standing issues is the role of magnetic âelds in the launch- ing, collimation, and propagation of young stellar object. (YSO) jets. The current consensus holds that ...
THE ASTROPHYSICAL JOURNAL, 524 : 947È951, 1999 October 20 ( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.
AMBIPOLAR DIFFUSION IN YOUNG STELLAR OBJECT JETS ADAM FRANK,1 THOMAS A. GARDINER,1 GUY DELEMARTER,1 THIBAUT LERY,2 AND RICCARDO BETTI3 Received 1999 January 29 ; accepted 1999 May 10
ABSTRACT We address the issue of ambipolar di†usion in Herbig-Haro (HH) jets. The current consensus holds that these jets are launched and collimated via MHD forces. Observations have, however, shown that the jets can be mildly to weakly ionized. Beginning with a simple model for cylindrical equilibrium between neutral, plasma, and magnetic pressures, we calculate the characteristic timescale for ambipolar di†usion. Our results show that a signiÐcant fraction of HH jets will have ambipolar di†usion timescales equivalent to, or less than, the dynamical timescales. This implies that MHD equilibria established at the base of an HH jet may not be maintained as the jet propagates far from its source. For typical jet parameters one Ðnds that the length scale at which ambipolar di†usion should become signiÐcant corresponds to the typical size of large (parsec) scale jets. We discuss the signiÐcance of these results for the issue of magnetic Ðelds in parsec-scale jets. Subject headings : ISM : jets and outÑows È MHD È stars : preÈmain-sequence 1.
INTRODUCTION
occur via di†usive processes. Large-format CCD mosaics have recently demonstrated that YSO jets can extend over multiparsec length scales (L B 3 pc). The dynamical age j in the range t \ 104È105 (t \ L /V ) for these jets falls dyn j j yr (Reipurth et al. 1997 ; Eisloffel & Mundtdyn 1997). Thus, even if di†usive processes are slow, they will have a relatively long time to a†ect the propagation of the beams. In plasmas with less than full ionization, ambipolar di†usion is one of the most e†ective means of driving the rearrangement of Ñux relative to the plasma. Observations of optical Herbig-Haro (HH) jets have shown them to be moderately ionized, with ionization fractions ranging from x \ 0.5È 0.01 (Hartigan, Morse, & Raymond 1995 ; Baccioti & Eisloffel 1999), where the ionization fraction is deÐned as x \ n /n , and n and n are the number densities of neutral i n n i and ionized particles, respectively. The ionization fractions are seen to fall as the plasma propagates away from internal shocks and may, therefore, fall to even lower values in the optically invisible regions of the beam. Given the potentially low ionization fraction and the extreme ratio of jet radius to length, R /L \ 10~3, it may be possible for ambij signiÐcantly the MHD equilibria polar di†usion to j alter established when the jet was launched. In this paper we wish to address the issue of ambipolar di†usion in YSO jets. Our present goal is simply to establish the characteristic timescales and length scales for ambipolar di†usion to operate with respect to the fundamental jet parameters. We leave its consequences to later papers. In ° 2 we establish the conditions for an MHD equilibrium in a simpliÐed model of a jet in terms of plasma, neutral, and magnetic pressures. In ° 3 we consider the equation for ion-neutral drift in this context, and in ° 4 we present an equation for the ambipolar di†usion timescale in YSO jets. In ° 5 we discuss our results in light of observations and other models.
