Edwards, giving biographical details and the place of the story in the entire canon of writer's work. It also points out where the story originally appeared, and how difficult it may have been to read it without this book. The author's range from Baroness Orczy, of Scarlet Pimpernel fame, with one of her "Lady Molly" stories, a woman apparently gifted with all sorts of intuitive skills appreciated by Scotland Yard in solving murder. While Jepson's story "By the Sword" sounds grand, it actually concerns a country house mystery of multiple motives. The eponymous story concerning a Christmas card has a large cast, an effective detective, and a railway journey affected by the wintery weather. The other stories cover a large range of situations; distantly described crimes from the oddest of motives, observations on families and relationships, and a complex tale of jewel robbery. Murder is rarely a straightforward issue in these tales, elegant schemes replace brutality, natural justice often wins out.
This book has some very welcome short stories by authors that have been largely ignored since the time of their greatest popularity; it is only with the start of this series that writers such as John Bude and E.C.R. Lorac have been rediscovered and enjoyed in the novels reprinted by the British Library. As with any collection of this nature from different authors written at different times, some novels work better than others and will appeal more to a particular reader differently at different times. However, the stories are all of a high quality and will appeal to anyone who is a fan of the Golden Age of Detection writings. Despite the festive title, these stories are not so Christmas based that they can only be read at that time of year; this book would make an excellent gift as every recipient would find stories that would appeal in this collection. While some crime novels are more suited to one reader or another, this collection has a broader range. I would recommend it has a gift or as a treat for anyone interested in mysteries written in a different age but with themes familiar even today.
Life in the Vicarage at this time is busy. The weekend was dominated by a display of nativity/ crib scenes in one of the churches. Happily we actually had more than fifty sets in the end, ranging from the tiny in a jewelry case to a large set of antique figures. Over a hundred and seventy people came to see them, including lots of small people who enjoyed spotting the main characters and various animals. It was a good day but a bit tiring. Thankfully virtually all of them have been collected intact; none of them had a huge monetary value but many had been in families for years, with children, grandchildren and even great grandchildren expecting to see them at home. No two were the same, but I am still looking for one we bought in September and put away safely…
4 Comments The Christmas Card Crime and other Stories Edited by Martin Edwards – A British Library collection
Blood on the Tracks – Railway Mysteries edited by Martin Edwards. An excellent British Library Crime Classic
Ranging from Arthur Conan Doyle to Michael Gilbert, this collection of stories edited by Martin Edwards represents a Golden Age of murder mysteries encapsulated in short stories. The common theme is railways; travel on them, as settings for untimely death, and even the use of trains as weapons. As pointed out by Edwards, they are enclosed spaces in which 'locked room' mysteries are contrived, as only a limited number of people are travelling in the carriages and coaches, thus providing a certain number of suspects. This collection of stories, all chosen as representative of the authors' output, are strong tales creating a world of steam trains, timetables and solutions. I was grateful to receive a copy of this book.
From the beginning the stories in this volume include mysteries of bodies found with interesting clues. Not all the bodies are found on a train; in some the bodies are found in or near the lines. Weapons must be found, means of killing established, bodies horribly mangled to conceal other wounds. Rarely are these gentle deaths by poison or other sophisticated means, but often well thought out and dependent on the accuracy of time tables. Other crimes such as theft are facilitated by the use of trains; depending on frequency and predictability, opportunity for deceit. There are also inverted tales to be found, as the murderer is described from the start and the suspense is to be found in the possibility of detection. Sophisticated stories prevail here, as men and women plot and plan but circumstances intervene. Not all the mysteries here are sorted out by detectives; the truth emerges in various ways and there is always a satisfactory ending. I particularly enjoyed the story by Dorothy L. Sayers, as the detection and resolution is not the obvious eventual solution. At least one of these stories reveal accurate knowledge of incidents on railways which ended in death; all depend on understanding of trains, signals and the way they were run. A few of the stories relate to the supernatural or non human agencies, they are literally haunting in their use of the supernatural and atmospheric reality of trains cloaked in steam. The power and predictability of trains is well examined here, not as dreary modes of commuter transport but scenes of struggle and emotion in so many ways.
