## 2. Structure of magnetic jetsThroughout the paper we apply the following basic assumptions:
We emphasise that the term ## 2.1. The force-free cross-field force-balanceWith the assumption of axisymmetry, a magnetic flux function can be defined, measuring the magnetic flux through a surface element with radius
with the toroidal component (index ) of the current density . The poloidal current, defined similarly to the magnetic flux function, flows within the flux surfaces, . The projection of the force-free, relativistic equation of motion (where inertial effects of the plasma are neglected), (with the electric field and the charge density ) perpendicular to the magnetic flux surface provides the toroidal current density, is the angular velocity of the field lines and is conserved along the flux surfaces, . Both the current distribution and the rotation law of the field, , determine the source term for the GSS equation and govern the structure of the magnetosphere. Combining Eqs. (5) and (2) the cross-field force-balance can eventually be written as which is called the modified relativistic GSS equation. At the light surface with the rotational velocity of the field lines equals the speed of light. Here, the GSS equation becomes singular. For differentially rotating magnetospheres the shape of this surface is not known a priori and has to be calculated in an iterative way together with the 2D solution of the GSS equation. For constant field rotation the light surface is of cylindrical shape. We choose the following normalisation, For the length scale the radius of the
With the normalisation applied, Eq. (6) can be written dimensionless,
in the case of AGN, and for protostellar parameters. Note that ## 2.2. Where is the asymptotic light cylinder located?We define the All asymptotic flux surfaces within rotate slower than the speed of light and . Flux surfaces outside may rotate faster than the speed of light, here . Despite a possible degeneration of the GSS equation for a special rotation law (see below), there is only a single physical asymptotic light cylinder possible. Therefore, . It should be noted that the introduction of a light cylinder
also relies on the Ideal MHD assumption. For a
non-infinite plasma conductivity a ## 2.2.1. Stellar magnetosphereIn the case of a constant field rotation the light cylinder radius
just follows from the rotational velocity of the field (and does
This radius is of the order of the observationally resolved asymptotic jet radius of about cm (Mundt et al. 1990; Ray et al. 1996). HST observations indicate on slightly smaller jet radii of 20 AU (Kepner et al 1993). For neutron stars the light cylinder is at ## 2.2.2. Disk magnetosphereFor disk magnetospheres the rotation law is determined by the flux
distribution along the disk surface together with the disk rotation.
If the foot point of a flux surface on the disk (here the term (here for protostellar parameters) with the mass of the central
object (again for a protostellar disk magnetosphere). Is the central object a neutron star, this ratio decreases by a factor of about 100. For AGN we can estimate and in general where is the Schwarzschild radius of the black hole. The question, whether or not a relativistic description is required
for the jet magnetosphere, depends on the In principal, the asymptotic field distribution is a result of the two-dimensional force-balance of the jet, and therefore should follow from the solution of the two-dimensional GSS equation. We hypothesise that the asymptotic force-free solution will uniquely be determined by the disk flux and current distribution (and vice versa). Our results for differentially rotating jets can hardly deliver a
statement about the ## 2.3. The asymptotic force-balanceIn the asymptotic regime of a highly collimated jet structure we reduce Eq. (7) to a one-dimensional equation, equivalent to the assumption . Then, , and the conserved quantities
and can be expressed as
functions of With the assumptions made above, the GSS Eq. (7) reduces to an
ordinary differential equation of Since is related to the magnetic pressure of the poloidal field, Eq. (9) can be rewritten as The magnetic flux function then follows from integration of with . At the singular point the solution must satisfy the regularity condition We mention that Eq. (10) can also be derived from the equation for the asymptotic force-equilibrium perpendicular to the flux surfaces, where indicates the gradient perpendicular to the flux surfaces, and where poloidal field curvature and the centrifugal force are neglected (Chiueh, Li & Begelman, 1991; ACa). ## 2.4. Discussion of the force-free assumptionOne may question the assumption of a force-free asymptotic jet.
Indeed, in a self-consistent picture of jet formation, the asymptotic
jet is located beyond the collimating, non force-free wind region and
beyond the fast magnetosonic surface. The asymptotic jet parameters
are determined by the critical wind motion and thus, the poloidal
current and the angular velocity of the field are The essential point here is the assumption of a where is the particle flow rate per flux
surface, is the conserved total energy,
The centrifugal term, which was neglected in Eq. (13), is , with the plasma density and plasma angular velocity (see ACa). This term may be important for small plasma densities , where might be large, as well as for high densities, where the toroidal plasma velocity is supposed to be small. We can estimate the importance of this term by normalising and introducing a coupling constant which is of the order of one tenth for a jet mass loss rate and other parameters typical for AGN. For protostellar jet parameters and a mass loss rate , increases by a factor of 1000. However, in this case we may expect that the Alfvén surface of the plasma motion is located well inside the light cylinder. Thus, the plasma, rotating with constant angular momentum beyond the Alfvén surface, has a decreasing and low angular velocity (which is normalised to the ). The centrifugal term may become comparatively small. We emphasise that the latter arguments are rather (simplifying) assumptions than keen conclusions, as long as the true non force-free jet equilibrium is not investigated. Contopoulos & Lovelace (1994) and Ferreira (1997) constructed
## 2.5. Solution of the asymptotic GSS equationEq. (10) can be solved by the method of the variation of constants. The integrating factor of the differential equation is with the formal solution Using Eq. (12), the general solution can be evaluated, As already mentioned by ACa, the solution is determined by the regularity condition (12). The magnetic flux function is In the case of a constant field rotation, ACb found an analytical, non-linear solution to the asymptotic GSS equation, the flux distribution together with the current distribution leading to a certain relationship between the current distribution
parameter © European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |