Protostellar jets and winds are probably driven magnetocentrifugally from the
surface of accretion disks close to the central stellar objects. The exact
launching conditions on the disk, such as the distributions of magnetic flux
and mass ejection rate, are poorly unknown. They could be constrained from
observations at large distances, provided that a robust model is available to
link the observable properties of the jets and winds at the large distances to
the conditions at the base of the flow. We discuss the difficulties in
constructing such large-scale wind models, and describe a novel technique which
enables us to numerically follow the acceleration and propagation of the wind
from the disk surface to arbitrarily large distances and the collimation of
part of the wind into a dense, narrow ``jet'' around the rotation axis. Special
attention is paid to the shape of the jet and its mass flux relative to that of
the whole wind. The mass flux ratio is a measure of the jet formation
efficiency.Comment: 6 pages, figures included. To appear in "The Origins of Stars and
  Planets: The VLT View". J. Alves and M. McCaughrean, editor

Blandford, Roger D.

Krasnopolsky, Ruben

Li, Zhi-Yun

English

arXiv

Protostellar jets and winds are probably driven magnetocentrifugally from the surface of accretion disks close to the central stellar objects. The exact launching conditions on the disk, such as the distributions of magnetic flux and mass ejection rate, are poorly unknown. They could be constrained from observations at large distances, provided that a robust model is available to link the observable properties of the jets and winds at the large distances to the conditions at the base of the flow. We discuss the difficulties in constructing such large-scale wind models, and describe a novel technique which enables us to numerically follow the acceleration and propagation of the wind from the disk surface to arbitrarily large distances and the collimation of part of the wind into a dense, narrow “jet” around the rotation axis. Special attention is paid to the shape of the jet and its mass flux relative to that of the whole wind. The mass flux ratio is a measure of the jet formation efficiency

Caltech Authors - Main

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1
Structure of Magnetocentrifugal Disk-Winds:
From the Launching Surface to Large Distances
Ruben Krasnopolsky1, Zhi-Yun Li2, and Roger D. Blandford3
1 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL
60637, USA
2 Department of Astronomy, University of Virginia, Charlottesville, VA 22903, USA
3 Theoretical Astrophysics, Caltech, Pasadena, CA 91125, USA
Abstract. Protostellar jets and winds are probably driven magnetocentrifugally from
the surface of accretion disks close to the central stellar objects. The exact launching
conditions on the disk, such as the distributions of magnetic flux and mass ejection rate,
are poorly unknown. They could be constrained from observations at large distances,
provided that a robust model is available to link the observable properties of the jets
and winds at the large distances to the conditions at the base of the flow. We discuss the
difficulties in constructing such large-scale wind models, and describe a novel technique
which enables us to numerically follow the acceleration and propagation of the wind
from the disk surface to arbitrarily large distances and the collimation of part of the
wind into a dense, narrow “jet” around the rotation axis. Special attention is paid to
the shape of the jet and its mass flux relative to that of the whole wind. The mass flux
ratio is a measure of the jet formation efficiency.
1 Basic Mechanism and Previous Work
The magnetocentrifugal mechanism is the leading candidate for producing the
jets and winds observed around young stellar objects. The basic principle is rel-
atively simple, and has been understood for a long time [1]. It envisions parcels
of fluid element being lifted off and accelerated centrifugally along rapidly ro-
tating open magnetic field lines anchored firmly on an accretion disk. Beyond a
certain point where the energy densities in the bulk flow motion and magnetic
field are comparable, the field lines can no longer enforce rigid rotation, and the
field becomes increasingly toroidal. It is the “hoop stress” associated with the
toroidal field that is thought to be responsible for the wind collimation and jet
production. The quantitative properties of the jet expected from this mechanism
are poorly determined however, even though the MHD equations that govern the
wind structure and jet formation are well known. Our understanding of the jet
properties has been hampered to a large extent by the mathematical difficulties
associated with obtaining wind solutions.
1.1 Time-Independent Wind Solutions
There are two basic approaches in obtaining wind solutions. The first is to solve
for the steady-state wind structure directly from the time-independent MHD
2 Krasnopolsky et al.
equations. These equations can be cast into a second order partial differential
equation (the Grad-Shafranov equation). It is well known that, for a cold wind
that we are interested in here, the equation changes its type from being elliptic
inside the fast (magnetosonic) surface to hyperbolic outside. Computationally,
the structure of the inner part of the wind inside the fast surface can be solved
first by relaxation, and that beyond the fast surface later by the method of
characteristics [2,3]. The fact that the position of the fast surface, where the
poloidal flow speed matches the fast magnetosonic speed, is unknown a priori
poses a problem. To obtain a converged solution, one needs to have a good initial
guess of the fast surface position, which is generally difficult to obtain.
1.2 Time-Dependent Numerical Simulations
A more flexible approach is to numerically follow the time evolution of a wind
to steady-state, if such a state exists. This approach has been taken by several
groups [4,5,6]. It is also the approach that we took [7]. Our simulations are based
on the ZEUS MHD code, and treat the Keplerian disk as a (lower) boundary,
on which an open magnetic field of a prescribed distribution is anchored and
from which cold material is injected into the wind at a prescribed rate. A novel
feature of our simulation is the treatment of the region near the rotation axis,
where the magnetocentrifugal mechanism is ineffective. The reason is that to
launch a cold parcel of fluid element centrifugally from a Keplerian disk the field
line must incline an angle of at least 30◦ away from the disk normal [1]. This
condition is not met in the axial region since the field line along the axis must
be exactly vertical by symmetry. In reality, the axial region may be filled with a
normal stellar wind from the central object or bundles of open field lines from
the stellar magnetosphere. We are thus motivated to inject a light fluid with
little mass flux at a speed fast enough to escape from the potential well along
those field lines that fail to operate magnetocentrifugally. The light axial flow
provides a plausible inner boundary to the magnetocentrifugal disk-wind, the
focus of our investigation.
A typical example of the steady-state disk-wind solutions obtained from time-
dependent simulations is given in Fig. 1. The wind is driven off all of the (equato-
rial) disk surface. Flow acceleration is apparent along all field lines except those
near the axis where a fast initial injection is imposed. Note that the field (and
stream) lines collimate gradually, as expected. What was not expected was the
great care that went into designing the shape of the simulation box, so that a
steady state could be reached at all. If we were to cut the box shown in Fig. 1
in half or to elongate the box horizontally instead of vertically, and restart the
simulation with the same initial and boundary conditions, the wind would be-
come chaotic. The sensitive solution dependence on the simulation box has also
been noted by others [5]. It is a major concern for the time-dependent approach
to finding disk-wind solutions.
Magnetocentrifugal Disk-Winds 3
0 20 40
0
20
40
60
80
Fig. 1. A representative magnetocentrifugal wind launched from a Keplerian disk (in
arbitrary units; taken from [7]). Shown are the magnetic field lines (light solid lines),
velocity vectors (arrows), density contours (shades), and the fast magnetosonic surface
(solid line of medium thickness). The thickest solid line divides the portion of the
wind that becomes super fast-magnetosonic inside the simulation box (above) from the
portion that does not (below).
2 Magnetocentrifugal Winds From Inner Accretion Disks
We believe that the simulation box dependence described above comes from the
fact that a large fraction of the wind remains completely sub fast-magnetosonic
in the computational domain, as shown in lower-right corner of Fig. 1. Informa-
