The Galactic Center Companion, page 13
All these observations suggest a model for the entire region invoking electrodynamical coupling and formation of large current paths. The Arc, Arch, threads, and filaments remind one powerfully of plasma discharges in the laboratory which are heated by currents and are shaped by both external and self-generated magnetic fields. Given the evidence for a strong poloidal field within 50 pc of Galactic center (Yusef-Zadeh and Morris 1987), it seems natural to suppose currents run along these ordered fields, illuminating individual paths by synchrotron emission in the filaments and by thermal radiation in the regions nearer the large molecular clouds. For convenience let us term all such structures “strands,” implying that they may have deeper, smaller structure, but all are woven by similar effects.
There is further suggestive evidence. The gas temperature inferred from radio recombination lines is ~ 1 eV, but ionization in some regions requires 10-100 eV. This implies a wide range of local temperatures or else some local ionizing agency, such as current flow. Polarization measurements show that all filaments thus far studied are aligned along the field. Along with evidence of helical structure in some southern filaments, these morphologies suggest strong magnetic ordering of the energy release. (Heyvaerts, Norman, and Pudnitz 1988).
The low-energy jet emanating from the Galactic nucleus (Yusef-Zadeh et al. 1986) suggests that relativistic particle acceleration and ordered flows are common near Galactic center (Jacobson 1982). The slim “threads” (Morris and Yusef-Zadeh 1985) speak for extremely localized heating which is naturally explained by field lines which carry current but may have been separated from the filament families by the nearby intervening molecular clouds. With these hints, we attempt here a discussion in terms of an overall circuit equation for the region.
II. Basic Circuit
Consider an electrodynamic plasma region in which currents J flow, electrostatic potentials ϕ and vector potentials A exist, with resistivity η and electron pressure Pe.
The circuit equation for electrodynamically connected regions is
where we neglect electron inertia and write
The integrations here are performed over any path in the plasma. The left side of equation (1) is the applied voltage V, and the terms on the right side are, respectively, the resistive, capacitive, and inductive voltage drops. The quantity E0 contains both solenoidal and electrostatic terms, but a small displacement of electrons carrying negligible currents will make the capacitive part in equation (1) cancel the electrostatic term, which is proportional to the gradient of electron pressure, ∇Pe.
The circuit equation then is
Local conditions contribute to the integrals, and evolution of the circuit depends also on global aspects of the current. We can write
Here the inductance Li is that of a loop of length L and cross-sectional area A = πa2. The capacitance C is that of N current paths of length Ld (
Fig. 1.—Schematic of a circuit with currents driven along ordered magnetic field lines by induction from a partially ionized molecular cloud. Lightly shaded irregular clouds are thermal radio emitters. Synchrotron-emitting linear filaments are ~ 1 lt-yr wide. A molecular cloud with velocity r drives the circuit current I with inductive field E. Local ordered magnetic field lines are deformed by the cloud. Galactic center is at position X. We presume all features are roughly in the plane of the paper.
The basic picture appears in Figure 1. A giant molecular cloud moving opposite to the Galactic rotation can induce powerful electric fields by v xB motion throughout the partially ionized cloud. The plasma component of the cloud can push the poloidal field B out of the way, locally deforming it. Electric fields will drive currents which can become attached to neighboring, straighter field lines, through scattering of the electrons by current-induced turbulence. Once on the ordered field lines, the currents proceed around a large circuit which leads through the Galactic plane. Conditions in the plane are unclear, so we cannot judge whether ionization is sufficient there to return the currents through the plane to field lines lying closer to Galactic center, and thence back up into the Arch. If not, the currents may proceed to negative Galactic latitudes, forming the somewhat weaker filamentary structures there. Presumably the circuit eventually closes by turning near the polarized lobes at the ends of the filaments and moving back toward Galactic center. The circuit probably passes through the Arch. Figure 1 illustrates this last possibility. Along a given current path there may be several molecular clouds which induce electric fields, i.e., several “batteries.”
