% First you see macros invented from an author , % up to === are only macros, so % Find ======= to see the beginning from the article % % UNITS % \def\cm{\,{\rm cm}} \def\mG{\,{\rm mG}} \def\dyn{\,{\rm dyn}} \def\erg{\,{\rm erg}} \def\kpc{\,{\rm kpc}} \def\yr{\,{\rm yr}} \def\sec{\,{\rm sec}} \def\Msol{\,{\rm M_\odot}} %\def\Msyr{\,{{{\rm M_\odot}\over{\rm yr}}} % % SYMBOLS % \def\sun{\odot} \def\quarter{\textstyle{1\over4}} \def\half{\textstyle{1\over2}} \def\fiveforth{\textstyle{5\over4}} \def\sixteenth{\textstyle{1\over{16}}} % % REFERENCES % \def\aua{{\it Astron. Astrophys.} } \def\apj{{\it Astrophys. J.} } \def\phfl{{\it Phys. Fluids} } \def\phrev{{\it Phys. Rev.} } \def\rprph{{\it Rep. Prog. Phys.} } \def\araa{{\it Ann. Rev. Astron. Astrophys.} } %\def\mnras{{\it Mon. Not. Roy. Astr. Soc.} } \def\mnras{{\it Monthly Notices Roy. Astron. Soc.} } \def\rmph{{\it Rev. Mod. Phys.} } \def\jplph{{\it J. Plasma Phys.} } \def\jmph{{\it J. Math. Phys.} } \def\spscirev{{\it Space Sci. Rev.} } \def\jgeores{{\it J. Geophys. Res.} } \hyphenation {non-re-la-ti-vi-stic} %%==================================================================%% %% %% %% Springer International %% %% %% %% %% %% THE %% %% %% %% A S T R O N O M Y %% %% %% %% AND %% %% %% %% A S T R O P H Y S I C S %% %% %% %% %% %% TeX Support Version 1988,1 %% %% %% %%==================================================================%% % Welcome to THE ASTRONOMY AND ASTROPHYSICS REVIEW document style. % This file will automatically format your document according to % the requirements of the publisher. You may run it on any system % based on Plain TeX by Donald E. Knuth. % Please follow the instructions carefully. % % !!! 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Investigation of the structure of hot spots in extragalactic radio jets } % %-------------------------------------------- % % Now the NAME(S) OF THE AUTHOR(S) must be given: % the first name followed by the surname. If there is more than % one author, the order is up to you. However, if there are more % than two authors, you must separate the names by commas. % If the authors have different affiliations, each % name has to be followed by @1, @2, ... , the numbers referring to the % different addresses to be listed in \INSTITUTE. % If you have done this correctly, the entry now reads, for example % \AUTHOR={ H. Sexner@1, B. Gustofsson@2, and T. Mayer@3 } % \AUTHOR={ S.\ts T. Laurel@1 and O.\ts L. Hardy@2} % % The author running head (on left-hand pages) will contain a % maximum of two authors' names. Thus, one author, one name; % two authors, two names. This will be done automatically using the % variable \AUTHOR which you have already filled in. 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To remind % you there is a message that appears in the TeX log as well as in your % output. % \RECDATE={October 24} % \ACCDATE={December 3, 1986} % If the two dates are in the same year, type the year in \ACCDATE only; % otherwise code both years. % \RECDATE={ December 16, 1988} \ACCDATE={ March 7, 1989 } %---------------------------------------------------------------------- % % Each paper MUST have a SUMMARY. Do not type the heading "Summary" again; % this will be automatically generated by the macro. % \SUMMARY={ The theory of special relativistic magnetohydrodynamic shock waves is analyzed for relativistic jets consisting of a perfect plasma of infinite conductivity. All the postshock physical quantities are expressed in terms of the relativistic compression ratio. This compression ratio is the solution of a polynomial of seventh degree, which has to be solved simultaneously with an equation for the polytropic index for the shocked plasma. The downstream state of the shocked plasma is then determined by the upstream state of the jet as specified by the Lorentz factor, the ratio of the electromagnetic to the material energy