Blandford–Znajek Jet Power Calculator

How the Blandford–Znajek calculator works

Introduction to Blandford–Znajek jet power

Relativistic jets from quasars, radio galaxies, and compact binaries are a reminder that a black hole can do more than simply swallow matter. In the Blandford–Znajek picture, magnetic flux anchored in the surrounding plasma threads the event-horizon region of a rotating Kerr black hole. That rotation twists the field lines, drives currents, and pushes electromagnetic energy outward in a jet.

The full problem is messy. Jet power depends on flux accumulation, disk state, field geometry, plasma loading, and relativistic corrections that only show up cleanly in full simulations. This calculator strips those complications down to a compact scaling so you can see the direction and relative size of the effect without pretending to model every detail.

That simplification is still useful. It lets you explore why spin matters nonlinearly, why the horizon field usually dominates the uncertainty budget, and why the gravitational radius makes supermassive black holes especially large engines even when the local field is modest. If you are reading a paper that quotes jet power in erg/s, this page is meant to give you a quick way to test whether the order of magnitude makes sense before you dive into a longer derivation.

How to use the Blandford–Znajek calculator

To use the Blandford–Znajek calculator, enter the three quantities that feed the scaling directly. First choose the black hole mass in solar masses. Then choose the dimensionless spin parameter a*, which must stay between 0 and 1. Finally enter the magnetic field strength near the horizon in tesla. When you press the compute button, the results panel updates with the gravitational radius, jet power in SI and cgs units, and the ratio of the estimate to the Eddington luminosity.

1. Enter the black hole mass M in solar masses (M☉). Stellar-mass black holes are often around 5 to 50 M☉, while supermassive black holes in galactic nuclei can range from about 10⁶ to 10¹⁰ M☉.
2. Enter the dimensionless spin a* with 0 ≤ a* < 1. Values such as 0.5, 0.9, or 0.99 are useful starting points when you want to see how quickly spin changes the answer.
3. Enter the magnetic field strength B near the horizon in tesla. This is often the least certain input, so it is worth trying a range of values instead of trusting a single guess.
4. Click Compute Power to generate the estimate. If you want to reuse the result elsewhere, the copy button appears after a successful calculation.

Many astronomy papers quote jet energetics in cgs units, so the calculator shows both watts and erg/s. The conversion is straightforward: 1 W = 10⁷ erg/s. If you are comparing your result with an AGN cavity estimate, a radio jet fit, or a line in a log jet-power plot, the erg/s value is usually the one you will read first.

Blandford–Znajek formula, units, and assumptions

For Blandford–Znajek jet power, the calculator uses a compact scaling that keeps the main dependencies visible:

Jet power: $P\approx \kappa a^2{B}^2{r}^{2}c$

• a* is the dimensionless spin parameter entered in the form.
• B is the magnetic field strength at the horizon in tesla.
• r_g is the gravitational radius: $r_g=\frac{GM}{c^2}$
• c is the speed of light.
• κ is an efficiency factor. This page uses κ = 0.05 as a typical ballpark value.

The shorthand power formula above is intentionally compact. In this estimate, the radius term is the gravitational radius r_g, so mass enters through the size of the horizon-scale engine rather than through a separate luminosity term. At fixed field strength, a larger black hole has a larger characteristic area, which is why the estimate grows strongly with mass.

The calculator also reports an Eddington fraction using $L_E=1.26\times {10}^{31}(M/M_{☉})$ watts.

That ratio is a convenient comparison scale, but it is not a statement about whether the jet is literally Eddington-limited. A jet is a collimated electromagnetic outflow, not a spherical radiation field, so it can be beamed, radiatively inefficient, or energetically dominated by spin extraction in ways that the Eddington luminosity does not capture. The calculator keeps the comparison because it is a familiar yardstick, not because the two quantities obey the same physics.

As a practical rule, think of the three inputs like this: mass sets the engine size, spin sets how much rotational energy is available to tap, and magnetic field strength sets how hard the horizon couples to the surrounding magnetosphere. Because the estimate depends on both a*² and B², moderate changes in either one can produce surprisingly large shifts in jet power. That quadratic sensitivity is the main reason the page is useful for scenario testing.

Assumptions built into this simplified estimate matter. The magnetic field is treated as one characteristic value instead of a full spatial structure. Field geometry, flux saturation, and plasma loading are absorbed into κ. The spin dependence is approximated as a*², which is a helpful scaling but not a precision fit across every spin and accretion state. The output should therefore be read as an order-of-magnitude estimate, not as a source-specific GRMHD prediction.

Blandford–Znajek worked examples across three parameter regimes

The rows below compare three very different Blandford–Znajek setups: one supermassive case, one stellar-mass case, and one intermediate case. Each value is evaluated from the same scaling used by the calculator, so the table is meant to show how the result moves when mass, spin, and field strength shift into different regimes.