Strong electric fields induced by v x B motions can explain some of these features. The induced field is
where β is v/c, v is the velocity of mass motion, B-3 = (B/10-3 G), and ψ is the mean angle between v and B. In a complex plasma-filled environment, laboratory experiments and nonlinear theory can be a useful guide. A plasma moving across a strong magnetic field immediately experiences the induced E produced by the plasma motion. The plasma edges experience charge separation, and small “capacitors” form transverse to the plasma velocity (Peter and Rostoker 1982). Using equation (6), we require that the energy lodged in the electric field across a width h be less than the incident plasma kinetic energy eEh ≈ mv2/2. This implies that h < ρiv/2vi, with ρi the ion cyclotron radius and vi the ion thermal velocity. Since v ~ 10vi, we expect the cloud to shred into filaments wherein electric fields comparable to equation (6) occur, driving charge separation sheaths (Schmidt 1966). The plasma cloud slows very slightly as it breaks into separate regions of strong E fields. There is some evidence in numerical simulations of structures larger than h (Mitchell et al. 1985). In any case, ρi ≈ 105 cm is very small compared with other dimensions, so the setting up of the “battery” is microscopic in the larger circuit picture. The circuit forms because nearby plasma will immediately move to compensate the charge separations which form from the induced E. Although the induced E is perpendicular to B, currents will flow along B to neutralize charge. A complex pattern of resulting electric fields on larger scales drives the overall circuit, as is seen in experiment (Wessel and Robertson 1981). Note, though, that h approximately less than 10ρi can be comparable with the Bennett pinch radius we discuss shortly. The induced field will occur where the external magnetic field begins to diffuse into the molecular cloud. This process is usually very slow if only classical Coulomb scattering occurs, but there is an added effect from the strong magnetic field. The appropriate conductivity is reduced because of the pinning of electrons to the field lines, and this enhances inward diffusion of the magnetic field. A rough estimate is (Felber et al. 1982)
where x is the distance diffused, and the ion temperature Ti and the electron temperature Te are in eV. This implies that considerable mixing of the ambient field into the cloud can occur during one Keplerian orbit about the Galactic center. The induced electric fields will be strongest at the edges of such clouds, where shredding and magnetic diffusion are going on. We should therefore expect to see circuits set up preferentially at the margins of molecular clouds which are being heated by their soft “collisions” with the strong, ordered magnetic field. This fits well the observations of long, luminous features near clouds. In particular, the Sgr C linear filament at negative latitudes (Liszt 1985) and two parallel filaments at positive latitudes (Bally and Yusef-Zadeh 1988) appear at cloud edges. The entire “Continuum Arc” complex seems to lie at the edges of a 40 km s-1 Sgr A molecular cloud. Cloud heating is also indirect evidence of the strong coupling. The Keplerian velocity at distance r from an enclosed mass M is
and large molecular clouds with twice this speed appear near some linear filaments. A field of B-3 ≈ 1 would provide rough equilibrium with the ram pressure of thermal gas in the area, which moves at a few tens of km s-1 and has density n ≈ 400 cm-3. Only slight bending of a filament interacting with the “sickle” (Yusef-Zadeh and Morris 1987) by such gas implies B-3 approximately greater than 1. There will be a steady electron drift in the mean field of equation (6),
with the appropriate collision frequency v set by the level of electrostatic turbulence, Ee, described by W≡
We do not need outright, constant instability to produce a low level of electrostatic turbulence, since in a steady state levels with W<<1 can persist and still dominate over classical resistivity, which for our conditions would yield v≈10-8ωp. We shall take a form suggested by extensive simulation and theory (Boris et al. 1970; Schrijver 1973; Papadopoulos 1977)
where W includes only Ee(ω/k
Further, we find, with the ion temperature Ti in units of eV,
The first condition insures that drifting electrons cannot resonate with the Alfvén velocity, vA, and suffer severe pitch angle scattering. The second implies that vD > vti the ion thermal speed, can excite ion instabilities in regions where Ti (in eV) ~ 1, n ~ cm-3, and the other quantities are of order unity. The condition vD > 10vi for the ion cyclotron instability thus occurs if W-2 is approximately less than 0.3. Low levels of turbulence, W ~ 10-5, thus yield high vD, self-consistently producing the resultant W. How the plasma strikes a steady state, fixing W, depends on macroscopic losses as well. We cannot plausibly include such considerations, given our level of ignorance of conditions at Galactic center, and so leave W an open parameter.
III. ENERGETICS
A self-constricted pinch achieves a balance between inner thermal pressure and the Bθ2/8π confining pressure. The radius of a cylindrical pinch is then fixed by specifying the net drift speed and the plasma pressure to be confined (Krall and Trivelpiece 1973; Spitzer 1962). This yields
Here n(0) is the plasma density on axis, where the current density is n(0)evD. Using equation (10) for vD yields
This is a very small object for astrophysical conditions, and it suggests immediately that any self-pinched discharges will be discernible only if they congregate, contributing to a larger structure. The narrowness of the pinch means its resistance is large, as equation (5) yields
With many such pinch paths available, the voltage derived from equation (1) will doubtless flow through a large number of small pinches in parallel. The driving voltage is
The total resistance of N pinches in parallel is Rt = R/N, and total current flowing is
while the magnetic field seen near any small pinch, Be, is still small,
Note that Be < B ~ 10-3 G, which is necessary for stable, long-lived pinches, as we shall discuss.