content in the jet and the orientation of the magnetic field. These results are applied to the beam cap of relativistic magnetized jets, where the beam shock is mostly perpendicular to the jet direction. We show in particular that already a moderate magnetic field can lead to weak shocks and that for small angles the toroidal component of the magnetic field in the jet will strongly be amplified in the shock transition. We discuss applications of these results for the structure of hot spots in extragalactic radio jets. } % %---------------------------------------------------------------------- % % The next entry concerns the keywords. 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HERE TYPE IN YOUR ARTICLE: % % For example: % % \titlea {Introduction} % Relativistic plasmas have been extensively studied in ... . %----------------------------------------------------------------------- % % If you miss the instruction for coding with the % % THEAAR MACRO PACKAGE 1988, % % please TeX it's source file again with: % % tex instruct.tex % % (you will find this file on the macro disk) %---------------------------------------------------------------------- %--------------------------------------------------------------------- % % If you have already written your text using TeX: % ---------------------------------------------------- % 1. You must code (for titles etc., see Instructions) with our codes. % 2. If you have layout codes in your text, such as % hsize, vskip, noindent, spaces, or special fonts, % these must be taken out. % 3. If you have your own macros, you must % a) take out the layout commands; % b) input theaar.macros AFTER your own macros, % so some of them are replaced. %---------------------------------------------------------------------- % \titlea {Introduction} Relativistic plasmas have been extensively studied in the context of pulsar physics (see, e.g., the review by Michel, 1982). Over the past ten years, it has become apparent that relativistic plasma flows are also involved in extragalactic jets (Scheuer, 1987). The most extensively studied class of jets is obviously associated with the extragalactic radio sources (Bridle and Perley, 1984, and Perley, 1986). Prominent jets are common amongst low luminosity radio galaxies and high luminosity quasars. The jets associated with quasars are nearly all one-sided and far more radio luminous than the jets associated with strong radio galaxies. \titlea {Characteristic speeds for magnetized jets} The characteristic speeds in a relativistic magnetized flow follow from the fundamental differential equations of relativistic MHD. These are particle number conservation % $$(n u^\alpha)_{;\alpha} = 0 , \eqno(1)$$ % magnetic flux conservation % $$(u^\alpha\,B'^\beta - u^\beta\,B'^\alpha )_{;\alpha} = 0 , \eqno(2)$$ % and conservation of total energy and momentum % $$T^{\alpha\beta}_{;\beta} = 0 , \eqno(3)$$ % where $T^{\alpha\beta}$ is the total energy-momentum tensor of the system % $$T^{\alpha\beta} = \left(\rho + P + {B'^2\over{4\pi}}\right) \,u^\alpha\,u^\beta-\left(P + {B'^2\over{8\pi}}\right) \, g^{\alpha\beta}- {1\over{4\pi}}\,B'^\alpha\,B'^\beta . \eqno(4)$$ % $n$ is the proper particle density, $\rho$ the total energy density, $P$ the pressure and $B'$ the magnetic field as seen in the plasma frame % $$B'_\alpha = u^\beta *F_{\beta\alpha} . \eqno(5)$$ % This magnetic field is a spacelike vector field, $B'^2 = - B'^\alpha B'_\alpha > 0 $ and satisfies $u^\alpha\,B'_\alpha = 0$. The corresponding electric field vanishes due to the MHD assumption. The tensor field $*F$ is the dual of the Maxwell field $F$. In MHD it has the form % $$*F_{\alpha\beta} = u_\alpha\,B'_\beta - u_\beta\,B'_\alpha . \eqno(6)$$ % The equations of perfect MHD form a first order quasi-linear system of partial differential equations. For this system, the characteristic surfaces $\Phi = const$ have been rigorously