Bipolar Outflows and Herbig-Haro Objects

A forming star has a disposal problem before it has a luminosity problem. The gas spiralling toward it out of a collapsing cloud core carries far more angular momentum than a star can hold — spin it up with all of that infalling $j$ and it would fly apart before it finished assembling. Nature's remedy is not to slow the accretion but to throw part of it back out: as matter grinds inward through a disk, magnetic field lines threading the inner disk fling a fraction of the gas outward along the rotation axis, carrying off the excess angular momentum and letting the rest fall on. The exhaust is a pair of narrow, oppositely directed jets punching out of the poles of the star–disk system. These jets are ferociously supersonic — Mach numbers of tens relative to the ambient gas — so they cannot slide through the cloud quietly. They drive shocks: internally, where a faster pulse of ejecta rams slower gas ahead of it, and at the leading edge, where a bow shock ploughs the cloud. Each shock heats gas to $10^4$ – $10^6\ \mathrm{K}$ , and as that gas cools it radiates the optical and infrared lines we photograph. The bright shock-excited condensations near the source are the Herbig-Haro (HH) objects, catalogued by George Herbig and Guillermo Haro at the start of the 1950s. Farther out, the jet drags cold cloud gas into a slow, wide, swept-up molecular outflow, mapped in CO as twin blue- and red-shifted lobes straddling the star. This article builds the physics in three linked layers, held together by the conservation laws any correct model must obey: magnetocentrifugal launching and the angular-momentum budget of the jet; the radiative shock physics that turns kinetic energy into HH line emission; and the momentum-driven entrainment that inflates a molecular outflow and feeds back on the cloud.

Key takeaways

A protostar cannot accrete without shedding angular momentum; a magnetocentrifugally launched disk wind removes it, ejecting only $\dot M_{\rm jet}/\dot M_{\rm acc}\sim0.1$ of the accreted mass at speeds $v_{\rm jet}\sim100$ – $400\ \mathrm{km\,s^{-1}}\approx v_{\rm esc}$ .
Where the supersonic jet shocks the cloud, a strong $J$ -shock heats gas to $T_2=\tfrac{3}{16}(\mu m_H/k_B)v_s^2\approx1.4\times10^5\ \mathrm{K}$ at $v_s=100\ \mathrm{km\,s^{-1}}$ ; the cooling gas radiates $\mathrm{H\alpha}$ , $\mathrm{[S\,II]}$ , $\mathrm{[O\,I]}$ , and $\mathrm{[Fe\,II]}$ — a Herbig-Haro knot.
HH knots are thin bright sheets because the post-shock cooling length is short ( $\ell_{\rm cool}\sim10^{13}$ – $10^{15}\ \mathrm{cm}$ ); line ratios like $\mathrm{[S\,II]}/\mathrm{H\alpha}$ diagnose shock speed and preshock density.
The jet acts as a snowplough, sharing its momentum with a much larger entrained mass, so the CO outflow is slow and massive ( $v_{\rm out}\sim$ a few–tens of $\mathrm{km\,s^{-1}}$ , $M_{\rm out}v_{\rm out}\approx\dot M_{\rm jet}v_{\rm jet}\,t$ ) — momentum-driven, not radiation-driven, carrying hundreds of times more momentum than stellar photons could supply and capping core-to-star efficiency near $\sim30\%$ .
Jets are episodic; chains of knots record accretion outbursts, and proper motions ( $\sim0.1$ – $1''\,\mathrm{yr^{-1}}$ ) give distance-independent dynamical ages $t_{\rm dyn}=\theta/\mu_{\rm pm}\sim10^2$ – $10^4\ \mathrm{yr}$ .
Key idea: a protostellar jet is a magnetized wind that carries off the angular momentum accretion must shed; its supersonic impact drives radiative shocks whose cooling emission is a Herbig-Haro knot, while its momentum sweeps up a molecular outflow.

The core insight

A protostellar jet is a magnetized wind that removes the angular momentum accretion must shed; its supersonic impact on the cloud drives radiative shocks whose cooling emission we see as Herbig-Haro knots, while momentum sweeps up a molecular outflow.

Where this lives today

Protostellar jets and outflows are now studied at every wavelength that can reach them. Hubble and JWST image classic systems such as HH 34, HH 111, and HH 47 as strings of knots along dead-straight axes capped by bow shocks; multi-epoch astrometry measures the knots' proper motions of $100$ – $300\ \mathrm{km\,s^{-1}}$ and, through equation (6), their dynamical ages, while the regular knot spacing reads out a fossil record of episodic accretion. ALMA maps the molecular side in CO and SiO at sub-arcsecond resolution, resolving the launch region and testing whether the wind is loaded from the inner disk (a disk wind) or the star–disk boundary (an X-wind) — a live debate the mass-loading and rotation signatures of equations (1)–(2) are designed to settle. For deeply embedded Class 0 sources whose optical HH lines are extinguished, JWST traces jets in $\mathrm{[Fe\,II]}$ ( $1.26,1.64\ \mu\mathrm{m}$ ) and $\mathrm{H_2}$ ( $2.12\ \mu\mathrm{m}$ ), mapping the shock hierarchy from fast atomic $J$ -shocks to slow molecular $C$ -shocks. On cloud scales, outflow-driven turbulence and cavity-clearing are now standard ingredients in star-formation simulations — the open problem being how much of a cluster's turbulent budget outflows actually supply.