Narrow hypersonic jets are a ubiquitous phenomenon associated with star formation. In spite of the large database of multiwavelength observations and numerous theoretical studies, a number of fundamental questions remain unanswered about these jets. Paramount among the outstanding issues is the role of magnetic Ðelds in the launching, collimation, and propagation of young stellar object (YSO) jets. The current consensus holds that the jets are formed on small scales (L \ 10 AU) via magnetocentrifugal processes associated with accretion disks (Burrows et al. 1996 ; Shu et al. 1994 ; Ouyed & Pudritz 1997). At larger scales, where jet propagation rather than collimation is the issue, there is an implicit assumption that the magnetic Ðelds involved in the launching process will remain embedded in the jets as they traverse the intercloud medium. Observations of narrowly collimated jets, or chains of bow shocks, extending out to parsec-scale distances (Reipurth, Bally, & Devine 1997) have strengthened the viewpoint that the jet beams must ““ carry their own collimators.ÏÏ This is based on the possibility that without dynamically signiÐcant magnetic Ðelds conÐning the beam, the jets may be disrupted by hydrodynamical instabilities (Stone, Xu, & Hardee 1997 ; Hardee, Clarke, & Rosen 1997). Do the magnetic Ðelds remain embedded in the jets ? That is the question we address in this paper. If we accept that jets are created via MHD forces, then the only way to lose the imposed Ðelds is via reconnection or through di†usive processes. The helical topology expected for magnetocentrifugally launched jets would not be likely to lead to large-scale reconnection throughout the beam (though reconnection may have important e†ects on the radiative properties of the jets ; Gardiner et al. 1999). Thus, if magnetic Ðelds can be cleared out of a jet, this is more likely to
2. 1 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171. 2 Department of Physics, QueenÏs University, Kingston K7L 3N6, Ontario, Canada. 3 Department of Mechanical Engineering, Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14627-0171.
947
INITIAL CONFIGURATION
We begin by considering the simplest model of a magnetized jet. We imagine a cylindrical column of material of radius R . The column is composed of ions, electrons, and j neutral species and is initially held together internally by a radial balance of pressure (P , P , P ) and magnetic (J Â B) i e n
948
FRANK ET AL.
forces. We further assume that the column is in force balance with a (possibly magnetized) ambient medium. Thus, initially, the jet does not expand laterally (i.e., in the radial direction). To simplify our calculation we assume that the temperature of all species is the same (T \ T \ T ). i e n We further assume that the magnetic Ðeld in the jet is purely poloidal, B \ B (r)eü . Figure 1 shows a schematic of our z z model for the magnetized jet. 2.1. Global Force Balance Our calculation is similar to derivations of the ambipolar di†usion timescale in collapsing molecular clouds (a description of those models can be found in Spitzer 1978 and Mouschovias 1991). Molecular clouds are cold, gravitationally bound systems of extremely low ionization (x B 10~6). This implies that terms involving gas pressure and ion density can be dropped from consideration when calculating the e†ects of ambipolar di†usion. Gravity does not, however, play a role in maintaining the cross-sectional properties of a YSO jet, and pressure forces cannot be ignored. In addition, given the range of ionization fractions in these systems, the ion inertia should not, a priori, be dropped from calculations. In what follows we derive a simple expression for the ambipolar di†usion timescale (based on the ion-neutral drift velocity) including the e†ects of gas and plasma pressure as well as the ion density. We begin with the three Ñuid force balance equations for electrons (e), ions (i), and neutrals (n), which can be written
Vol. 524
as
A A
(1)
¿ d¿ o i \ [$P ] n Ze E ] i  B ] F ] F , ie in i dt i i c
(2)
d¿ n \ [$P ] F ] F , o n dt n ne ni
(3)
where the operator d/dt represents the convective derivative, and F represents the drag force imparted on species 12 1 by colliding with species 2. Note that F \ [F . 12 21 In what follows we will ignore the electron inertia term. This is valid when the timescales of interest are long compared with the response time of the electrons. Formally this means that the dynamical time t is longer than either the dyn electron cyclotron period 2n/) or the electron plasma ec period 2n/u (Elliott 1993), as is the case in YSO jets with ep Ðelds greater than the microgauss level. We will also neglect the electron collisional coupling to the neutrals, F , given en the di†erence in masses, M > M , M (see footnote 4 e i n of Mouschovias 1996 ; G. Ciolek 1999, private communication). With these caveats the addition of equations (1) and (2) yields the following expression : d¿ o i \ [$(P ] P ) ] (n Ze [ n e)E i dt i e i e ]