derived by Lichnerowicz (1967). According to his analysis, the characteristic surfaces are defined by two equations % $$Q^{\alpha\beta}\, n_\alpha \, n_\beta = 0 \eqno(7)$$ % and % $$P^{\alpha\beta\rho\sigma}\, n_\alpha n_\beta n_\rho n_\sigma = 0 , \eqno(8)$$ % where $n_\alpha = \Phi_{,\alpha}$ is the wave normal. The two tensors $Q$ and $P$ are defined in the following way (see Lichnerowicz, 1967) % $$Q^{\alpha\beta} = n\,\mu\,\left(1 + {B'^2\over {4\pi\mu n}}\right)\, u^\alpha u^\beta - {1\over {4\pi}}\, B'^\alpha\,B'^\beta ,\eqno(9)$$ % $$\eqalignno{{c_{\rm s}^2\over n}\,P^{\alpha\beta\rho\sigma} &= (1 - c_{\rm s}^2)\,u^\alpha u^\beta u^\rho u^\sigma - \left(c_{\rm s}^2 + {{B}'^2\over {4\pi\mu n}} \right)\, g^{(\alpha\beta} u^\rho u^{\sigma )}\cr &\qquad + {c_{\rm s}^2\over {4\pi\mu n}}\, g^{(\alpha\beta}\, B'^\rho B'^{\sigma)}.&(10)}$$ % $c_{\rm s}$ is the sound speed given by $c_{\rm s}^2 = \Gamma P/n\mu $ and $\mu = (\rho + P)/n$ the relativistic specific enthalpy. The first equation (7) defines the Alfv\'en phase speed, since the quantity % $$v^2 = {{\omega_{{\rm plasma}}^2}\over { {\bf k}^2 }} = {{(u^\alpha n_\alpha)^2}\over {-h^{\alpha\beta}\, n_\alpha n_\beta }} \eqno(11)$$ % is the wave velocity in the plasma frame. The projection tensor % $$h^{\alpha\beta} = g^{\alpha\beta} - u^\alpha u^\beta \eqno(12)$$ % defines the metric in the plasma rest space. With the introduction of the magnetic field $B'_n$ in the spatial direction of propagation of the waves, % $${B'}_n^2 = {{(B'^\alpha\,n_\alpha)^2} \over {- h^{\alpha\beta}\, n_\alpha n_\beta }} , \eqno(13)$$ % we find the Alfv\'en speed $v_{\rm A}$ as measured in the plasma rest frame (in units of {\it c}) % $$v_{\rm A}^2 = {{{B'}_n^2}\over {4\pi\mu n (1 + {B'}^2/4\pi\mu n)}} .\eqno(14)$$ % Since ${B'}_n^2 \leq {B'}^2$, the Alfv\'en speed $v_{\rm A}$ is always less than the speed of light. In a similar way, we obtain from the second Eq. (8) the definition of the slow and fast magnetosonic speeds, $v_{{\rm SM,FM}}$, % $$\eqalignno{&\left(1 + {B'^2\over {4\pi\mu n}}\right)\, v_{{\rm SM,FM}}^4-\left(c_{\rm s}^2 + {B'^2\over {4\pi\mu n}} + c_{\rm s}^2\,{{B'}_n^2\over {4\pi\mu n}} \right) \,v_{{\rm SM,FM}}^2 \cr &\qquad + c_{\rm s}^2\,{{B'}_n^2 \over {4\pi\mu n}} = 0 , &(15)}$$ % which satisfy the inequalities $v_{{\rm SM}}\leq v_{\rm A} \leq v_{{\rm FM}} \leq c$. In the following we work out the shock conditions with magnetic fields given in the shock frame and not in the proper plasma frame. The proper magnetic fields follow then from the definition given in Eq. (5). \titlea {The jump conditions} \titleb {Basic shock equations} The most prominent shocks in extragalactic jets occur at the front of the jets in the form of hot spots, which are considered as the downstream flow of a strong shock produced by the interaction of the jet with the intergalactic medium (Blandford and Rees, 1974). These shocks are not collision-dominated, but involve a collisonless plasma (for the discussion of Newtonian collisionless shocks, see Tidman and Krall, 1971; Wu, 1982). The downstream plasma of the hot spots consists in general of three different components, the shocked thermal plasma, magnetic fields and a relativistic component generated from suprathermal particles. % $$\eqalignno{[n\,u^\alpha]\,n_\alpha &= 0 , &(16) \cr [T^{\alpha\beta}]\,n_\alpha &= 0 , &(17) \cr [\vec{E}] \wedge \vec{n} &= 0 , &(18) \cr [\vec{B}]\cdot\vec{n} &= 0 . &(19) \cr} $$ % We consider a shock located in the $(s,\phi )$-plane in cylindrical coordinates $(t,s,\phi ,z)$. The upstream flow velocity is normal to the shock front. Without loss of generality, the upstream magnetic field has no radial component % $$\eqalign{u_1^\alpha &= (\gamma_1,0,0,u_1) , \quad u_1 = \beta_1\,\gamma_1 , \cr u_2^\alpha &= (\gamma_2,u_s,u_\phi ,u_z) = \gamma_2 \,(1,\beta_s,\beta_\phi,\beta_z) , \cr \vec{B_1} &= (0,B_{1\phi},B_z) , \cr \vec{B_2} &= (B_s,B_{2\phi},B_z) , \cr \vec{E_1} &= (E_s,0,0) , \cr \vec{E_2} &= (E_s,0,E_z) , \cr } $$ % where Eqs. (18) and (19) have been used. Equations (16) and (17) evaluated in the shock rest frame yield the following relations % $$\eqalignno{n_1 u_1 &= n_2 u_z , &(20) \cr n_1 \mu_1 \gamma_1 u_1 + {1\over {4\pi}} E_s B_{1\phi} &= n_2 \mu_2 \gamma_2 u_z + {1\over {4\pi}} E_s B_{2\phi} , &(21) \cr 0 &= n_2 \mu_2 u_s u_z &(22) \cr &\qquad - {1\over {4\pi}} (E_s E_z + B_s B_z) , \cr - {1\over {4\pi}} B_{1\phi} B_z &= n_2 \mu_2 u_\phi u_z - {1\over {4\pi}} B_{2\phi} B_z , &(22{\rm a}) \cr n_1 \mu_1 u_1^2 + P_1 + {1\over {8\pi}} B_{1\phi}^2 &= n_2 \mu_2 u_z^2 + P_2 &(22{\rm b}) \cr &\qquad + {1\over {8\pi}} (B_s^2 + B_{2\phi}^2 - E_z^2 ) . \cr }$$ % The electric and magnetic fields are connected through the MHD condition $$\eqalignno{\beta_1\,B_{1\phi} = E_s &= \beta_z\,B_{2\phi} - \beta_\phi\, B_z , &(23{\rm a}) \cr 0 &= \beta_s\,B_z - \beta_z\,B_s , &(23{\rm b}) \cr E_z &= \beta_\phi\,B_s - \beta_s\,B_{2\phi} . &(23{\rm c}) \cr } $$ % \titleb {Method of solution} We define the following dimensionless parameters % $$\eqalign{r &= {{n_2\,\gamma_2}\over {n_1\,\gamma_1}} , \quad q = {B_{2\phi}\over B_{1\phi}} , \quad \alpha ={{\mu_2\,\gamma_2}\over {\mu_1\,\gamma_1}} , \cr \sigma &= {{T^{0z}_{1{\rm em}}}\over {T^{0z}_{1{\rm mat}}}} = {B_{1\phi}^2\over {4\pi n_1\mu_1\gamma_1^2}} , \cr x &= {B_z^2\over {4\pi n_1\mu_1 u_1^2}} = {1\over M^2} .\cr}\eqno(24)$$ % $r$ is the shock frame compression ratio, $q$ the amplification factor in the transverse magnetic field, $\sigma$ is the upstream Poynting flux relative to the total mass-energy flux, while $M$ is a quasi-Newtonian Alfv\'en Mach number of the jet. It is related to the proper relativistic Alfv\'en Mach number $M_{\rm A}$ through % $$M_{\rm A}^4x - M_{\rm A}^2\Bigl(1 + \sigma + (x - 1)\beta_1^2 \Bigr) + \sigma\beta_1^2 = 0 .$$ % $M$ and $M_{\rm A}$ have the same Newtonian limit. From Eqs. (22c) and (23a) one obtains % $$\eqalignno{q &= r\,{{\alpha - x}\over {\alpha - r\,x}} , &(25) \cr \beta_\phi^2 &= {{\sigma\,x}\over\alpha^2}\,\,(q - 1)^2 , &(26) \cr} $$ % and from Eqs. (23c) and (22b) % $$B_s \bigl (\alpha^2 + \alpha\,(\sigma\,q - r\,x) - \sigma\,r\,x (q - 1) \bigr ) = 0 . \eqno(27) $$ % It has been shown by Lichnerowicz (1967) that the 2-plane defined by the tangential 4-vectors $t^\alpha = B'^\alpha - (B'^\beta n_\beta)n^\alpha$ and $w^\alpha = u^\alpha - (u^\beta n_\beta)n^\alpha$ remains invariant across the shock. If $t^\alpha$ and $w^\alpha$ have no radial components, the corresponding 3-vectors do not have one either. Therefore we can exclude solutions with $B_s \ne 0$, and in the following we only will have to consider the case $B_s = 0$ and $u_s = 0$. The normalization of the 4-velocity gives an equation for $\gamma_2$ % $$\gamma_2^2\,\left(1 - {\beta_1^2 \over r^2} - {{\sigma\,x} \over \alpha^2}\,(q - 1)^2 \right) = 1 , \eqno (28)$$ % and Eqs. (21), (22a) and (23a) yield % $$1 - \alpha = \sigma\,(q - 1) . \eqno (29)$$ % We restrict ourselves to cold jets $(P_1 = 0)$ and get for (22d) % $$1 - {\alpha \over r} = \Pi + {\sigma \over {2\beta_1^2}} \, (q^2 - 1) \eqno(30)$$ % with % $$\Pi = {P_2 \over {n_1mc^2u_1^2}} = 1 - {1\over r} + {\sigma \over r} \, (q - 1) - {\sigma \over {2\beta_1^2}} \, (q^2 - 1) . \eqno (31)$$ % Using the definition of $\alpha$ and the equation of state for a Boltzmann gas with polytropic index $\Gamma$ % $$\eqalign{\mu_2 &= mc^2 \, \left(1 + {\Gamma \over {\Gamma - 1}} \, {P_2 \over {n_2mc^2}} \right) ,\cr \mu_1 &= mc^2 , \cr}\eqno (32)$$ % we find with Eq. (31) % $$\eqalignno{&1 - {\gamma_2 \over \gamma_1} = \sigma (q - 1) &(33) \cr &\qquad + {\Gamma \over {\Gamma - 1}} \, {{\gamma_2^2\beta_1^2} \over r^2} \, \left(r - 1 + \sigma (q - 1) - {{\sigma r} \over {2\beta_1^2}} \, (q^2 - 1) \right) .