Bondi Spherical Accretion Radius (RB) & Black-Hole Gas Capture Formula

Find the sonic capture sphere where gravity overcomes interstellar gas pressure!

How far away can a black hole, neutron star, or young protostar of mass M capture warm ambient interstellar gas with sound speed c_s? Hermann Bondi (1952) solved steady spherically symmetric hydrodynamic accretion to identify the gravitational capture radius R_B = (2 * G * M) / (c_s^2), where gravitational escape speed SQRT(2*G*M / r) equals the ambient sound speed c_s—accelerating infalling gas smoothly through a transonic point into supersonic free-fall!

When the compact object also moves at supersonic relative speed v_rel through the gas cloud, Bondi-Hoyle-Lyttleton accretion generalizes the denominator to R_BHL = (2 * G * M) / ((c_s^2) + (v_rel^2))! Expressing M in solar masses (M_sun = 1.98847e30 kg), speeds in km/s (* 1,000 m/s), G = 6.6743e-11 m^3/(kg*s^2), and dividing meters by 1 AU = 1.4959787e11 m: a 10 M_sun stellar black hole in warm ionized gas (c_s = 15 km/s, v_rel = 0) captures gas across R_B = 78.86 AU—shrinking 4-fold to 19.71 AU if gas heats to c_s = 30 km/s!

Stationary Bondi Accretion Radius RB = (2 * G * M) / c_s^2 in Astronomical Units (AU)

Evaluates (2 * 6.6743e-11 * M_solar * 1.98847e30) / ((c_s_kms * 1000)^2) / 1.4959787e11.

=IF(OR(B2 <= 0, B3 <= 0), "Invalid Bondi Inputs", ROUND(((2 * 6.6743e-11 * B2 * 1.98847e30) / ((B3 * 1000) ^ 2)) / 149597870000, 2))

Bondi-Hoyle-Lyttleton Moving Capture Radius R_BHL in AU

Includes bulk relative velocity v_rel (B4, km/s) via (c_s^2 + v_rel^2) in the denominator.

=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Bondi Inputs", ROUND(((2 * 6.6743e-11 * B2 * 1.98847e30) / (((B3 ^ 2) + (B4 ^ 2)) * 1000000)) / 149597870000, 2))

Further reading

• Microsoft Support — POWER function