What this calculator estimates
A black hole does not glow like a normal star, yet it can still cast a visible silhouette. Light from hot plasma, an accretion flow, or a bright background source can pass near the hole and be strongly bent by gravity. Some photon paths escape and reach the observer, while others are trapped. The dividing line between escaping rays and captured rays creates a dark patch on the sky called the black hole shadow. In practical images, especially those discussed in connection with the Event Horizon Telescope, that shadow is usually surrounded by a bright ring or crescent produced by lensed emission rather than by a solid physical surface.
The most important intuition is that the observed size depends mainly on the ratio M/D: mass divided by distance. A larger mass makes the shadow larger, while a larger distance makes it smaller. That is why two very different objects can look surprisingly similar on the sky. Sagittarius A* at the center of our own galaxy is modest compared with the giant black hole in M87, but it is also vastly closer, so both targets end up in the same rough tens-of-microarcseconds range. This simple scaling is exactly what the calculator helps you explore.
The page assumes a Schwarzschild black hole, meaning a non-rotating, spherically symmetric solution. That choice is deliberate: it gives a clean analytic estimate that is widely used for back-of-the-envelope work. Real black holes can spin, tilt relative to the observer, vary in brightness, and sit inside turbulent plasma, but the Schwarzschild result remains a very useful reference size. If you are asking a question like “Is this target even in the right ballpark for horizon-scale imaging?” this simplified model is often the right place to start.
How to use it. Enter the black hole mass M in solar masses and the distance D in parsecs, then choose Compute Angular Size. The result table shows the gravitational radius rg, the physical shadow diameter dsh, the angular diameter in µas, and a short observing note that compares the answer with a rough Earth-sized 230 GHz very-long-baseline interferometry resolution scale. Scientific notation is accepted, so values such as 4.1e6 or 1.68e7 work well. If your distance is given in kiloparsecs or megaparsecs, convert it first: 1 kpc = 1,000 pc and 1 Mpc = 106 pc.
Formula, units, assumptions, and worked examples
The calculation begins with the gravitational radius, often written as rg = GM/c2. For a Schwarzschild black hole, the apparent shadow radius is √27 · rg, so the full physical shadow diameter becomes dsh = 2√27 · (GM/c2), or about 10.392 · GM/c2. Once the physical diameter is known, the sky size follows from the small-angle approximation: physical size divided by distance. Because the angles involved are tiny, that approximation is excellent for this purpose.
where D is the distance to the black hole. The calculator then converts radians into microarcseconds using 1 rad = 206,265,000,000 µas. The mass input is converted with M☉ = 1.98847×1030 kg, the distance input is converted with 1 pc = 3.08567758×1016 m, and the constants used are G = 6.6743×10−11 m3·kg−1·s−2 and c = 299,792,458 m/s.
Why the units matter. Astronomical sizes often span absurd ranges, so a clean input convention prevents hidden mistakes. If you accidentally enter megaparsecs as though they were parsecs, the angular size will be too large by a factor of one million. The same caution applies to light-years: a common conversion is 1 pc ≈ 3.26 ly. The linear scaling is simple but unforgiving. Double the mass and the angular size doubles; double the distance and the angular size halves. There is no complicated threshold hidden inside this calculator—just a very compact relation expressed in physically meaningful units.
Worked example: Sagittarius A*. Take M = 4.1×106 M☉ and D = 8,200 pc. The gravitational radius comes out to millions of kilometers, the shadow diameter is about ten times larger than GM/c2, and the final angular diameter lands near 50 µas. That is exactly why Sgr A* became one of the prime Event Horizon Telescope targets: it is not the most massive known black hole, but it is close enough for its shadow to appear large on the sky.
Worked example: M87*. For M87*, use M ≈ 6.5×109 M☉ and D ≈ 16.8 Mpc, which is 1.68×107 pc. Despite the enormous mass, the much larger distance pushes the angular diameter down to roughly 42 µas. The result is still horizon-scale and imageable, but it makes the same point in a different way: giant mass helps, distance hurts, and the balance between those two quantities is the whole story for a first estimate.
Worked example: a stellar-mass black hole in the Milky Way. A 10 M☉ black hole at 1 kpc gives an angular diameter on the order of 0.1 µas. That is dramatically smaller than the supermassive examples above. It explains why current horizon imaging efforts focus on very massive black holes rather than nearby X-ray binaries: even when a stellar-mass system is close in galactic terms, its gravitational length scale is so small that the shadow remains far below the angular resolution available to Earth-sized millimeter arrays.
