Abstract Abstract The magnetic field structure and the collimated outflow of rapidly rotating Young Stellar Objects (YSOs) are calculated from the stellar source to the asymptotic region. The calculations are based on ideal MHD and the further simplifying assumptions of stationarity and axisymmetry. The star-disk-jet system can be completely defined by the Grad-Shafranov (GSS) equation, describing the structure of the magnetospheres, and the wind equation, which are given by equilibrium perpendicular and parallel to the field lines. Both equations must be solved simultaneously to obtain a self-consistent solution. General solutions of the Grad-Shafranov equation are not yet available. Here we discuss an analytical model for the magnetic flux surfaces which is a solution for small and large radii. This model assumes a stellar dipole field and a gap between the star and a disk at a distance of a few stellar radii. Due to the features of the disk no field can penetrate the disk and the resulting opening of the field lines close to the polar cap is obtained as a computational result. In addition, our model guarantees the collimation of the outflow into a cylindrical shape at asymptotic jet radii of several thousand stellar radii. This model for collimated outflows reproduces all essential properties of magnetospheres for rapidly rotating stars. The result is used as input to the wind equation. This problem is completely integrable, determined by five constants of motion: the total energy E, the total angular momentum L, the total mass flux η along a flux surface, and the total entropy S in a flux surface, together with the rotation Ω F of the field lines. For adiabatic plasma flows, this problem is algebraic and can easily be solved. Pressure is neglected in our computation. Abstract The theory of axisymmetric magnetospheres around rapidly rotating stellar sources is outlined including electric fields due to the rapid rotation. Gravity of the central object is consistently built into this theory. Due to the injection of plasma either from the stellar surface or by interaction with a surrounding disk, these magnetospheres are neither vacuum solutions, nor force-free. A consistent Abstract Correspondence to: neil99@example.com wind theory is developed which contains the Newtonian theory as a classical limit. Abstract Current-carrying plasma flows will lead to a collimation of the magnetospheric structure into a cylindrical shape. Particular solutions are discussed for the asymptotic collimation. We show that the asymptotic structure is essentially determined by electric forces in the pinch equation, and not by centrifugal and pressure forces. Introduction The phenomenon of highly collimated high-velocity bipolar outflows occurs frequently in Astrophysics for a large range of objects such as protostars and Active Galactic Nuclei. The formation of jets seems to be closely related to the presence of magnetic fields and the existence of a gaseous disk around a rapidly rotating central object (Camenzind 1990) . It is widely believed that due to these rotating magnetospheres, the mass flow can be accelerated and collimated on scales of a few tens of AU as shown by HST observations of Young Stellar Objects (Ray et al. 1996) . Nevertheless, their origin is still under discussion. Protostellar jets are most probably disk winds or stellar winds. The structure of these magnetospheres follows from solutions of the Grad-Shafranov equation for force-balance perpendicular to the field lines. Magnetospheres generated by isolated stars are typically three dimensional objects (e.g., the Earth's magnetosphere, pulsar magnetospheres or stellar magnetospheres) and as such are very complicated to model consistently. There are a few types of systems where the magnetosphere is constrained to be axisymmetric. Accretion disks will produce highly axisymmetric structures and stars in close interaction with accretion disks are also sources of axisymmetric magnetospheres. Magnetized rotators are natural sources of energetic plasma flows. We investigate the dynamics of the wind in these collimated magnetospheres for axisymmetric and stationary flows. The shape of the magnetic flux function determines whether the wind is accelerated or decelerated. Introduction In this paper we develop a general theory for steady axisymmetric ideal plasma flows driven by rapidly rotating magnetospheres of stellar sources including contributions from electric fields. This theory also includes gravity, such that outflows driven by neutron stars and black holes are covered. With the elements developed in this paper one can calculate plasma flows driven by young stellar sources, where the central object is rotating near its breakup velocity, as well as winds driven away by rapidly rotating neutron stars. This is an extension of the work by Mobarry & Lovelace (1986) including new insights into this problem from the last ten years. Introduction An overview of the attempts to approach the problem of solving the Newtonian Grad-Shafranov equation and the wind equation can be found in Lery et al. (1998) . Lery et al. (1998) also present a model for stationary, axisymmetric MHD winds without the need for self-similar assumptions as is done here. Whereas our model gives the structure of the magnetospheres for all regions, they find a solution of the Grad-Shafranov equation for the Alfvén surface from the Alfvén regularity condition and at the base of the flow. Their magnetic field structure for small radii is conical in shape, which does not take a stellar dipole field or the influence of a disk into consideration. We find that the shape of the flux surfaces close to the stellar source is important for the acceleration of the plasma. The Alfvén point and the fast magnetosonic point are rather close to each other. The fast magnetosonic surface lies at radii smaller than the inner disk radius. The collimation of the magnetospheres into