Engineering & Science
Neutron Star Compactness (2GM/Rc^2), Gravitational Redshift & Buchdahl Limit
Calculate general-relativistic stellar compactness Xi = (2 * G * M) / (R * c^2), exact Schwarzschild surface gravitational redshift z_grav = ((1 - Xi)^-0.5) - 1, and verify Buchdahl's stability bound Xi < 8/9.
Weigh pulsars with X-ray gravitational redshift and test Buchdahl's 8/9 collapse limit!
A canonical neutron star packs 1.4 solar masses into a city-sized sphere of radius R = 12.0 km—so dense that Newtonian gravity fails by over 30%! General relativity characterizes relativistic gravity via dimensionless compactness Xi = R_s / R = (2 * G * M) / (R * c^2), where R_s = 2.95334 * (M / M_sun) km is the Schwarzschild radius! An X-ray photon climbing out of that deep spacetime well loses energy, arriving at NASA's NICER telescope with exact gravitational surface redshift z_grav = (1 / SQRT(1 - Xi)) - 1!
Even before reaching a black-hole horizon (Xi = 1.0000), Hans Buchdahl (1959) proved that ANY static isotropic fluid sphere with Xi >= 8/9 = 0.8889 suffers infinite central pressure and MUST collapse into a black hole! At M = 1.40 M_sun and R = 12.0 km, Xi = 0.3446 and z_grav = 0.2352 (23.5% wavelength stretch!); for a massive 2.00 M_sun pulsar at R = 11.5 km, Xi jumps to 0.5136 and z_grav reaches 0.4339!
| Compact Object Configuration | Compactness Xi = 2GM/(Rc^2) | Surface Redshift z_grav | General-Relativistic Regime |
|---|---|---|---|
| Canonical Pulsar (1.4 M_sun, R = 12.0 km) | 0.3446 | z = 0.2352 (+23.5%) | Stable Neutron Star (< 8/9) |
| Heavy Pulsar (2.0 M_sun, R = 11.5 km) | 0.5136 | z = 0.4339 (+43.4%) | Stiff EOS Neutron Star (< 8/9) |
| Buchdahl Collapse Violated (2.0 M_sun, R = 6.0 km) | 0.9844 (>= 8/9!) | Unstable Collapse! | "Exceeds Buchdahl Stability Limit (8/9)!" |
Relativistic Compactness Parameter Xi = (2 * G * M) / (R * c^2)
Evaluates (2 * 6.6743e-11 * M_solar * 1.98847e30) / (R_km * 1000 * 299792458^2).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Star Inputs", ROUND((2 * 6.6743e-11 * B2 * 1.98847e30) / (B3 * 1000 * (299792458 ^ 2)), 4))Gravitational Surface Redshift z_grav with Buchdahl Stability Check (Xi < 8/9)
Flags relativistic collapse if compactness Xi >= 8/9, otherwise evaluates (1 / SQRT(1 - Xi)) - 1.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Star Inputs", IF(((2 * 6.6743e-11 * B2 * 1.98847e30) / (B3 * 1000 * (299792458 ^ 2))) >= (8 / 9), "Exceeds Buchdahl Stability Limit (8/9)!", ROUND((1 / SQRT(1 - ((2 * 6.6743e-11 * B2 * 1.98847e30) / (B3 * 1000 * (299792458 ^ 2))))) - 1, 4)))Try the example
Use the original values or change them to test your understanding.