What matters most is not memorizing the numbers, but noticing which parameter dominates each row. In the high-mass example, the gravitational radius helps the power rise even with a relatively modest field. In the stellar-mass example, the field has to work much harder to compensate for the smaller size scale. In the intermediate case, the answer lands between those two extremes, which is exactly the kind of comparison this calculator is meant to make easy.

If you are using the page for a paper reading session, use the table as a guide for intuition rather than as a fixed benchmark. BZ power is sensitive enough that one change in spin or field strength can move the estimate by a wide margin, so the safest habit is to ask which input is doing the heavy lifting before you compare the output with an observed jet.

How to interpret the Blandford–Znajek outputs

The Blandford–Znajek output panel is easiest to read as a set of linked diagnostics rather than as one naked power number. The gravitational radius tells you the size scale of the hole, the watt and erg/s lines tell you the same estimate in two unit systems, and the Eddington fraction gives you a familiar comparison point for discussing how ambitious the jet power is.

• Gravitational radius r_g: a length scale set directly by mass.
• Jet power in W: the estimated electromagnetic extraction rate in SI units.
• Jet power in erg/s: the same quantity in the cgs units common in astrophysics literature.
• Eddington fraction: the ratio of estimated jet power to the Eddington luminosity.

The Eddington comparison is especially useful as a rough sanity check, but it should not be overinterpreted. A jet is a collimated outflow rather than isotropic radiation, so values near or even above unity do not automatically make the estimate unphysical. Instead, they tell you that the chosen field strength, spin, or efficiency factor implies an energetically ambitious jet. In some systems, that is exactly what the Blandford–Znajek mechanism is supposed to allow.

A good way to build intuition with Blandford–Znajek jet power is to vary one input at a time. Because the power scales with a*² and B², doubling either one does not merely double the result. Likewise, because r_g scales with mass and the formula uses r_g², increasing mass by a factor of 10 raises the power by roughly a factor of 100 when the other inputs are held fixed.

Blandford–Znajek limitations and common pitfalls

A quick Blandford–Znajek estimate is ideal for intuition, but it compresses a lot of jet physics into one number. That is useful when you want to compare scenarios quickly, yet it also means there are several places where a reader should slow down before treating the value as definitive.

• Magnetic field uncertainty: the horizon field is rarely measured directly and can vary by orders of magnitude depending on the assumed accretion state.
• Efficiency factor κ: the chosen value is a representative ballpark, not a universal constant.
• Near-extremal spin: the simple square-law scaling becomes less trustworthy very close to a* = 1.
• Jet power versus observed luminosity: a jet can be radiatively inefficient or strongly beamed, so observed light is not the same thing as total jet power.
• Eddington comparison: a jet is not constrained in exactly the same way as isotropic radiation.
• Input formatting: scientific notation such as 1e8 is accepted, but a small typo in B can produce a large change because the dependence is quadratic.

In practice, the most common mistake is to treat the magnetic field as a known number when it is often the least certain input. If you are using this page to interpret a source from the literature, it is usually better to explore a plausible range of B values and watch how the jet-power band moves than to lean too heavily on a single estimate.

Practical Blandford–Znajek FAQ

What magnetic field values are reasonable? There is no single correct field strength. The useful range depends on whether you are imagining an AGN, a stellar-mass binary, or another compact engine, and the calculator is most helpful when you test a plausible band instead of one guess. Because Blandford–Znajek power scales as B², the result can shift dramatically even when the input range looks modest.

Why does mass matter so much? Mass enters through the gravitational radius, so the calculator turns M into a size scale first and then squares that scale in the power estimate. That is why the answer climbs quickly as the black hole gets larger, even before you change spin or field.

Can jet power exceed the accretion luminosity? Yes. Blandford–Znajek jets can tap black-hole spin energy through magnetic flux, so the outflow does not have to track the disk's radiative luminosity one-for-one. In strongly magnetized systems, the jet can therefore look more powerful than the light you see.

What should I cite if I use this calculator? For scholarly work, cite Blandford and Znajek 1977 and any modern jet-launching or GRMHD paper that matches your source. This page is only a convenience scaling, not a substitute for the primary literature.

Further Blandford–Znajek exploration

If you want to place a Blandford–Znajek estimate in context, the next useful step is to compare it with another compact-object diagnostic. Continue with the black hole shadow calculator to think about horizon-scale geometry, compare another energy-extraction idea with the Penrose process energy extraction tool, or move to rotating neutron-star systems with the pulsar spin-down parameters calculator. A natural workflow is to compute a jet power here, compare it with an observed luminosity or cavity-power estimate, and then ask what field strength or effective κ would make the two line up.