a cylindrical outflow is obtained by imposing an external pressure in Lery et al. (1999) . That is not necessary for our model. When plasma flows reach radii near the light cylinder, electric forces become important for the modelling of the underlying magnetosphere for fast rotators (Camenzind 1986a; Fendt et al. 1995; Fendt & Camenzind 1996) . This does not mean that plasma flows are accelerated to relativistic energies, only that electric fields contribute to the forces as much as magnetic fields do. The electric field can only be neglected, at radii much smaller than the light cylinder radius (at least only a few percent of the light cylinder). The poloidal current follows from the MHD wind theory. For Lery et al. (1999) the light cylinder is of no importance. In Lery et al. (1998) the wind equation is solved from the source to the fast magnetosonic point, whereas our wind solution is global and goes up to the asymptotic region of the jet. In contrary to our cold wind solution they use a polytropic equation of state. For a cold wind our fast magnetosonic surface does not go to infinity, because we impose a dipole-like structure on the flux surfaces instead of a monopole-like one. In addition, their wind solution represents the Newtonian limit of our derivation. Newtonian MHD is perfect for modelling the magnetic structure of the solar wind (Mac Gregor 1996) . But winds ejected from protostars rapidly rotating with periods of a few days could pass the light cylinder given as (Camenzind 1997) R L = c Ω F = 27 AU P * days . Introduction Such a radius is interestingly close to the observed collimation radii. While the light cylinder is not essential for the plasma kinematics, it does influence the collimation processes at large radii by means of electric fields due to field line rotation. The paper is structured as follows. In Sect. 2 a consistent wind theory for rapidly rotating objects is presented. Axisymmetric fields and stationary flows are investigated. The Newtonian case follows quite naturally as a limit of the special relativistic MHD. The polytropic equation of state is included. A general expression for the wind equation is given and the special solution for a cold wind is discussed. In Sect. 3 the Grad-Shafranov equation is studied. Sect. 4 gives a discussion of the collimation processes for spherical outflows. The importance of the electric fields close to the light cylinder is shown. In Sect. 5 a model is presented to reproduce the characteristics of collimated outflows near to and far from the star. A calculation of the magnetic flux surfaces and the resulting wind is given in Sect. 6 for a typical young, low-mass star. Also the influence of various parameters of the model is studied. We draw our conclusions in the last section. A consistent wind theory for rapidly rotating stars A modern treatment of magnetohydrodynamics including the effects of gravity must be based on a general relativistic approach in the 3+1 split of space-time (see e.g. Thorne et al. 1986; Beskin & Pariev 1993; Camenzind 1996) . When the rotation of the central object is not important, the line element is given by the Schwarzschild decomposition A consistent wind theory for rapidly rotating stars Here α = 1 − 2GM * /c 2 r is the redshift factor which accounts for the gravitational redshift between a local observer and infinity. It describes the effects of gravity due to the presence of the central star of mass M * . In the Newtonian limit we have A consistent wind theory for rapidly rotating stars where Φ is the Newtonian potential. In the following we also use the cylindrical radius R = r sin θ. The time t is the global time. A consistent wind theory for rapidly rotating stars The spin J * of the central object is also a source of gravity in general relativity (the so-called gravitomagnetic effect). This spin would produce an off-diagonal element g tφ in the above line element, g tφ ∝ J * /r 3 , which can be important for neutron stars rotating with millisecond periods (Camenzind 1986b ). This effect is crucial for the treatment of plasma flows on rapidly rotating black holes (Camenzind 1996) . Since we are interested here for applications to stellar sources, we will neglect these effects. Local fiducial observers (FIDOs) are represented by orthonormal tetrads 3+1 split of electrodynamics On a curved spacetime background electromagnetic fields must be given in terms of the Faraday tensor F µν , and Maxwell's equations are given in tensorial form. With respect to the particular observer field defined above, we can however split the Faraday tensor as in special relativity into an electric and magnetic part 3+1 split of electrodynamics Similarly, we get a charge density 3+1 split of electrodynamics and a current density 1 J given by 3+1 split of electrodynamics In terms of the electric fields E and magnetic fields B, Maxwell's equations assume the familiar form (Thorne et al. 1986 ) 3+1 split of electrodynamics Charge conservation is given as 3+1 split of electrodynamics In this treatment, the redshift factor α only appears as a form factor, but there are no additional couplings between electromagnetic fields and gravitational forces. This is in contrast to the treatment on rapidly rotating backgrounds, where the gravitomagnetic force couples directly into Maxwell's equations (Camenzind 1996) . Axisymmetric fields As in the Newtonian theory of plasma confinement, the above equations can be considerably simplified for axisymmetric structures. Here, we can define the poloidal flux function Ψ(r, θ) Axisymmetric fields and the axial current flowing downward along the central axis Axisymmetric fields A denotes an upward directed surface around the central axis. The level surfaces of Ψ are the magnetic surfaces which are generated by rotating field lines. The level surfaces of I are the surfaces where the poloidal currents flow. Due to stationarity and axisymmetry, the electromagnetic fields