}$$ % Combining Eqs. (25) and (29) gives the quadratic equation for $\alpha$ % $$\alpha^2 - \alpha \, \bigl (1 + \sigma - (\sigma - x)\,r\bigr ) + rx = 0 . \eqno (34)$$ % We are now able to eliminate $\gamma_2$ and $q$ from Eq. (33) and after considerable manipulations we end up with a polynomial of seventh degree in $r$ % $$\sum_{i=0}^7\, c_i(\gamma_1,\sigma,x,\Gamma)\,r^i = 0 . \eqno (35)$$ % The coefficients $c_i$ are themselves polynomials in the parameters $\gamma_1, \sigma, x, \Gamma$. These expressions are quite lengthy and are given in the appendix. The quantities $\alpha$, $q$, $\gamma_2$, $b_z$, $b_\phi$ and $\Pi$ can then be obtained from Eqs. (34), (25), (28), (21), (26) and (31). The coefficients $c_i$ are dependent only on upstream parameters, except for $\Gamma$, which is a function of the downstream temperature involving modified Bessel functions (Synge, 1957) % $$\Gamma (\Theta) = 1 + \left({1 \over \Theta} \,{\left( {{K_1(1/\Theta)} \over {K_2(1/\Theta)}} - 1 \right)}+ 3 \right)^{-1} \eqno (36)$$ % with $\Theta = kT/mc^2$. $\Theta$ is related to $r$ through % $$\Theta = \Pi\,\gamma_1\,\gamma_2\,\beta_1^2\,/r . \eqno(37)$$ % Thus Eqs. (35) and (36) form a system of two nonlinear equations for $r$ and $\Gamma$, which will be solved numerically. The polynomial (35) has in general several real solutions corresponding to the different shock transitions. For computational reasons, instead of Eq. (36) we use a polynomial approximation given by Service (1986), which has a high accuracy over the whole range in $\Theta$. With this approximation we could in principle merge the two polynomials into one. In practice, however, this expression became far too complex. \titleb {Limiting cases} In the case of a pure transverse magnetic field, Eq. (35) reduces to a fifth order polynomial, and having only a longitudinal component, the magnetic field drops out of the jump conditions. The corresponding equation is the following cubic in $r$ % $$\eqalign{&\gamma_1^2 {(\Gamma - 1)}^2 \, r^3 - \gamma_1^2\, (\Gamma^2 - 1)\,r^2 \cr &\qquad + (\gamma_1^2 - 1)(2\Gamma - 1) \, r - (\gamma_1^2 - 1) = 0 .}$$ % Both in the Newtonian and the ultrarelativistic limit the familiar relations for strong shocks, $r = (\Gamma + 1)/ (\Gamma - 1)$ and $r = 1/(\Gamma - 1)$ respectively, are recovered. The ultrarelativistic shock with only transverse magnetic field has been treated by Kennel and Coroniti (1984). They make an expansion in $\sigma $ for small values and in $1/\sigma $ in the opposite limit. Their approximations $r = 3(1 - 4\sigma )$ and $r = 1 + 1/(2\sigma )$ are verified to be good for $\sigma \la 0.01$ and $\sigma \ga 10$ respectively, for $\beta_1 = 1$ and $\Gamma = 4/3$. \titleb {Parameter space} Although $\beta_1$, $\sigma$ and $M$ are natural parameters for the description of relativistic jets, which were chosen to solve the system consisting of Eqs. (21) and (22), we give the numerical results in terms of $\beta_1$, the ratio $b'$ of the magnetic to the material energy content transported in the jet, $b'^2 = B'^2/4\pi n\mu$, and the angle $\theta'$ between the magnetic field and the propagation direction in the comoving frame, $\theta' = \arctan (B_{1\phi }/\gamma_1 B_z)$. These parameters are independent, while the others are not. This makes it easier to interpret the results. But as we will use in the discussion also $\sigma$ and $M$, the relation of the two parameter sets is given below. They are related through % $$b'^2 = \sigma + x\,u_1^2 , \quad \theta' = \arctan \sqrt {{\sigma \over{xu_1^2}}} , \eqno(38)$$ % $$\sigma = b'^2 \sin^2\theta' , \quad x = b'^2 \cos^2\theta' {1\over u_1^2} .