Given the range and variety of these stories and the fifteen different authors who wrote them, it would be reasonable to expect one or two weak efforts, but actually they are all strong in their way. None have the depth and subtlety of the novels set on trains as they are all brief works, but all are clever working out of mysteries set either on trains or the lines on which they run to time and significantly, signals. This book would appeal to fans of mysteries in the grand traditions of the Golden Age, even if some are earlier and later than the inter war period. It would also be of great interest to the many who appreciate the preserved trains and lines throughout Britain, and are fascinated by the railways of previous times.
Of course , Northernvicar has shown great interest in this particular volume in the British Library Crime Classics series! I imagine he will find time to polish off a few of these stories; he has already checked some of the accuracy of the circumstances. A great value for money book….
Leave a comment Blood on the Tracks – Railway Mysteries edited by Martin Edwards. An excellent British Library Crime Classic
The Story of Classic Crime in 100 Books by Martin Edwards – the Review
This is a very, very good book. It works on so many levels; as a readable text, a history of Crime writing, and a reference book for anyone who collects and reads British Library Crime and Dean Street Press classic crime books. I was thrilled to receive a review copy of this book for the reasons above, and it proved an enjoyable and in depth read.
Basically this book is a run through the major fictional crime books of the twentieth century. Although there a hundred books mentioned in the title, there are dozens, probably hundreds mentioned in the text to explain and reinforce the headings. The chapter headings, which include such phrases as "Making fun of murder" and "Playing politics" demonstrate a lively style which carries on through a detailed examination of such sub-genres as 'inverted', 'impossible' and 'locked – door' crime stories, as well as the popularity of using real life crimes as the basis for fictional treatment. These headings are also familiar to anyone who has read such collections of stories from the British Library series as "Murder at the Manor".
I am happy to report that the book is full of information/commentary/references to women authors, which is not always the case with 'books about books'. I have not counted yet, but the stars such as Christie, Allingham, Simpson and others are well represented, with many reviews of Christie's books in particular. Dorothy L Sayers not only gets credited for her books, but also her role in founding the Detection Club and her influential reviews collected by Edwards in "Taking Detective Stories Seriously". Edwards also makes clear where women have adopted a male non de plume, or have written books jointly. I have not done the maths but I am fairly confident that all the brilliant women writers are fully recognised!
Edwards has obviously a great breath of knowledge of his subject and the confidence to write critically and persuasively about the history of these books. Such diverse groups as clergy get honourable mentions, as they spend hours puzzling over "a variety of matters which would puzzle many a businessman". Books such as "Gaudy Night" are not in the hundred, but get mentioned as "a love letter to Oxford", which really sums up the book.
Edwards is trying to flesh out and produce evidence to back up his argument for the nature, importance and popularity of classic crime that he set out in "The Golden Age of Murder". The hundred titles effectively receive mini reviews, with references to other relevant books, and authors are mentioned in terms of sadness on occasion that they did not produce more stories, and comments which show their development of themes. I think that one of the achievements is to write about all these books without revealing the end, which is a huge success as he provides enticements to read the stories without spoiling them. This is a useful book for anyone interested in fictional crime as a reference book, and an excellent read for those who just want to expand their enjoyment of this popular genre.
So, do get hold of a copy of this book if you possibly can. It is great to recognise so many books from the British Library series, as well as look forward to many to come. Thank you, Martin, for letting me see a copy of this book, for being such a major part of this fantastic series, and providing a great guest post!
1 Comment The Story of Classic Crime in 100 Books by Martin Edwards – the Review
The Story of Classic Crime in 100 Books by Martin Edwards – A Guest Post
Detective stories from the "Golden Age of murder" between the world wars are being discovered all over again. Thanks to enterprising publishers, and the advent of digital publishing, readers around the world now have the chance to enjoy many long-forgotten mysteries of the past. They are also finding that the critics who for decades tended to write off these books as dated and facile were way off the mark. Novels by writers such as John Bude and Christopher St John Sprigg, who were far from household names even in their hey-day are now enjoying a wholly unexpected renaissance.