assume a simple form Axisymmetric fields Infinite conductivity requires B · E = 0, and thus the integration constant Ω F (Ψ) represents the angular velocity of field lines. This is not the angular velocity of a plasma particle. Plasma motion We treat plasma motion only in the one-fluid approach (see Khanna 1998) . For a one-component non-viscous plasma the energy-momentum tensor has the form Plasma motion ρ is the total energy of the plasma including internal energy, P the proper pressure, u the four-velocity of the plasma and g the metric tensor. We decompose the energy-momentum tensor T into horizontal and vertical components with respect to fiducial observers Plasma motion ǫ is the energy-density with respect to local observers, S the momentum flux and ↔ t the stress tensor. With the decomposition of u = γ(O + v) we obtain the following components with respect to FIDOs Plasma motion Here, ρ 0 is the rest mass density, µ = (ρ + P )/n the relativistic specific enthalpy and γ the Lorentz factor measured by fiducial observers. S em represents the Poynting flux measured by FIDOs and ↔ t em the Maxwell stresses Plasma motion Using the 3+1 split of the affine connection, one can now derive the 3+1-split of the hydrodynamic equations ∇ · T = 0. Energy conservation, given by O · (∇ · T ) = 0, can be written as (Thorne et al. 1986 ) Plasma motion The right-hand side in the energy conservation is the familiar divergence of the energy flux, with one factor of α inside the divergence to account for the gravitational redshift of the energy and the other to convert the differential proper time element in the definition of the flux into to a differential universal time element. Similarly, Euler's equations, given by h · (∇ · T ) = 0, where h is the projector into the plasma rest frame, assume the form Plasma motion The quantity −∇ ln α represents the local gravitational force measured by observers. Stationary flows The plasma flow is given by the continuity equation Stationary flows Similarly to the Newtonian theory, one also obtains for the poloidal velocity Stationary flows and, from the frozen-in condition, that the plasma is guided by the structure of the axisymmetric flux surfaces Stationary flows As a consequence of Faraday's induction law, the rotation of the field lines remains constant at Ω F (Ψ). The above relation tells us that the particle flux in a flux surface remains constant. Stationary flows For stationary and axisymmetric flows the energy equation (26) is now simply given as Stationary flows and similarly for the momentum equation (27) 1 Stationary flows By means of the MHD relations the energy flux can be written as Stationary flows In the stationary case, this provides the following expression for the energy equation Stationary flows Using flux conservation ∇ · B p = 0 and (B p · ∇)η = 0 we find conservation for the total energy per particle Stationary flows A similar conservation law follows from the angular momentum equation Stationary flows Due to axisymmetry, all other terms drop out. Since Stationary flows we get Stationary flows or the total angular momentum conservation Stationary flows with the specific angular momentum j = R 2 Ω/α 2 . In the total energy, the kinetic part is redshifted. Together with the above relation (30) for the toroidal motion, Stationary flows this provides three equations for the unknowns αγ, j (or u φ ) and I. In addition to these relations, an adequate equation of state is needed. For n = n(P, s) and T = T (P, s) the first law of thermodynamics implies Stationary flows In particular, for P = K 0 (s) n Γ with a polytropic index Γ, one finds for the specific enthalpy Stationary flows With this expression, we can derive the Newtonian limit for the total energy, Stationary flows which is the standard energy conservation used in Newtonian MHD. Lorentz factor, angular momentum and poloidal current For given integrals of motion E(Ψ), L(Ψ), Ω F (Ψ), η(Ψ) and K(Ψ) the conservation laws for stationary and axisymmetric plasma flows can be reduced to give relations for the Lorentz factor, the angular momentum and the poloidal current flux function I(R, Ψ) Lorentz factor, angular momentum and poloidal current where the quantity M , defined as Lorentz factor, angular momentum and poloidal current represents the (poloidal) Alfvén Mach number. We normalize radii in units of the light cylinder R L , x = R/R L . The redshift factor α corrects for the singular behaviour of the poloidal velocity at the horizon of the black hole. 2 The Alfvén Mach number M is a well-behaved quantity, e.g. at the horizon. The quantity ǫ defined by 2 is a measure for the amount of energy carried by the electromagnetic fields. In special relativity, ǫ ≤ 1, and 2 L is a direct measure of the position of the Alfvén surface in terms of the light cylinder radius R L . For relativistic flows, R A → R L , and therefore ǫ → 1. For the solar wind ǫ ⊙ ≃ 10 −8 and for winds ejected by rapidly rotating young stellar objects we find ǫ * ≃ 10 −6 . In the case of the Crab pulsar ǫ p ≃ 0.999 and the wind dynamics is essentially relativistic. 