B
B B
d¿ ¿ e \ [$P [ n e E ] e  B ] F ] F , o e dt e e ei en c
A
B
n Ze¿ [ n e¿ i i e eÂB ]F . in c
(4)
Assuming charge neutrality (n Z \ n ) yields the following e form for the current density, i
Rj
J \ n Ze¿ [ n e¿ \ n e(¿ [ ¿ ) , (5) i i e e e i e and the term with the electric Ðeld drops out of equation (4). Making use of AmpereÏs law, we can decompose the Lorentz force into two terms, 1 B2 B2 (J Â B) \ [$ ] nü , (6) M 8n 4nR c c where nü is a unit vector directed toward the local center of curvature of the Ðeld line, R is the local radius of curvature, c to Ðeld lines. and $ is the gradient normal M Our choice of B \ B (r)eü implies that the second term in equation (6) is zero. Wez arezleft, therefore, with only the Ðrst term, which can be identiÐed as the gradient of magnetic pressure, P \ B2/8n. Finally, ifB we add the momentum equations for all three species, again ignoring the electron inertia, we arrive at the following equation : d¿ d¿ n \ [$(P ] P ] P ] P ) . o i]o i dt n dt i e n B
(7)
In what follows we need only consider the radial component of the force equation. Thus, if FIG. 1.ÈCartoon of jet model. Shown is a section of a magnetized jet composed of a plasma, neutral gas, and magnetic Ðeld oriented along the axis of the jet. The pressure and Ðeld distributions are a function of cylindrical radius only.
L (P ] P ] P ] P ) \ 0 , e n B Lr i
(8)
and v \ v \ v \ 0 (where the v values refer to radial i,r atn,rt \ 0, e,r then the entire conÐguration r velocities) begins in
No. 2, 1999
AMBIPOLAR DIFFUSION IN YSO JETS
equilibrium. It is, however, what we shall call a quasiequilibrium, since each species will be accelerated separately by the terms on the right-hand sides of equations (1)È(3). The plasma and the neutrals will attempt to move past each other as each responds to its own forces. It is only the collisional drag F between species that holds the initial ij conÐguration together. This is exactly the situation that occurs in a molecular cloud where the cloud can be supported for some period of time by its magnetic Ðeld. As in the molecular cloud case, the quasi-equilibrium in the jet cannot persist indeÐnitely, and the initial conÐguration will change on a timescale t \ R /v , where v is the ionad j d d neutral drift velocity v \ v [ v . We seek to calculate this d n i timescale. Note that equation (8) implies that the pressure distributions of the ions and neutrals are not independent : L L P \ [ (P ] P ] P ) . e B Lr n Lr i
3.
THE ION-NEUTRAL DRIFT
If we divide equation (4) by o and subtract from it equai tion (3) (itself divided by o ), we Ðnd n d¿ d¿ 1 i [ n \ [ $(P ] P ] P ) i e B dt dt o i o ]o 1 n i . (10) ] $P ] F n in o o o n i n If we now assume that the collisions will quickly bring the ion-neutral drift to a steady velocity, i.e., the relative accelerations approach zero after a few collisions, we have the following radial balance equation between collisions and hydromagnetic forces :
A
A
B
B
o ]o 1 L 1 L n i \ (P ] P ] P ) [ P . (11) e B in o o o Lr i o Lr n n i i n Substituting equation (9) into the above expression, we Ðnd that the densities cancel, leading to F
L F \ (P ] P ] P ) . in Lr i e B
where SpwT B 3 ] 10~9 cm3 s~1 is the average collision in rate (Spitzer 1978). We can now solve for the ion-neutral drift velocity :
(12)
We now make the approximation that the scale of the gradients in the jet is such that LP P kB k , (13) Lr R j where P is the characteristic scale for the pressure of k the kth component of the conÐguration. We also use the deÐnition of the plasma b parameter, b \ P /P , where g B P \ P ] P . Thus g i e P B2 F \ B (b ] 1) \ (b ] 1) . (14) in R 8nR j j To reduce this equation further we must consider the form of the collisional force. F can be written as in SpwT in (¿ [ ¿ ) , F \o o (15) in i n M ]M n i i n
A
B
M ]M 1 B2 i n (b ] 1) . (16) 8n SpwT o o R in i n j Thus the timescale for changes in the initial conÐguration can be written as v \ d
A B
n R 2 8nSpwT M M in n i x n j (b ] 1)~1 , (17) B M ]M i n where we have used x \ n /n . The above expression gives i n the ambipolar di†usion timescale in terms of the fundamental parameters in the jet (x, n , R , and B). n j t \ ad
4.