The physical picture

Picture a spinning turntable with a rigid wire pinned to it and a bead threaded on the wire. If the bead sits beyond the radius where the centrifugal push beats gravity, the wire flings it outward, forcing it to corotate and gain angular momentum until the wire can no longer hold it in step. Swap the wire for a magnetic field line anchored in the inner disk, and the bead for a parcel of ionized gas, and you have the magnetocentrifugal launch of a jet. The field enforces corotation out to the Alfvén radius $r_A$ — the lever arm — so every gram ejected carries away $\Omega r_A^2$ of angular momentum, far more than it started with at the footpoint $r_0$ . The disk pays for the ejection by losing exactly the angular momentum it needed to shed, and it accretes. Once launched, the jet is a supersonic bullet through cold molecular gas. Its nose drives a bow shock; internal "burps" from variable ejection send faster pulses crashing into slower ones, forming internal working surfaces. Every shock heats gas, which promptly radiates the energy back out as line emission — the HH knot. Because radiative cooling behind a fast shock is quick, the emitting region is a thin luminous sheet, a cooling zone only $10^{13}$ – $10^{15}\ \mathrm{cm}$ deep rather than a fat glowing volume. The trick to reading the whole system is to think in two currencies. Energy is spent almost instantly — radiative shocks dump it into photons that leave the cloud — but momentum is conserved. So the molecular outflow is built by momentum transfer: the jet is a snowplough sharing its thrust with a growing pile of swept-up gas, which is why the CO lobes are slow and heavy while the jet that drives them is fast and light.

Figure

Anatomy of a bipolar outflow: an accreting protostar and disk launch two opposed collimated jets at 100 to 400 km/s; internal working surfaces glow as Herbig-Haro knots and the leading bow shocks ram the cloud, while the jet sweeps ambient gas into slow wide CO lobes moving a few to tens of km/s — only about 10 percent of the accreted mass is ejected, yet it carries the angular momentum

The two exhausts differ in every property except the momentum they share, and conflating them is the single most common error in the subject: the narrow, fast, atomic/ionized jet makes HH knots, while the broad, slow, molecular CO shell is almost entirely ambient cloud gas the jet has swept up, not stellar material. The quantitative heart of the HH phenomenon is what the shock does to temperature — the relation an observer uses to turn a measured shock speed into a physical temperature, and to see why HH knots glow in the lines they do.

Figure

Post-shock temperature versus shock speed for a strong shock, T₂ = (3∕16)(μ m_H ∕ k_B) v_s² with μ = 0.6: a 100 km/s shock heats gas to 1.4×10⁵ K, hot enough to collisionally ionize hydrogen, which then cools back down through the temperatures that emit Hα and the forbidden lines we see as an HH knot

The core equations

Everything begins with a budget: accretion proceeds only if angular momentum leaves, and a magnetized wind is the accountant. Setting the angular momentum lost by the accreting gas equal to that carried away by the wind — which corotates to the Alfvén radius $r_A$ — gives the master balance.

\dot M_{\rm acc}\,\Delta j \;\approx\; \dot M_{\rm jet}\,\Omega\,r_A^2 \qquad (1)

where: $\dot M_{\rm acc}$ — mass accretion rate onto the star; $\Delta j$ — specific angular momentum the accreting gas must shed; $\dot M_{\rm jet}$ — mass-loss rate in the jet/wind; $\Omega$ — angular velocity at the wind footpoint $r_0$ ; $r_A$ — Alfvén radius, the magnetic lever arm. Reading it: because each ejected gram removes $\Omega r_A^2$ rather than $\Omega r_0^2$ , a small mass loss disposes of a large angular momentum. Rearranging (1) with $\Delta j\approx\Omega r_0^2$ gives the mass-loading ratio — the single number that says the jet is light, depending only on the square of the lever-arm ratio.