2 This quantity ǫ determines the position of the Alfvén surface (positions, where the denominators in the above equations vanish) 2 This equation has in general, for given rotation Ω F , only one solution, the outer Alfvén surface corresponding to the special relativistic one. The inner Alfvén surface only exists in black hole magnetospheres. The light cylinder surfaces are special solutions of this equation for M 2 A = 0. The regularity condition at the Alfvén point also determines the total angular momentum L as a function of the position of the Alfvén point 2 which becomes in the Newtonian case simply L = R 2 A Ω F . The Newtonian limit There is much confusion in the literature as to the Newtonian limit of the special relativistic MHD. The two equations (45) and (46) are well-known in Newtonian MHD (α = 1, ǫ ≪ 1 and R L → ∞), where they form the basis for a treatment of axisymmetric MHD winds in stellar systems (Heyvaerts & Norman 1996) The Newtonian limit In Newtonian MHD, the Mach number is simply The Newtonian limit A . The first equation indicates that plasma is corotating with the field lines for low Mach numbers, M 2 ≪ 1, and that the specific angular momentum j is equal to the total angular momentum L for high Mach numbers, M 2 ≫ 1. These equations tell us that Newtonian MHD is only valid inside the light cylinder R L = c/Ω F . Since for stellar objects, ǫ ≪ 1, a comparison between Eqs. (45-46) and (51-52) tells us that the Newtonian limit is correct for the wind equation, as long as M 2 ≫ x 2 for all relevant radii. This approximation is therefore not justified e.g. for rapidly rotating magnetic surfaces generated by accretion disks around black holes. Even in the case of magnetic surfaces generated by rapidly rotating young stars slight modifications occur in the wind equation. The Newtonian limit In the Newtonian limit, the energy conservation at the injection point can be written as The Newtonian limit Using the solution for the current at the injection point, x 2 * ≪ ǫ, M 2 * ≪ 1 and p * = kT * /mc 2 , The Newtonian limit this shows the various contributions to the total energy The Newtonian limit The parameter ǫ is therefore a real measure of the Poynting flux at the injection point in units of rest mass energy flux. For the solar wind one finds ǫ ⊙ = 10 −8 and p * = 10 −6 , and therefore the solar wind is thermally driven and not Poynting flux driven (see also MacGregor 1996) . For rapidly rotating protostars, the Poynting flux increases, ǫ ≃ 10 −6 which demonstrates that these winds are Poynting flux driven, even if they would be ejected from a hot corona with p * ≃ 10 −6 . The ratio ǫ/p * is the true measure for the question of Poynting flux driven outflows. For relativistic winds, ǫ → 1, and the Poynting flux can then even exceed the rest mass energy flux. The wind equation In ideal MHD, plasma flows along the magnetic surfaces Ψ = const. Since the Lorentz factor, the angular momentum (or angular velocity Ω) and the poloidal current function I are essentially only functions of the radius along the flux surface and the Mach number, the normalisation of the plasma 4-velocity, u α u α = −1, The wind equation leads to the equation The wind equation Using the solutions for γ and j, this can be rearranged into the form The wind equation where the subscript S is for 'Schwarzschild'. With radii measured in units of the light cylinder radii R L = c/Ω F , the potential function V S can be expressed as The wind equation with the definitions The wind equation This relation is also correct near the Schwarzschild horizon, since αu p remains finite there. The above equation represents an algebraic relation for M 2 for each position along a flux surface. This follows from the relation between u p and B p The wind equation Multiplying the equation by x 4 we obtain The wind equation The last term can be written in terms of the flux function The wind equation which is normalized by the footpoint R * and its magnetic field strength B p * . In addition, we introduce the dimensionless magnetisation parameter The wind equation which is the analogue of the magnetisation parameter introduced in the Newtonian discussion. This altogether leads to the equation The wind equation This fundamental equation clearly shows that the solutions of the wind equation, when formulated for the Mach number M , depends on the following parameters: The wind equation -the parameter ǫ, which defines the position of the Alfvén point -the dimensionless energyĒ = E/mc 2 , -the magnetisation parameter σ * , -the flux tube function function Φ Ψ -two additional parameters hidden in the specific enthalpy The wind equation These are essentially 5 parameters for each flux surface Ψ = const. It can be shown that the relativistic wind equation has the same critical points as the nonrelativistic one: the slow magnetosonic point, the Alfvén point D S (R A ) = 0 and the fast magnetosonic point. It is very important in this regard that the light cylinder is not a critical point 3 . The requirement that a wind solution passes through all three critical points fixes therefore three of the five parameters. We may consider σ * and p * as free parameters which are fixed by injection physics. The wind equation When pressure is as important as Poynting flux, p * ≃ ǫ, we can apply the expansion of the wind equation The wind equation Since protostellar outflows are probably nearly isothermal (Paatz & Camenzind 1996) , Γ ≃ 1.2, the energy equation can also be transformed into a polynomial form for the square of the Mach number The wind equation with a suitable power n p (n p = 9 for Γ = 1.2). This form of the wind equation can then be used to study all real branches (for a Newtonian study, see Paatz & Camenzind 1996) . The wind equation 2.7.1. The cold wind equation The wind equation In the cold limit, p * ≪ ǫ, we may neglect the pressure terms in the specific enthalpy, µ ≃ mc 2 , and the wind equation then provides a 4th degree polynomial in the square of the Mach number for the poloidal velocity The wind equation with the following polynomial coefficients The wind equation and with m 2n = m 2n (E, ǫ, σ * , Φ; x). Without gravity (α = 1), these expressions are identical with the flat space expansion (Camenzind 1986b ). In the case of a cold wind u p (x inj ) = 0, where x inj is the injection point of the plasma from the star, and m 0 (x inj ) = 0. Consequently, we get a relation between injection radius, energy and Alfvén radius The wind equation hence m 2n = m 2n (x inj , σ * , Φ; x). A cold wind has only two critical points, the fast magnetosonic one x F M , where the wind velocity must equal the fast magnetosonic velocity, and the Alfvén point. The slow magnetosonic point x SM disappears, with the consequence that the mass flux or the magnetisation along