Jet Collimation and Acceleration

A relativistic jet is not fully characterized by the power injected at its base. Magnetic stress must collimate the flow, external pressure must supply or fail to supply transverse confinement, and Poynting flux must be converted into bulk kinetic energy without losing lateral causal contact too early. In cold ideal MHD these processes are linked by the conserved energy per unit rest-mass flux, $\mu=\Gamma(1+\sigma)$. Acceleration lowers the magnetization $\sigma$ as the Lorentz factor $\Gamma$ rises, but the rate and terminal value depend on field-line shape, mass loading, pressure profile, and dissipation. A measured jet outline is therefore a boundary condition on a model, not a direct measurement of $\Gamma$ or of the launch mechanism.

Required background. Classical Field Theory supplies electromagnetic stress-energy and ideal flux freezing.

Helpful background. Black-Hole Geometry and Mechanics supplies the rotating compact-object geometry at the inlet; Relativistic Matter and Compact Objects supplies relativistic thermodynamics and inertia; and Fluid and Kinetic Theory supplies relativistic MHD, shocks, and causal characteristics.

Energy conversion along a magnetic surface

Consider a stationary, axisymmetric, cold ideal-MHD outflow far enough from the hole that gravitational redshift is negligible. Each magnetic surface carries conserved field-line angular velocity $\Omega_F(\Psi)$, mass flux per unit magnetic flux $\eta(\Psi)$, angular momentum per unit rest-mass flux, and total energy per unit rest-mass flux $\mu(\Psi)$. Splitting that energy between matter, including rest mass, and electromagnetic transport gives

$$\mu=\Gamma(1+\sigma), \qquad \sigma=\frac{P_{\rm EM}}{P_{\rm matter}}.$$

This relation is a diagnostic identity for the declared cold, lossless streamline. It implies $\Gamma\le\mu$, with equality only after nearly complete conversion, $\sigma\to0$. If the inlet has $Γ_{\rm in}\simeq1$ and $\sigma_{\rm in}\gg1$, then $\mu\simeq\sigma_{\rm in}$. Mass loading lowers the maximum attainable Lorentz factor even at fixed electromagnetic power.

Why does geometry matter? Beyond the light cylinder $R_L=1/\Omega_F$, poloidal field lines act as a magnetic nozzle. Collimation changes the spacing of neighboring flux surfaces and allows toroidal magnetic pressure to do work on the plasma. In an equilibrium-collimated, Poynting-dominated region a useful asymptotic relation is

$$\Gamma\sim\frac{R}{R_L},$$

where $R(z)$ is cylindrical jet radius. The ideal-MHD integrals and asymptotic transfield equation are derived in Lyubarsky 2009, §§2–5. The scaling is not a kinematic prescription valid through shocks or current sheets: it follows from a particular steady, cold, axisymmetric, causally connected regime.

External confinement and two asymptotic regimes

Write the confining pressure as

$$p_{\rm ext}(z)=p_0\left(\frac{z}{z_0}\right)^{-\kappa}.$$

For $κ<2$, pressure decreases slowly enough that a magnetically dominated jet can remain in transverse equilibrium. Balancing the boundary magnetic pressure against $p_{\rm ext}$ gives the asymptotic scalings

$$R(z)\propto z^{\kappa/4}, \qquad \Gamma(z)\propto R(z),$$

until finite $\mu$, dissipation, or a pressure transition intervenes. For $κ>2$, confinement declines too quickly to maintain the same cylindrical equilibrium; the jet approaches a conical, non-equilibrium regime and ideal acceleration is generally less efficient. The $κ=2$ case is marginal and depends on pressure normalization, so assigning it to one branch without that normalization is incomplete. These regimes and their distinct acceleration laws are developed in Lyubarsky 2009, §§6.1–6.4, with terminal Lorentz factors and opening angles in §§7.1–7.2.

Collimation is not produced by toroidal “hoop stress” in isolation. In a relativistic flow, the electric force nearly cancels much of the toroidal magnetic pinch. The residual transverse force, poloidal-field pressure, inertia, and external pressure must be balanced together. This is why importing a nonrelativistic pinch argument can predict acceleration and shape incorrectly.

Lateral causal contact supplies an independent check. If $\theta_j=dR/dz$ is the local opening angle and $\theta_M$ the fast-magnetosonic Mach-cone angle, a signal can cross the jet only if $\theta_j\lesssim\theta_M$. In a cold magnetized flow, a useful estimate is

$$\Gamma\theta_j\lesssim\sqrt{\sigma},$$

which reduces to the familiar order-unity $\Gamma\theta_j$ condition near matter domination. Once the inequality fails, opposite sides cannot continually establish transverse equilibrium. Recollimation shocks or a change in the ambient pressure can restore local communication, but then the smooth ideal asymptotics no longer describe the whole flow.

Observed jet shapes provide valuable tests only through a forward model. The parabolic-to-conical profile of M87 is a prominent example, yet core shift, opacity, deprojection, time variability, and the distinction between emitting sheath and dynamical boundary all enter. Likewise, a moving brightness knot measures a pattern speed; it need not equal the bulk plasma Lorentz factor. The observational and dynamical qualifications are reviewed in Blandford, Meier, and Readhead 2019, §§2.2.3.4–2.2.3.5 and §3.3.

