text:
\section{Introduction}
\label{sec_intro}

Accretion onto black holes is frequently accompanied by collimated outflows. Although many of the physical properties of these flows proved difficult to measure, strong evidence exists that they are relativistic in a variety of environments. This evidence includes the one-sided appearance of jets and observations of apparent superluminal motions in Galactic black-hole binaries \citep{mira99} and in active galactic nuclei (AGN; e.g., \citealt{jors05}), as well as the rapid gamma-ray variability in AGN \citep{dond95, ahar05}.

Jets are also magnetized. The polarization fraction of synchrotron radiation from AGN jets approaches its theoretical maximum value in some objects (e.g., \citealt{zava05, list05, push05}), indicating that the magnetic field is highly ordered. The maps of rotation measure of 3C273 \citep{asad02, zava05} and of several BL Lac objects \citep{gabu04} show strong gradients of the rotation measure across the jet which are expected if the toroidal component dominates. The polarization position angles indicate toroidal fields in more than half of all BL Lac objects \citep{gabu00, list05, push05}, but such observations may be subject to projection effects and relativistic aberration \citep{pari03, lyut05}. The strength of the magnetic field is poorly constrained, so its true dynamical importance cannot be evaluated from observations.

The current theoretical paradigm is that most relativistic jets are probably accelerated by large-scale electromagnetic stresses \citep{blan82, li92, cont94}. Acceleration is rapid until the flow passes the fast magnetosonic surface and attains rough equipartition between kinetic energy flux and Poynting flux. Further increase in the jet Lorentz-factor tends to be more gradual, but eventually it is possible for most of the Poynting flux to be converted to kinetic energy \citep{bege94, vlah03a, vlah03b, besk06, komi07}. Ultimately, the flow is confined laterally by either the inertia of the disk at the base of the jet, in which case the collimating force is transmitted along the jet by magnetic tension, or the pressure of the ambient medium along its walls \citep{bege95}.

In this paper we investigate the interior of laminar ultrarelativistic jets whose boundary pressures are matched to their surroundings. The flux-freezing condition implies that the poloidal component of the magnetic field decays faster than the toroidal component in an expanding jet \citep{bege84}, and we consider the structure of the jet far enough from its initial acceleration site that the toroidal component dominates. \citet{cont95} showed formally that at large distances down the jet the poloidal magnetic field is dynamically unimportant and discussed the critical points of such solutions.

As forces exterior to the jet gradually collimate it, curving the outermost streamlines slightly toward the axis, some of the material in the jet interior runs into the wall and piles up. Thus, jets may develop an edge-brightened or `hollow-cone' structure, with much of the jet energy, momentum and emissivity concentrated into a narrow sheath along the interface between the jet and its surroundings. The pinching effect due to magnetic tension has the opposite effect, tending to focus jet energy toward the axis \citep{usty95, usty00, ston00, komi07}. We find that for some functional dependencies of the ambient medium pressure $p_a(z)$ such jets can develop self-similar structures that admit a wide range of prescribed entropy and toroidal magnetic field distributions, and we investigate the interplay of pressure and magnetic field for the self-similar solutions.

In Section \ref{sec_solutions} we describe the basic equations, our assumptions and the family of self-similar solutions that we find. In Section \ref{sec_sync} we discuss the appearance of the synchrotron radiation from our solutions. We conclude in Section \ref{sec_conclusions}. In Section \ref{sec_solutions}, unless otherwise noted we use units in which $c=1$ and the magnetic field is redefined to eliminate factors of $4 \pi$ from Maxwell's equations. The magnetic field in Gaussian units is $\sqrt{4 \pi}$ times our magnetic field ${\bf B}$ and the current (and its density) in Gaussian units is $1/\sqrt{4\pi}$ times our current (and density). Our Greek indices run from 0 to 3 and the Latin ones from 1 to 3, and summation is assumed over repeating indices (using the Minkowski metric for Greek indices). We use Gaussian and CGS units in Section \ref{sec_sync}.

\section{Self-similar solutions}
\label{sec_solutions}

\subsection{Main equations}
\label{sec_maineq}

We start from the covariant form of the equations of relativistic MHD in flat space-time \citep{dixo78, anil89, komi99, gamm03}, assuming that there exists a frame in which the pressure tensor is isotropic:
\begin{eqnarray}
\nabla_{\nu} (\rho u^{\nu})=0; \label{covar_1}\\
\nabla_{\nu} T^{\mu\nu}=0; \label{covar_2}\\
\nabla_{\nu} (b^{\mu} u^{\nu}-b^{\nu} u^{\mu})=0.\label{covar_3}
\end{eqnarray}
These equations are, from top to bottom, the continuity equation, the energy and momentum conservation equations, and the source-free part of Maxwell's equations. Here, $\rho$ is the mass density, $u^{\mu}=(\gamma, {\bf v}\gamma)$ is the 4-velocity of a fluid element,
\begin{equation}
T^{\mu\nu}=(\rho+\varepsilon+P+b^2)u^{\mu}u^{\nu}+(P+\frac{1}{2}b^2)g^{\mu\nu}-b^{\mu}b^{\nu} \label{en_tensor}
\end{equation}
is the energy tensor, $\varepsilon$ is the internal energy density per unit volume, $P$ is pressure, $b^2=b^{\mu}b_{\mu}$ is the electromagnetic energy density and $g^{\mu\nu}$ is the space metric (flat in our case). If ${\bf B}$ is the magnetic field (in the observer's frame) of the fluid element moving with $u^{\mu}$, the components of the 4-vector $b^{\mu}$ are $b^0=B^i u^i$ and $b^i=(B^i+b^0u^i)/u^0$, so the source-free Maxwell's equations (\ref{covar_3}) are the same as in the non-relativistic case: $\nabla \cdot {\bf B}=0$ and $\partial {\bf B}/\partial t - {\bf \nabla \times [v \times B]}=0$ (flux-freezing condition). Equations (\ref{covar_1})-(\ref{covar_3}) need to be supplemented by the equation of state of the gas $P=P(\rho,\varepsilon)$. The rest of Maxwell's equations determine the current given $b^{\mu}$ and $u^{\mu}$:
\begin{equation}
J^{\mu}=\nabla_{\nu}(e^{\mu\nu\alpha\beta}u_{\alpha}b_{\beta}),\label{eq_current}
\end{equation}
where $e^{\mu\nu\alpha\beta}$ is the Levi-Civita tensor (related to the completely antisymmetric symbol as described in \citealt{gamm03} and references therein). Equations (\ref{covar_1})-(\ref{covar_3}) are obtained under the standard assumption of ideal MHD -- that is, that the electric field in the fluid frame vanishes. The electric field in the observer's frame is non-zero, but the ideal MHD condition allows us to eliminate it from the equations (it satisfies ${\bf E}+{\bf v \times B}=0$), and we do not discuss it further.

We now consider an ultrarelativistic gas in which the particle rest mass is small compared to the thermal energy; alternatively, our analysis is applicable to radiation-dominated flows in which the optical depth is high enough to ensure local thermodynamic equilibrium. The equation of state is then $P=\varepsilon/3$ and the mass density can be neglected in equation (\ref{en_tensor}). In this case, equations (\ref{covar_2}) and (\ref{covar_3}) do not contain the mass density which can be determined from the continuity equation given the solution $u^{\mu}$.

We further specialize to stationary flows ($\partial/\partial t=0$). Using the condition $\nabla\cdot {\bf B}=0$ and the explicit expression $\gamma=1/\sqrt{1-{\bf v}^2}$, we rewrite the linear combination of the components of equation (\ref{covar_2}), $\gamma^4 v^i \nabla^j T^{ij}-\gamma^4 v^i v^i \nabla^j T^{0j}=0$, as
\begin{equation}
v^i \nabla^i (P\gamma^4) + \frac{1}{2}\gamma^2 v^i \nabla^i ({\bf B}^2) - \gamma^4 v^i \nabla^j \frac{B^i B^j}{\gamma^2}=0.\label{eq_bernoulli}
\end{equation}
In the absence of a magnetic field, equation (\ref{eq_bernoulli}) implies that the quantity $S=P\gamma^4$ is conserved along any streamline and can therefore be identified with entropy, by analogy with the non-relativistic Bernoulli equation. This property is the consequence of our choice of the equation of state. In addition, in the non-magnetized case the linear combination $\gamma v^i\nabla^jT^{ij}-\gamma \nabla^j T^{0j}=0$ can be rewritten as $\nabla^i(P^{3/4}\gamma v^i)=0$, so according to equation (\ref{covar_1}) mass density in the non-magnetic flow is $\propto P^{3/4}$.

We now consider a stationary axisymmetric ($\partial/\partial\varphi=0$) flow with a purely toroidal magnetic field ($B=B_{\varphi}$) and a purely poloidal fluid velocity. Equations (\ref{covar_2}) and (\ref{covar_3}) can be then written in cylindrical coordinates $z$ (along the axis) and $r$ (perpendicular to the axis):
\begin{eqnarray}
\frac{1}{r}\frac{\partial}{\partial r}\left[r v_r (4P\gamma^2+B^2)\right]+\frac{\partial}{\partial z}\left[v_z(4P\gamma^2+B^2)\right]=0;\label{cyl_1}\\
(4P\gamma^2+B^2)\left(v_r\frac{\partial v_z}{\partial r}+v_z\frac{\partial v_z}{\partial z}\right)+\frac{\partial}{\partial z}\left(P+\frac{B^2}{2\gamma^2}\right)=0;\label{cyl_2}\\
(4P\gamma^2+B^2)\left(v_r\frac{\partial v_r}{\partial r}+v_z\frac{\partial v_r}{\partial z}\right)+\frac{\partial}{\partial r}\left(P+\frac{B^2}{2\gamma^2}\right)+\frac{B^2}{\gamma^2 r}=0;\label{cyl_3}\\
\frac{\partial}{\partial r}(v_r B)+ \frac{\partial}{\partial z}(v_z B)=0.\label{cyl_4}
\end{eqnarray}
Equation (\ref{cyl_1}) is the 0-component of (\ref{covar_2}), whereas (\ref{cyl_2}) and (\ref{cyl_3}) are the $z$ and $r$ components of the equation $\nabla^j T^{ij}-v^i \nabla^j T^{0j}=0$ and (\ref{cyl_4}) is the $\varphi$-component of equation (\ref{covar_3}). The remaining components of equations (\ref{covar_2})-(\ref{covar_3}) are automatically satisfied given our geometric assumptions.