a flux surface becomes a free parameter. The solution of Eq. (77) is uniquely determined by the critical energy E crit (Ψ). Eq. (71) is a 4th order polynomial. For this reason the wind equation has 4 solutions u p,i (x), (i = 1, ..., 4) at each radius x. We consider negative or complex solutions to be 'unphysical'. We obtain two positive branches of which the 'physical' solution is the one which passes through the critical points and starts with zero poloidal velocity. In order to determine the critical solution, we use a bracketing method varying the energy parameter E(Ψ) until the critical solution is found. For energies E < E crit , the wind ceases before being able to reach the fast magnetosonic point or the asymptotic radius, hence there is no continuous wind solution for all x. In the case E > E crit , the two branches of the solution do not meet at the fast magnetosonic point (see also Fendt & Camenzind 1996) and no wind solution is found either. We fix the free parameter ǫ by choosing the values for the injection radius x inj and the magnetisation parameter σ * . For small σ * the energy must be determined up to the order 10 −2 σ * . The transfield equation and its action integral Ampère's equation determines the form of the magnetic flux surfaces, known as the Grad-Schlüter-Shafranov The transfield equation and its action integral ∇ is here the covariant derivative with respect to 3-space. This divergence-operator can explicitly be written as The transfield equation and its action integral and is equivalent to the flat space analog except for the redshift factor α. The split-monopole The transfield equation and its action integral is still a solution of the vacuum equation. A dipole The transfield equation and its action integral would be modified somewhat by the gravitational potential. The Grad-Schlüter-Shafranov equation When the magnetosphere is filled with plasma, currents will deform the vacuum solutions. The current density j φ follows from the transverse component of the equation of motion The Grad-Schlüter-Shafranov equation The explicit form of this equation has been derived in various papers (Mobarry & Lovelace 1986; Camenzind 1987; Beskin et al. 1995; Fendt et al. 1995 ) The Grad-Schlüter-Shafranov equation with D S as defined in Eq. (60) (Blandford & Znajek 1976; Mobarry & Lovelace 1986; Okamoto 1992; Fendt 1997 ) The Grad-Schlüter-Shafranov equation In this case, D S = α 2 − x 2 and the source of the magnetic flux is only given by the poloidal current I(Ψ). Even in this limit, the equation is highly non-linear and the Alfvén surface now coincides with the light cylinder. Such solutions are interesting for black hole driven outflows. The Grad-Schlüter-Shafranov equation Another limit which has been discussed quite often in the literature is the Newtonian limit without gravity, α = 1 and R ≪ R L (Heinemann & Olbert 1978; Pelletier & Pudritz 1992; Rosso & Pelletier 1994 ) The Grad-Schlüter-Shafranov equation Rosso & Pelletier (1994) have used this limit for the discussion of disk magnetospheres. They have derived a Lagrangian description of the form The Grad-Schlüter-Shafranov equation depending on a potential function U 0 (M 2 , Ψ; R, z). It is then evident that this equation can be derived from the Lagrangian The Grad-Schlüter-Shafranov equation and the proper action of the system is simply The Grad-Schlüter-Shafranov equation The above formulation is indeed only a special case of the action discussed by Mobarry & Lovelace (1986) . On the causal structure of self-consistent magnetospheres As shown above, the concept of light cylinders appears naturally in ideal relativistic MHD. This condition is largely satisfied in thin outflowing plasmas, in particular when plasma will be reheated by some processes (shocks e.g.). The light cylinder is not a critical surface of the Grad-Schlüter-Shafranov equation (see e.g. Beskin & Pariev 1993) except in the force-free limit. As in the Newtonian case, the relativistic Grad-Schlüter-Shafranov equation has the same critical surfaces, the slow and the fast magnetosonic surface, as well as the Alfvén surface (Heinemann & Olbert 1978 . The system becomes hyperbolic beyond the fast magnetosonic surface, so that no back-reaction with the star is allowed from this region. Collimation of spherical outflows The collimation of magnetized winds into a conical outflow is very similar to the plasma confinement in a z-pinch. In this region we can neglect gravity. The equilibrium condition for a non-rotating axisymmetric current carrying pinch configuration (Freidberg 1987) κ B Collimation of spherical outflows has to be generalized to include various relativistic effects (Appl & Camenzind 1993a,b) κ B Collimation of spherical outflows Pressure gradients acting perpendicular to the magnetic surfaces are in equilibrium with electric forces, centrifugal forces, pinch forces produced by poloidal currents, as well as with curvature forces expressed at the left hand side. This latter force is modified by the Mach number M and the light cylinder, and µ = (ρ + P )/n. A comparison between the Newtonian expansion (89) and the special relativistic expression (90) clearly demonstrates the influence of the light cylinder in the force-balance equation. The major distinction is found in the first expression on the right-hand side, (1−x 2 ) ∇ ⊥ B 2 p /8π. This term changes its sign at the position of the light cylinder, x = 1, as a consequence of the existence of electric fields in Maxwell's stress tensor ↔ t em , Eq. (25). Since the observed collimation radii are larger than the light cylinder, field lines anchored in the star will cross the light cylinder, at least when the stellar rotation periods are of the order of a few days. Collimation of spherical outflows The specific angular momentum of the plasma j = R 2 Ω = Ru φ /γ is now given by the wind theory Collimation of spherical outflows and depends on the Mach number M . Outside the