\eqno(39)$$ % Except for small angles, $\theta' \la 20\degr$, $\sigma$ and $b'^2$ are of the same order. The characteristic speeds and Mach numbers are now given by (for $c_{\rm s} = 0$) % $$\eqalignno{\beta_{\rm A}^2 &= {{b'^2 \cos^2 \theta'}\over{1 + b'^2}} ,\quad M_{\rm A}^2 = {\beta_1^2\over \beta_{\rm A}^2} , &(40) \cr % \beta_{{\rm FM}}^2 &= {{b'^2}\over{1 + b'^2}} , \quad M_{{\rm FM}}^2 = {\beta_1^2\over{\beta_{{\rm FM}}^2}}. &(41) \cr}$$ % Results in terms of $\beta_1$, $\sigma$ and $M$ are given in Hardy (1988). % \begfig 4 cm \figure{1} {Shock frame compression ratio for a fast magnetosonic shock, as a function of the jet velocity for different magnetic field configurations} \endfig % \begfig 5 cm \figure{2} {Shock strength defined as the ratio of downstream thermal pressure and upstream pressure. Parameters as in Fig. 1} \endfig \titlea {Results for fast sonic shocks} In this paper we restrict ourselves to fast magnetosonic shocks, being the relevant modes for jets. Figures 1--5 show different physical quantities for certain sets of parameters that are of interest in the context of extragalactic radio sources. Figure 6 shows the hydrodynamical case for comparison. The thresholds in Figs.\ts 1--5 come from the fact that for a fast shock to occur the fast magnetosonic Mach number has to exceed unity, and are thus located at $M_{{\rm FM}} = 1$. Please note that the abscissae in Figs.\ts 1--5 is linear in $M_{{\rm FM}}$. % \begtab 3.2 cm \tabcap{1}{ Plasma-$\beta$ in the downstream region} \endtab % For $\theta' = 90\degr$ the amplification of the transverse magnetic field $q$ equals the compression ratio, but this is no longer true in the general case. Instead, $q$ exceeds the compression ratio in fast shocks. This effect becomes dramatical for small angles $\theta'$ and is reflected in the jump of the shock frame angle $\theta$ between $\vec{B}$ and $\vec{n}$ across the shock (Fig. 3). Here, the solid line corresponds to $\theta_1$ and the dotted line to $\theta_2$. The parameter $\theta'$ is equal to $\theta_1$ for non-relativistic speeds. Simultaneously, a considerable refraction of the plasma flow is observed in Fig. 4. As the downstream plasma speeds are still very high, for an axisymmetric configuration this would produce a relativistically rotating plasma cone. \begtab 5 cm \tabcap{2}{Hydrodynamic shock} \endtab % \titlea {Astrophysical implications for hot spots in jets} Relativistic magnetized jets are characterized by four parameters: the Lorentz factor, the Alfv\'en Mach number, the $\sigma $-parameter for the influence of the toroidal field and the internal pressure, which will be neglected against the ram pressure and the magnetic pressure. The most probable source for these jets is a rapidly rotating magnetosphere kept up by an accretion disk in the center of a galaxy (Blandford and Payne, 1982; Laurel, 1987b; Kundt, 1987). In this picture, jets are collimated on a scale smaller than about one light year. The propagation of these jets from the parsec-region to the hot spot domain (at least 10 kiloparsecs) is beyond the presently available numerical techniques. We can, however, estimate the typical parameters for these jets, when they enter into the hot spot, i.e. just upstream of the terminal shock. For this we assume that hot spots are downstream flows of a strong shock produced by the interaction of the jet with the intergalactic medium. The typical density in the jet at the position of the hot spot (HS) is then given by % $$\eqalignno{n_1^S &= {\dot M_{\rm j} \over {\pi R_{{\rm HS}}^2 m c \beta_1 }} &(42) \cr &\qquad = 0.4 \, 10^{-4} \cm^{-3}\, \left({\dot M_{\rm j}\over {\Msol /\yr}} \right)\, (R_{{\rm HS}}/\kpc)^{-2} \, \beta_1^{-1} .