tion on the sub-fast outer boundary can propagate upstream all the way to the
disk surface and interfere with the wind launching. The reason for the region to
remain sub fast is simple: the wind coming off the outer part of the disk encoun-
ters the edge of the simulation box too soon; it simply does not have enough
room to get accelerated to the fast speed. This situation remains as long as the
wind is driven off from all of the (equatorial) disk plane (as assumed in Fig. 1
and other previous time-dependent disk-wind simulations) regardless of the box
size. It motivates us to restrict the wind launching to only the inner region of
an accretion disk, and focus on inner-disk driven winds for which the simulation
box dependence disappears.
4 Krasnopolsky et al.
2.1 Inner-Disk Driven Winds: Simulation Setup
Physically, the wind launching may be limited to the inner region of a protostellar
disk where the temperature is high enough (greater than ∼ 103 K) that thermal
ionization of alkali metals can provide enough charges to couple the magnetic
field to the disk matter. For typical parameters, this occurs inside a radius of
order 1 AU. Numerically, we set up the simulation as sketched in Fig. 2. To fill
all available space above (and below) the equatorial plane, we demand the last
field line anchored at the outer radius of the launching region R0 to lie exactly
on the equatorial plane. Wind plasma sliding along this last (horizontal) field
line will become super fast-magnetosonic in the computational realm, provided
that the size of the simulation box is sufficiently large. Once the whole fast
surface is completely enclosed inside the simulation box, the size and shape of
the box would have minimal effects on the structure of the wind, especially near
the launching surface, since information cannot propagate upstream in a super
fast region. In this way, we should be able to study the wind structure up to
arbitrarily large distances from the source region, limited only by computer time.
Fig. 2. Schematic view of a cold magnetocentrifugal wind launched from a limited,
inner disk region
2.2 Large-Scale Wind Structure: Numerical Results
For illustration, we consider a specific example. We adopt as the launching con-
ditions on the disk a power-law distribution of the vertical field strength with
radius as Bz ∝ R
−1.5 and a mass injection rate (per unit area) j ∝ R−0.5 be-
tween 0.1 and 0.8 AU. The inner radius is chosen to mimic the disk truncation
radius due to the stellar magnetosphere. Inside this radius, we inject a fast light
flow as described earlier. At the edge of the wind launching region, taken to be
R0 = 1 AU for simplicity, we impose the condition that Bz = j = 0 since the
Magnetocentrifugal Disk-Winds 5
last field (and stream) line must be horizontal. A cubic polynomial is used to
connect smoothly the values of Bz (or j) between 0.8 and 1 AU. As before, we
follow the time evolution of the wind numerically to a steady state. The steady
wind solution, from the launching surface (inside 1 AU) all the way to a large
distance of 102 AU, is displayed in Fig. 3 on two scales.
10 6
 cm −3
10 7cm −3
0 1 2 3 4 5 6 AU
0
1
2
3
4
5
6
10 4
 cm
−3
10 5
cm
−3
0 20 40 60 80 100 AU
0
20
40
60
80
100
Fig. 3. Streamlines (light solid lines) and density contours (heavy solid lines and
shades) of a representative steady wind driven from the inner region of an protostellar
disk in a 6 AU (top panel) and 102 AU (bottom panel) box. The fast surface (dashed
line) is also plotted in the smaller box. The streamlines divide the wind into 10 zones
of equal mass flux, with a total mass flux of 10−8M⊙yr
−1 per side of the disk. The
gray scale shows the log of the hydrogen number density, with three shades per decade.
The density contours correspond to 104, 105, 106 and 107 in units of cm−3.
6 Krasnopolsky et al.
Several features are worth noting. First, the fast surface shown in the smaller
box closes on the equatorial plane, as advertised. This closure enabled us to con-
tinue the wind solution on to large distances without having to worry about the
effects of box size. Second, the wind speed is anisotropic, with a value roughly
3 times higher in the axial region than in the equatorial region. The anisotropy
appears to be even stronger in density, which is stratified more or less cylin-
drically (or jet-like) near the rotation axis, as expected. In the more equatorial
region, the density contours bulge outward prominently, retaining some memory
of the nearly horizontal shape of the initial density contour. This non-cylindrical
shape of density contours is significant because the wind emission in forbidden
lines such as [SII]λλ6716,6731 is sensitive to the density [8], and the shape of
the jet may resemble to a zeroth order the shape of the density contour at some
fiducial value. We choose to represent the outer boundary of a “jet” by a fidu-
cial density contour of 104 cm−3 (the outermost contour in the larger box of
Fig. 3). The “jet” so defined has a width of ∼ 30 AU at a height of 102 AU,
comparable to that observed in HH 30 jet. The bulging out at the “jet” base
is not observed, however. Furthermore, the “jet” contains only about a quarter
of the total wind mass flux, making its formation rather inefficient. These un-
desirable “jet” features demonstrate that not all combinations of the launching
conditions are capable of producing cylindrical jets that contain the majority of
the wind mass flux. We find that one way to improve the jet shape and increase
its mass flux fraction is to make the mass injection rate j on the disk decrease
more steeper with radius. The details will be presented in a forthcoming paper.
3 Conclusion
To summarize, by limiting the wind launching to the inner part of an accretion
disk, we are able to obtain using time-dependent simulation steady-state wind
solutions that extend from the launching surface to large distances. Combined
with a detailed calculation of the thermal structure and emission properties,
these large-scale wind solutions can be used to yield constraints on the launching
conditions from the properties of jets and winds observed at the large distances.
The constraints may provide clues to the origin of the disk magnetic fields that
launch the jets and winds.
References
1. R. D. Blandford, D. G. Payne: MNRAS 199, 883 (1982)
2. T. Sakurai: AA, 152, 121 (1985)
3. J. R. Najita, F. H. Shu: ApJ, 429, 808 (1994)
4. R. Ouyed, R. E. Pudritz: ApJ, 482, 712 (1997)
5. G. V. Ustyugova, A. V. Koldoba, M. M. Romanova, V. M. Chechetkin, R. V. E.
Lovelace: ApJ, 516, 221 (1999)
6. S. Bogovalov, K. Tsinganos: MNRAS, 305, 211 (1999)
7. R. Krasnopolsky, Z.-Y. Li, R. D. Blandford: ApJ, 536, 631 (1999)
8. H. Shang, F. H. Shu, A. E. Glassgold: ApJ, 493, 91 (1998)


