YSO Jets

Consider a YSO where the poloidal dipole field of the YSO has been wrapped up into a toroidal field () in the accretion disk surrounding the young stellar object (Fig. 2). The sign of the field changes as one passes through the disk. This magnetic structure suggests that the midplane of the inner accretion disk can become a zone of magnetic reconnection (Freeman 1977, Grosso et al. 1997). As the dipole field sweeps across the disk it generates a radial Freeman current in the disk. The Lorentz force generated from the Freeman current tends to compress the disk. The electric field in the disk's corona interacts with the toroidal field to drive the flow via drift.

  
Figure 2: Interaction of the dipole magnetic field from a YSO with its accretion disk.

To put this scenario on a slightly more quantitative basis, and to show how the toroidal field is created, we assume that the magnetic field rotates rigidly with the star, so at the disk surface, the velocity of the object's B field () is

where r is the distance, in the plane of the disk, from the centre of the star and is the angular rotational frequency of the star. If the disk rotates with a Keplerian velocity, then the disk surface has an angular velocity

where G is the gravitational constant, and is the mass of the central object.

The magnetic field velocity relative to the disk is simply
 
So when , we have reached the corotation radius , where the angular velocity of the stellar magnetic field matches the Keplerian velocity of the disk, and has the form
 
with G being the gravitational constant, the mass of the star and the rotation period of the star, where we have taken the typical rotation period of a YSO as our normalization value.

For the magnetic field will induce an electric field (E), in the disk, of the form
 
where is the strength of the star's magnetic field, in the plane of the disk, at distance r from the centre of the object

Combining Eqs (10), and (12) gives
 
For a disk with finite conductivity (), the induced electric field drives a Freeman current in the disk with a current density of the form
 
This current, in turn, induces a toroidal field. The form of which can be deduced from Ampere's Law
 
where is the permeability of free space.

  
Figure 3: Current density and toroidal field in the top half of the disk.

We integrate Eq. (15) over the section of cylindrical surface shown in figure 3. The bottom of this surface lies on the midplane of the accretion disk, where the toroidal field () has zero magnitude. The top line lies at a height z above the midplane of the disk, where the toroidal field has a non-zero magnitude. The disk current flows in a direction perpendicular to the integration surface, so the magnitude of the toroidal field is simply:
 

Equations (14) and (16) were first obtained by Bardou and Hayvaerts (1996), (see also Bardou 1997). These authors deduced these equations by analyzing the electric circuit made up of the disk, star and the magnetic fields. We have given an alternative derivation to illustrate the veracity of these results.

With our toroidal field in place, our hypothetical flow scenario is shown in Fig. 4. The wind arises from the toroidal magnetic field embedded in the disk, and expands in a roughly conical shape. Our domain of investigation is a flow-tube with a variable thickness and initial radius , where the surface of this tube is constituted from neighbouring streamlines.

  
Figure 4: The jet flow equation is derived for a thin, approximately, conical sheet, which is embedded within the jet flow.

For a flow tube of fixed average radius and variable thickness, one can show (Liffman and Siora 1997) that
 
where G is the gravitational constant and M is the mass of the star. We concentrate on the case and , so Eq. (17) has the simple form
 
which is again the MHD nozzle equation, Eq. (1). So, the results for the one-dimensional flow hold for this astrophysical case. Note that the Alfvn speed at the throat and from Eq. (6) the exhaust velocity for the jet is simply

For a magnetically confined plasma one can show (see Appendix A) that , in which case Eqs (18) and (5) integrate to an analytic flow solution:
 
where we have replaced by u, with , and , and being, respectively, the values of and at the throat. This equation is also true in the cold plasma limit ( see Schoenberg et al. 1991).

In Fig. 5(a) we show the shape of the nozzle as given by Eq. (20). At the start of the flow , which gives . An infinitely wide nozzle is clearly unphysical, so we can expect that the initial value of will be greater than zero in a real jet, i.e., gas has to be injected into the magnetic nozzle for it to work. At the throat and , as expected. Finally, at the exit, , i.e., to obtain the maximum possible jet speed, we again require the nozzle to have infinite width. Because this is impossible, we should expect that a real jet will have a normalized exit speed somewhere between 1 and .

From Eqs (3) and (4) we have
 
and
 
where , , , and are the values of , B, , and normalized by their values at the throat of the nozzle.

At the exit of an ideal nozzle, , , , and . The behaviour of and is shown in Fig. 5(b). The magnetic field and gas density start with an initial normalized value of 1.5. The values of and decrease to a value of 1 at the throat of the nozzle and to a value of 0 at the end of the nozzle. Thus, an ideal MHD nozzle will produce a gas flow with a very small magnetic field and a very low gas density. This result calls into question the popular idea that toroidal fields collimate jet flows, since we may produce a jet with little or no magnetic field. We also note that a nearly ``ideal'' YSO jet may be difficult to observe, because the low density of the exhaust gas may cause the jet to fall below detection limits.