From equations (\ref{cyl_1})-(\ref{cyl_4}) we see that the relativistically invariant magnetization parameter
\begin{equation}
\beta_B=\frac{B^2}{2P\gamma^2}\label{eq_betaB}
\end{equation}
(the inverse of the standard ratio of gas pressure to magnetic pressure $\beta$) quantifies the relative dynamical importance of the magnetic field. This value is also the ratio of the magnetic to thermal energy, so that $\beta_B=1$ corresponds to energy equipartition in the rest-frame of the fluid element. Indeed, for a fluid element moving with $\gamma$, the rest-frame magnetic field is $B/\gamma$ and the electric field is zero (the condition of ideal MHD), whereas pressure $P$ is a relativistic invariant. An additional factor of $4\pi$ appears in the denominator of (\ref{eq_betaB}) once the magnetic field is written in Gaussian units. In the observer's frame, the magnetic field is $B$, but there is also an electric field which counteracts the tension of the magnetic field lines, so the dynamical importance of the magnetic tension is decreased by a factor of $\gamma^2$.

We now look for self-similar solutions of equations (\ref{cyl_1})-(\ref{cyl_4}) in the paraxial ($r\ll z$) relativistic ($\gamma\gg 1$) limit, and in particular we use the approximation $v_z\simeq 1-1/(2\gamma^2)-v_r^2/2$. For streamlines, the condition of self-similarity implies that fluid elements move along the curves $r\propto z^k$. Instead of cylindrical coordinates $r$ and $z$, it becomes convenient to use the similarity variable (which is constant on a streamline) $x=r/z^k$ along with $z$. To the lowest order in $1/\gamma$, the shape of the streamline uniquely determines the radial velocity:
\begin{equation}
v_r(x,z)=k x z^{k-1}. \label{eq_vr}
\end{equation}
We then look for all solutions of the form
\begin{equation}
\gamma(r,z)=z^{k_g}g(x), \quad P(r,z)=z^{k_p}p(x), \quad B(r,z)=z^{k_b}u(x).\label{eq_self}
\end{equation}
After transforming equations (\ref{cyl_1})-(\ref{cyl_4}) into new variables $x$ and $z$ (derivatives change as $\partial/\partial r \rightarrow z^{-k}\partial/\partial x$, $\partial/\partial z\rightarrow \partial/\partial z - (k x/z) \partial/\partial x$) we substitute the four unknown functions in the form (\ref{eq_vr})-(\ref{eq_self}) and require that $x$ and $z$ variables separate. To satisfy equations (\ref{cyl_1})-(\ref{cyl_4}), the $z$ dependence of the dominant terms should be the same. After this condition is satisfied, we write down the $x$ component of all equations. These requirements constrain combinations of $k$, $k_b$, $k_g$ and $k_p$. For example, the presence of terms $P\gamma^2+KB^2$ (where $K$ is 1/4 or 1/2) implies that $\beta_B$ (defined in eq. \ref{eq_betaB}) can depend only on $x$, so that in self-similar solutions magnetization is constant on streamlines (i.e., $k_p+2k_g=2k_b$). We then consider three cases for $v_z$: $v_r\ll 1/\gamma$, $v_r\sim 1/\gamma$ (that is, $k-1=-k_g$) and $v_r\gg 1/\gamma$. We find that there are no self-similar solutions for $v_r\gg 1/\gamma$, and we discuss this issue further in section \ref{sec_boundary}. The case $v_r\ll 1/\gamma$ yields two solutions, one with $k=0$ and one with $k=1$. The case $v_r\sim 1/\gamma$ yields another solution with $k=1/2$. In all three cases, $P\gamma^4$ is conserved along the streamlines, as in an unmagnetized case. In section \ref{sec_cyl} we discuss the solutions with straight streamlines $k=0$ (cylindrical flow) and $k=1$ (conical expansion), and in section \ref{sec_parab} the parabolic flow $k=1/2$.

\subsection{Solutions with straight streamlines}
\label{sec_cyl}

In the case of the cylindrical jet, $k=0$, all streamlines are parallel to the axis, the radial velocity is $v_r=0$, and the corresponding self-similar solution of (\ref{cyl_1})-(\ref{cyl_4}) is $\partial P/\partial z=0$, $\partial \gamma/\partial z=0$ and $\partial B/\partial z=0$. Equations (\ref{cyl_1})-(\ref{cyl_4}) then reduce to the well-known (e.g., \citealt{komi99}) condition of balance between pressure gradients and magnetic tension forces:
\begin{equation}
\frac{{\rm d}}{{\rm d} r}P+\frac{1}{r^2}\frac{{\rm d}}{{\rm d} r}\left(\frac{r^2 B^2}{2\gamma^2}\right)=0.\label{eq_cyl}
\end{equation}
In the non-magnetic case, this condition is simply ${\rm d} P/{\rm d} r=0$, and if it is not satisfied the radial pressure gradients would create radial velocities in the flow destroying the initially cylindrical flow. The Lorentz-factor $\gamma(r)$ can vary from one streamline to another arbitrarily in the non-magnetized solution.

In a magnetized flow given two of the functions $B(r)$, $\gamma(r)$, $P(r)$ (or two independent combinations of them) we can determine the third using equation (\ref{eq_cyl}). We assume that the magnetic field of the jet does not penetrate into the ambient medium (which we consider conductive enough). Since the total current flowing through a cross-section of radius $r$ is $I(r)=2\pi r B$, this assumption is equivalent to requiring that the net current carried by the jet is 0. The initial condition for integrating equation (\ref{eq_cyl}) is then
\begin{equation}
\left(P+\frac{B^2}{2\gamma^2}\right)\Big |_{r_j}=P_a, \label{eq_boundary}
\end{equation}
where $P_a$ is the ambient pressure.

As an example, we consider a flow which has a constant magnetization parameter $\beta_B(r)={\rm const}$ and is additionally homogenized so it has the same entropy on all streamlines: $S(r)=P\gamma^4={\rm const}$. In this case the solution of (\ref{eq_cyl}) is formally divergent, with infinite pressure $P(r)\propto r^{-2\beta_B/(1+\beta_B)}$ and current density $j(r)=(Br)'_r/r\propto r^{-(2+3\beta_B)/(2+2\beta_B)}$ on the axis, and the symmetry condition $B(r=0)=0$ is not satisfied. However, the divergence is weak -- the enclosed current $I(r)=2\pi r B$ and the enclosed energy flux are both zero on the axis. For a jet with a constant magnetization parameter, another singularity occurs at the jet boundary $r_j$, where a current $I_{sc}=-2\pi r_j B(r_j)$ flows with an infinite current density along the cylindrical boundary to screen the magnetic field $B(r_j)$ inside the jet. In Figure \ref{pic_cyl} we illustrate typical solutions of equation (\ref{eq_cyl}). The effect of the magnetic field is to pinch jet material toward the axis.

The singularity on the axis arises because our equations (\ref{cyl_1})-(\ref{cyl_4}) assume that gas pressure gradient is the only force that balances the pinch due to the toroidal field. Adding other pressure components (for example, a poloidal magnetic field) would remove the singularity. Alternatively, current density and pressure singularities can be removed by assuming a finite current density everywhere and using the current $I(r)$ as a free function (along with another one, e.g., the same constant entropy condition $S(r)={\rm const}$ as above) to determine the solution of equation (\ref{eq_cyl}). Such solutions automatically satisfy the symmetry condition $B(r=0)=0$. Since there is no current singularity on the boundary of the jet in this case, the initial condition (\ref{eq_boundary}) reduces to $P(r_j)=P_a$. Solutions with finite current density are shown in Figure \ref{pic_cyl} with dashed lines. While they produce a smooth pressure profile near the axis and near the boundary of the jet, they are otherwise similar to the analytic solution.

For the solution with $\beta_B(r)=$const, the total energy flux carried by the jet is
\begin{equation}
\dot{E_j}=\int_0^{r_j} T^{0z} 2\pi r {\rm d} r = \int_0^{r_j} 4P\gamma^2\left(1+\frac{\beta_B}{2}\right) 2\pi r {\rm d} r = 4\pi \sqrt{SP_a(1+\beta_B)} r_j^2=4\pi P_a \gamma_j^2 r_j^2, \label{eq_enflux}
\end{equation}
which needs to be multiplied by $c$ to get the energy flux in dimensional units (cf. \citealt{lain02}). Remarkably, under our assumptions (constant entropy and magnetization across the jet) the energy flux is independent of magnetization. For AGN jets, this gives
\begin{equation}
r_j=140\mbox{ pc}\times \left(\frac{\dot{E_j}}{5\times 10^{44} \mbox{ erg sec}^{-1}}\right)^{1/2} \left(\frac{\gamma_j}{5}\right)^{-1} \left(\frac{P_a}{3\times 10^{-10}\mbox{ erg cm}^{-3}}\right)^{-1/2}.\label{eq_energy}
\end{equation}
The value for the pressure is taken from the observations of thermal X-ray emission in the center of 3C31 \citep{hard02}; it is much higher than the typical interstellar pressure in the disk of the Milky Way ($\simeq 5\times 10^{-13}$ erg cm$^{-3}$) and higher yet by comparison with intergalactic pressures. In other words, a typical intergalactic medium cannot provide enough pressure support for collimation of energetic jets. \citet{bege89} showed that such high pressures may be supplied by the cocoon which forms as the jet interacts with the ambient medium.

Another self-similar solution of equations (\ref{cyl_1})-(\ref{cyl_4}) can be obtained for $k=1$ and corresponds to a conical flow. The streamlines are straight and the variable $x$ (which is constant on a streamline) is simply the angle this streamline makes with the axis, so that $v_r(z,x)=x$. The solution is $P(z,x)=z^{-4}p(x)$, $\gamma(z,x)=z g(x)$ and $B=z^{-1} u(x)$. The current density is also directed along the streamlines, with $j_r(z,x)=z^{-2}(xu)'_x$ and $j_z(z,x)=z^{-2}(xu)'_x/x$. The condition for maintaining the conical flow is
\begin{equation}
\frac{{\rm d}}{{\rm d} x}p+\frac{1}{x^2}\frac{{\rm d}}{{\rm d} x}\left(\frac{x^2 u^2}{2g^2}\right)=0.\label{eq_cyl1}
\end{equation}
It is identical to equation (\ref{eq_cyl}), so formally the same conclusions apply for the lateral structure of the conical jet as for the radial structure of the cylindrical jet. However, in this geometry the jet is expanding fast enough that the outer parts of it are causally disconnected from the inner parts. We discuss the applicability of our solutions to this case in Section \ref{sec_boundary}.