Structure of Magnetocentrifugal Disk-Winds: From the Launching Surface to Large Distances

Crossref

ar
X
iv
:a
st
ro
-p
h/
01
07
31
5v
1 
 1
7 
Ju
l 2
00
1
Structure of Magnetocentrifugal Disk-Winds:
From the Launching Surface to Large Distances
Ruben Krasnopolsky1, Zhi-Yun Li2, and Roger D. Blandford3
1 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL
60637, USA
2 Department of Astronomy, University of Virginia, Charlottesville, VA 22903, USA
3 Theoretical Astrophysics, Caltech, Pasadena, CA 91125, USA
Abstract. Protostellar jets and winds are probably driven magnetocentrifugally from
the surface of accretion disks close to the central stellar objects. The exact launching
conditions on the disk, such as the distributions of magnetic flux and mass ejection rate,
are poorly unknown. They could be constrained from observations at large distances,
provided that a robust model is available to link the observable properties of the jets
and winds at the large distances to the conditions at the base of the flow. We discuss the
difficulties in constructing such large-scale wind models, and describe a novel technique
which enables us to numerically follow the acceleration and propagation of the wind
from the disk surface to arbitrarily large distances and the collimation of part of the
wind into a dense, narrow “jet” around the rotation axis. Special attention is paid to
the shape of the jet and its mass flux relative to that of the whole wind. The mass flux
ratio is a measure of the jet formation efficiency.
1 Basic Mechanism and Previous Work
The magnetocentrifugal mechanism is the leading candidate for producing the
jets and winds observed around young stellar objects. The basic principle is rel-
atively simple, and has been understood for a long time [1]. It envisions parcels
of fluid element being lifted off and accelerated centrifugally along rapidly ro-
tating open magnetic field lines anchored firmly on an accretion disk. Beyond a
certain point where the energy densities in the bulk flow motion and magnetic
field are comparable, the field lines can no longer enforce rigid rotation, and the
field becomes increasingly toroidal. It is the “hoop stress” associated with the
toroidal field that is thought to be responsible for the wind collimation and jet
production. The quantitative properties of the jet expected from this mechanism
are poorly determined however, even though the MHD equations that govern the
wind structure and jet formation are well known. Our understanding of the jet
properties has been hampered to a large extent by the mathematical difficulties
associated with obtaining wind solutions.
1.1 Time-Independent Wind Solutions
There are two basic approaches in obtaining wind solutions. The first is to solve
for the steady-state wind structure directly from the time-independent MHD
2 Krasnopolsky et al.
equations. These equations can be cast into a second order partial differential
equation (the Grad-Shafranov equation). It is well known that, for a cold wind
that we are interested in here, the equation changes its type from being elliptic
inside the fast (magnetosonic) surface to hyperbolic outside. Computationally,
the structure of the inner part of the wind inside the fast surface can be solved
first by relaxation, and that beyond the fast surface later by the method of
characteristics [2,3]. The fact that the position of the fast surface, where the
poloidal flow speed matches the fast magnetosonic speed, is unknown a priori
poses a problem. To obtain a converged solution, one needs to have a good initial
guess of the fast surface position, which is generally difficult to obtain.
1.2 Time-Dependent Numerical Simulations
A more flexible approach is to numerically follow the time evolution of a wind
to steady-state, if such a state exists. This approach has been taken by several
groups [4,5,6]. It is also the approach that we took [7]. Our simulations are based
on the ZEUS MHD code, and treat the Keplerian disk as a (lower) boundary,
on which an open magnetic field of a prescribed distribution is anchored and
from which cold material is injected into the wind at a prescribed rate. A novel
feature of our simulation is the treatment of the region near the rotation axis,
where the magnetocentrifugal mechanism is ineffective. The reason is that to
launch a cold parcel of fluid element centrifugally from a Keplerian disk the field
line must incline an angle of at least 30◦ away from the disk normal [1]. This
condition is not met in the axial region since the field line along the axis must