\frac{\dot M_{\rm jet}}{\dot M_{\rm acc}}\;\approx\;\left(\frac{r_0}{r_A}\right)^2 \qquad (2)

where: $r_0$ — wind footpoint radius in the disk; $r_A$ — Alfvén radius. Reading it: $r_A/r_0\approx3$ gives $(r_0/r_A)^2\approx0.11$ , so only $\sim10\%$ of the accreted mass need be ejected — the observed $\dot M_{\rm jet}/\dot M_{\rm acc}\sim0.1$ . The same machinery sets the terminal speed: gas launched near a few stellar radii is accelerated to of order the local escape speed, which is why jets emerge at hundreds of $\mathrm{km\,s^{-1}}$ rather than tens or thousands.

v_{\rm jet}\;\sim\;v_{\rm esc}(R_*)=\sqrt{\frac{2GM_*}{R_*}} \qquad (3)

where: $v_{\rm jet}$ — terminal jet velocity; $M_*,\,R_*$ — protostellar mass and radius; $G$ — gravitational constant. Reading it: $M_*=0.5\,M_\odot$ , $R_*=3\,R_\odot$ give $v_{\rm esc}\approx250\ \mathrm{km\,s^{-1}}$ , in the observed HH jet range. Downstream, the jet's supersonic impact converts bulk motion into heat: the Rankine-Hugoniot jumps for a monatomic gas in the high-Mach limit give the immediate post-shock temperature.

T_2=\frac{3}{16}\frac{\mu m_H}{k_B}\,v_s^2 \qquad (4)

where: $T_2$ — immediate post-shock temperature; $v_s$ — shock speed in the frame of the unshocked gas; $\mu$ — mean molecular weight ( $\approx0.6$ once ionized); $m_H$ — hydrogen mass; $k_B$ — Boltzmann constant. Reading it: $T_2\propto v_s^2$ , so $v_s=100\ \mathrm{km\,s^{-1}}$ gives $T_2\approx1.4\times10^5\ \mathrm{K}$ — the HH thermostat of the second figure. On large scales the jet no longer heats gas but pushes it. Because the interface radiates its energy away, the outflow conserves momentum, not energy, and the swept-up lobe momentum equals the integrated jet thrust.

P_{\rm out}=M_{\rm out}\,v_{\rm out}\;\approx\;\dot M_{\rm jet}\,v_{\rm jet}\,t_{\rm dyn} \qquad (5)

where: $P_{\rm out}$ — momentum in the swept-up CO lobe; $M_{\rm out}$ — outflow mass (mostly ambient gas); $v_{\rm out}$ — lobe velocity; $t_{\rm dyn}$ — dynamical age. Reading it: since $M_{\rm out}\gg\dot M_{\rm jet}t_{\rm dyn}$ , we get $v_{\rm out}\ll v_{\rm jet}$ — slow, massive lobes. The dynamical age is measurable without knowing the distance, because both the projected separation and the tangential velocity scale with distance and it cancels.

t_{\rm dyn}=\frac{d}{v_t}=\frac{\theta D}{\mu_{\rm pm}D}=\frac{\theta}{\mu_{\rm pm}} \qquad (6)

where: $d$ — projected knot–source distance; $v_t$ — tangential velocity; $\theta$ — angular separation; $\mu_{\rm pm}$ — proper motion; $D$ — distance (cancels). Reading it: proper motions of $0.1$ – $1''\,\mathrm{yr^{-1}}$ give ages of $10^2$ – $10^4\ \mathrm{yr}$ from astrometry alone. Finally, the thickness of the glowing shocked layer is set by how fast it cools — the post-shock cooling length, which is what makes HH knots thin.

\ell_{\rm cool}\;\sim\;\frac{\tfrac32 n_2 k_B T_2\,v_2}{n_e n_H\,\Lambda(T)} \qquad (7)

where: $\ell_{\rm cool}$ — thickness of the radiative cooling zone; $n_2,\,v_2$ — post-shock density and speed; $\Lambda(T)$ — cooling function; $n_e,\,n_H$ — electron and hydrogen densities. Reading it: because $\Lambda(T)$ is large across $10^4$ – $10^5\ \mathrm{K}$ , $\ell_{\rm cool}$ is small ( $10^{13}$ – $10^{15}\ \mathrm{cm}$ ) and the emission collapses into a sheet.

Derivation AThe angular-momentum problem and the magnetized disk wind

The goal is to show that a protostar cannot swallow its infalling gas without first getting rid of angular momentum, and that a magnetocentrifugal wind disposes of the excess so efficiently that ejecting only a tenth of the accreted mass suffices. We compare the specific angular momentum of core gas with what a star can retain, show a disk is unavoidable, and close the budget with a corotating wind to recover the observed mass-loading ratio.

Step 1. Compare reservoirs. A dense, slowly rotating core delivers gas with $j_{\rm core}\sim10^{20}$ – $10^{21}\ \mathrm{cm^2\,s^{-1}}$ . A star cannot spin faster than breakup, $j_{\rm break}=\sqrt{GM_*R_*}\approx3.7\times10^{18}\ \mathrm{cm^2\,s^{-1}}$ for $0.5\,M_\odot$ and $3\,R_\odot$ , and observed young stars rotate well below that, at $j_*\sim10^{17}\ \mathrm{cm^2\,s^{-1}}$ . The infalling gas carries $10^2$ – $10^4$ times more $j$ than the star can accommodate. Why: this mismatch, not gravity, is the true bottleneck — accretion is throttled by angular momentum, not by lack of fuel.