\subsection{Parabolic solution}
\label{sec_parab}

There is only one more solution of equations (\ref{cyl_1})-(\ref{cyl_4}) in the form (\ref{eq_vr})-(\ref{eq_self}), and it occurs at $k=1/2$, so that the streamlines are parabolae ($r=xz^{1/2}$). In this case, the functions vary along the streamlines as $v_r(z,x)=z^{-1/2}x/2$, $P(z,x)=z^{-2}p(x)$, $\gamma(z,x)=z^{1/2}g(x)$, $B(z,x)=z^{-1/2}u(x)$. The lateral structure of the jet satisfies
\begin{equation}
\frac{{\rm d}}{{\rm d} x}p+\frac{1}{x^2}\frac{{\rm d}}{{\rm d} x}\left(\frac{x^2 u^2}{2g^2}\right)-x\left(pg^2+\frac{u^2}{4}\right)=0.\label{eq_par}
\end{equation}
Just like in the cylindrical and conical cases, the first two terms describe the balance between pressure gradients and magnetic tension forces, whereas the last term represents centrifugal forces which arise because the fluid moves along curved streamlines in the parabolic case. Again, there are two free functions and the third can be determined from equation (\ref{eq_par}). In the case when the free functions are $s(x)=p g^4={\rm const}$ and $\beta_B(x)=u^2/2pg^2={\rm const}$, the solutions can be obtained analytically, with the initial condition (\ref{eq_boundary}) at the boundary of the jet $x_j=r_j/z^{1/2}$. Current density singularities similar to those discussed in the previous section are also present in this case, but since they are weak (in the sense that the total current and energy flowing through a finite cross-section are finite) and can be easily eliminated by an appropriate choice of the current density, we do not concern ourselves with them any further. The external pressure must decline as $P_a(z)=p_a z^{-2}$ as the distance $z$ from the source increases in order for the parabolic solution to be realized. Example solutions for the lateral structure are shown in Figure \ref{pic_parab}.

In the non-magnetized case (left panel in Figure \ref{pic_parab}), the first free function is $u=0$, and we again consider an isentropic flow $s(x)=p g^4={\rm const}$. The solution of equation (\ref{eq_par}) is then
\begin{eqnarray}
\sqrt{\frac{p(x)}{p_a}}=1-\frac{g_j^2 x_j^2}{4}\left(1-\frac{x^2}{x_j^2}\right). \label{eq_pressure}
\end{eqnarray}
These solutions are shown in Figure \ref{pic_parab} with solid lines. However, for $g_j>2/x_j$ (or alternatively $s>s_{\rm cr}=16 p_a/x_j^4$) the right-hand side of this equation becomes negative near the axis, and we conclude that for these values of the Lorentz-factor (entropy) a global self-similar solution of MHD equations (\ref{cyl_1})-(\ref{cyl_4}) does not exist. The reason for this behavior is clear when we rewrite this condition as $\gamma_j(z)>2 z/r_j(z)$. In other words, the global solution in the parabolic case does not exist if $\gamma$ is larger than, roughly, the inverse of the opening angle of the jet $1/\theta_j$. In this case the outer parts of the jet are not causally connected with the inner parts, and the lack of a global solution is unsurprising.

In the magnetized case (right panel in Figure \ref{pic_parab}), the main difference from the purely hydrodynamic case is the pinching effect of the magnetic field. The constant entropy and magnetization solutions are
\begin{equation}
\sqrt{\frac{p(x)}{p_a/(1+\beta_B)}}=\left(\frac{x}{x_j}\right)^{-\beta_B/(1+\beta_B)}\left(1-\frac{g_j^2 x_j^2}{4}\frac{1+\beta_B/2}{1+3\beta_B/2}\right)+\frac{g_j^2 x^2}{4}\frac{1+\beta_B/2}{1+3\beta_B/2},\label{eq_pressure_mhd}
\end{equation}
which turn into solution (\ref{eq_pressure}) for $\beta_B=0$. The entropy of this solution is $s=p_ag_j^4/(1+\beta_B)$. The pinching effect does not eliminate the same critical behavior as we see in the non-magnetized case, and for
\begin{equation}
g_j>g_j({\rm cr})=\frac{2}{x_j}\sqrt{\frac{1+3\beta_B/2}{1+\beta_B/2}} \label{eq_unconnected}
\end{equation}
the global solution does not exist.

The total energy carried by the jet can be calculated the same way as in equation (\ref{eq_enflux}):
\begin{equation}
\dot{E}_j=4\pi p_a g_j^2 x_j^2-\frac{\pi}{2}p_a\frac{1+\beta_B/2}{1+\beta_B}g_j^4 x_j^4\label{eq_enflux_parab}
\end{equation}
(to be multiplied by $c$ to get the energy flux in CGS units; $p_a$, $g_j$ and $x_j$ are all dimensional values). We now fix the values of $\beta_B$, $p_a$ and $x_j$ and consider the solutions for different values of $g_j$. The behavior of the solutions is illustrated in Figure \ref{pic_parab}, with $g_j$ increasing from the top curve to the bottom curve in both panels. In the hydrodynamic case ($\beta_B=0$; Figure \ref{pic_parab}, left) as we increase $g_j$ the total energy of the jet $\dot{E}_j$ increases all the way up to the point described by condition (\ref{eq_unconnected}), when the jet becomes too fast to maintain causal contact and solutions (\ref{eq_pressure_mhd}) become inapplicable. For the magnetized jet ($\beta_B>0$; Figure \ref{pic_parab}, right), as we consider solutions with increasing values of $g_j$, the total energy of the jet also increases at first, but then reaches its maximum value of
\begin{equation}
\dot{E}_{\rm max}=8\pi p_a\frac{1+\beta_B}{1+\beta_B/2}\label{eq_emax}
\end{equation}
(times $c$ in CGS units) and then decreases slightly before the jet becomes causally disconnected. For energies close to $\dot{E}_{\rm max}$, most of the jet material is pushed against the wall of the jet, whereas for small energies most of the jet material is in the pinched core. One can think of increasing the Lorentz-factor of the flow as analogous to increasing the flow's inertia, so that the greater the inertia, the harder it is to pinch the jet toward the axis.

\subsection{Comments on flow geometry and causality}
\label{sec_boundary}

While we do see a pinching effect of the magnetic field in the self-similar parabolic solution, we need to make a distinction between our solution and the electromagnetic collimation seen in 2D numerical simulations. In the numerical simulations by \citet{komi07} and others the core is almost cylindrical when the outer streamlines are parabolic, so the ratio $\theta_{\rm core}/\theta_j$ decreases away from the base. The magnetic fields act to provide an additional collimation of the jet matter (the poloidal component of the magnetic field is dynamically important), not just to redistribute the matter close to the axis. In our case all streamlines are parabolic, and the pinched core also has a parabolic shape (albeit with a smaller opening angle than the outer streamlines), so that $\theta_{\rm core}/\theta_j=$const. The difference between the geometry of our solutions and those obtained in numerical simulations is a direct consequence -- and the limitation -- of the self-similarity assumption, in that no additional collimation beyond what is provided by the ambient pressure can be obtained in the self-similar solutions. In our solution the $z$-dependence of all functions, including $\gamma(x,z)=g(x)z^{1/2}$, is identical in the magnetized and purely hydrodynamic cases, indicating that acceleration by electromagnetic forces is matched to the thermal acceleration (their ratio is a constant on a streamline).

The lack of causal connection for $\theta_j\gamma>1$, which we suggested in Section \ref{sec_parab} based on the lack of a global solution of the lateral equilibrium equation, can be understood if one considers the expansion of the jet cross-section in the frame moving with the Lorentz-factor $\gamma$. In this frame, the condition $\theta_j\gamma<1$ is the same as the requirement that the cross-section expands slower than the speed of light. In the observer's frame, the condition $\theta_j\gamma<1$ comes from requiring that the time-scale for signal propagation across jet at distance $z$ is smaller than the time scale for the fluid elements to get from $z$ to $2z$.

Whether the condition $\theta_j\gamma<1$ is satisfied depends not just on the physical parameters such as energy or magnetization, but also on the general geometry. For the cylindrical solution, the value $\theta_j\gamma$ decreases with the distance from the origin of the jet, since $\gamma=$const. Therefore, far enough from the origin the cylindrical jet is in causal contact. For the conical jet, $\gamma\propto z$ and $\theta_j=$const, so far enough from the origin the conical jet always loses causal contact. When that happens, the condition $v_r\ll 1/\gamma$ used to derive the conical solution (end of section \ref{sec_maineq}) fails, since $v_r=\theta_j$. Therefore, the self-similar conical flow whose lateral structure satisfies equation (\ref{eq_cyl1}) is no longer a solution of the MHD equations downstream of $\theta_j\gamma\sim 1$. The parabolic case is unique in that the product $\theta_j\gamma$ remains constant as the jet propagates. In other words, in the parabolic case the sound-crossing time of the jet cross-section is matched to its expansion and propagation rate.

Our critical condition (\ref{eq_unconnected}) was obtained for a specific choice of free functions, namely, $s(x)=$const and $\beta_B(x)=$const. We now consider how this choice affects the presence of a critical condition and we take as an example a solution with a different set of free functions: $g(x)=$const and $\beta_B(x)=$const. The solution is then
\begin{equation}
p(x)=\frac{p_a}{1+\beta_B}\left(\frac{x}{x_j}\right)^{-2\beta_B/(1+\beta_B)}\exp\left(\frac{1+\beta_B/2}{2(1+\beta_B)}(x^2-x_j^2)g^2\right).
\end{equation}
The qualitative features of this solution are the same as those of solution (\ref{eq_pressure_mhd}), as both the edge pile-up (exponential term) and the magnetic pinch (power-law term) are present. Although the global solution formally exists for all $x_j$ and $g$, for large values of $x_j g$ all of the matter is piled up into the layer of thickness $\Delta x \sim 1/g$ near the edge and the pressure is exponentially small elsewhere.