be exactly vertical by symmetry. In reality, the axial region may be filled with a
normal stellar wind from the central object or bundles of open field lines from
the stellar magnetosphere. We are thus motivated to inject a light fluid with
little mass flux at a speed fast enough to escape from the potential well along
those field lines that fail to operate magnetocentrifugally. The light axial flow
provides a plausible inner boundary to the magnetocentrifugal disk-wind, the
focus of our investigation.
A typical example of the steady-state disk-wind solutions obtained from time-
dependent simulations is given in Fig. 1. The wind is driven off all of the (equato-
rial) disk surface. Flow acceleration is apparent along all field lines except those
near the axis where a fast initial injection is imposed. Note that the field (and
stream) lines collimate gradually, as expected. What was not expected was the
great care that went into designing the shape of the simulation box, so that a
steady state could be reached at all. If we were to cut the box shown in Fig. 1
in half or to elongate the box horizontally instead of vertically, and restart the
simulation with the same initial and boundary conditions, the wind would be-
come chaotic. The sensitive solution dependence on the simulation box has also
been noted by others [5]. It is a major concern for the time-dependent approach
to finding disk-wind solutions.
Magnetocentrifugal Disk-Winds 3
0 20 40
0
20
40
60
80
Fig. 1. A representative magnetocentrifugal wind launched from a Keplerian disk (in
arbitrary units; taken from [7]). Shown are the magnetic field lines (light solid lines),
velocity vectors (arrows), density contours (shades), and the fast magnetosonic surface
(solid line of medium thickness). The thickest solid line divides the portion of the
wind that becomes super fast-magnetosonic inside the simulation box (above) from the
portion that does not (below).
2 Magnetocentrifugal Winds From Inner Accretion Disks
We believe that the simulation box dependence described above comes from the
fact that a large fraction of the wind remains completely sub fast-magnetosonic
in the computational domain, as shown in lower-right corner of Fig. 1. Informa-
tion on the sub-fast outer boundary can propagate upstream all the way to the
disk surface and interfere with the wind launching. The reason for the region to
remain sub fast is simple: the wind coming off the outer part of the disk encoun-
ters the edge of the simulation box too soon; it simply does not have enough
room to get accelerated to the fast speed. This situation remains as long as the
wind is driven off from all of the (equatorial) disk plane (as assumed in Fig. 1
and other previous time-dependent disk-wind simulations) regardless of the box
size. It motivates us to restrict the wind launching to only the inner region of
an accretion disk, and focus on inner-disk driven winds for which the simulation
box dependence disappears.
4 Krasnopolsky et al.
2.1 Inner-Disk Driven Winds: Simulation Setup
Physically, the wind launching may be limited to the inner region of a protostellar
disk where the temperature is high enough (greater than ∼ 103 K) that thermal
ionization of alkali metals can provide enough charges to couple the magnetic
field to the disk matter. For typical parameters, this occurs inside a radius of
order 1 AU. Numerically, we set up the simulation as sketched in Fig. 2. To fill
all available space above (and below) the equatorial plane, we demand the last
field line anchored at the outer radius of the launching region R0 to lie exactly
on the equatorial plane. Wind plasma sliding along this last (horizontal) field
line will become super fast-magnetosonic in the computational realm, provided
that the size of the simulation box is sufficiently large. Once the whole fast
surface is completely enclosed inside the simulation box, the size and shape of
the box would have minimal effects on the structure of the wind, especially near
the launching surface, since information cannot propagate upstream in a super
fast region. In this way, we should be able to study the wind structure up to
arbitrarily large distances from the source region, limited only by computer time.
Fig. 2. Schematic view of a cold magnetocentrifugal wind launched from a limited,
inner disk region
2.2 Large-Scale Wind Structure: Numerical Results
For illustration, we consider a specific example. We adopt as the launching con-
ditions on the disk a power-law distribution of the vertical field strength with
radius as Bz ∝ R