  
Figure 5: (a) The width of the nozzle, , in terms of the flow speed . (b) Gas density () and magnetic field strength () in terms of the flow speed.

In recent years, some authors have made the suggestion that dust and small silicaceous spheres may be ejected by YSO jets (e.g., Liffman and Brown 1996). As a first step to investigate the plausibility of such a hypothesis, we take a macroscopic test particle, e.g., a small silicate sphere, and place the test particle into the flow at different places along the nozzle. In this way, we can map out the initial drag force experienced by a macroscopic particle that is simply dropped into the flow at various points along the nozzle.

The drag force () experienced by our test particle is given by

where is the drag coefficient and A is the cross sectional area of the test particle. If we divide the value of by its value at the throat of the nozzle (), we obtain the normalized drag force () which (using Eqs (20) and (21)) has the form
 
where we have assumed that . From Eq. (24), obtains a maximum value of 9/8 when . The behaviour of is shown in Fig. 6, where we see, as expected, that the maximum drag occurs for . Since the gas speed at the throat of the nozzle is the Keplerian speed, it is possible that a particle carried by the flow may reach escape speed and be ejected from the YSO system.

  
Figure 6: The normalized drag force on a macroscopic particle placed in the MHD nozzle flow.

In the discussion so far, we have explored some of the properties of the MHD nozzle flow, but what produces the nozzle in the first place? We believe the nozzle width, , is primarily determined by the balance between the centrifugal force of the gas in the disk and the magnetic pressure gradient in the toroidal field, . To see how this could arise, we consider the steady state form of the MHD momentum equation:
 
where p is the pressure and the acceleration due to gravity. In cylindrical coordinates, the r component of Eq. (25) is simply
 
From Liffman and Siora (1997), we know that

where is the initial value of r for a parcel of gas that starts its journey from the midplane of the accretion disk. Sample calculations suggest that is small relative to the other terms, so we neglect this term. The magnitude of is given by Eq. (16), except we now have
 
where
 

We are required to modulate by Eq. (29), because in the disk, . Above the disk, however, we have a perfectly conducting corona and so the radial current that generates in the finitely conducting disk cannot do the same in the corona. Thus, we should expect to decrease as some function of z. In this purely illustrative example, we assume that decreases linearly with z and disappears completely when we reach z = H + l. In the illustrative example that we give below, we have set l = H.

We can now differentiate Eq. (28) with respect to r (assuming ) and obtain
 

From Eqs (28) and (30), we see that the toroidal field starts its wind-up at the corotation radius, , and the magnitude of increases with distance from the star until it reaches a certain distance . The exact value of is dependent on the r dependence of , but we have assumed a stellar dipole field (i.e., ) and so, from Eq. (30), has the form

Between and the magnetic pressure gradient is pointing towards the star. If an element of gas, located in this region, moves away from the central (z = 0) plane of the disk. This gas element, initially, feels no magnetic force, because the central plane is a magnetic reconnection region and will have no toroidal field. However, as the particle approaches the surface of the disk, it will feel the magnetic force from the gradient in the toroidal field and move inwards towards the star until the centrifugal force balances the magnetic and gravitational forces.

A gas element that initially has will move only slightly towards the star, since in the near neighbourhood of the corotation orbit, . A gas element with , will have a near its maximum value and the motion of the gas element will be perturbed significantly towards the star. Above the plane of the disk, decreases with distance along the nozzle, allowing the nozzle to expand. This flow property is shown in Fig. 7, illustrating the formation of the nozzle. We obtained Fig. 7 by finding the values of r such that Eq. (26) was satisfied. The parameter values used in this simulation were, H = 0.0001 AU, Siemens, stellar radius AU, and Teslas.

  
Figure 7: Path of a gas particle in the jet flow. z is in units of ``magnetic scale height'' (H), where . Two elements of gas are released from the central plane of the disk (z = 0) at distances of 0.042 and 0.057 AU from the star. Magnetic pressure from the toroidal field causes these gas elements to move in towards the star. The toroidal field decreases in magnitude for z > H allowing the gas elements to return to their initial r positions, thereby producing the jet nozzle.

As is well known, toroidal magnetic fields are intrinsically unstable. This would tend to limit the applicability of toroidal fields as a driving mechanism for YSO jet flows. However, a simple analysis suggests that the ram pressure of the jet may ensure the stability of the toroidal magnetic fields (Liffman and Siora 1997).

© Copyright Astronomical Society of Australia 1997