}$$ % %The mass flux $\dot M_j$ in the jet is measured in units %of solar masses per year, and the radius $R_{HS}$ of the hot spot in units of %kiloparsecs. This jet density $n_1^S$, as measured in the frame of reference of the shock, is in agreement with the mean density $n_2 = 5\, 10^{-4} \cm^{-3}$ obtained for the thermal matter inside hot spots (Kerr et al., 1981). The typical magnetic fields in the hot spots are of the order of a fraction of a milli-Gauss. From these jet parameters we finally obtain the typical values for $b'$, $\sigma $ and $M$ % $$\eqalignno{b'^2 &= {{B'}_1^2\over{4\pi n_1\mu_1}} &(43)\cr &\qquad = 5.3\, {10}^{-3}\,\left({{B'}_1\over{0.1\mG}}\right)^2 \, \left({n_1\over{{10}^{-4}\cm^{-3}}}\right)^{-1} , \cr % \sigma &= 0.005 \,\left({B_{1\phi }\over {0.1\mG}} \right)^2\, \left({n_1^S\over {{10}^{-4}\cm^{-3}}} \right)^{-1}\, {1\over \gamma_1} , &(44) \cr M &= {{(4\pi n_1\mu_1)^{1/2}\, u_1}\over B_z} &(45) \cr &\qquad = 13.7\,\,\left({B_z\over {0.1\mG}}\right)^{-1}\, \left({n_1^S\over {{10}^{-4}\, \cm^{-3}}}\right)^{1/2}\, \beta_1\,\sqrt{\gamma_1}. }$$ % \begfig 3 cm \figure{3} {Angle in degrees between the magnetic field and the shock-normal, as seen in the rest frame of the shock front. Parameters as in Fig. 1} \endfig For $\sigma \ga 0.05 $, the dynamic pressure decreases in the postshock domain with a similar behaviour for the compression ratio (see Figs. 1 and 2). Shocks are then no longer prominent and would even disappear, when $\sigma$ is of order unity. Jets with a high Poynting flux could not provide strong hot spots. This fact could explain the absence of hot spots in jets associated with weak radio galaxies, which are called FR 1 sources (Fanaroff and Riley, 1974). These jets could carry relatively more magnetic energy with a higher dissipation rate along the jet. Alternatively, the jet flow could be super-Alfv\'enic in FR 1 sources, but {\it not supermagnetosonic}. These jets could then not pass through supermagnetosonic shock transitions. According to Eqs. (40) and (41), FR 1 jets would satisfy the inequality ${b'}^2\cos^2 \theta' < (1 + {b'}^2)\beta_1^2 < {b'}^2$. %$\beta_1 < \sqrt{\sigma /(\sigma + 1 - 1/M^2)}$ (see Fig. 1). This fact agrees with the generally accepted opinion that the jet velocity in FR 1 sources should be sub-relativistic (Scheuer, 1987). It was mentioned in Sect. 4 that, in this one-fluid approximation, the polytropic index for the postshock plasma stays closer to 5/3 than to 4/3, except for extremely high Lorentz factors. This conclusion would be weakened somewhat by taking into account a high pressure in synchrotron electrons. \titlea {Conclusions} We have analyzed the jump conditions for relativistic magnetized jets. Since these jets carry an appreciable amount of magnetic energy, the compression ratio in fast magnetosonic shocks depends on two additional parameters, the Alfv\'en Mach number of the jet and the relative Poynting flux in the jet expressed in terms of the $\sigma$-parameter, or, alternatively, the magnetic to material energy ratio and the orientation of the magnetic field. Already a moderate Poynting flux leads to weak shocks for all values of the Alfv\'en Mach number. As an important result, we obtained a threshold for the jet velocity, when the jet goes through a fast magnetosonic shock. In our calculation of the shock conditions for relativistic jets, we neglected the influence of the relativistic suprathermal particles on the nature of the postshock plasma. When these particles dominate pressure and energy in classical shocks, the compression ratio increases to a value of 7 (Pelletier and Roland, 1986). The effect of this component in the postshock plasma must be studied, if particle acceleration is very effective in the postshock region. This problem will be solved in future work. It will be interesting to see, whether the inclusion of this component works against the weakening effect of the Poynting flux. Prominent emission features are also visible in the parsec-scale jets of quasars. In particular, superluminal sources show strong emission knots propagating along fixed directions. If shocks are behind these structures, the relative Poynting flux should also be weak in the nuclear jets of these sources. This fact is in disagreement with the relation between the Lorentz factor and the value of $\sigma$, Eq. (48), for magnetically driven jets. We guess, therefore, that these emission knots have to be associated with instabilities in strongly magnetized jets. \ack{This research was partially supported by the Sonderforschungsbereich 328. We would also like to thank H.