In this paper we investigate the structure of the jet for three special cases of the ambient pressure ($p\propto z^0$, $p\propto z^{-2}$ and $p\propto z^{-4}$). In general, the interactions between the jet and the ambient medium may result in a formation of a shocked layer, such as that described by \citet{brom07} for a hydrodynamic stationary jet. Self-similar jets should be considered special cases in which a solution without a shocked layer is possible.

\section{Synchrotron emission}
\label{sec_sync}

\subsection{Constructing emission and polarization maps}

Our models relate the distribution of pressure and the magnetic field within the jet. In this setup, the relativistic particles that produce the pressure emit synchrotron radiation. If the total synchrotron losses are negligible relative to the energy flux of the jet, this emission does not affect the dynamical structure of the jet, and we can calculate the spatial distribution of synchrotron emission given $P(r,z)$ and $B(r,z)$.

We assume that all emission is optically thin and that the pressure in our model is due to relativistic particles with the distribution function $N(E){\rm d}E=K_e E^{-2}{\rm d}E$ between $E_{\rm min}$ and $E_{\rm max}$, so the spectral index of the synchrotron emission is $\alpha=0.5$ (defined as $S_{\nu}\propto\nu^{-\alpha}$) and the pressure is $P=\varepsilon/3=K_e\ln(E_{\rm max}/E_{\rm min})/3$. Since the pressure is rather insensitive to the poorly known value $E_{\rm max}/E_{\rm min}$, the normalization of the distribution function $K_e$ follows the pressure during the adiabatic expansion as the jet propagates. Assuming a purely leptonic jet, the rest-frame emissivity per unit frequency and unit solid angle is
\begin{equation}
j'_{\nu'}(\nu', {\bf n'})=4.86\times 10^{-14} |{\bf b \times n'}|^{3/2} K_e \nu'^{-0.5}\mbox{ erg s}^{-1}\mbox{ ster}^{-1}\mbox{ cm}^{-3}\mbox{ Hz}^{-1},\label{eq_emis}
\end{equation}
where $b$ is the rest-frame magnetic field in Gauss, $K_e$ is in erg cm$^{-3}$ and rest-frame frequency $\nu'$ is in Hz (e.g., \citealt{ginz79}). The unit vector in the direction of propagation of a photon is ${\bf n}$ in the observer's frame, but the same photon propagates with ${\bf n'}$ in the rest-frame of the fluid element. The rest-frame magnetic field is $b=B/\gamma=\sqrt{8\pi\beta_B P}$, where we are now using the Gaussian units. For a constant magnetization parameter, the emissivity would have traced pressure as $j_{\nu}\propto P^{(3+\alpha)/2}=P^{7/4}$ (as in the classical equipartition case, \citealt{bege84}) if not for the relativistic beaming and aberration effects, which are different for every volume element since they all move with different velocities at different angles to the line of sight.

We integrate emissivity (\ref{eq_emis}) along the line of sight (parametrized by $\zeta$) in the observer's frame and at a fixed observer's frequency $\nu$ (corresponding to different frequencies $\nu'$ in the rest-frame). The observed flux density from a piece of jet with coordinates $\xi$, $\eta$ as projected on the plane of the sky subtending a solid angle of $\Delta\Omega$ is
\begin{equation}
S_{\nu}(\xi, \eta)=\Delta\Omega \int \delta^{2+\alpha}(\xi,\eta,\zeta)j'_{\nu'}(\nu, \xi,\eta,\zeta, {\bf n'}) {\rm d}\zeta,\label{eq_flux}
\end{equation}
where the Doppler factor is $\delta=[\gamma(1-{\bf v \cdot n}/c)]^{-1}$. The aberration factor (the angle $\psi'$ between ${\bf b}$ and ${\bf n'}$ in eq. \ref{eq_emis}) can be calculated using the Lorentz-transformation of the propagation vector (Appendix C of \citealt{lyut03}) and turns out to be $\sin^2 \psi'=1-\delta^2\cos^2\psi$.

In addition to the intensity map (\ref{eq_flux}), we construct maps of polarization position angle and polarization fraction. To this end, we calculate the emissivity into the two independent modes of linear polarization, that is, Stokes parameters $j_Q$ and $j_U$ emitted by a unit volume within the jet \citep{ginz79, lyut03}. These expressions are similar to (\ref{eq_emis}), but contain additional dependence on the orientation of the magnetic field through the $E$-vector of polarization. As detailed in \citet{lyut03}, this latter vector also suffers relativistic aberration which we take into account in our calculations. The maps of $Q(\xi,\eta)$ and $U(\xi,\eta)$ are then obtained just like the total intensity (\ref{eq_flux}). Polarization fraction is $\sqrt{Q(\xi,\eta)^2+U(\xi,\eta)^2}/S_{\nu}(\xi,\eta)$ and polarization position angle is $\arctan [U(\xi,\eta),Q(\xi,\eta)]/2$.

Five parameters appear in equations (\ref{eq_pressure_mhd}), (\ref{eq_emis}) and (\ref{eq_flux}): $\phi$ is the angle between the jet axis and the line of sight; $p_a$ is the normalization of the ambient pressure; $\beta_B$ is the magnetization parameter; $x_j$ is the normalization of the opening angle; and $g_j$ is the normalization of the bulk Lorentz-factor. Solution (\ref{eq_pressure_mhd}) has a linear dependence on $p_a$, so changing this parameter simply results in scaling the total intensity $\propto p_a^{7/4}$. Furthermore, because of the self-similarity of our solutions $x_j$ is a geometrical factor, and increasing $x_j$ results in zooming in on the region near the basis on the jet. Therefore, if $g_j$ is expressed as a fraction of its critical value given by (\ref{eq_unconnected}) then $p(x/x_j)$ (eq. \ref{eq_pressure_mhd}) does not depend on $x_j$. As a result, only $\phi$, $\beta_B$ and $g_j/g_j({\rm cr})$ need to be specified to produce maps of synchrotron emission, and  $p_a$ is necessary for determining the overall flux normalization.

\subsection{Edge-brightened jet}

As an example, we consider the jet in the nucleus of a nearby elliptical galaxy M87. The large mass of the supermassive black hole which powers the jet ($\sim 3\times 10^9 M_{\odot}$) and its relatively small distance (16 Mpc) imply that the Schwarzschild radius $r_s$ subtends a few micro-arcsec on the sky, bringing the immediate environment of the black hole within reach of modern radio observations. The VLBI images on scales of few hundred $r_s$ show that the jet is wide at the base and collimated at large distances from the nucleus \citep{juno99, ly07, kova07}; in fact, the jet is rather close to being parabolic in shape (Figure \ref{pic_m87}a). Additionally, it is strongly edge-brightened, which is usually interpreted in terms of the Kelvin-Helmholtz instability acting on the boundary between the jet and the surrounding medium (\citealt{aloy00, peru07}; see also discussion of various models in \citealt{kova07}). We propose that the edge pile-up seen in our models of pressure-matched jets may provide an alternative explanation for edge-brightening in the sub-pc jet of M87.

Among our models, the most strongly edge-brightened one corresponds to the critical solution $g_j=g_j({\rm cr})$ (eq. \ref{eq_unconnected}; model A in Figure \ref{pic_parab}) which we adopt as a model for the jet in M87. In this case the magnetization parameter $\beta_B$ enters the expression for $g_j({\rm cr})$ and therefore weakly affects the Doppler factor, but the pressure profile (\ref{eq_pressure_mhd}) does not explicitly depend on $\beta_B$. Therefore, the spatial distribution of emission is almost independent of $\beta_B$, while the total intensity scales as $\propto\beta_B^{3/4}$ (eq. \ref{eq_emis}). We take $\beta_B=1$. The last parameter needed to specify the model is the angle between the jet axis and the line of sight $\phi$. In M87 it is constrained to be between $30^{\rm o}$ and $50^{\rm o}$ by the combination of apparent velocities of clumps (assumed to be representative of bulk fluid motions) and the brightness ratio of the approaching main jet to the receding counterjet \citep{ly07}. If we further assume that the jet is relativistic ($v/c\simeq1$) by the time it reaches the projected distance of 3.1 mas and if we take the brightness ratio measurement of \citet{kova07} at face value ($R>200$ at this distance) we obtain $\phi<38^{\rm o}$. We use $\phi=30^{\rm o}$ in our models. The observed emission from the jet of M87 is optically thin ($\alpha=0.4\pm 0.3$; \citealt{ly07}), in agreement with the assumptions we made when constructing emissivity maps.

The maps of total intensity, polarization fraction and polarization position angle for this setup are shown in Figure \ref{pic_map}A. The brighter right side is the component of the jet moving toward the observer. The contrast in intensity between the edge and the emission observed on-axis is similar in our model and in the jet of M87 (Figure \ref{pic_m87}b). The brightness of the jet in the intensity map declines away from the base of the jet roughly as $\propto \xi^{-3.8}$, where $\xi$ is the coordinate along the projected axis. This decline is due to the combined effects of the adiabatic expansion and the changes in aberration and the Doppler factor as the jet accelerates. The observed emission declines somewhat slower, $\propto \xi^{-2.4}$ as measured between the clumps from the map provided by \citet{ly07}. One possible explanation for the difference in decline rates, if they are taken at face value, is that our models are developed in the ultrarelativistic approximation and are not strictly applicable to the jet in M87 which has a typical Lorentz-factor of just a few, especially near the base. Comparing the model intensity to the observed intensity at about 2 mas from the base of the jet (projected distance of 0.16 pc$=500 r_s$) leads to an estimate of the ambient pressure ($\sim 10^{-3}$ erg cm$^{-3}$), of the total energy flux carried by the jet (a few times $10^{44}$ erg sec$^{-1}$; eq. \ref{eq_enflux_parab}) and of the magnetic field ($\sim 1$ Gauss).

