Engineering & Science
Poynting-Robertson Effect: Interplanetary Dust Spiral Decay Lifetime & Beta Ratio
Calculate the solar radiation-to-gravity ratio beta = 0.574 * Qpr / (rho * s) and Poynting-Robertson orbital spiral decay lifetime t_PR = 700 * s * rho * a^2 / Qpr (years) for zodiacal dust grains.
Calculate how fast sunlight sweeps micron-scale dust into the Sun!
Micron-to-millimeter asteroidal and cometary dust grains orbiting the Sun absorb radially outward sunlight, but in the Sun's rest frame they re-emit thermal photons slightly beamed forward along their orbital velocity vector (relativistic aberration of light)! Burns, Lamy, and Soter (1979) showed this tangential Poynting-Robertson drag drains orbital angular momentum continuously, causing circular orbits at heliocentric distance a (in AU) to spiral into the Sun in t_PR = 700 * s_um * rho_gcm3 * (a_AU^2) / Qpr years!
Meanwhile, radial radiation pressure opposes solar gravity with dimensionless ratio beta = F_rad / F_grav = 0.574 * Qpr / (rho_gcm3 * s_um), where s_um is grain radius in micrometers, rho_gcm3 is bulk density in g/cm^3, and Qpr (~1.0) is radiation pressure efficiency! Note that grains released from a circular parent body are blown onto unbound hyperbolic trajectories ("beta-meteoroids") whenever beta >= 0.50 (s_um <= 0.46 um for silicate rho = 2.5 g/cm^3)! A 10 um silicate grain at 1 AU has beta = 0.023 and spirals into the Sun in just 17,500 years—proving the zodiacal cloud must be continually replenished!
| Silicate Dust Grain (rho = 2.5 g/cm^3, Qpr = 1) | Radius s & Orbit a | Radiation Ratio beta | Poynting-Robertson Lifetime t_PR |
|---|---|---|---|
| Zodiacal Grain near Earth | s = 10 um, a = 1.0 AU | beta = 0.0230 (Bound) | 17,500 years to reach Sun |
| Main-Belt Asteroid Dust | s = 10 um, a = 2.0 AU | beta = 0.0230 (Independent of a!) | 70,000 years (Scales as a^2!) |
| Sub-Micron Blowout Grain | s = 0.4 um, a = 1.0 AU | beta = 0.5740 (>= 0.5 Blowout!) | Ejected as beta-meteoroid |
Poynting-Robertson Orbital Decay Lifetime t_PR (Years)
Evaluates 700 * s_um * rho_gcm3 * (a_AU^2) / Qpr in years for circular initial heliocentric distance a_AU.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Dust Grain Inputs", ROUND((700 * B2 * B3 * (B4 ^ 2)) / B5, 1))Solar Radiation-Pressure-to-Gravity Ratio beta = F_rad / F_grav
Computes Burns-Lamy-Soter beta = 0.574 * Qpr / (rho_gcm3 * s_um); beta >= 0.5 triggers hyperbolic blowout from circular parent orbits.
=IF(OR(B2 <= 0, B3 <= 0, B5 <= 0), "Invalid Dust Grain Inputs", ROUND((0.574 * B5) / (B3 * B2), 4))Try the example
Use the original values or change them to test your understanding.