-J. R\"oser and K. Meisenheimer for valuable discussions on the observed structure of hot spots in radio galaxies.} % %------------- Appendix --------------- % \app{}{With the abbreviations $g = \gamma_1^2$ , $h = \gamma_1^2 - 1$ , $k = \Gamma / (\Gamma - 1)$ we give the coefficients $c_i$ of the polynomial in {\it r} of Eq.(35).} % %------- linksbuendig ---------------- % \newdimen\mathindent \mathindent=0pt \def\eqno{$\hfill$} \long\def\leftdisplay#1$${\line{\hskip\mathindent $\displaystyle#1$\hfil}$$} \everydisplay{\leftdisplay} % % $$\eqalign{ c_0 &= 4\, {h^2\over g}\, (k - 1)^2\, z_0 \cr\cr}$$ $$\eqalign{ c_1 &= {h\over g}\, (k - 1)\, (k^2xz_1 - 4kz_2 + 4z_3) \cr\cr}$$ $$\eqalign{ c_2 &= k^2\,z_4 - k^3\,x\,z_5 + k^4\,x\,z_6 - 4\,k\,h\,z_7 + 4\,h\,z_8 \cr\cr}$$ $$\eqalign{ c_3 &= k^4\,x\,z_9 - k^2\,z_{10} - k^3\,x\,z_{11} - 4\,k\,z_{12} + 4\,h\,z_{13} \cr\cr}$$ $$\eqalign{ c_4 &= g\,\,(-k^2\,z_{14} + 4\,k\,z_{15} + k^3\,x\,z_{16} - 4\,z_{17})\cr\cr}$$ $$\eqalign{ c_5 &= g\,\,(k^2\,z_{18} - 4\,k\,z_{19} + k^3\,x\,z_{20} - 4\,z_{21})\cr\cr}$$ $$\eqalign{ c_6 &= g^2\,(\sigma - x)^2 \,(k - 2) \, (k\,z_{22} - k^2\,x\,z_{23} - 2\,z_{24})\cr\cr}$$ $$\eqalign{ c_7 &= - g^2\, (\sigma - x)^4 \,(k - 2)^2 \,(\sigma + xh) \cr\cr\cr }$$ % $$\eqalign{ z_0 &= \sigma^5 g + \sigma^4 \bigl[x(g^2 + h) + 4g \bigr] \cr &\qquad + 2\sigma^3 \bigl[x^2gh + x(2g - 1)(g + 1) + 3g \bigr] \cr &\qquad + \sigma^2 \bigl[x^2h(xh + 4g) + x(3g - 1)(2g + 1) + 4g \bigr] \cr &\qquad + \sigma g \bigl[2x(xh + 2g) + 1 \bigr] \cr &\qquad - kx \bigl[\sigma xh + g(\sigma + 1)^2 \bigr] \bigl[2\sigma xh + (2\sigma + g)(\sigma + 1) \bigr] \cr &\qquad + \quarter k^2x \bigl[2\sigma xh + (2\sigma + g)(\sigma + 1) \bigr]^2 + xg^2 \cr\cr }$$ % % $$\eqalign{ z_1 &= 4\sigma^4(6g^2 - 3g - 1) \cr &\qquad + 4\sigma^3 \bigl[ g(3g^2 + 13g - 11) + 2xh(3g^2 - 1) \bigr] \cr &\qquad + \sigma^2 \bigl[29g^3 + 19g^2 - 36g + 4 \cr &\qquad\quad + 4x^2h^2(3g - 1) + 52xg^2h\bigr] \cr &\qquad + 2\sigma g\bigl[2(5g^2 - 3g - 1) + xh(21g + 8) + 16x^2h^2\bigr] \cr &\qquad + g^2h(20x + 3) \cr\cr }$$ % $$\eqalign{ z_2 &= \sigma^5 g(4g - 3) \cr &\qquad + \sigma^4 \bigl[x(4g^3 + g^2 - 5g + 1) + g(13g - 10) \bigr] \cr &\qquad + \sigma^3 \bigl[x(15g^3 - 5g^2 - 9g + 2) \cr &\qquad\quad + 2x^2h(2g - 1)^2 + 3g(5g - 4) \bigr] \cr &\qquad + \sigma^2 \bigl[x(18g^3 - 12g^2 - 4g + 1) \cr &\qquad\quad + x^2h(15g^2 - 4g + 2) +x^3h^2 + g(7g - 6) \bigr] \cr &\qquad + \sigma g \bigl[x^2h(14g + 5) + xg(7g - 6) + 10x^3h^2 + h \bigr] \cr &\qquad + 8x^2g^2h \cr\cr }$$ % % % % % ---------- References ------------- % \begref \ref Anderson, E.: 1963, {\it Magnetohydrodynamic Shock Waves}, MIT Press % % \ref Ardavan, H.: 1976, \apj {\bf 206}, 822 % \ref Blandford, R.D., Payne, D.G.: 1982, \mnras {\bf 199}, 883 % \ref Blandford, R.D., Rees, M.J.: 1974, \mnras {\bf 169}, 395 % \ref Bridle, A.H., Perley, R.A.: 1984, \araa {\bf 22}, 319 % \ref Laurel, S.T.: 1987a, \aua {\bf 162}, 32 % \ref Carioli, S.M.: 1986, \phfl {\bf 29}, 672 % \ref Courvoisier, T.J.-L., Laurel, S.T.: 1987, \aua {\bf 183}, 167 % \ref De Hoffman, F., Teller, E.: 1950, \phrev {\bf 80}, 692 % \ref Drury, L.O'C.: 1983, \rprph {\bf 46}, 973 % \ref Drury, L.O'C., Axford, W.I., Summers, D.: 1982, \mnras {\bf 198}, 833 % \ref Fanaroff, B., Riley, J.M.: 1974, \mnras {\bf 167}, 31 % \ref Kennel, C.F., Coroniti, F.V.: 1984, \apj {\bf 283}, 694 % \ref Kerr, A.J., Birch, P., Conway, R.G., Davis, R.J., Stannard, D.: 1981, \mnras {\bf 197}, 921 % \ref Kirk, J.G., Schneider, P.: 1987, \apj {\bf 315}, 425 % \ref Kirk, J.G., Schlickeiser, R., Schneider, P.: 1987, preprint % \ref Kundt, W.: 1987, in {\it Astrophysical Jets and their Engines}, ed. 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