Alfvén surface, M 2 > 1, this can be expressed as Collimation of spherical outflows Outside the Alfvén surface, the specific angular momentum is essentially constant and has the value Collimation of spherical outflows Similarly, the pinch current I is not a free quantity, but also follows from the wind theory Collimation of spherical outflows Using the expression for the Lorentz factor, the total energy E can be replaced by, α = 1, Collimation of spherical outflows For radii x 2 > ǫ and Mach numbers M 2 > 1 , i.e. essentially outside the Alfvén surface, the current can be approximated by Collimation of spherical outflows With the definition of η this gives the universal expression for the current Collimation of spherical outflows always valid outside the Alfvén surface. β p = v p /c is the dimensionless poloidal velocity of the plasma. Asymptotic collimation We discuss in the following the solutions for a cylindrical pinch, where the curvature forces vanish, κ = 0. In this case, the equilibrium condition reduces to its onedimensional form Asymptotic collimation The pressure of the toroidal field and the pinch force can be combined into one single expression. In addition, we normalize the poloidal magnetic field in units of its value on the central axis, Asymptotic collimation Electric fields, E ⊥ = (R/R L )B p , due to the rotation of the magnetic surfaces modify the classical pinch equilibrium in two ways. First, the equation has a singular point at the light cylinder, and secondly, the additional term 4xy is crucial for the form of the equilibrium. In contrast to force-free models (Appl & Camenzind 1993b) , the specific angular momentum j of the plasma and the poloidal current I are not arbitrary, but follow from the MHD wind theory. Asymptotic collimation In particular, the last term can be written as Asymptotic collimation where V A is the local Alfvén speed and v φ,A = R A Ω F the rotational velocity at the Alfvén point. We also expressed the specific angular momentum in terms of its characteristic value, j =j R 2 A Ω F . This term contributes therefore to the second term in the pinch equation (99). This shows therefore that the centrifugal force is not important near the light cylinder and beyond, whenever the Alfvén radius is far inside the light cylinder. In this domain the electric fields provide the important forces. Far inside the light cylinder, however, centrifugal forces are important. Pinch solutions without angular momentum We first discuss solutions of the relativistic pinch equilibrium neglecting the influence of the centrifugal force. In this case the equilibrium is determined by the condition Pinch solutions without angular momentum The toroidal current follows to a good approximation Pinch solutions without angular momentum and we prescribe the plasma pressure in terms of P = βB 2 p /8π with a constant plasma beta. This leads finally to the equation, Pinch solutions without angular momentum This homogeneous equation has the remarkably simple solution Pinch solutions without angular momentum with Pinch solutions without angular momentum as the core radius of the jet with β p ≃ 10 −3 and γ p = 1 for protostellar outflows. Inside the core, given by the light cylinder radius and the poloidal velocity β p , the poloidal magnetic field is essentially homogeneous. It decays as a monopole field outside the core, Pinch solutions without angular momentum This is not unexpected since the jet is formed by a supermagnetosonic wind. The current function I increases quadratically inside the core and saturates towards a constant value outside the core, I ∞ ≃ cB 0 R L β p . This is exactly the structure of the force-free solutions found previously (Appl & Camenzind 1993b ). The flux function has the same form as in the forcefree solutions Pinch solutions without angular momentum This is not unexpected, since in the collimated regime the magnetic structure is force-free, I = I(Ψ). MHD jets always have a core-envelope structure, along the axis the magnetic field is predominantly longitudinal, beyond the core it becomes dominated by the toroidal component. The light cylinder R L ≃ 100 AU is the basic scale for the jets, the core is extremely small, R c ≃ 10R * . Pinch solutions including angular momentum In this section we investigate the behaviour of the pinch equation inside the light cylinder including centrifugal forces Pinch solutions including angular momentum The centrifugal force must be regularized inside the Alfvén surface. A suitable function which guarantees corotation inside the Alfvén point is given bȳ Pinch solutions including angular momentum Using the expression for the current and neglecting the pressure Pinch solutions including angular momentum with solutions as obtained by a simple integration Pinch solutions including angular momentum or ln y = ln Pinch solutions including angular momentum The second integral only contributes in the region of the Alfvén radius. For the case R c = R A andV = const, the solution is simply given by Pinch solutions including angular momentum L , angular momentum modifies the solution only near the Alfvén surface. These effects drop out very rapidly beyond the Alfvén radius, and we are left with the solution found previously. A model for collimated outflows Fully self-consistent solutions of the Grad-Shafranov equation are not yet available. We want to demonstrate the main features of collimated outflows in terms of a model which satisfies the inner boundary conditions and the asymptotic flux distribution for cylindrical outflows. The existence of a disk surrounding the star will modify the dipolar field in the innermost few stellar radii (see Fendt et al. 1995) . Beyond the inner edge of the disk the resulting magnetosphere is however largely spherical and cylindrical in the asymptotic region. The model The flux distribution should be a solution of the GradShafranov equation. As a first approximation we use the