Such estimates are based on the standard assumption that the observed synchrotron emission from the jet reflects bulk motions and bulk pressures of the material in the jet, but it is not clear that this is indeed the case. An extreme example is the radio jet in Centaurus A, in which practically all of the jet emission is contained in compact clumps \citep{hori06}. Although the individual exposures of the jet in M87 are clumpier than the time-averaged composite that we used for our analysis \citep{ly07}, they all show the same edge-brightened structure in the process of collimation, lending some support to the bulk origin of this emission. Because the nature of clumps and their relationship to the bulk motions of the fluid are not understood, in our model we do not attempt to reproduce the somewhat contradictory data on proper motions of these features \citep{ly07, kova07}.

Polarization measurements for the jet in M87 are now being obtained \citep{ly07}, and we present polarization maps expected from our models in Figure \ref{pic_map}A. The polarization fraction of emission from every volume of fluid is the same and equals $(1+\alpha)/(\frac{5}{3}+\alpha)\simeq 0.69$, but the polarization position angles of emission from different elements of the volume vary, so the polarization signal washes out when magnetic fields with different directions are present along the same line of sight. The polarization fraction and the polarization position angle are not the same in the jet and in the counterjet because of the effects of aberration mentioned above, even though the purely geometrical projection effects are the same for the main jet and for the counterjet. In the edge-brightened case the polarization is expected to be high almost everywhere. The orientation of the magnetic field changes along every line of sight, but only a thin region near the edge contributes most of the emission, and the change in the direction of ${\bf B}$ across this layer is small. Stripes of low polarization seen in the map occur because the superposition of emission from the near and far walls of the jet leads to depolarization in these areas. On the projected axis of the jet, the magnetic field in the emitting region lies in the plane of the sky and is perpendicular to the axis, so the polarization $E$-vector is parallel to the axis. On the edge of the jet, the projected magnetic field is parallel to the axis (Fig. \ref{pic_loop}), so the polarization $E$-vector is perpendicular to the axis. In other words, even though the magnetic field is purely toroidal, the projection effects result in the so-called `spine-sheath' polarization angle pattern \citep{attr99,push05}. This pattern is sometimes interpreted as evidence for a poloidal component of the magnetic field near the edge of the jet (for example, due to shearing between the jet and ambient medium). But as long as the jet does not lie exactly in the plane of the sky, projection effects lead to the `spine-sheath' pattern even for a purely toroidal field.

\subsection{Core-dominated jet}

As another example, we now consider the other extreme in which most of the emission is coming from the pinched core. In this case the surface brightness on the axis formally diverges for $\beta_B>2/5$. One could eliminate this singularity by considering different profiles of $\beta_B(x)$, such as those that we describe in Section \ref{sec_cyl}. Alternatively, it can be shown that the optical depth for synchrotron self-absorption (opacity is $\propto K_e \nu'^{-3} b^2$) diverges faster than emissivity for these values of $\beta_B$. So the emission from the pinched core is composed of a geometrically thin optically thick core surrounded by an optically thin jet. If we look at the jet at higher frequencies, we see deeper into the jet, so the emissivity near the optically thick core rises steeply, but at the same time the surface area of the optically thick core shrinks. The result is that one can always find a frequency at which the optically thick core contributes a negligible amount to the total emission. Observed jets are typically optically thin, so we assume that this condition is satisfied. The value of surface brightness exactly on axis is affected by the effects of optical depth, so in our calculations we assume that the core is smaller than a resolution element of our map and do not consider the lines of sight going directly through the axis.

The resulting maps of total intensity, polarization fraction and polarization position angle are presented in Figure \ref{pic_map}B. As expected, the intensity rises toward the axis. The polarization position angle map still displays the `spine-sheath' structure in the main jet, albeit with a reduced `spine', but in the counterjet the dominant orientation of the $E$-vector is now perpendicular to the axis. The `spine' shrinks because the emissivity rises toward the axis, so the largest contribution to the surface brightness is acquired where the line of sight passes closest to the axis, where the magnetic field projected on the plane of the sky is entirely parallel to the axis (tangential points in Figure \ref{pic_loop}). The naive conversion of the polarization position angle into the orientation of the magnetic field would imply that the field is mostly parallel to the axis, but in fact the fields are purely toroidal. This example illustrates that not only the orientation of the jet is important in inferring the direction of the magnetic field, but the emissivity profile as well.

In the core-dominated case, the polarization fraction is small in the center, where multiple field directions contribute, resulting in depolarization. The polarization fraction is close to its theoretical maximum on the edge, where only a thin layer of the jet material contributes to the observed emission. This rise is reminiscent of the puzzling polarization fraction profiles in the parsec-scale jets of 3C273 \citep{zava05} and in Mkn501 \citep{push05}. We suggest that in these objects the emissivity is rising toward the axis, producing the observed polarization pattern. In both these objects the orientation of the $E$-vector of polarization is perpendicular to the jet in the regions where the emission is highly polarized, in agreement with our predictions. To further strengthen the case for the toroidal fields in these objects, Mkn501 shows a well-defined `spine-sheath' polarization pattern, while 3C273 shows strong gradients of the rotation measure across the jet.

The core-dominated case is only mildly relativistic (the highest Lorentz-factor on the edge of the jet is only $\gamma \sim 3$ in Figure \ref{pic_map}B), so our solutions for the lateral equilibrium are not, strictly speaking, very accurate in this case. However, the effects of the geometric projection and emissivity profile on emission and polarization maps are rather generic and are well-illustrated by this approximate model.

\section{Conclusions}
\label{sec_conclusions}

We have considered axisymmetric relativistic MHD jets with an ultrarelativistic equation of state and toroidal magnetic fields. We have then found all self-similar solutions of relativistic MHD equations in this setup assuming that the jet propagates in a pressure equilibrium with the surrounding matter. There are three geometrically self-similar solutions which occur in the medium with pressure scaling as $P(z)\propto z^0$ (cylindrical jet), $\propto z^{-2}$ (parabolic jet) and $\propto z^{-4}$ (conical jet). The lateral structure of the jet is determined by the balance of the radial pressure gradient, the tension of the toroidal magnetic field lines and the centrifugal force. The last term is relevant only when the streamlines are curved. Our models represent a special case of a more general situation in which interactions between the jet and the medium lead to shocked layers \citep{brom07}; in our case, solutions with no shocks are possible.

While the self-similarity condition is quite restrictive, in that only a limited set of geometries can be studied, the lateral structure of the parabolic jet displays some generic features which we investigate in detail. The pressure gradients act to push the jet material against the walls of the jet, whereas the magnetic field tension acts to pile the material closer to the axis. The relative importance of the two effects is determined by the physical parameters of the jet, such as its magnetization $\beta_B$ and the total energy. The pinched core component is stronger for a higher value of $\beta_B$, whereas the pile-up component is stronger for a jet with a higher Lorentz-factor (or higher total energy, to put it another way).

Our models describe jets whose Lorentz-factors are smaller than the critical value $\gamma_{\rm cr}\simeq 1/\theta_j$, where $\theta_j$ is the opening angle of the jet. We find that for larger Lorentz-factors $\gamma\gg\gamma_{\rm cr}$ either the formal solution of the lateral equilibrium equation does not exist at all, or it exists in the form of a thin pile-up of the material along the edge of the jet. The condition $\gamma<\gamma_{\rm cr}$ is equivalent to requiring that the jet maintains causal contact in the lateral direction as it propagates.

Edge pile-up is a generic feature of a jet with curved streamlines and it might explain edge-brightening of the jet in M87 in the acceleration region \citep{ly07, kova07}. We also find that even purely toroidal magnetic fields can produce the `spine-sheath' polarization structure \citep{push05} when projection effects are taken into account. We find that for core-dominated jets the polarization fraction is expected to be near zero on the projected jet axis and rise toward the edge, as observed in 3C273 \citep{zava05} and Mkn501 (B1652+398; \citealt{push05}). The orientation of the polarization position angle is determined not just by the geometric projection effects, but also by the emissivity profile. For emissivity rising toward the axis, the polarization perpendicular to the axis may become the dominant orientation even for purely toroidal magnetic fields. These effects significantly complicate the interpretation of the observed polarization position angle in terms of the intrinsic direction of the magnetic field.

\acknowledgments
The authors are very grateful to Chun Ly and collaborators for providing the map of radio intensity in M87 in electronic form (Figure 3 of \citealt{ly07}). NLZ would like to thank Anatoly Spitkovsky and Paul Wiita for discussions, the members at JILA (University of Colorado at Boulder) for hospitality, and Serguei Komissarov and Vasily Beskin for the comments on the manuscript.

NLZ is supported by NASA Spitzer Space Telescope Fellowship Program, through a contract issued by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. MCB acknowledges support from NSF grant AST-0307502 and NASA Astrophysics Theory Program grant NNG06GI06G.