The British Library's highly popular Crime Classics series has led the way. The series features authors who were once very highly regarded and successful – like Anthony Berkeley and Freeman Wills Crofts – and also the likes of John Rowland and Charles Kingston, who never hit the heights but were capable of telling a good story. Bude in particular has become a real readers' favourite – five of his books have now reappeared as Crime Classics, with two more in the pipeline. And now plenty of other publishers – including Harper Collins with their revived "Detective Story Club" series, and Dean Street Press, who have revived authors as diverse as Sir Basil Thomson, once a kingpin of Scotland Yard, and former naval commander Peter Drax – are following suit.
Of course, nostalgia plays a part in this revival. And the gorgeous period artwork of the British Library paperback covers has led many people to collect the whole set. But there's much more to it than fascination with the past and high production values. The fundamental appeal of Golden Age detective fiction is that the leading authors knew how to entertain their readers.
Yet if entertainment was their priority, their books still tell us a great deal about life during the Twenties and Thirties. Read Antidote to Venom by Crofts, for instance, and you'll be presented with an interesting picture of life in a provincial zoo, as well as a tricky murder method, and an interesting moral at the heart of the story. Sprigg was a poet and a Marxist, but his playful Death of an Airman offers a glimpse of the workings of a small Thirties airfield that is not only authentic (Sprigg was an expert on aeronautics) but also highly engaging. A visiting bishop from Australia does the detective work – you don't find sleuthing bishops nowadays!
Crofts and Sprigg are two of the writers whose work is discussed in my new book for the British Library, The Story of Classic Crime in 100 Books. It's a companion to the Crime Classics series, but it's not meant merely to be a reference book – though you can dip into it if you don't immediately have time to read it from cover to cover, and sample some of the themes I discuss, such as country house mysteries and police stories.
The book follows my The Golden Age of Murder, which gave me a pleasant surprise by winning awards here and in the US, and earning gratifying reviews worldwide (it's currently being translated into both Japanese and Chinese). Here my approach has been different, because the canvas – the first half of the last century – is much broader. I've chosen to discuss in depth books which seem to me to illustrate the evolution of the genre in an interesting way. But as with The Golden Age of Murder, I've endeavoured to use techniques I've honed as a novelist to tell a story that is much more – I hope! – than a recitation of endless facts.
What both books particularly have in common is that writing them has been a labour of love. I've been thrilled by the number of readers who have contacted me to express thanks for having their attention drawn to new titles. And if by any chance, you're casting round for fresh reading, have no fear. The Story of Classic Crime in 100 Books can help – never mind the title. In all about seven hundred books feature!
Thank you so much, Martin! I'm not sure I've ever had an author guest post on Northernreader before; I would love some more.
My copy of "The Story" has travelled with me to Orkney and back, and tomorrow I hope to post my review. Needless to say it was a brilliant read, and yes, I am compiling a list of books I must read as a result…
4 Comments The Story of Classic Crime in 100 Books by Martin Edwards – A Guest Post
The Golden Age of Murder by Martin Edwards
More fictional Murder! The title "Golden Age of Murder" sounds dated, but I think we still enjoy the Agatha Christies and so many other murder mysteries presented in books, television and film that this book is a fascinating insight into how it all began or at least how it got to be so popular.
This is the ultimate source book for anyone interested in the crime writers of the interwar period in Britain. It has depth; I used some of the information for a talk on one of the writers it has large sections on, Dorothy L Sayers. It has breadth; many of the authors currently being reprinted by the British Library Crime Classics and the excellent Dean Street Press are mentioned in detail here. It is, perhaps most importantly, a very readable book about the writers of novels and stories with very few spoilers for the actual books. Edwards has published a very clearly written, detailed guide to an entire era in publishing of books that most of us have encountered at some point, and for real fans of the genre it gives so many starting points for further reading it provides a reason why it has taken me a while to read.
The subtitle of this book is "The Mystery of the Writers who Invented the Modern Detective Story". This book features the writers, such as Sayers and Agatha Christie, who were the mainstay of The Detection Club formed by those British writers who wanted to meet to provide mutual support in the financially unpredictable pursuit of writing novel which featured mysteries, usually murder. Ritual and rules assumed different levels of support in a Society which invited members on the basis of their achievements in a genre which was emerging with the twentieth century, and gained its greatest popularity in the interwar years and in the beginnings of the Second World War. There are so many writers mentioned in this book I would suggest that most could find their own favourites among the ranks of Allingham, Berkeley, Doyle, Punshon, Jerrold and dozens more. Their interrelationships, personal issues and so forth are discussed here, as well as their relationships with other writers and publishers. The human stories behind the creation of the classics of criminal endeavour in terms of what the stories were based on, as well as highlighting why and how the stories were constructed provides some fascinating reading.
