🕳️Black Hole Calculator - Schwarzschild Radius, Hawking Radiation, Orbits, Lensing

How to use Black Holes & Gravitational Lensing

1. 1In 'Black hole calculator' pick a preset or enter a mass, then set the distance, body length and tidal test position
2. 2Read the characteristic radii, shadow size, Hawking radiation figures and tides, and see whether you would be spaghettified before the horizon
3. 3In 'Time dilation' move the position slider to see clock rates, redshift, orbital and escape speeds
4. 4In 'Orbits & light' choose a preset or set r₀ and angular momentum and press Play; switch to light rays to vary the impact parameter
5. 5In 'Gravitational lensing' drag the lens over the background galaxy to form an Einstein ring, and compute θ_E for real scenarios

FAQ

How is the Schwarzschild radius calculated?

r_s = 2GM/c² is the event horizon radius of a non-rotating black hole. One solar mass gives about 2.95 km, Earth's mass only about 8.9 mm, and M87* at 6.5 billion solar masses about 128 AU, larger than Pluto's orbit. Because the radius grows in proportion to mass, the "mean density" of a supermassive black hole is low: for M87* it is less than half the density of air.

Why is the shadow larger than the horizon, and does it match the EHT images?

Near the photon sphere (1.5 r_s) light can circle the hole, and every ray with impact parameter below b_c = (3√3/2) r_s ≈ 2.6 r_s is captured. So the dark region seen from afar has a radius of about 2.6 r_s. For M87* (6.5 billion M☉ at 16.8 Mpc) the shadow is about 40 µas across and for Sgr A* about 53 µas; the EHT rings measure about 42 µas and 52 µas, slightly larger because the ring is glowing gas just outside the shadow.

What do the Hawking temperature and evaporation time mean?

Quantum effects make a black hole radiate like a blackbody at T = ħc³/(8πGMk_B), hotter for smaller masses. Stellar black holes sit at a few to a few tens of nanokelvin, far colder than the 2.7 K cosmic microwave background, so today they absorb more than they emit. The evaporation time t = 5120πG²M³/(ħc⁴) scales with mass cubed: about 2×10⁶⁷ years for one solar mass, and only a primordial hole of about 1.7×10¹¹ kg would be finishing now (photons only; including all particles shortens it).

Why do the orbits precess when Newtonian orbits do not?

In Schwarzschild spacetime the orbit equation gains one term: d²u/dφ² = M/L² − u + 3Mu² with u = 1/r. It shifts the periapsis forward every orbit; it explains Mercury's 43 arcseconds per century, and near a black hole the shift can exceed a hundred degrees per orbit. The same term puts a peak in the effective potential: below L = √12 GM/c there are no circular orbits and everything plunges, and 6GM/c² is the innermost stable circular orbit. The tool overlays the Newtonian ellipse with the same starting conditions.

When does an Einstein ring appear?

When source, lens and observer are almost perfectly aligned, light bends around every side of the lens and the source is stretched into a ring of radius θ_E = √(4GM/c² · D_ls/(D_l D_s)). A small offset breaks the ring into two images at θ± = (β ± √(β² + 4))/2 in units of θ_E, with total magnification μ = (u² + 2)/(u√(u² + 4)); microlensing surveys use this to find exoplanets. A 0.5 M☉ star in our galaxy gives θ_E under a milliarcsecond, while galaxy clusters reach tens of arcseconds.