Reference values. The table below gives a few benchmark targets using the same Schwarzschild estimate. Treat the numbers as orientation rather than as final published values. In real work, you would want the latest mass measurement, the right distance definition, and a model of source brightness and scattering, but a table like this is useful when you need a sanity check before going deeper.
| Object | Mass (M☉) | Distance (pc) | Angular Diameter (µas) |
|---|---|---|---|
| Sgr A* | 4.1×106 | 8,200 | 50 |
| M87* | 6.5×109 | 1.68×107 | 42 |
| Stellar-mass BH (10 M☉) at 1 kpc | 10 | 1,000 | 0.1 |
How to read the answer and avoid common mistakes
After you press the button, the result table gives four pieces of information. First comes the gravitational radius rg, which is the natural length scale associated with the black hole mass. Second comes the physical shadow diameter, which is the size of the shadow in meters if you could somehow place a ruler across it. Third comes the angular diameter in microarcseconds, which is the quantity observers care about most because it says how large the feature appears on the sky. Finally, the page adds a short observational note based on a rough 20 µas benchmark associated with Event Horizon Telescope class baselines at 230 GHz. That note is intentionally simple. It is a rule of thumb, not a guarantee of image quality.
If your answer is comfortably above about 20 µas, the target is at least in the broad size range where present or near-term horizon-scale VLBI becomes plausible. If it is much smaller, the source may still be scientifically interesting, but resolving the shadow itself becomes far more difficult. Brightness, variability, interstellar scattering, wavelength, baseline coverage, calibration quality, and source geometry all matter in addition to angular size. A large predicted shadow can still be hard to image if the emission is faint or distorted, while a target close to the nominal limit might become more accessible with shorter observing wavelengths, extra stations, or future space baselines.
Assumptions and limitations. This calculator is a first-order estimate. It assumes a Schwarzschild black hole, so it ignores spin and viewing inclination. Real Kerr black holes can produce shadows that are slightly distorted rather than perfectly circular, and the center of the shadow can shift relative to the bright ring. The apparent image also depends on the accretion flow, magnetic fields, and radiative transfer. Toward the Galactic center, interstellar scattering can blur fine structure. Even before any of those astrophysical complications enter, uncertainties in M and D feed directly into the result because the angular size scales as M/D. A 10% change in mass changes the angular size by roughly 10%, and the same direct dependence applies to distance.
For the same reason, this tool is not a parameter-fitting engine and not a substitute for relativistic image synthesis. It does not choose between luminosity distance and angular-diameter distance for high-redshift objects, and it does not incorporate cosmological models automatically. If you are working with a distant active galactic nucleus, you must decide which distance measure is appropriate before entering a value. If you are preparing a paper or proposal, think of this page as a fast screening tool: excellent for intuition and comparison, but intentionally lightweight.
Is the shadow the same as the event horizon? No. The event horizon is the true causal boundary in spacetime. The shadow is an apparent dark region on the sky created by gravitational lensing and photon capture. In the Schwarzschild case, the shadow radius corresponds to √27 rg, which is larger than the horizon radius 2rg.
Why use microarcseconds? Because these objects are unbelievably small on the sky. One arcsecond is already only 1/3600 of a degree, and one microarcsecond is a millionth of that. Shadows large enough to image with Earth-sized baselines still measure only a few tens of µas.
What if I know distance in light-years, kiloparsecs, or megaparsecs? Convert first, then enter the value in parsecs. A handy rule is 1 pc ≈ 3.26 ly. For example, 26,000 light-years is roughly 7,975 pc, while 16.8 Mpc becomes 1.68e7 pc. Scientific notation is accepted by the form, so those conversions can be typed directly.
Can this page handle cosmology or Kerr ray tracing? Not directly. It is intentionally focused on the simplest and most transparent estimate. That is a strength when you need to understand the scaling, but it also means you should move to a more detailed model if spin, inclination, redshift, or detailed image structure are central to your question.
Related calculators. If you want to keep building intuition, continue with the Schwarzschild radius calculator, explore time-rate effects with the gravitational time dilation tool, or compare gravitational speeds using the escape velocity calculator. Together, those tools connect the same mass scale to size, timing, and orbital behavior, which makes the shadow estimate easier to interpret in a broader relativistic context.
Mini-game: Shadow Lock Observatory
This optional canvas game turns the same mass-to-distance idea into a quick telescope challenge. Each target presents a black hole mass and distance. Your job is to resize the orange observation ring until it matches the glowing blue shadow ring and hold steady until the lock meter fills. Larger mass or smaller distance means a larger apparent shadow, so the fastest way to score is to read the numbers, anticipate the size change, and keep your aim calm as the observing conditions get more chaotic.
Click Start game to begin your observing run.
Educational takeaway: the calculator and the game both rely on the same scaling, θ ∝ M/D. A more massive black hole casts a larger shadow, but increasing the distance shrinks that apparent size just as directly.