Vol. Science · Astronomy · Horizon

Black holes
one radius

A non-rotating black hole has one length that sets the rest. The Schwarzschild radius is r_s = 2GM/c². Light can circle, unstably, at the photon sphere, 1.5 r_s. The innermost stable circular orbit for a massive particle is the ISCO, 3 r_s. Slide the mass from one solar mass to ten billion. The drawing is in units of r_s, so the circles keep their shape and the kilometers change. A second control places a hovering observer. That clock runs at √(1 − r_s/r) compared with a clock far away. The stretch across a 2 m body at the horizon is 2GM L / r³. Hawking temperature and an evaporation time are order-of-magnitude figures. Newtonian escape speed reaches c at this same radius — the escape desk. A high-mass star’s rough end is the stellar lifetime desk. Curvature, named without these radii, is gravity. A wavelength shift from motion or expansion is redshift.

Schwarzschild geometry. Educational approximation. Real astrophysical black holes spin, so the spacetime is Kerr and the photon orbit and the ISCO move. Bardeen, Press, and Teukolsky 1972 put the photon orbit at 3M when the spin is zero, and at M (prograde) or 4M (retrograde) when the spin is maximal. Their zero-spin stability condition is r > 6M, which is 3 r_s. This page holds the two orbits at 1.5 r_s and 3 r_s. It does not draw a disk, a jet, or a shadow. The 1 and 10 solar-mass buttons are computed anchors. Named masses are cited central values. Radii, the tide, and the temperature use the IAU nominal GM☉. Density and the evaporation time also divide by the CODATA value of G.

Literacy · the numbers on the readout

Schwarzschild radius

r_s = 2GM/c². In geometric units the horizon is r = 2M, so M = r_s/2. One nominal solar mass uses the IAU 2015 mass parameter GM☉ = 1.3271244 × 10²⁰ m³ s⁻² and c = 299792458 m/s. That product is 2.953 km. G does not appear separately, because the mass parameter is already GM.

Photon sphere

The unstable circular orbit for light, at 1.5 r_s (3M) when the hole does not spin. A slight nudge and the ray falls in or escapes. The shadow a telescope might see is larger than this circle, because of light bending. This page draws the orbit, not the shadow.

ISCO

Innermost stable circular orbit. For Schwarzschild it is 3 r_s (6M). Inside it, a circular orbit is unstable and a slight inward nudge becomes a plunge. Gas in a thin disk is often described as drifting in until it reaches this orbit. Spin moves the orbit.

Clock factor

√(1 − r_s/r) is dτ/dt for a static observer, relative to proper time at infinity. At 10 r_s it is √0.9 = 0.9487. The same factor is a gravitational redshift, z = 1/f − 1. The redshift desk uses z = (λ_obs − λ_rest)/λ_rest for motion and for cosmic expansion. That z is a different measurement.

Horizon tide

2GM L / r³ at r = r_s, L = 2 m. Because r_s scales with M, the tide scales as 1/M². The “g” column divides by 9.80665 m/s², the conventional standard gravity. It is a comparison, not a second measurement of the hole.

Mean density

M / (⁴⁄₃ π r_s³), with M = GM☉ / G and G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², the 2022 CODATA central value. “× water” divides by 1000 kg/m³. The quotient falls as 1/M². It is not the density of any material at the horizon.

Hawking temperature and evaporation

T = ħ c³ / (8 π k GM). ħ = h/(2π), with h and k the exact SI constants. The lifetime row is t = 5120 π G² M³ / (ħ c⁴), reported in Julian years of 31,557,600 s. That coefficient is the horizon-area Stefan–Boltzmann estimate. It is an order of magnitude. A hole colder than the cosmic microwave background gains energy from that radiation while the bath is warmer than T.

1 and 10 solar masses

Computed. The Sun’s nominal mass parameter, and ten times it. NASA’s classroom split leaves a white dwarf below about 8 solar masses and a neutron star from about 8 to 20. Neither button is a claim that such a hole exists at that mass.

Cited holes

Cygnus X-1, 21.2 ± 2.2 M☉, Miller-Jones et al. 2021. Sagittarius A*, (4.297 ± 0.012) × 10⁶ M☉ statistical, systematics about 40,000 M☉, GRAVITY Collaboration 2022. M87*, 6.5 ± 0.2 (stat) ± 0.7 (sys) × 10⁹ M☉, EHT 2019 Paper VI. The readout uses the central value.

Sources · the constants and the masses

Classroom Schwarzschild arithmetic. r_s = 2GM/c² with the IAU 2015 nominal GM☉ and the exact speed of light. Photon sphere 1.5 r_s. ISCO 3 r_s. The clock factor is √(1 − r_s/r). The horizon tide is 2GM L/r³ across 2 m. Hawking temperature uses ħ, c, k, and GM. The evaporation time is the horizon-area order of magnitude, in Julian years. Density uses 2022 CODATA G. Named masses are the cited central values. Real holes spin. Earthrise preview: NASA / Bill Anders (public domain).

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