Step 2. A disk is inevitable. Conserving $j$ , infalling gas cannot reach the star; it stalls where centrifugal support balances gravity, at the centrifugal radius $r_{\rm cf}=j_{\rm core}^2/(GM_*)$ . For $j_{\rm core}=10^{21}\ \mathrm{cm^2\,s^{-1}}$ this is $r_{\rm cf}\approx1000\ \mathrm{AU}\gg R_*$ . The gas piles into a rotationally supported disk and must transport $j$ outward to accrete further. Why: the disk is the direct geometric consequence of the angular-momentum excess, and the platform from which the wind is launched.

Step 3. Load a corotating wind. Field lines anchored at $r_0$ behave like rigid wires, forcing loaded gas to corotate out to the Alfvén radius $r_A$ , where the wind's kinetic energy density equals the magnetic energy density and the field can no longer enforce corotation. Each gram released therefore removes $\Omega r_A^2$ of angular momentum — the lever arm $r_A$ , not the footpoint $r_0$ , sets the payload. Why: this magnetic lever lets a trickle of mass carry a torrent of angular momentum — the crux of magnetocentrifugal launching.

Step 4. Close the budget. Setting the angular momentum lost by accreting gas equal to that carried off by the wind, $\dot M_{\rm acc}\,\Omega r_0^2=\dot M_{\rm jet}\,\Omega r_A^2$ , and solving gives $\dot M_{\rm jet}/\dot M_{\rm acc}=(r_0/r_A)^2$ . With the favoured lever arm $r_A/r_0\approx3$ this is $\approx0.11$ — the boxed result and the measured value. The same corotation also predicts $v_{\rm jet}\sim(r_A/r_0)\,v_{\rm Kep}(r_0)$ , giving hundreds of $\mathrm{km\,s^{-1}}$ for launch a few stellar radii out. Why: one geometric input reproduces both the observed one-tenth mass loading and the observed $100$ – $400\ \mathrm{km\,s^{-1}}$ jet speed — a nontrivial double success of the picture.

\frac{\dot M_{\rm jet}}{\dot M_{\rm acc}}\approx\left(\frac{r_0}{r_A}\right)^2\sim0.1

Go deeper · Derivation B · 2 minThe radiative shock and the birth of a Herbig-Haro knot

Derivation BThe radiative shock and the birth of a Herbig-Haro knot

The goal is to derive the post-shock temperature of a strong shock from the conservation laws, and then show why the heated gas is seen not as a hot diffuse glow but as a thin, sharply defined emission knot. We apply the Rankine-Hugoniot jumps in the strong-shock limit for the temperature, then follow the shocked gas as it radiates, so the HH line spectrum and the thinness of the emitting sheet both drop out of one calculation.

Step 1. Write the jumps. In the shock frame, mass and momentum conservation read $\rho_1 v_s=\rho_2 v_2$ and $\rho_1 v_s^2+P_1=\rho_2 v_2^2+P_2$ , with subscripts $1,2$ for pre- and post-shock gas. For a strong shock the upstream pressure $P_1$ is negligible against the ram pressure $\rho_1 v_s^2$ . Why: dropping $P_1$ is the defining high-Mach simplification and makes the jumps depend on $v_s$ alone.

Step 2. Take the strong-shock limit. For a monatomic gas ( $\gamma=5/3$ ) and Mach number $\mathcal M\gg1$ , the density compression saturates at $\rho_2/\rho_1=(\gamma+1)/(\gamma-1)=4$ , so the gas decelerates to $v_2=v_s/4$ and the post-shock pressure is $P_2\simeq\tfrac34\rho_1 v_s^2$ . Why: the compression cannot exceed 4 for $\gamma=5/3$ however fast the shock — a hard ceiling the unbounded temperature does not share.

Step 3. Extract the temperature. Combining $P_2=n_2 k_B T_2$ with $\rho_2=4\rho_1$ and $P_2=\tfrac34\rho_1 v_s^2$ gives $T_2=\tfrac{3}{16}(\mu m_H/k_B)v_s^2$ . Numerically, with $\mu=0.6$ , this is $1.4\times10^5\ \mathrm{K}\,(v_s/100\ \mathrm{km\,s^{-1}})^2$ , hot enough to collisionally ionize. Why: this formula converts a measured shock speed into a temperature, and its $v_s^2$ scaling is why faster bow shocks glow in higher-excitation lines.

T_2=\frac{3}{16}\frac{\mu m_H}{k_B}\,v_s^2\approx1.4\times10^5\ \mathrm{K}\ \Big(\frac{v_s}{100\ \mathrm{km\,s^{-1}}}\Big)^2

Step 4. Let it cool, and count the thickness. As the ionized gas flows downstream it radiates, recombines, and passes back through the temperatures where $\mathrm{H\alpha}$ , $\mathrm{[S\,II]}\,\lambda\lambda6716,6731$ , $\mathrm{[O\,I]}\,\lambda6300$ , and $\mathrm{[N\,II]}$ are excited. The distance it travels while cooling is $\ell_{\rm cool}\sim\tfrac32 n_2 k_B T_2 v_2/(n_e n_H\Lambda)$ , which is short — $10^{13}$ – $10^{15}\ \mathrm{cm}$ — because $\Lambda(T)$ is large here. Why: a small cooling length concentrates the luminosity in a razor-thin layer, which is why HH knots image as sharp filaments and bow-shock caps, not fuzzy blobs.