(9)
We will use this relation below in our calculation of t . ad
949
AMBIPOLAR DIFFUSION IN YSO JETS
Of the four variables in equation (17), only the Ðrst three (x, n , R ) are well characterized for HH jets. From obsern j typical values are 0.01 ¹ x ¹ 0.1 ; 103 ¹ vations, n /cm~3 ¹ 104 ; 1 ] 1015 ¹ R /cm ¹ 5 ] 1015 (Baccioti & n ffel 1999). Magnetic Ðelds,j however, are not so easily Eiso categorized. To date there have been only a handful of measurements of Ðelds in YSO jets (Ray et al. 1997). Thus, it would be better to cast equation (17) in terms of a parameter such as the temperature that has been measured in many YSO jets. Typical values in optical jets are 5 ] 103 K \ T \ 3 ] 104 K (Baccioti & Eisloffel 1999). To express equation (17) in terms of T , we use our assumption that the electron and ion temperatures are the same and, furthermore, assume that the plasma is mainly composed of hydrogen atoms. Thus M \M , i n n \n , i e P \ n kT \ n kT \ P . e e i i The magnetic Ðeld can now be expressed as
(18) (19) (20)
P ]P 1 i \ 16nn kT , B2 \ 8n e i b b
(21)
and the ambipolar di†usion timescale becomes
A BA B
b M SpwT n R2 n in n j . (22) T b]1 4k j Upon normalization the ambipolar di†usion timescale can be written as t \ ad
A
n n t \ 28,904 ad 103 cm~3 Q(b) \
A B
b . b]1
BA
BA
B
R 2 104 K j Q(b) yr, (23) 1015 cm T j (24)
Note that Q(b) is positive with an asymptotic value of 1 as b ] O. This shows that the ion-neutral drift can be driven by gas pressure forces alone if the individual speciesÏ pressure gradients have di†erent signs, an artiÐcial situation not likely to be encountered in real jets. Detailed calculations of equilibrium magnetically conÐned jets arising from accretion
950
FRANK ET AL. 10
10
10
3
2
1
β 10
10
10
0
−1
−2
0
0.5
1
FIG. 2.ÈPlasma b vs. jet radius for MHD equilibrium jet. Jet model taken from calculations of Lery et al. 1999. Note that the lowest values of b occur within the jet.
disks indicate that b can become quite low in some regions of the jet. As an example, we provide in Figure 2 a plot of b versus radius for an equilibrium MHD jet. This model was calculated via the Given Inner Geometry method of Lery et Beta = 1.
6 5.5 5
Log T_ad y
4.5
0.5
2 1
4 6
Density 10^3 cc
1.5 8
2 10
FIG. 3.ÈAmbipolar di†usion timescale as a function of temperature (in units of 104 K) and density (in units of 103 cm~3) when b \ 1. The jet radius is 2 ] 1015 cm.