True life crime and moral questions as well as the detailed working out of seemingly impossible crimes are detailed here, especially in brief notes at the end of each chapter which can trigger many ideas for further reading. It is possible to read this book straight through as a study of people who wrote books, why and how, but it is also an extensive resource for more detailed study of an entire genre. Sometimes dismissed as cosy, dated storiess, this book covers an enormous range of writing which provides an invaluable social history and mysteries which still confuse and are popular today in their many tv and film outings.
My particular reason for enjoying this book, aside from the detailed notes and indexes, is the fascinatingly human stories of writers such as Sayers and Christie in their context. Women writers and indeed readers have been the mainstay of murder mysteries for decades. Edwards, together with the British Library and Dean Street Press, together with so many other publishers and groups, are keeping fictional murder alive (!) in all its complex and entertaining "Golden Age".
Leave a comment The Golden Age of Murder by Martin Edwards


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This Halloween bag makes a cute trick-or-treat tote! There's plenty of room for all the candy the kids will be getting, but you could also just carry it as a fun everyday tote with a Halloween message on it. Becca from Simply Edged shares a sewing tutorial showing how to make it.


Good morning! Emily Midgett with you today sharing some inspiration that I gleaned from one of my favorite new shows…. Outlander. I have always gotten loads of inspiration from fashion and fabrics, and the Paris episodes from Outlander are full of inspiration… color, pattern, aesthetic… and I couldn't wait to translate one of the beautiful dresses from Versailles into a card! Let's get started with the details!
One of my favorite outfits from the Paris episodes is a gown of chocolate brown silk with stunning pink and yellow flowers embroidered on it. I began by stamping some of the blooms from Winter Anemone on some chocolate brown card stock using white pigment ink. I then colored them using Polychromos pencils, and die cut them with the coordinating dies. I didn't want a white outline to ruin the effect of the beautiful colors on the chocolate brown, which is why I chose to color the images on the chocolate brown card stock.
Next, I die cut a hexagon frame from the same chocolate brown card stock using the Hexagon Solid Stack dies. I also used the Hexagon Stitched Stack to create a quilted pattern on another chocolate brown A2 panel. I also stamped a greeting on strips of chocolate brown card stock from Grateful and embossed them with white embossing powder. I also die cut a coordinating "grateful" die cut from the Grateful Honey Cuts set from some buttery yellow card stock.
I adhered all of the die cuts to the quilted card front using dimensional foam adhesive, then added a few Cupcake Rhinestones.
Well, that's all for me today! From what unexpected source will you get your next inspiration? It's everywhere! 😉 Thanks so much for stopping by, and have a marvelous weekend!


The proceedings of the CIB W65 Symposium on the Organization and Management of Construction conference are presented here and in the companion volumes as state-of-the-art papers documenting research and innovative practice in the field of construction.
The volumes cover four broad themes: business management, project management, risk management, IT development and applications. Each volume is organized to provide easy reference so that the practitioner can speedily extract up to date information and knowledge about the global construction industry.
Managing the Construction Enterprise (Volume One): Covers the firm and its business environment, markets and marketing, human resource management strategic planning, and quality management.
Managing the Construction Project (Volume Two): focuses upon productivity, procurement, international projects and human issues in relation to management performance of construction organisations.
Managing Risk (Volume Two): incorporates discussion of risk away from regulation by government and those safety risks inherent in the construction process.
Managing Construction Information (Volume Three, published in conjunction with Construct IT Centre of Excellence): incorporates material on information systems and methods, application of IT to the design and construction processes and how IT theory and applications are best transmitted to students and practitioners.
The work represents a collation of wide ranging ideas and theory about construction and how research has contributed to the development of the industry on a global application of research to the problems of the construction industry.