Step 5. Read the diagnostics. Because the line ratios are set by $v_s$ and the preshock density, the observed $\mathrm{[S\,II]}/\mathrm{H\alpha}$ ratio and the $\mathrm{[S\,II]}$ doublet ratio invert directly to shock speed and density; strong forbidden lines relative to $\mathrm{H\alpha}$ distinguish a shock-excited HH knot from a photoionized H II region. Why: this makes HH objects quantitative laboratories — the spectrum measures the shock, it is not just pretty.

Go deeper · Derivation C · 1 minThe momentum-driven molecular outflow as a snowplough

Derivation CThe momentum-driven molecular outflow as a snowplough

The goal is to explain why CO outflows carry enormous momenta yet move slowly, by treating the jet as an engine that continuously injects momentum into a growing shell of swept-up ambient gas. We argue that the conserved quantity is momentum, not energy, integrate the jet thrust to get the shell momentum, and read off why the lobe is massive and slow while the jet is light and fast — and what it implies for feedback on the cloud.

Step 1. Decide the conserved currency. At the jet–cloud interface, radiative shocks drain the shocked gas of its thermal energy on a timescale short compared to the flow's age, so the swept-up material cannot build an adiabatic pressure-driven bubble. What survives is momentum, which no radiation carries away. Why: choosing momentum- over energy-conservation is the whole physical content of the snowplough model, justified only because the shocks are radiative.

Step 2. Integrate the thrust. The jet delivers momentum at the constant rate $\dot P_{\rm jet}=\dot M_{\rm jet}v_{\rm jet}$ , and this accumulates in the shell: $\tfrac{\rm d}{{\rm d}t}(M_{\rm out}v_{\rm out})=\dot M_{\rm jet}v_{\rm jet}$ , so after a time $t$ the shell momentum is $M_{\rm out}v_{\rm out}=\dot M_{\rm jet}v_{\rm jet}\,t$ . Why: this is Newton's second law for a body gaining mass, tying the observable lobe momentum to the integrated jet history.

M_{\rm out}(t)\,v_{\rm out}(t)=\dot M_{\rm jet}\,v_{\rm jet}\,t

Step 3. Explain slow and massive. The shell mass is dominated by entrained cloud gas, $M_{\rm out}\gg\dot M_{\rm jet}t$ , while the shared momentum is fixed; dividing, $v_{\rm out}=\dot M_{\rm jet}v_{\rm jet}t/M_{\rm out}\ll v_{\rm jet}$ . The outflow is heavy and slow exactly because it is mostly stolen cloud gas, not ejected star-stuff. Why: this resolves the paradox that CO lobes have momenta far exceeding anything the star could launch — the momentum is the jet's, but the mass is the cloud's.

Step 4. Estimate feedback. With $\dot M_{\rm jet}\approx0.1\,\dot M_{\rm acc}$ , the deposited momentum is $P_{\rm out}\sim0.1\,\dot M_{\rm acc}v_{\rm jet}\,t$ ; over a $\sim10^5\ \mathrm{yr}$ lifetime this can accelerate a mass comparable to the final stellar mass out of the core, and it beats the radiative supply $L_{\rm bol}/c$ by two to three orders of magnitude. Why: mechanical outflow feedback, not radiation, is the leading candidate for limiting core-to-star efficiency to $\sim30\%$ and stirring cloud turbulence.

Limiting cases

Case	Result	Meaning
Radiative (fast-cooling) shock	$\ell_{\rm cool}\ll R_{\rm shock}$	Cooling length short compared to the shock scale: energy radiates locally, momentum is conserved, and a thin bright HH sheet forms — the usual HH regime.
Adiabatic (energy-conserving) shock	$\ell_{\rm cool}\gg R_{\rm shock}$	Fast, low-density limit; heat is retained in a hot Sedov-like bubble. Relevant very early in the flow or for the fastest shocks before cooling switches on.
Strong (high-Mach) jump	$\dfrac{\rho_2}{\rho_1}\to\dfrac{\gamma+1}{\gamma-1}=4$	For $\gamma=5/3$ , $\mathcal M\gg1$ the compression saturates at 4 while $T_2\propto v_s^2$ keeps rising — density is capped, temperature is not.
Momentum-driven outflow	$P_{\rm out}=\dot M_{\rm jet}v_{\rm jet}\,t$	Swept-up mass grows while total momentum tracks the integrated jet thrust; explains the large CO-lobe momenta and slow speeds.
Molecular (C-)shock, $v_s\lesssim40\ \mathrm{km\,s^{-1}}$	$\mathrm{H_2}$ survives, no dissociation	Magnetic cushioning spreads the transition; gas stays molecular and glows in $\mathrm{H_2}$ $2.12\,\mu\mathrm{m}$ not atomic lines.