−1.5 and a mass injection rate (per unit area) j ∝ R−0.5 be-
tween 0.1 and 0.8 AU. The inner radius is chosen to mimic the disk truncation
radius due to the stellar magnetosphere. Inside this radius, we inject a fast light
flow as described earlier. At the edge of the wind launching region, taken to be
R0 = 1 AU for simplicity, we impose the condition that Bz = j = 0 since the
Magnetocentrifugal Disk-Winds 5
last field (and stream) line must be horizontal. A cubic polynomial is used to
connect smoothly the values of Bz (or j) between 0.8 and 1 AU. As before, we
follow the time evolution of the wind numerically to a steady state. The steady
wind solution, from the launching surface (inside 1 AU) all the way to a large
distance of 102 AU, is displayed in Fig. 3 on two scales.
10 6 cm −3
10 7cm −3
0 1 2 3 4 5 6 AU
0
1
2
3
4
5
6
10 4
 cm −3
10 5
cm −3
0 20 40 60 80 100 AU
0
20
40
60
80
100
Fig. 3. Streamlines (light solid lines) and density contours (heavy solid lines and
shades) of a representative steady wind driven from the inner region of an protostellar
disk in a 6 AU (top panel) and 102 AU (bottom panel) box. The fast surface (dashed
line) is also plotted in the smaller box. The streamlines divide the wind into 10 zones
of equal mass flux, with a total mass flux of 10−8M⊙yr
−1 per side of the disk. The
gray scale shows the log of the hydrogen number density, with three shades per decade.
The density contours correspond to 104, 105, 106 and 107 in units of cm−3.
6 Krasnopolsky et al.
Several features are worth noting. First, the fast surface shown in the smaller
box closes on the equatorial plane, as advertised. This closure enabled us to con-
tinue the wind solution on to large distances without having to worry about the
effects of box size. Second, the wind speed is anisotropic, with a value roughly
3 times higher in the axial region than in the equatorial region. The anisotropy
appears to be even stronger in density, which is stratified more or less cylin-
drically (or jet-like) near the rotation axis, as expected. In the more equatorial
region, the density contours bulge outward prominently, retaining some memory
of the nearly horizontal shape of the initial density contour. This non-cylindrical
shape of density contours is significant because the wind emission in forbidden
lines such as [SII]λλ6716,6731 is sensitive to the density [8], and the shape of
the jet may resemble to a zeroth order the shape of the density contour at some
fiducial value. We choose to represent the outer boundary of a “jet” by a fidu-
cial density contour of 104 cm−3 (the outermost contour in the larger box of
Fig. 3). The “jet” so defined has a width of ∼ 30 AU at a height of 102 AU,
comparable to that observed in HH 30 jet. The bulging out at the “jet” base
is not observed, however. Furthermore, the “jet” contains only about a quarter
of the total wind mass flux, making its formation rather inefficient. These un-
desirable “jet” features demonstrate that not all combinations of the launching
conditions are capable of producing cylindrical jets that contain the majority of
the wind mass flux. We find that one way to improve the jet shape and increase
its mass flux fraction is to make the mass injection rate j on the disk decrease
more steeper with radius. The details will be presented in a forthcoming paper.
3 Conclusion
To summarize, by limiting the wind launching to the inner part of an accretion
disk, we are able to obtain using time-dependent simulation steady-state wind
solutions that extend from the launching surface to large distances. Combined
with a detailed calculation of the thermal structure and emission properties,
these large-scale wind solutions can be used to yield constraints on the launching
conditions from the properties of jets and winds observed at the large distances.
The constraints may provide clues to the origin of the disk magnetic fields that
launch the jets and winds.
References
1. R. D. Blandford, D. G. Payne: MNRAS 199, 883 (1982)
2. T. Sakurai: AA, 152, 121 (1985)
3. J. R. Najita, F. H. Shu: ApJ, 429, 808 (1994)
4. R. Ouyed, R. E. Pudritz: ApJ, 482, 712 (1997)
5. G. V. Ustyugova, A. V. Koldoba, M. M. Romanova, V. M. Chechetkin, R. V. E.
Lovelace: ApJ, 516, 221 (1999)
6. S. Bogovalov, K. Tsinganos: MNRAS, 305, 211 (1999)
7. R. Krasnopolsky, Z.-Y. Li, R. D. Blandford: ApJ, 536, 631 (1999)
8. H. Shang, F. H. Shu, A. E. Glassgold: ApJ, 493, 91 (1998)


Structure of Magnetocentrifugal Disk-Winds: From the Launching Surface to Large Distances

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