Introduction
Quasars, radio galaxies, microquasars, and some transient engines produce narrow relativistic jets that carry energy across distances far larger than the central black hole itself. One of the most important questions in high-energy astrophysics is where that energy comes from. Accretion can certainly power an outflow, but in many systems the jet seems especially sensitive to how fast the black hole spins. The Blandford–Znajek mechanism offers a simple and influential answer: if a rotating Kerr black hole is threaded by magnetic field lines, the hole can transfer part of its rotational energy into an outgoing electromagnetic jet.
In the membrane-paradigm picture, the horizon acts somewhat like a rotating conductor. Frame dragging forces nearby field lines to rotate, currents are induced, and those currents apply an electromagnetic torque. Energy and angular momentum are then carried outward in a Poynting-flux dominated flow. Detailed general-relativistic magnetohydrodynamic simulations include magnetic geometry, plasma loading, disk state, and nonlinear feedback. This page does not try to reproduce all of that complexity. Instead, it gives you a compact scaling law that captures the main dependence on black hole size, spin, and magnetic field strength.
That makes the calculator useful for intuition. You can explore how much power changes when the spin parameter approaches unity, how uncertain magnetic field estimates can swing the answer by orders of magnitude, and how a larger black hole increases the effective horizon scale. In other words, this is a back-of-the-envelope physics tool: simple enough to use quickly, but grounded enough to support paper reading, classroom discussion, and sanity checks when observational jet powers are quoted in erg/s.
How to use the calculator
Start with the three inputs that matter in the simplified BZ estimate. First enter the black hole mass in solar masses. Then enter the dimensionless spin parameter a*, which must stay between 0 and 1, with values near 1 representing rapid rotation. Finally enter the magnetic field strength near the horizon in tesla. Press the compute button and the results panel will report the gravitational radius, jet power in SI and cgs units, and the ratio of the estimated jet power to the Eddington luminosity.
- 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☉.
- Enter the dimensionless spin a* with 0 ≤ a* < 1. Values such as 0.5, 0.9, or 0.99 are common exploratory choices.
- Enter the magnetic field strength B near the horizon in tesla. This is often the least certain parameter, so it is wise to test a range rather than a single guess.
- Click Compute Power to generate the estimate. If you need to paste the result elsewhere, the copy button appears after a successful calculation.
Many astronomy papers quote powers in cgs units, so the page shows both watts and erg/s. The conversion is straightforward: 1 W = 107 erg/s. If you are comparing your result with a radio jet scaling, an AGN cavity estimate, or a plot of log jet power, the erg/s line is usually the one to read first.
The calculator uses an order-of-magnitude Blandford–Znajek scaling:
Jet power:
- a* is the dimensionless spin parameter entered in the form.
- B is the magnetic field strength at the horizon in tesla.
- rg is the gravitational radius:
- 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 the context of this page, the radius term is the gravitational radius rg, so the mass dependence enters through the horizon size. 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
watts.
A practical way to read the scaling is this: mass sets the size of the engine, spin tells you how much rotational energy is available to tap, and magnetic field strength tells you how strongly the horizon is coupled to the surrounding magnetosphere. Because the estimate depends on both a*2 and B2, moderate changes in those parameters can produce surprisingly large shifts in jet power.
Assumptions baked into this simplified estimate are important. The magnetic field is treated as a single 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 useful 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 a source-specific GRMHD prediction.
Worked example and scenario table
Consider a supermassive black hole with M = 108 M☉, spin a* = 0.9, and horizon field B = 0.1 T. The simplified scaling gives a jet power near 1037 W, or about 1044 erg/s. This is exactly the sort of range often discussed for active galactic nuclei. If you raise the spin from 0.9 to 0.99 while keeping everything else fixed, the power changes by roughly the square of the ratio, so the increase is noticeable but not enormous. If you instead raise the magnetic field from 0.1 T to 1 T, the power jumps by a factor of 100 because the field enters quadratically.
That comparison shows why this calculator is helpful. The absolute result depends on uncertain astrophysical inputs, especially B and the hidden physics folded into κ, but the relative scalings are often robust enough to guide intuition. If an observed jet seems too powerful for your starting guess, you can ask whether the tension is more naturally resolved by a stronger field, a higher spin, a larger mass, or some combination of the three.
Example jet power estimates for three parameter sets
| Case |
M (M☉) |
a* |
B (T) |
P (W) |
P (erg/s) |
| A |
1e8 |
0.9 |
0.1 |
— |
— |
| B |
10 |
0.99 |
1e4 |
— |
— |
| C |
1e4 |
0.5 |
1 |
— |
— |
How to interpret the outputs
The results panel includes four lines, and each one answers a slightly different physical question. The gravitational radius gives the characteristic size scale of the black hole. For a stellar-mass object it is measured in kilometers; for a very large supermassive black hole it can approach the scale of the Earth–Sun distance. In the BZ estimate this radius appears squared, so even without changing spin or field, a more massive black hole dramatically enlarges the engine.
- Gravitational radius rg: 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. The Eddington luminosity is a radiative concept for spherical accretion balance, while a jet is a collimated outflow. So values near or even above unity do not automatically mean the estimate is unphysical. Instead, they tell you that your chosen field strength, spin, or efficiency factor implies an energetically ambitious jet.
A good way to build intuition 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 power. Likewise, because rg scales with mass and the formula uses rg2, increasing mass by a factor of 10 raises the power by roughly a factor of 100 when the other inputs are held fixed.
Limitations and common pitfalls
This page intentionally favors speed and transparency over full realism. That is a strength for learning and quick estimates, but it also means there are several places where readers should slow down and think before treating the number 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 very small or very large field choices can lead to huge swings 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 power band moves than to lean too heavily on a single estimate.
Practical FAQ
What magnetic field values are reasonable? The answer depends entirely on the system and the model you have in mind. A toy AGN estimate might use much smaller fields than a compact transient engine. Because the output scales as B2, range testing is usually more informative than a one-number guess.
Why does mass matter so much? In this scaling, the effective engine size is tied to the gravitational radius. Since rg is proportional to mass and the power depends on rg2, the estimate grows roughly with M2 when the field is held fixed.
Can jet power exceed the accretion luminosity? Yes. In strongly magnetized flows, especially in magnetically arrested disk scenarios, the jet can tap black hole spin energy and need not track the radiative output one-to-one.
What should I cite if I use this calculator? For formal scientific writing, cite Blandford and Znajek 1977 and relevant modern GRMHD work. This page is a convenience tool built around a standard scaling, not a substitute for the primary literature.
Further exploration
Continue modeling extreme environments with the black hole shadow calculator, 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 next step 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 reconcile the two.