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Non-Pitcher Wilson Valdez Throws MLB Game
Wilson Valdez, a journeyman infielder for the Philadelphia Phillies, was pressed into emergency service as a pitcher. Teammates laughed. The Cincinnati Reds licked their chops and waited for their chance to tee off on some batting-practice fastballs and end the 19-inning game versus the Phillies at Citizens Bank Park. Instead of getting meatballs, Wilson Valdez threw hot cheese, pitching a scoreless inning, hitting 90 miles-an-hour on the radar gun, and getting the victory. Valdez, who has never pitched at the Major League level, picked up a win after pitching a scoreless, hitless inning.
The last time this weird feat happened was 11 years ago. Brent Mayne, a catcher for the Colorado Rockies, pitched one inning in a 7-6 victory over the Atlanta Braves. The game went into extra innings, and Mayne's scoreless inning was the winning inning. That was August 22, 2000, according to Stats Inc.
As for this current game, the 19-inning game is the longest in Citizens Bank Park history. The Phillies used 21 players; The Reds used 20. The Phillies won the game, beating the Reds by a score of 5-4.
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BDCC has $350,000 to loan to local small businesses.
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Published September 27, 2017 | By Will Tuttle
By Dr. Will Tuttle
As we deepen our inquiry into the nature of our culture, and how we can best help liberate animals, our Earth, and ourselves from the harmful actions and attitudes causing conflict and destruction, we may begin to realize that the problem is not just other people. We ourselves are part of the problem also. How many people can we actually and substantively change for the better? Quite honestly that number is one.
This is not to say that we can't help move our world and other people in a positive direction, but rather to emphasize that this capacity is, paradoxically, a result essentially of our authentic efforts to move ourselves in a positive direction. Most of us, when encountering someone who would like to "improve" us will wisely act to block that effort. Thus, in the end, the most effective way to change others and the world is to endeavor to embody and exemplify in ourselves whatever changes we'd like to see in others.
Substantive positive personal change requires inner effort to question the obsolete and harmful attitudes living within us, to understand them, and to heal them through the effort to cultivate a higher and more inclusive and realistic awareness. We have all been wounded from infancy be being born into a herding culture based on the exploitation and abuse of the weak by the strong. In order to build a movement capable of transforming our culture, we are called to undertake the most challenging of all efforts: the effort to transform ourselves.
Grassroots vegan advocacy starts with us as individuals in our communities, endeavoring to become more aware and proficient in our understanding, and in our ability to communicate and cooperate with others. This is the indispensible foundation of the movement to liberate animals, and all of us, from the status quo culture of domination that steals and destroys the sovereignty of both animals and people.
Large non-profit organizations, unlike grassroots efforts, tend to be reductionist and authoritarian in nature. Ironically, herderism, the living core of our cultural mindset and root of our problems, is similarly reductionist and authoritarian, but in the extreme. (If we were born as cows or chickens, we would have no doubt of this.) Thus the large animal protection organizations exist ironically as products of the herding culture and tend to mirror and support its underlying mentality and way of functioning. These organizations compete with one another for funding, and operate under centralized hierarchical structures of authority. The living transformation of vegan awakening tends in their hands to be reduced to a commodity to be marketed by compliant leafletters and spokespeople whose job is to deliver a message that has been crafted by public relations "experts." Such an approach is ironically an expression of the same mind-set that is creating the problem of animal exploitation in the first place. It tends to muffle the authentic creative voices of individuals who have unique talents and gifts to share.
Thus, to continually invent and renew a movement for the liberation of animals, we are each called, as individuals, to embody as best we can the change we are working for, and diligently monitor and refresh our organizations to ensure they reflect this same spirit. While going vegan is a liberating, healing, and empowering step for us as individuals, and naturally helps heal and improve our interconnected society, it is not the last step. In many ways, it is the first step of a life where we awaken from being a mere culturally-programmed bio-robot, and step out of the prison of the imposed food narrative of disease and destruction and its many manifestations. Our journey beckons us to continue ever on, so that our inner world, like our outer behavior, ever more fully reflects our core values of respect, kindness, freedom, and harmony.
This is the real challenge we all face. It's much easier and more tempting to blame others (such as political figures, financial elites, and so on) for our problems than it is to work with our internalized attitudes and wounds. And yet, it is only to the degree that we transform ourselves—so that our lives are radiant expressions of the message we would like others to hear—that we will experience our full potential to bring healing and positive change to our world. As the vegan movement becomes increasingly a movement of people who are authentically doing the inner work necessary to explore and purify our individual consciousness, our movement, through this, is becoming increasingly unstoppable.
Satyagraha, the "truth power" that Gandhi described, is ultimately triumphant, and the best way to spread truth is through embodying it so that we plant seeds of change naturally and effectively, with means that reflect the ends we envision. While we don't try to change others, we can definitely do our best to plant seeds of positive change in others by making these changes congruent in ourselves, so our actions, words, values, thoughts, feelings, and gestures are all aligned within us and people sense this congruency.
How do we make these positive internal changes? With practice. Like with anything else, we become what we practice. The ancient wisdom traditions have many practices that have helped people heal and renew their minds and lives. These practices can be especially helpful for us in contributing to a vegan world of respect and kindness for all.
One example is the Six Paramitas from the Buddhist tradition, also called the Six Perfections. Paramita in Sanskrit means "crossing over," so these six Paramitas are qualities that we can cultivate every day to help us cross over to the other shore of greater awareness and freedom. Although the Six Paramitas is an ancient practice, some contemporary teachers have come up with the idea of practicing them by focusing on a different paramita on each day of the week. What follows is the Six Paramitas practice for the seven days of the week, with a short explanation for each one. It's recommended to recite and contemplate each day's paramita in the morning to help guide and focus awareness for the day.
Monday, The Day of Giving: Dana Paramita. Let us have all-pervading love, not only for people but for all sentient beings, and give of ourselves and our possessions freely and without regret, with a heart of lovingkindness. Let our giving be rooted in transcendent awareness, knowing the three elements involved are of one essence. Finally, let us remember that the teachings contain no greater wisdom than the wisdom of letting go. This is also called Dana. (Note: the three elements mentioned here are giver, gift, and receiver)
Tuesday, The Day of Ethics: Sila Paramita. Let us maintain standards of discipline, etiquette, and decorum so that our actions are honorable in every situation, and thinking of others always, let us be careful not to harm them in any way. Let us cultivate the natural integrity, diligence, and straightforwardness that fundamental aspects of clear seeing and real virtue.
Wednesday, The Day of Patience: Kshanti Paramita. Let us treat all beings as the Buddha, seeing through the spell of appearance to the essential reality in which there is no fundamentally separate self. Let us remember that patience is humility, a basic acceptance of this present moment as the perfect unfoldment of life and love, and that it is also an accurate mirror of our present state of mind, and thus continuously bears great opportunities to grow in wisdom and compassion.
Thursday, the Day of Zealous Effort: Virya Paramita. Whatever is appropriate, let us perform it diligently and persistently push forward. Let us be aware of the Source of all energy and activity, and live in harmony with it.
Friday, the Day of Stillness and Stability of Mind: Dhyana Paramita. Let us function in tranquil one-pointedness of mind, free from agitation arising through the senses, and dualistic conceptions of this and that, self and other. Let us let go of tension and separateness, and realize the equanimity that is born of awakening to the truth underlying phenomena.
Saturday, the Day of Wisdom: Prajna Paramita. Let us act so that the mind is always pure and bright, and behave so that we are never uncertain about the rightness of our actions. By the brightness of inherent wisdom, let us dispel the delusion of separateness, and greed and hatred, which spring from ignorance about the real nature of sense objects.
Sunday, the Day of Service: Simultaneous Practice of all the Paramitas. Let us put all forms of wholesome action into practice, by rendering service to others and practicing generosity, loving speech, beneficial action, and cooperativeness. May lovingkindness and compassion emanate from our every thought, word, and deed, and bless all sentient beings, and may they all realize the original brightness of their minds and be instantly reborn in the Pure Land of clear seeing.
In conclusion, with this Six Paramitas practice, the idea is daily to recite and feel into the essential nature of the quality of mind and heart that is being evoked by this day's Paramita, and to hold it in consciousness throughout the day as best we can. Gradually, and with persistent practice, all six Paramitas will begin to suffuse their healing presence into our awareness in all seven days. We begin to realize that vegan living, nonviolence, respect for animals and other humans, and spiritual liberation are all profoundly interconnected. As we endeavor to embody the universal principles indicated by the six Paramitas, we are working at the roots to help build a more kind and harmonious world for all.
(Paintings by Madeleine W. Tuttle)
Posted in 2017 Essays, Current Blogs, Uncategorized
11 Responses to The Six Paramitas
Nancy Poznak says:
Insightful, brilliant, and loving as always. Thank you.
Barbara Olson says:
Thank you. This is a good workbook for myself. I will try to practice these and become this.
Mark Caponigro says:
What a handsome recommendation for a truly enlightened vegan practice! This way of focusing every day of our week is most helpful.
It's a shame that there can be such confusion caused by the advocacy of the animal-protection non-profits. They do some very good work, but leave their supporters often without a clear understanding of what they're up to, why they do what they do. As was said a long time ago, "Discernment is the mother of all the virtues."
It's certainly a fundamental truth, which we can't emphasize enough, shared by a number of wisdom traditions, that we are all one: ourselves, other human beings, nonhuman animals of all kinds, and all living creatures. And therefore, it's not just a matter of what these others deserve from us, i.e. respect, justice, even love, though that is true enough; but also, whatever good we do for others, and feel toward others, we also do and feel for ourselves.
Edward Rutledge says:
Fantastic article… thank you Will!
vivian tucciarone says:
Nancy Poznak says it beautifully. This is Insightful, brilliant, and loving as always. Thank you.
Susan Christiansz says:
Thank you for this gift, Will. Much appreciated.
Susan Mitchell says:
Thank you Dr. Tuttle for this very helpful instruction which is so beautifully accentuated by Madeline Tuttle's paintings. I heard you speak in Provincetown, MA, last year, and am so grateful for your radical message and your immense courage to stay loyal to the purity of your message. You have set the bar very high and we have a long way to go to cross over from our wounded selves to our shared healings with each other. Let me be guided on my own path by your dedication and compassion and faith in the inevitability of our unified hearts.
Brenda Glasgow says:
Clarity and love, such wisdom always, now for action
in the coming days. Beautiful pictures.
Many thanks, Will and Madelaine
Jacqueline Jakle says:
Dr. Will….as always, you bring compassion and clarity with your words. I recently purchased your book "The World Peace Diet" and have started reading it. I hope your seminars are going well and hope you will sometime come to Southern California.
Daniela Bachmaier says:
Thank you so very much for the wonderful writing and the beautiful pictures 💖💖💖🐝
I feel the truth and the power and the beauty in these words and concepts. Thank you for your work.
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Limerick will be the place to be if you want to be part of the biggest gathering of Antique & Art dealers anywhere in Ireland – and we mean bigger than any other Fair in Ireland! No matter what your interest there will be something for you at the Fair. Antique Furniture, Irish Art, Jewellery & Silver, Porcelain & China, Books, Coins & Banknotes, Clocks, Vintage Fashion & Accessories – you name it and it will be here. Up to 100 Dealers from all over Ireland will be selling their wares to the public over the two days.
There has never been a better time to collect or invest in Antiques & Art. You cannot fail to pick up a bargain no matter what your interest may be.
Admission: Adults €3.50 – children FREE.
*Or just €5 (for two adults + their children) if you print out and bring the voucher below.


Good Smile Company Teases The Arrival Of Nendoroid One-Punch Man Genos!
Since the debut of the Nendoroid One-Punch Man Saitama, I'm sure many have wondered if or when his apprentice, Genos, would be arriving. Well today Good Smile Company teased fans with an image featuring artwork of a Nendoroid Genos and said, "Nendoroid Genos coming soon!" This is definitely going to make a great addition to the Nendoroid Saitama figure. Hopefully we'll get some official images of the figure soon.


