2. Structure of magnetic jets
Throughout the paper we apply the following basic assumptions: axisymmetry, stationarity, and ideal MHD. We use cylindrical coordinates or, if normalised, . The notation is similar to that of Fendt et al (1995) and ACa, b.
We emphasise that the term 'asymptotic' always denotes the limit and that jets with finite radius, , are considered.
2.1. The force-free cross-field force-balance
With the assumption of axisymmetry, a magnetic flux function can be defined,
measuring the magnetic flux through a surface element with radius R, threaded by the poloidal component (index 'P') of the magnetic field . With Eq. (1) the toroidal component of Ampère's law leads to the GSS equation
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 asymptotic light cylinder (see below) is selected. In order to allow for an immediate comparison to rigidly rotating magnetospheres, the normalisation is chosen such that at .
With the normalisation applied, Eq. (6) can be written dimensionless,
g is a coupling constant describing the strength of the current term in the GSS equation,
in the case of AGN, and
for protostellar parameters. Note that g in this paper is in accordance with the definitions in Fendt et al. (1995) and differs from the definition in ACa, b by a factor of two, . A coupling constant, defined in a similar way for the differential rotation term, would be equal to unity, indicating on the important role of this effect.
2.2. Where is the asymptotic light cylinder located?
We define the asymptotic light cylinder, , as the asymptotic branch of the light surface . Asymptotically, this quantity plays the same role for the GSS equation as the light cylinder does in the case of a rigid rotation of the magnetosphere.
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 conserved angular velocity of the field lines cannot be defined. However, even in this case, the field may move with relativistic speed. The mathematical formalism, of course, becomes more complicated and its solution is beyond the scope of this paper. An estimate of diffusion and dynamical times scales for protostellar jets, respectively, leads to the conclusion that the Ideal MHD assumption may be appropriate (Fendt 1994). For AGN this assumption would be even more valid.
2.2.1. Stellar magnetosphere
In the case of a constant field rotation the light cylinder radius just follows from the rotational velocity of the field (and does not depend on the flux distribution ). Under the assumption that the field is anchored in the stellar surface, the field rotation follows from the stellar rotational period . The rotational period of many protostellar jet sources is not known, but in the case of T Tauri stars it is of the order of days. Thus, we estimate the light cylinder radius
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 magnetosphere
For 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 foot point denotes the position along the field line, where ideal MHD sets in) at a radius rotates with Keplerian speed, the flux surfaces intersect the light surface at the radius
(here for protostellar parameters) with the mass of the central object M. The ratio between the position radius of the light surface and the light cylinder for rigid rotation is then
(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 asymptotic radius of the flux surface, . If for any flux surface , a relativistic description of the magnetosphere is required. In the contrary, if for all flux surfaces , the Newtonian description is appropriate. Note that even then, for two arbitrary flux surfaces and with , is possible.
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 absolute value of the asymptotic jet radius, but only in terms of the asymptotic light cylinder jet radius .
2.3. The asymptotic force-balance
In 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 x. If we further assume a monotonous flux distribution , the derivatives . Note that this excludes hypothetical solutions with a return current from our treatment (see also Sect. 3.4.1).
With the assumptions made above, the GSS Eq. (7) reduces to an ordinary differential equation of first order in the derivative ,
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,
2.4. Discussion of the force-free assumption
One 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 not functions free of choice.
The essential point here is the assumption of a cylindrical shape of the asymptotic jet, an assumption, however, which is clearly indicated by the observations. The general, non force-free expression for the poloidal current is
where is the particle flow rate per flux surface, is the conserved total energy, M the Alfvén Mach number, and the Alfvén radius of the flux surface. For cylindrical flux surfaces, all quantities on the r.h.s. are functions of , and thus, also is a function of . Although is not equal to the force-free current, it enters Eqs. (10) and (13) in a similar way.
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 self-similar solutions including centrifugal forces showing that the magnetic terms indeed may dominate the centrifugal term for large radii leading to a recollimation of the outflow.
2.5. Solution of the asymptotic GSS equation
Eq. (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 b, the core radius a, the coupling constant g, and the asymptotic jet radius .
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998