Common trapConfusing the fast atomic jet with the slow molecular outflow

Do not treat the HH jet and the CO outflow as the same object at different speeds. The jet is narrow, atomic/ionized, and fast ( $100$ – $400\ \mathrm{km\,s^{-1}}$ ); the CO outflow is wide, molecular, and slow (a few to tens of $\mathrm{km\,s^{-1}}$ ). Crucially, the outflow is overwhelmingly swept-up ambient cloud gas, so its mass is not the mass the star has lost — reading $M_{\rm out}$ as an ejection mass overestimates the true mass-loss rate by orders of magnitude. And once radiative shocks drain the thermal energy the flow is momentum-conserving, not energy-conserving, so do not model it as an adiabatic wind bubble. Keep the two exhausts, and their two conservation laws, distinct.

Worked examples

1 · Launch speed and mass loading (confidence builder)

A protostar has $M_*=0.5\,M_\odot$ and $R_*=3\,R_\odot$ . Estimate the jet speed it should launch, and the fraction of accreted mass it must eject if the wind's Alfvén lever arm is $r_A/r_0=3$ .

Step 1. The magnetocentrifugal terminal speed is of order the escape speed at the launch radius. With $M_*=0.5\,M_\odot=9.9\times10^{32}\ \mathrm{g}$ and $R_*=3\,R_\odot=2.09\times10^{11}\ \mathrm{cm}$ , $v_{\rm esc}=\sqrt{2GM_*/R_*}=\mathbf{2.5\times10^{7}\ cm\,s^{-1}=250\ km\,s^{-1}}$ — the middle of the observed HH jet range.

Step 2. The mass loading follows from the lever arm alone: $\dot M_{\rm jet}/\dot M_{\rm acc}=(r_0/r_A)^2=(1/3)^2=\mathbf{0.11}$ . So the star ejects only about one part in nine of what it accretes.

Step 3. Sanity-check the energetics. With $L_{\rm acc}=GM_*\dot M_{\rm acc}/R_*$ , $L_{\rm mech}=\tfrac12\dot M_{\rm jet}v_{\rm jet}^2$ , and $\dot M_{\rm acc}=10^{-6}\,M_\odot\,\mathrm{yr^{-1}}$ , one finds $L_{\rm acc}\approx5\,L_\odot$ and $L_{\rm mech}\approx0.5\,L_\odot$ , so $L_{\rm mech}/L_{\rm acc}\approx0.1$ — the outflow is a modest tap on the accretion power, not its rival. v_jet ≈ 250 km/s, one-tenth mass loading, one-tenth of the accretion luminosity into the jet.

2 · The temperature of an HH bow shock (fully worked)

The leading bow shock of an HH flow is measured, from optical proper motion, to advance at $v_s=100\ \mathrm{km\,s^{-1}}$ into ambient gas. Find the immediate post-shock temperature and say what emission to expect.

Step 1. Use the strong-shock temperature, equation (4), with $\mu=0.6$ (ionized), $m_H=1.67\times10^{-24}\ \mathrm{g}$ , $k_B=1.38\times10^{-16}\ \mathrm{erg\,K^{-1}}$ , $v_s=10^{7}\ \mathrm{cm\,s^{-1}}$ :

T_2=\frac{3}{16}\frac{(0.6)(1.67\times10^{-24})}{1.38\times10^{-16}}(10^{7})^2\ \mathrm{K}.

Step 2. Evaluating, $T_2=\mathbf{1.4\times10^{5}\ K}$ . The immediate post-shock gas is collisionally ionized. Because $T_2\propto v_s^2$ , a faster $200\ \mathrm{km\,s^{-1}}$ portion of the same bow shock would reach $\sim5.4\times10^{5}\ \mathrm{K}$ — higher-excitation lines appear where the shock is fastest.

Step 3. Trace the emission. As the gas streams downstream it cools through $10^5\to10^4\ \mathrm{K}$ , recombining and radiating $\mathrm{H\alpha}$ and the forbidden lines $\mathrm{[S\,II]}$ , $\mathrm{[O\,I]}$ , $\mathrm{[N\,II]}$ . The cooling length is short ( $\ell_{\rm cool}\sim10^{14}\ \mathrm{cm}$ ), so all of this light comes from a thin sheet just behind the front — an HH knot. T₂ ≈ 1.4×10⁵ K; a thin cooling sheet radiating Hα and forbidden lines.

3 · Momentum and age of a CO outflow

A CO map shows a lobe of mass $M_{\rm out}=0.1\,M_\odot$ moving at $v_{\rm out}=10\ \mathrm{km\,s^{-1}}$ , extending $0.1\ \mathrm{pc}$ from a protostar of bolometric luminosity $L_{\rm bol}=10\,L_\odot$ . Find the dynamical age, the outflow momentum, and compare its thrust to radiation pressure.

Step 1. Dynamical age: $t_{\rm dyn}=R/v_{\rm out}=(0.1\ \mathrm{pc})/(10\ \mathrm{km\,s^{-1}})=(3.1\times10^{17}\ \mathrm{cm})/(10^{6}\ \mathrm{cm\,s^{-1}})=3.1\times10^{11}\ \mathrm{s}=\mathbf{9.8\times10^{3}\ yr}$ — a young, embedded (Class 0/I) source.

Step 2. Outflow momentum: $P_{\rm out}=M_{\rm out}v_{\rm out}=\mathbf{1.0\ M_\odot\,km\,s^{-1}}$ , and the outflow force is $\dot P=P_{\rm out}/t_{\rm dyn}\approx1.0\times10^{-4}\ M_\odot\,\mathrm{km\,s^{-1}\,yr^{-1}}=6.4\times10^{26}\ \mathrm{dyn}$ .

Step 3. Compare to radiation. The most momentum stellar photons can supply is $L_{\rm bol}/c=(10\,L_\odot)/c=1.3\times10^{24}\ \mathrm{dyn}$ , so $\dot P/(L_{\rm bol}/c)\approx\mathbf{5\times10^{2}}$ — the outflow carries some five hundred times more momentum than radiation pressure could provide. That is the quantitative proof that outflows are momentum-driven by the jet, not pushed by starlight. Age ≈ 10⁴ yr, P_out ≈ 1 M_⊙ km/s, thrust ≈ 500× the radiative limit.

Physics of the portrait

Herbig-Haro objects are named for two astronomers who, working an ocean apart and unaware of each other, first realized they were something new.

Origins · 1 minGeorge Herbig and how this was discovered

George Herbig

1920–2013 · Herbig-Haro objects and pre-main-sequence stars

George Herbig, at Lick Observatory in California, and Guillermo Haro, at Tonantzintla in Mexico, independently drew attention around 1951–1952 to small, bright emission nebulae embedded in dark clouds near young stars — compact patches glowing in $\mathrm{H\alpha}$ and forbidden lines with no ordinary star to illuminate them. (One such object had been noted by Sherburne Wesley Burnham as early as 1890, near T Tauri, but its nature went unrecognized for six decades.) Their spectra showed strong low-ionization forbidden lines of $\mathrm{[S\,II]}$ and $\mathrm{[O\,I]}$ that ruled out photoionization by a hot star. The decisive reinterpretation came in the mid-1970s, when Richard Schwartz argued that HH objects are shock-excited — gas heated not by starlight but by a supersonic wind crashing into the cloud. That picture, married to the 1980 discovery of bipolar molecular outflows and the magnetocentrifugal disk-wind theory of Blandford and Payne in 1982, turned HH objects from a spectroscopic oddity into the visible tracers of protostellar accretion. Herbig spent a career on the pre-main-sequence stars that drive them — the Herbig Ae/Be stars also carry his name — and lived to see Hubble resolve his nebulae into jets, knots, and bow shocks moving measurably across the sky.

History

1890

— Burnham notes a small emission nebula beside T Tauri, later HH 255

1951

— George Herbig describes bright nebulous knots in star-forming clouds

1975

— Schwartz interprets HH objects as shock-excited by a stellar wind

1980

— Snell, Loren, and Plambeck discover bipolar CO outflows in L1551

1982

— Blandford and Payne formulate magnetocentrifugal disk-wind launching

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 12 (Star Formation) · Draine, Physics of the Interstellar and Intergalactic Medium, Ch. 35–36 (shock waves) · Binney & Tremaine, Galactic Dynamics, 2nd ed. (angular-momentum transport in disks) · Reipurth & Bally 2001, ARA&A 39, 403 (Herbig-Haro flows) · Frank et al. 2014, Protostars and Planets VI (Jets and Outflows) · Blandford & Payne 1982, MNRAS 199, 883 (magnetocentrifugal launching).

Provenance. Escape-speed and mass-loading numbers follow Carroll & Ostlie Ch. 12: $v_{\rm esc}(0.5\,M_\odot,3\,R_\odot)=252\ \mathrm{km\,s^{-1}}$ and $(r_0/r_A)^2=0.11$ for $r_A/r_0=3$ were recomputed here. The Rankine-Hugoniot jumps and boxed $T_2=\tfrac{3}{16}(\mu m_H/k_B)v_s^2$ are standard (Draine Ch. 35); the numerical $T_2=1.36\times10^5\ \mathrm{K}$ at $v_s=100\ \mathrm{km\,s^{-1}}$ , $\mu=0.6$ was verified, as were $j_{\rm break}=\sqrt{GM_*R_*}=3.7\times10^{18}\ \mathrm{cm^2\,s^{-1}}$ , $r_{\rm cf}\approx1000\ \mathrm{AU}$ , and the CO-outflow example ( $t_{\rm dyn}\approx10^4\ \mathrm{yr}$ , $P_{\rm out}=1\,M_\odot\,\mathrm{km\,s^{-1}}$ , thrust $\approx500\times L_{\rm bol}/c$ ), consistent with Frank et al. 2014. All quoted velocities, proper motions, and ages are standard observed ranges; no non-textbook numbers were invented.