following ansatz for the flux function, where Ψ consists of a homogeneous and a special solution The model with The model The vacuum solution (see Appendix A) is given by The model where The model The quantity ρ is dimensionless due to the normalization of the spherical radius r in units of the inner radius of the disk a disk , ρ := r/a disk . We assume the existence of a gap between the star and the disk due to the magnetic pressure. The dipole moment is given by µ * = R 3 * B * . For our calculations the stellar radius is R * = 3R ⊙ and the maximum magnetic field strength on the stellar surface B * = 1000 G. The last term in Ψ hom denotes the dipole magnetic flux generated by the star itself, while the remaining terms denote the flux contribution from the screening of the disk (Kundt & Robnik 1980) ; Aly 1980; Paatz & Camenzind 1996) . The integration constant ψ equals the flux on the open field lines for the homogeneous solution. For an integration constant ψ = 0 all flux is forced to go through the hole between star and disk and cannot escape to infinity (Kundt & Robnik 1980) . The term ψ/ρ does not contribute to the magnetic field, but guarantees Ψ = 0 on the z-axis. In the collimation region we find for large radii from Eq. (116) The model This means that Ψ hom behaves like a monopolar type function for large radii. Ψ spec describes the magnetic flux produced by a toroidal current j φ The model Apart from the normalisation factor A, which will be discussed in Sect. 6.3, it contains two essential radii. R c is the core radius discussed in the last section, since for large radii we find then The model as required by cylindrical collimation. For small radii, the cylindrical radius drops out of the expansion and we get a monopolar type solution The model The parameter a is a kind of jet radius, since the last magnetic surface that can be closed with the star is given by ln(a/R c ) 2 . In the case of Eq. (120), the flux surfaces can be parameterized by means of The model The angle θ extends from θ = 0 (asymptotic region) to a maximal value θC , where the flux surface intersects with the stellar surface, r(θC ) = R * . The flux surface labelC is therefore given by this anglẽ The model For the total flux function the collimation part will finally always dominate the homogeneous part. Comparing Eq. (119) with Eq. (121) we see that the first does not depend on the radius, hence the latter containing the square of the radius will determine the solution for sufficiently large radii. The model Note furthermore that we need all parts of Eq. (114) to reproduce the dipole influence of the star at small radii and the collimation of the field lines for the asymptotic radii. Excluding the monopolar type part of Eq. (120) we cannot obtain realistic jet radii and the magnetosphere will collimate much too quickly. Ψ spec alone cannot reproduce the influence of the dipole near the star. The surfaces of constant flux for the total flux function Eq. (114) are calculated with the Newton-Raphson method. The resulting magnetic fields can be found in Appendix B. The amount of flux through the hole of the disk The magnetic flux Ψ hole through the hole of the disk at z = 0 consists of the flux belonging to the disk field Ψ hom disk and Ψ spec between star and disk, the whole flux of the dipole Ψ Dip minus the part of the dipole on the disk (Riffert 1980) The amount of flux through the hole of the disk For ψ = 0 all other terms except for the flux of the dipole Ψ Dip and Ψ spec will vanish (see also Riffert 1980) . Then plasma can only escape to infinity due to the special solution. This characteristic is discussed in Sects. 6.3 and 6.4. For ψ = 0 and A = 0 the amount of flux through the hole is determined by the integration constant ψ and the disk radius a disk via Ψ hole ∝ −ψ/a disk . This means that for ψ = 0 and A = 0 all flux goes through the gap between the star and the disk. This amount will decrease with increasing ψ and A. Results and discussion We now present solutions for the magnetospheres and the resulting winds which are calculated along the collimating jet flux surfaces. This leads us to a 2D velocity distribution. We discuss the impact of the integration constant ψ, the disk radius a disk , the normalisation factor A of Ψ spec and the redshift factor α on the wind solutions. Table 1 summarizes the values of all other parameters which are kept constant throughout this paper. Magnetisation σ * 10 Results and discussion The distribution of the magnetic flux surfaces Ψ(R, z) and the corresponding magnetic flux tube functions Φ(R) for each surface are shown in Fig. 1 . The critical surface Ψ crit is the first open flux surface and reaches the asymptotic jet radius R jet . There is no jet at radii larger than that given by Ψ crit . We choose our parameter a of Eq. Results and discussion (120) such that we get realistic jet radii of R jet ≈ (3−4)R L (Fendt & Camenzind 1996) . In our calculation the period of the star is P * = 1.14 days which is equivalent to a light cylinder at R L = 2250R * . Because of the influence of Ψ hom the jet radius is shifted to bigger values than those given by a. We get R jet = 7937R * = 3.5R L for a = 1430R * . The In contrast to calculations done by Fendt & Camenzind (1996) our analytical model is not limited by numerics, hence we can take a realistic weak magnetisation parameter of σ * = 10 −8 which corresponds to observations. Throughout the jet σ * is kept constant which is equivalent to a constant mass flow rate on each surface Ψ. Results and discussion We normalize the radius R to the light cylinder R L and the velocities to the Alfvén velocity on the stellar surface v A, * , where v 2 A, * = B 2 * /4πm p n * , and the flux surfaces to Ψ crit . The density on the stellar surface is n * = 10 13 cm −3 , corresponding to an outflow rate ofṀ jet ≃ 10 Results and discussion The flux tube function is a measure for the opening angle of the flux surfaces. Small