Beta = .1
5 5 Log T_ad y 4.5 4 0.5
2 1
4 6
Temp 10^4 K
al. (1998, 1999) for determining the asymptotic structure of magnetocentrifugally launched Ñows. Figure 2 shows that b can drop to values below b \ 1 in the inner regions of the jet (r \ R ). j In Figures 3 and 4 we show surface plots of t vs. b for ad b \ 1 and b \ 0.1. From equations (23) and (24) and these Ðgures it is clear that depending on the conditions in the jets, t can become as large as 106 yr and as small as 103 yr. ad For jet parameters in the middle of the expected range of variation we Ðnd t of order 105È104 yr. What is noteworthy about theseadresults is that a signiÐcant region of parameter space exists in which the dynamical timescale for YSO jets is of the order of, or greater than, the ambipolar di†usion time : t [ t . T hus ambipolar di†usion is likely dyn ad to play a role in the dynamics of large-scale Y SO jets and outÑows. We discuss the consequences of this conclusion in the next section. 5.
relative radius
Temp 10^4 K
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Density 10^3 cc
1.5 8
2 10
FIG. 4.ÈAmbipolar di†usion timescale as a function of temperature (in units of 104 K) and density (in units of 103 cm~3) when b \ 0.1. The jet radius is 2 ] 1015 cm.
DISCUSSION AND CONCLUSIONS
Our conclusion that ambipolar di†usion timescales can be comparable to jet dynamical lifetimes raises a number of intriguing issues. The Ðrst is the most obvious. Ambipolar di†usion will rearrange the mass-to-Ñux ratio in the jet and alter any initial equilibrium between the plasma, neutrals, and magnetic Ðeld. Our analysis is too simpliÐed to yield conclusions about how the jet will evolve in the presence of ambipolar di†usion. In our analysis we did not consider the e†ect of toroidal Ðelds. The hoop stresses associated with toroidal Ðelds will pull the ions toward the jet axis. Thus, depending on the orientation of the magnetic pressure gradients, magnetic forces (pressure and tension) can either compete or apply forces in the same direction. Including a toroidal component does not, in general, change the order of magnitude of the ambipolar di†usion timescale ; however, it can change the direction in which the ions are pushed. The plasma and magnetic Ðeld may bleed out of the jet leaving only the neutrals, or, conversely, strong hoop stresses could draw the plasma and Ðeld in toward the jet axis. In both cases, however, the neutrals will be free to contract or expand depending on their pressure distribution relative to the ambient medium. Thus one expects a potentially signiÐcant rearrangement of the jetÏs cross-sectional properties when ambipolar and jet timescales are comparable. The consequences of ambipolar di†usion on long-term jet propagation should, therefore, be investigated in detail. The possibility that many jets will have t B t is suggestive for the general issue of YSO outÑows.dyn Thereadremains considerable debate over the connection between HH jets and molecular outÑows (Masson & Chernin 1993 ; Cabrit 1997). The discovery of parsec-scale ““ superjets ÏÏ (Reipurth et al. 1997 ; Eisloffel & Mundt 1997) has strengthened the case for jets as the source of molecular outÑows by extending jet lifetimes. Still, most outÑows have larger dynamical timescales than even the superjets by a factor of at least a few. One may wonder then why the jets have shorter lifetimes than the molecular outÑows. Our results suggest that in some systems at least it is ambipolar di†usion that sets the lifetime of the visible collimated jet. Converting to consideration of length scales, if one takes the minimum ambipolar di†usion time from our calculation, t B 103 yr, and the minimum characteristic jet velocity, V adB 102 km s~1, j one Ðnds the minimum distance in which ambipolar di†usion becomes e†ective : D B 0.3 pc. Our results indicate ad that ambipolar di†usion should not be e†ective before a jet
No. 2, 1999
AMBIPOLAR DIFFUSION IN YSO JETS
reaches this distance. Moreover, for typical values t B 104 ad yr and V B 3 ] 102 km s~1 one Ðnds D B 3 pc. It is j ad indeed noteworthy that this value corresponds well with the distance at which the longest jets start to fade away. When t becomes comparable to t the beam may no longer be dyn ad conÐned by magnetic forces.
951
We wish to thank Glenn Ciolek, Larry Helfer, and Al Simon for their help with this project. We also acknowledge the helpful comments of Tom Ray, Francesca Bacciotti, and Alex Raga. This work was supported by NSF grant AST 97-8765
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