Prerequisites. Molecular Clouds and Giant Molecular Clouds · CO as a Molecular Gas Tracer and the X-Factor · Interstellar Gas Heating and Cooling Processes · The Jeans Instability and Jeans Mass · Interstellar Polarization and Magnetic Fields. Leads to. Virial Theorem for Molecular Clouds · Thermal Dust Emission and Infrared Radiation · Phases of the Interstellar Medium.

Ask the tutor: Ask me to walk through why a magnetized wind can dump a hundred times the angular momentum it carries in mass, or to work an HH line ratio backward into a shock speed.

Why does the star not just spin faster instead of driving a wind?

It cannot spin fast enough. A star has a hard ceiling on the specific angular momentum it can hold — the breakup value $j_{\rm break}=\sqrt{GM_*R_*}\approx4\times10^{18}\ \mathrm{cm^2\,s^{-1}}$ — beyond which it sheds material from its equator. Infalling core gas arrives with $10^{20}$ – $10^{21}\ \mathrm{cm^2\,s^{-1}}$ , a hundred to ten thousand times more, so even a star spinning at breakup could absorb only a tiny fraction. The excess must be exported, and a magnetized wind carrying $\Omega r_A^2$ per gram is the efficient way to do it. Spinning up is not an option; disposal is mandatory.

What is the difference between the jet, the outflow, and a Herbig-Haro object?

They are three views of one system. The jet is the narrow, fast ( $100$ – $400\ \mathrm{km\,s^{-1}}$ ), atomic or ionized beam launched from the inner disk. A Herbig-Haro object is a specific bright knot within or at the end of that jet, where a shock heats gas that radiates optical/IR lines — a location, not the whole flow. The molecular outflow is the much larger, slower (a few–tens of $\mathrm{km\,s^{-1}}$ ) shell of ambient cloud gas the jet has swept up, seen in CO. So the jet drives the HH knots and, separately, entrains the outflow: one engine, several manifestations at different scales and temperatures.

Why is the compression across a strong shock exactly a factor of 4?

The factor 4 is the strong-shock limit of the density jump $\rho_2/\rho_1=(\gamma+1)/(\gamma-1)$ for adiabatic index $\gamma=5/3$ , appropriate to a monatomic gas. As the Mach number grows the compression asymptotes to 4 and never exceeds it, because the heated post-shock gas resists further squeezing. The number is specific to $\gamma=5/3$ : a molecular gas with $\gamma=7/5$ compresses by 6, and a radiative shock that keeps cooling can compress far more once it becomes effectively isothermal. Factor 4 is the ceiling only for the adiabatic monatomic case.

If the outflow carries so much momentum, why is it not powered by the star's light?

Because starlight simply cannot supply that much momentum. The maximum thrust radiation can exert is $L_{\rm bol}/c$ , obtained if every photon is absorbed once. For a typical embedded protostar this falls short of the measured outflow force by two to three orders of magnitude — in the worked example, by about five hundred. The momentum instead comes from the jet, launched magnetocentrifugally at the accretion power, not from photon pressure. This large excess over $L_{\rm bol}/c$ is itself the standard argument that outflows are mechanically, not radiatively, driven.

Why do jets show regularly spaced chains of knots?

The knots are internal working surfaces — places where a faster pulse of ejecta rams into slower material launched earlier, driving a shock that lights up. Their regular spacing records that the ejection velocity varies quasi-periodically, so the jet is episodic rather than steady. Each knot is a fossil of an accretion/ejection event, and the spacing divided by the outflow speed gives the interval between events, often $\sim10$ – $100\ \mathrm{yr}$ . Multi-epoch imaging then measures the proper motions directly, letting the whole ejection history be reconstructed from one well-studied source.

How do observers actually turn an HH spectrum into a shock speed?

Through the same shock physics that sets $T_2$ . A radiative-shock model predicts the full downstream cooling zone and hence the relative strengths of lines like $\mathrm{H\alpha}$ , $\mathrm{[S\,II]}\,\lambda\lambda6716,6731$ , and $\mathrm{[O\,I]}\,\lambda6300$ . Faster shocks reach higher $T_2$ and produce higher-excitation spectra, so measured line ratios map onto shock speed, while the $\mathrm{[S\,II]}$ doublet ratio measures the post-shock electron density. Comparing observed ratios to a grid of shock models yields $v_s$ and the preshock density — which is why HH objects serve as calibrated in-situ shock laboratories, not mere pretty nebulae.