Φ correspond to large openings, constant values to a conical structure of Ψ (Φ = const represents a monopolar type flux surface). For small radii we can reproduce a stellar dipole which is mainly responsible for the strong decrease of the flux tube functions. The more the surfaces show a dipolar structure instead of a conical one, the longer is the extent of the region of decrease of the flux tube function. This is equivalent to an acceleration of the plasma. There is a deceleration of the plasma for the outer flux surfaces for radii larger than the disk radius. The corresponding Φ-values then even catch up with those obtained by the more inward lying surfaces. Nevertheless, their total plasma acceleration always exceeds that of the latter ones. The deceleration process coincides with a sharp bending of these flux surfaces at the inner edge of the disk which is not found for the inner flux surfaces. From this we draw the conclusion that the inner dipole part of the star in our model is crucial for the plasma acceleration. The closer the flux surface is to the z-axis, the less the surfaces look dipole-like and show a conical structure instead. For intermediate radii the monopolar-type part of Ψ spec controls all flux surfaces, consequently all flux tube functions become constant. For large radii all surfaces collimate. The cold wind A physical cold wind solution has to start at zero velocity at the injection radius and, accelerating monotonically, pass through the Alfvén point and the fast magnetosonic point. Figure 2 shows the wind solution along the flux surface Ψ = 0.89. The maximum poloidal velocity is reached when Φ is lowest. As a general feature it can be seen that the acceleration and deceleration of the plasma along different flux surfaces varies. The acceleration for the outer surfaces is more efficient than for the inner ones and the asymptotic poloidal velocity u ∞ p is larger. Figure 3 shows the asymptotic velocity distribution over the cross-section of the jet. The velocity decreases for decreasing flux surfaces. In our calculations, the mass flow rate per flux surface is conserved, η(Ψ) = const. The cold wind The positions of the Alfvén point x A and the fast magnetosonic point x F M are plotted for several flux surfaces in Fig. 4 . The resulting value for ǫ is of the order of 10 −6 . The Alfvén point moves to smaller radii for decreasing flux surface values Ψ. This is equivalent to a smaller amount of angular momentum being transferred to the plasma. As expected for weakly magnetized flows (Fendt & Camenzind 1996) , the Alfvén point is very close to the injection radius and far from the light cylinder. For successively decreasing surfaces the fast magnetosonic points move linearly inward towards smaller radii. For the innermost flux surfaces no determinations of x F M and E crit are possible any longer. This means that for these surfaces the wind remains submagnetosonic. Wind solutions for flux surfaces lower than 50% of Ψ crit start oscillating. Concerning the question of inner wind solutions two different types of solutions are possible. Either we insist on every solution to be a critical one regardless of whether it is global or not, or we argue that the crucial feature of a wind is to be global regardless of being supermagnetosonic. The cold wind In the first case a flux surface can be found for which the acceleration is not sufficient for the plasma to reach the asymptotic regime (for Ψ = 0.42 Ψ crit , with the parameter set a disk = 2.5R * , A = 10, ψ = 0.1364). In this case, the wind still reaches the fast magnetosonic point but is then so much decelerated that it closes very quickly afterwards (upper panel in Fig. 5 ), where u The cold wind This implies that the inner part of the asymptotic jet is empty despite having a constant mass distribution for each Ψ at x inj . Even though the flux surface extends to some asymptotic radius, the plasma flow stops somewhere on its way along Ψ. Strictly speaking, the supermagnetosonic non-global solutions cannot be considered to be physical solutions. This behaviour is due to the fact that we do not take pressure into consideration and have no selfconsistent solutions. Especially close to the z-axis a wind can no longer be centrifugally driven and Lorentz-forces become negligible, hence pressure becomes more important. A mainly Poynting-flux driven cold wind (see Eq. (43)), where RB φ → 0 on the z-axis, will obviously not have any mechanism by which it can be accelerated away from the star. Only those winds can reach the asymptotic On the other hand, we can argue that because of the small influence of centrifugal and Lorentz-forces near the z-axis a wind becomes less accelerated so that it can no longer reach the magnetosonic point. This wind flows only with submagnetosonic velocities, but reaches the asymptotic jet radius. Hence we look for the smallest energy possible for which the wind can be global (Fig. 5) . In this case the physical wind solution is the lower branch. We claim that the energy continues to decrease smoothly from one flux surface to the next as is the case for the critical solutions. This feature is found for the supermagnetosonic, global solutions and should be maintained. For small σ * (Fendt & Camenzind 1996 ) the principal acceleration of the plasma occurs at radii smaller than the Alfvén-radius. There the plasma and the magnetic field corotate. This means that the acceleration is mainly due to the centrifugal force which decreases towards the z-axis. Consequently, less energy can be expected to be available near the z-axis to drive the wind. The final conclusion is that we have to investigate a hot wind in the future for describing the physics more consistently. The influence of
{'paper_id': '15993664', 'title': 'ASTRONOMY AND ASTROPHYSICS Collimated outflows of rapidly rotating young stellar objects', 's2_url': 'https://api.semanticscholar.org/CorpusID:15993664', 'source': 'S2ORC'}