"Once you accept someone for who and what they really are, they will really surprise you by being better than you ever expected."
"Know love, know happiness. No love, no happiness."
"Love is loving/ accepting a person with all his/her strength and weaknesses."
"Love is not for the one you love, it is for the one who makes you feel special, with whom you laugh, cry, share, with whom you are yourself, who makes a difference to your life, makes you feel....love."
"Lucky is the man who is the first love of a woman but luckier is the woman who is the last love of a man."
"Pain is to know that someone whom you love, is in love with someone else."
"You came from no where, and in no time got close.., not to me, but to my heart."
"Love is not something that happens on first sight, it happens when you start knowing each other and in turn end up needing each other, for every feeling, for every thought and for every moment."
"Love is seeing yourself in someone's eyes and finding yourself in somebody's heart."
and now that I've found it, I'm afraid that I will lose it."
I will change the order of alphabets and place U and I together."
the only one who is, will never make you cry. "
I would hold you in my arms forever."
we were the best we'll ever be, just for a moment, you and me."
"I may not get to see you as often as I'd like. I may not get to hold you in my arms all through the night. But deep inside my heart I know, this is for real And no matter what, I'll always love you."
"The stolen kisses are always the sweetest."
"I'd walk half way around the world... for just one kiss from you..."
"Distance can't keep us apart 'cause you'll always live within my heart."
"If you love someone, put their name in a circle, instead of a heart, because hearts can break, but circles go on forever."
"Every moment of life is so precious and every moment I'm not with you is a moment wasted!"
"You don't love a woman because she is beautiful, she is beautiful because you love her."
"Don't frown, because you never know who's falling in love with your smile!"
"It breaks your heart to see the one you love is happy with someone else ... but it's more painful to know that the one you love is unhappy with you."
"To be is to love, to love you is to be..."
"Some thought are better left unsaid, some feelings are better left kept to yourself, but love has it's way of expressing itself despite the silence."
"It's not being in love that makes me happy. It's being in love with YOU that makes me happy."
"You know you are in love when every time you close your eyes, you see that special someone."
"I don't miss you and you alone... I miss you and me together."
"You and love came into my life at the very same moment."
"When you're not here something's missing... my smile."
"Let me not love you, for the rest of your life, but, for the rest of mine."
"I do not have to see heaven, because I see heaven in your lovely heart."
"It happens that you are in love with someone for long, until you meet someone else and realize what love is."
"I hate the stars becauseI look at the same ones as you do, without you."
"Just always remember that even when we are apart I will be missing you!!"
"True love stories don't have endings."
"For the world, you're someone. For someone, you are the world..."
"One day I told god to give me all to enjoy life, and he gave me you to enjoy all."
"Love is when you wake up each morning knowing that you have this great person by your side."
The hardest part is being away from you"
"I don't want to look at you because, if I did, I would never turn away..."
"When you feel lonely, look to the sky and know that I am somewhere beneath that same sky thinking of you and wishing you were there with me."
"Missing you is only half the story. Hope the other half is you missing me."
"I wish you could live my life for just a moment. Then you would you see the difference you mean to me."
"If you think my eyes are beautiful, it's because their looking at you."
"'Its better to have never met you than to dream about you and find you not here."
"Love is how close you become to another, no matter how far apart the world may separate you."
and kiss your perfect lips."
"You have not lived... Until you've found someone... Worth dying for"
"A sad thing in life is when you meet someone who means a lot to you, only to find out in the end it was never meant to be and you just have to let go."
"Love is looking at him and knowing that even if you can't have him, you feel happy because he is happy with someone else."
"Never give up if you still want to try. Never wipe away your tears if you still want to cry. Never settle for the answers if you still want to know. Never say you don't love him if your heart won't let go."
"The worst way to miss someone is when they are right beside you but you know that you can't have them."
"Missing you isn't the hard part. Knowing I once had you is what breaks my heart."
"Sometimes the worst thing he could do is tell you that you can find someone better than him...when at that moment all you want is him."
"Everyone says give up on you, but they don't see you the way I do. You're the one who broke my heart, you're the reason why my world fell apart. You're the one who makes me cry; yet I still love you and I don't know why."
"When a guy says "the last thing I'd wanna do is hurt you" it just means he's gotta do other things first."
"I'll never forget the times we shared, and I'll never forget how much you once cared, now it's over but I have one more thing to say, If I had a four leaf clover, I'd wish for one more day."
"I'll love you forever until the very last day...but I sit and wonder...why did you go away?"
"I am everything you wanted, I am everything you need, I am everything inside of you that you wished you could be! I say the right things at exactly the right time...but I mean nothing to you and I don't know why."
"I know I shouldn't care or wonder how you are, but I can't hide this hurt inside my broken heart. I'm fighting back emotions that I've never fought before, all because I'm not supposed to love you anymore."
"You were there for me when I needed you, but know we're torn apart. I should have known from the start, that when you'd leave, you'd leave me with a broken heart."
"You never know what you have until you lose it, and once you lose it; you can never get it back."
"Tears are words that the heart can't say."
"It's hard to answer the question "what's wrong?" when nothing's right."
"I'm going to smile like nothing's wrong, talk like everything's perect, act like it's just a dream and pretend that he's not hurting me."
"Have you ever decided not to become a couple because you were so afraid of losing what you already had with that person?"
"You're only bad when you get caught...so that makes me a good girl, RIGHT?"
"Never be sad for what is over, just be glad it was once yours."
"It's hard to tell your mind to stop loving someone when your heart still does."
"My friend. my companion, through good times and bad, my friend, my buddy, through happy times and sad. Beside me you stand, beside me you walk, you're there to listen, you're there to talk, with happiness, with smiles, with pain and tears, I know you'll be there throughout the years!"
"Though time can change som many things, I know one thing is true. Time will never change how I feel about you."
"If you can't understand why we just can't be friends, if you don't understand that...then you don't understand me."
"Goodbyes make you think. They make you realise what you've had and what you've lost, and what you've taken for granted."
"I'll always love you through thick and thin, I hold your love deep within. Just to let you know my love is true, I'd go to hell and back to be with you."
"Of all the friends I've ever met you're the one I won't forget, and if I die before you do, I'll go to heaven and wait for you!"
"Guyz are like parking spaces. All the good ones are taken, and the rest are handicapped!"
"One out of every four people are retarded. Check three of your friends, if they're okay than you're it!"
"If love isn't a game then why do PLAYERS find the constant need to SCORE?"
"Never trust a playa, for all they do is play with your heart."
"Sometimes you have to let go of someone to see if there's anything worth holding on to."
"Don't cry over someone who won't cry over you."
"Remember when someone is annoying you, it takes 42 muscles to frown and only 4 muscles to punch their lights out!"
"You may laugh at me because I'm different, but I laugh at you because you're all the same."
"A Best Friend is someone who believes in you when you don't even believe in yourself."
"Always remember that in some way or another you are somebody's Angel, even if you don't know it."
"Angelz are there for you even when everyone else hates you."
"Somewhere there's someone who dreams of your smile and finds in your presence life is worth while. So whenever you're lonely, remember it's true, somebody, somewhere is thinking about you!"
"I've learnt that goodbyes will always hurt, pictures never replace having being there, memories- good or bad- will always bring tears, and words can never replace feelings."
"Why does a rose represent love, if a rose always dies?"
"If you're asking if I need you, the answer is forever. If you're asking if I'll leave you, the answer is never. If you're asking what I value, the answer is you. If you're asking if I love you, the answer is I do!"
"In every life there is one great love, and one unforgettable heartbreak."
"If the heart is one of the strongest muscles in the body, why is it so easy to break?"
"I would rather be physically hurt than emotionally, because you can put a band-aid on your finger, but you can't put one on your heart."
"Eight letters, three words, two people, one meaning... I love you!"
"Loving someone forever means loving them even if they love someone else."
"Why do we love the ones who hurt us, but hurt the ones who love us? It's so easy to fall in love, but it's hard to find the right person to catch us."
"I never felt true love until I was with you, I never felt true sadness until you left me. When we broke up I think you took half of my heart with you, you took the half I need the most, because the half you left is the half that only loves you."
"I never thought the person I love the most would be the person to hurt me the most."
"Girls are like fireworks their kiss lights up the night, their hugs make everything bright, when they get mad they explode, and when they're happy they make everything glow!"
"I wish my mom told me the same thing about guys as she did about scary movies "Don't worry honey they're all fake."
"Loving someone who doesn't love you is like reaching for a star, you know you'll never reach it but you got to keep trying!"
"One day you will love me as I have loved you. One day you will think of me as I thought of you. One day you'll cry for me as i have cried for you. One day you'll want me but I won't want you!"
"Fine guys open my eyes, smart guys open my mind but only a sweet guy can open my heart."
"To the world you maybe one person, but to one person you may be the world"
"I'm not supposed to love you, im not supposed to care, im not supposed to live my life wishing you were there. I'm not supposed to wonder where you are or what you're doing, but i can't help it, cause I'm in Love with you!"
"Friends are angels who lift us to our feet when our wings have problems remembering how to fly."
"If I had a star for each time I thought about you, I would have the entire sky in my hands."
"Guys are like stars, they're millions of them, but only one can make your dreams come true!."
"Don't fall in love with someone you can live with! Fall in love with someone you can't live without!"
"Missing you is'nt the hardest part, knowing I once had you and let go is breaking my heart!"
"Remember when you asked me who I liked and I said no one? What I really meant was no one but you!"
"If I could come back in life as anything I would be one of your tears. What more could I want than to be conceived in your heart, born in your eyes, live on your cheeks, and die on your lips?"
"You may laugh because I'm different, but I laugh because you're ALL the same."
"Last night I looked up at the stars and matched each one with a reason why I love you. I was doing great until I ran out of stars."
"When a door of happiness closes, another one opens, but often times we look so long at the closed door that we don't see the one which has been opened for us."
"It's true that we don't know what we've got until it's gone, but it's also true that we don't know what we've been missing until it arrives."
"The brightest future will always be based on a forgotten past, you can't get on well in life until you let go of past failures and heartaches."
"The best kind of friend is the kind you can sit on a porch and swing with, never say a word, and then walk away feeling like it was the best conversation you've ever had."
"You can complain because roses have thorns, or you can rejoice because thorns have roses."
"If everyone could hear what everybody else was saying about them, there wouldn't even be 5 friends in the world."
"I'd do anything, I'd give you my world I'd wait forever to be your girl Just call out my name, and I will be there Just to show you how much how I care."
"Maybe God wanted us to meet the wrong people before meeting the right one so that when we finally do meet the right person, we will know how to be grateful."
"Giving someone all your love is never an assurance that they will give it back. Don't expect love in return; just wait for it to grow in their hearts and if it doesn't, be content it grew in yours."
"The happiest of people don't necessarily have the best of everything; they just make the most of everything that comes their way."
But also know when it's time to let go- For each and every one of us must learn how to grow."
"Don't forget to be kind to strangers. For so many who have done this have entertainment angels without realizing it."
"A sad thing in life is when you meet someone who means alot to you, only to find out in the end it was never meant to be and you just have to let go."
"People may not remember exactly what you said or did, but they will always remember how you made them feel."
"What are you to do when the only person that can stop you from crying made you cry to begin with?"
"It's hard to answer what's wrong when nothing is right."
"It's the hardest thing to watch the one you love fall in love with someone else."
"Shoot for the moon. Even if you miss you'll land among the stars."
"A coincidence is when God performs a miracle and decides to stay anonymous."
"I tried to take the pain away by finding someone new, but then i realized no one compared to you. and even if i look around pretending not to cry, i'll always go back to the day you finally said goodbye."
"There are things you'd love to hear that you would never hear from the person whom you would like to hear them from, but dont be so deaf as not to hear it from the one who says it from his heart."
"Happiness lies for those who cry, those who hurt, those who have searched and those who have tried, for only they can appreciate the importance of people who have touched their lives."
"The brightest future will always be based on a forgotten past, you cant go on well in life until you let go of your past failures and heartaches."
"To see you is what my eyes long for, to touch you is what my body longs for, but to love you, my heart has done already for a long time."
"You dont have to hold on to the pain to hold on to the memory."
"I know you believe you understand what you think i said, but i am sure you realize that what you heard is not what i meant."
"To love someone when there is no chance of that love ever thriving...that is romance."
"To be your friend was all i ever wanted-to be your lover was all i ever dreamed."
"Why didnt she ever tell him that she loved him? because once you say i love you, you can never take it back."
"And what if you had to see this person everyday and he just does these things that make you so mad sometimes and you're just scared of what might happen..you just have to see if being away from that person scares you even more."
"If i call would you talk to me? if i'm cold, would you embrace me? if i need you, would you be there? if i'll open up, would you listen to me? if i tell you i love you, would you believe me? if you believe me, would you love me?"
"I'd be happy to come back to you except it was you that went away."
"Dont say we aren't right for eachother. the way i see it, we're not right for anybody else."
"You'd be surprise how often, if you knew, a joke, a song, a memory, will make me think of you."
"I'm just wondering, does it hurt you to know i'm afraid to talk to you? what do i say to someone who has hurt me, confused me, and broken me beyond measures?.. but whom i still love more than words can express."
"I dont know exactly where i stand with you, i dont know what i mean to you, or what i see in you, but i do know that all i dream about is you."
"It was a million tiny things like that, when you add them all up, that meant we were suppose to be together, and i knew it."
"I may not get to see you as often as i'd like, i may not get to hold you all through the night, but deep inside my heart, i know that this is true, no matter what i do, i will always be in love with you."
"And something has to be right about us being together, because if it wasnt, i dont think i would feel the way i do when you kiss me."
"I love it when i am just walking from kissing you goodbye and we both turn around at the same time to get just one more."
"If and when i find someone, it wont be someone who will give me the world, but someone who will share it with me."
"I'm glad i dont know everything about you...because everyday you amaze me."
"Our relationship was like this beautiful thing, and i dont think that you understood the power of it."
"And so i chose to be with you, if the choice was mine to take, but you can make decisions too, and you've decided you wanted this heart to break."
"Love cannot be measured in days or time. it can only be measured in the kisses you give him while he's there or the tears you cry when he's not."
"A memory of true love is like a favorite song: no matter how many times it plays again, you never get tired of it."
"Just because you're lonely, that doesnt mean you're not loved."