So far we have no idea how many small pinches comprise the ordered regions of illuminated strands. Equation (12) shows that a pinch supported by gas pressure with T ~ 1 is very small compared with the 0.3 pc observed width of the linear filaments. This means a filament is a congregation of self-pinched elements which may cooperate to form a larger scale (perhaps nearly force-free) equilibrium. We can confidently predict that higher resolution maps of the filaments will show finer, complex structure. To estimate how many pinches are in given features we can consider the circuit losses. The net energy dissipated in the total circuit is
The total observed thermal infrared luminosity ~5 x 1040 ergs s-1 comes principally from a large molecular cloud in which the inferred plasma density is ~500 cm-3 (Morris and Yusef-Zadeh 1988). (This estimate comes from assuming most of the molecular material is locally ionized.) To account for this wholly as ohmic dissipation requires
For the thermal emitters we estimate L= 10 pc, n = 100cm-3, T = l(eV), so with a ≈ ap we need NW = 1000. With turbulence levels of W = 10-3 this requires about a million current filaments.
Ohmic losses should occur where currents choose the highest available conductivity, which by equation (13) scales with n1/2. Luminosity should follow the visible density ridges, within the constraint that voltages occur at cloud edges, so circuits should include the cloud perimeter. Data of Yusef-Zadeh et al. (1987) seem to bear this out.
Nonthermal (synchrotron) emission observed in linear filaments is ~4 x 1033 ergs s-l, quite easily explained by only a few pinches.
Further, the voltage of equation (14) provides ample opportunity to accelerate electrons to the ~20 MeV needed to radiate the 5 GHz synchrotron emission in the 10-3 G field. If unscreened over a distance of ~108 cm, this voltage will provide the synchrotron electrons. This length is about a pinch radius when W = 0.01. Irregularities of flow on this scale are certainly plausible. The mean free path of a relativistic electron with collision frequency given by equation (9) is 5 x 107/W-2 n1/2 cm, so the scattering will not impede some electrons reaching MeV energies. Irregular conditions may arise naturally, as in double layers, which can produce substantial acceleration in concert with ohmic processes (Borovsky 1986). The few isolated “threads” are current paths, perhaps connected to other moving clouds. They could have been pulled away from the filaments by induction of passing conducting clouds, yet remain electrodynamically coupled to the governing potential.
IV. Stability
Thin, luminous strands suggest a self-pinched electrical flow, with the poloidal field Bz providing “backbone” while the flow confines itself with an ordered Bθ. Essentially, our envisioned circuit connects thermal gas regions which are at different potentials, so the linear luminous features are like lightning guided by the strong global fields.
Within a pinched structure, radial pressure P obeys
Where Bθ=2I/ca is the magnetic field at the surface of the flow, r = a, assuming j is independent of r for r < a and I = jπa2. For r > a, the “cocoon” region, external plasma can be held by the Bθ field lines (Benford 1981). The electron gyro-radius is
Here vG is the observed synchrotron frequency (in GHz) of the electron. This corresponds to a radiating electron which has a low pitch angle θ in the combined (Bz, Bθ) helical structure. The structure is liable to sidewise instabilities if Bθ is large, however.
A plasma flow in a helical magnetic field is unstable to pinching modes (sausage) if (Benford 1987)
where L is the structure length along z. It will kink if
We envision quite small pinches which have L>>a. Stability requires small Bθ, yet it must be large enough to enhance the luminosity of the core region, so Bθ cannot be very weak compared with Bz. To argue that pinches can survive kink instability for the dynamical time of the region (td ≈ 2 x 105 yr) we must include stabilizing effects. These are as follows:
1. The cocoon dragged by Bθ as it moves during kinking will slow growth to a time (Benford 1981).
where Mc(Mp) is the mass per unit length of the cocoon (pinched region). Here λ is the wavelength of the kink, which is comparable with the length of filaments. For large Mc/MP ~ 1000, long wavelengths may be suppressed for t≈ td. This requires a comoving cocoon of radius Rc ~ 30a, if densities in the two regions are comparable.
2. To suppress short-wavelength modes requires mirror forces on a relatively nearby surrounding, massive, conducting body at distance b. Then waves are stable with
This calls up our earlier picture of many small pinches, stabilized now by the inertia of surrounding clouds of plasma. These nearby “walls” by definition do not carry a net current (because of varying conditions of v x B) but can inhibit side-wise motions. The gross effect of many such small pinches can comprise a quasi-stable strand configuration of width 1 lt-yr.
However, this total “anthology” filament of pinches will then obey the same conditions as equations (20) and (21), with an effective radius a* equaling roughly the observed filament radius, ~1 lt-yr. Then, without a stabilizing “wall” nearby, equation (21) demands for L=100a*,
Presumably the synchrotron electrons are either brought into the pinches from other acceleration sites or are produced by local processes. The level of plasma turbulence provides easy, nonadiabatic scattering for fresh electrons to enter pinch configurations. The mean free path to scatter a relativistic electron of v0/c ≡ β0 ≈ 1 through one gyroradius is (Benford 1986).