Executable equilibrium-scaling benchmark

The simplest computational check integrates the equilibrium similarity law rather than assuming its answer in a plotted array. Put $x=\ln(z/z_0)$ and $y=R/R_0$. For fixed $κ<2$,

$$\frac{dy}{dx}=q y, \qquad q=\frac{\kappa}{4}, \qquad y(0)=1,$$

whose exact solution is $y=e^{qx}$. In the equilibrium benchmark set $\Gamma=y$, $\mu=20$, and recover $\sigma=\mu/\Gamma-1$ from energy conservation. The pinned environment used below is CPython 3.9.6 with NumPy 2.0.2. The numerical method is the explicit midpoint algorithm, which should converge at second order.

import numpy as np

kappa, x_end, mu = 1.5, 4.0, 20.0
q = kappa/4

def integrate(n):
    h = x_end/n
    gamma = 1.0
    max_energy_residual = 0.0
    for _ in range(n):
        k1 = q*gamma
        k2 = q*(gamma + 0.5*h*k1)
        gamma += h*k2
        sigma = mu/gamma - 1
        residual = abs(gamma*(1 + sigma) - mu)
        max_energy_residual = max(max_energy_residual, residual)
    exact = np.exp(q*x_end)
    return gamma, abs(gamma-exact), max_energy_residual, sigma

for n in (32, 64, 128):
    print(n, *integrate(n))

Grid $N$	Lorentz-factor result	Absolute error	Maximum energy residual	Final $\sigma$
32	4.4793128	$2.3763\times10^{-3}$	$3.55\times10^{-15}$	3.4649706
64	4.4810844	$6.0471\times10^{-4}$	$3.55\times10^{-15}$	3.4632054
128	4.4815366	$1.5252\times10^{-4}$	$3.55\times10^{-15}$	3.4627551

The error decreases by nearly four on each refinement, as expected. The energy residual is only an algebraic consistency check because $\sigma$ was defined from $\mu/\Gamma-1$; its smallness is not an independent test of energy conservation by an MHD solver. The script produces one result row per resolution. To rerun, save the executable block as parabolic_jet_benchmark.py and execute:

python3.9 -m venv .venv
.venv/bin/python -m pip install numpy==2.0.2
.venv/bin/python parabolic_jet_benchmark.py

This is an executable teaching check of one asymptotic scaling, not a claimed production simulation of jet launching or propagation. It prescribes the pressure exponent and equilibrium branch; it does not solve the transfield equation, determine mass loading, form shocks, reconnect fields, or fit an observed width profile. A production relativistic-MHD calculation must pin its code commit, injection surface and field geometry, $\mu$ and initial $\sigma$, mass loading, $p_{\rm ext}(z)$ and ambient density, dissipation closure, domain, boundaries, grid, and floor prescriptions. Resolution must be increased both across and along the jet, boundaries moved, and energy partition, shape, $\Gamma$, $\sigma$, and causal connectivity shown stable.

Failure modes and adversarial checks

Two tests expose common overinterpretations. First, repeat the calculation after changing $p_0$, $κ$, and the outer boundary. If a fitted shape is insensitive even when the solution should cross from equilibrium to non-equilibrium collimation, the ambient medium is not dynamically coupled as claimed. Second, vary density floors and inlet mass loading. Apparent acceleration caused by numerical removal of inertia is not conversion of resolved Poynting flux.

Axisymmetry can suppress kink-like modes and exaggerate long-lived ordered structure. Finite domains can mistake an intermediate power law for a terminal cone. Unresolved current sheets convert magnetic energy at a grid-dependent rate, while an ideal closure cannot predict nonthermal particles at all. Finally, agreement with $R(z)$ alone is underdetermined: pressure, magnetization, viewing angle, emitting layer, and pattern speed can trade off. A credible result closes the conserved energy flux and causal diagnostics in addition to matching morphology.

Exercises

Energy conversion and causal contact. A cold streamline has $\mu=20$. At one location $\Gamma=5$, $\sigma=3$, and the jet half-opening angle is $0.10$ rad. Check energy conservation and the magnetized causal-contact estimate. If the flow reaches $\Gamma=10$ without losses, find its new $\sigma$.

Solution

The energy check gives $\Gamma(1+\sigma)=5(4)=20=\mu$. The left side of the causal estimate is $\Gamma\theta_j=0.5$, while $\sqrt{\sigma}=\sqrt3\simeq1.73$, so lateral fast-mode contact is allowed by this order-of-magnitude criterion. At $\Gamma=10$, conservation requires

$$\sigma=\frac{\mu}{\Gamma}-1=1.$$

Half of the total energy per unit rest mass is then in matter and half in electromagnetic form under the cold-flow definition. This does not determine whether dissipation, rather than ideal acceleration, produced the change.

References

• Blandford, Roger, David Meier, and Anthony Readhead. “Relativistic Jets from Active Galactic Nuclei.” Annual Review of Astronomy and Astrophysics 57 (2019): 467–509. DOI. Open preprint.
• Lyubarsky, Yuri. “Asymptotic Structure of Poynting-Dominated Jets.” The Astrophysical Journal 698 (2009): 1570–1589. DOI. Open preprint.