The higher up you go in an organization's hierarchy, the value of your technical and functional skills decline while the value of your interpersonal skills increase.
Firms that direct senior executives to spend more time on their non-core job roles, see higher levels of performance, according to a recent survey by the University of Michigan's Ross School of Business. According to the survey of 378 Michigan-based executives, the average participant spend about 45 percent of his or her time on core job responsibilities and 55% in other areas: in project-based teams, enhancing individual career skills, developing innovative ideas or processes and working to support the overall company.
Those who spent just 36 percent of their time on core roles actually managed a higher-performing company than those who spent 50 percent of their time on core roles, according to the study.
"Leaders are pulled in various directions and that is why executive coaching is such a popular service these days, " says Jim Mitchell, senior vice president and general manager of Lee Hecht Harrison in Michigan. "If you don't take risks and are concentrating on those core roles that have an executive at his or her desk all day, you might be missing some good business opportunities."


I am trying to align road names in a row, or column as much as possible in QGIS. I know it can be done in Adobe Illustrator with a mapping plugin, but I don't have the $$ for that.
I tried something like this but it didn't didn't seem to help.
It doesn't need to be exactly like that, but I'd like to do something similar.
(1) Draw a line as the guide for the label position.
In this example this guideline layer has an id field "fid" with only one value (1).
(2) Go to the Layer properties | Labels | Placement.
There you will find Data defined | Coordinate X and Y. Select Edit from each pull-down menu.
(4) As the final step, please hide the guideline. Select the guideline layer and change the symbology to No Symbols.
Using Maplex, how do you keep labels from overlapping a non-labeled layer?
Correct syntax for loading virtualxl spatialite table?


text:
Q: Font Size scaling depending on length of the text and parent div React/Javascript
I have a div* it contains a paragraph. This is a question box so it can be longer or shorter. I would like to resolve if this paragraph is longer (number of text)then scale down the font size in order to the text fits in div. If the paragraph is shorter scale up the font size.
-font size clamp is not the best way. I tried using the Scaletext, but i could not resolve. (i think it can not be used for my problem)
*the parent div has max width and height and responsive

A: You could compare the divs boundaries (set by it's css width and height properties) with it's current scrollHeight:
So we're actually setting a fixed height and hide overflows via overflow:hidden.

*

*Then we're looping through all question divs

*check if they have an overflow

*if so – decrease font-size (within another loop) until scrollHeight <=divHeight

let questions = document.querySelectorAll(".question");
let resize = document.querySelector(".resize");

//initial font size adjustment
fixOverflow();

//resize observer
const resizeObserver = new ResizeObserver(() => {
  resetFontSizes();
  fixOverflow();
});
resizeObserver.observe(resize);

function fixOverflow() {
  questions.forEach(function (question, i) {
    let divHeight = question.getBoundingClientRect().height;
    let scrollHeight = question.scrollHeight;
    let style = window.getComputedStyle(question);
    let fontSize = parseFloat(style.fontSize);
    //shrink font size if text has overflow
    if (scrollHeight > divHeight) {
      for (let i = 0; scrollHeight > divHeight; i++) {
        fontSize--;
        question.style.fontSize = fontSize + "px";
        scrollHeight = question.scrollHeight;
      }
    }
    //optional: increase font size if free space available
    if (scrollHeight < divHeight) {
      for (let i = 0; scrollHeight < divHeight; i++) {
        fontSize++;
        question.style.fontSize = fontSize + "px";
        scrollHeight = question.scrollHeight;
      }
    }
  });
}

//reset previously changed font sizes
function resetFontSizes() {
  let questions = document.querySelectorAll(".question");
  questions.forEach(function (question, i) {
    question.style.removeProperty("font-size");
  });
}
.resize{
  overflow:auto;
  padding:1rem;
  border:1px solid #ccc;
  resize:both;
}

.question p{
  margin-top:0;
}

.question{
  font-size:20px;
  line-height:1.2em;
  max-width:calc(100% - 2rem);
  width:500px;
  height:120px;
  border:1px solid #ccc;
  overflow:hidden;
  padding:1rem;
}

One morning, when Gregor Samsa woke from troubled dreams, he found himself transformed in his bed into a horrible vermin.

One morning, when Gregor Samsa woke from troubled dreams, he found himself transformed in his bed into a horrible vermin. He lay on his armour-like back, and if he lifted his head a little he could see his brown belly, slightly domed and divided by arches into stiff sections. The bedding was hardly able to cover it and seemed ready to slide off any moment. His many legs, pitifully thin compared with the size of the rest of him, waved about helplessly as he looked.

Resize parent div

You will also need either an ResizeObserver() or a simple resize eventListener to update/reset the font-size after resizing the window.

A: If understand your question correctly you want to adjust the font size based on the screen width.
If that's the case - Have you tried to use the 'resize' javascript event and then scale up the font-size based on the innerWidth of that parent element?
Probably this would do the trick.
Reference to MDN Docs - resize event

A: maybe you can use "vw" - A unit of measure equal to a percentage of the width of the document
div{    font-size: 10vw; white-space: nowrap; }

A: First of all, it's good to know that there is a font-size CSS property and you could have something like

function setFontSize(element) {
    let map = [
        {start: 0, end: 10, fontSize: "60pt"},
        {start: 10, end: 20, fontSize: "40pt"},
        {start: 20, end: 30, fontSize: "20pt"},
        //...
    ]; //The actual values can be defined by you.

    let length = element.innerText.length;
    element.style.fontSize = map.filter(item => (item.start <= length && item.end > length))[0].fontSize;
}

function resizeAndSet(text) {
    let element = document.getElementById("youritem");
    element.innerText = text;
    setFontSize(element);
}

<input type="button" value="Big" onclick="resizeAndSet('abc')">
<input type="button" value="Average" onclick="resizeAndSet('abc0123456789')">
<input type="button" value="Small" onclick="resizeAndSet('abc01234567890123456789')">

Click through the buttons to see this in action. Alternatively you could use svg with its font-size, for example:
<svg viewBox="0 0 200 30" xmlns="http://www.w3.org/2000/svg">
  <text y="20" font-size="smaller">smaller</text>
  <text x="100" y="20" font-size="2em">2em</text>
</svg>

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