Eddington Radiation Force Balance

Worked example: Balanced forces

Let's calculate the forces for a model star where Mass = 1, Opacity = 1, Scaled luminosity = 1, and Radius = 1. The equations use reduced units where G=c=1.

1. Calculate the inward gravity: Gravity magnitude is M / r². Using M = 1 and r = 1, the inward gravity is 1 / 1² = 1.0.
2. Calculate the outward radiation acceleration: Acceleration is k · L_scaled / r². Using k = 1 and L_scaled = 1, the outward acceleration is (1 · 1) / 1² = 1.0.
3. Calculate the net outward acceleration: (k · L_scaled - M) / r². The net force is (1.0 - 1.0) = 0.0.
4. Calculate the critical scaled luminosity: M / k. This is 1 / 1 = 1.0.
5. Calculate the Eddington ratio (Γ): k · L_scaled / M. This is (1 · 1) / 1 = 1.0.

Since the net outward acceleration is 0.0 and Γ = 1.0, the forces are exactly balanced at this radius.

Experiment with radiation pressure and gravity

Open Eddington Radiation Force Balance to explore how these parameters affect the star's stability.

1. Set Mass to 1, Opacity to 1, and Scaled luminosity to 1. Notice that the Eddington ratio is 1.0 and the force regime is "Balanced". The inward blue arrows and outward red arrows in the visualization have the same length.
2. Increase the Scaled luminosity to 2. The Eddington ratio becomes 2.0, and the force regime changes to "Radiation dominates". The outward radiation acceleration (2.0) now exceeds the inward gravity (1.0).
3. Reset the Scaled luminosity to 1, and decrease Opacity to 0.5. The Eddington ratio drops to 0.5. With less opacity, the radiation exerts less force on the matter, and the force regime becomes "Gravity dominates".
4. Keep Mass at 1, Opacity at 0.5, and Scaled luminosity at 1. Change the Radius from 1 to 2. Both inward gravity and outward radiation decrease by a factor of 4 (from 1.0 to 0.25, and from 0.5 to 0.125, respectively) due to the inverse-square law. However, the Eddington ratio remains 0.5. The balance between radiation pressure and gravity is independent of the radius.

Assumptions and Limitations

This is an idealized model. At Γ = 1, the forces cancel at every radius, but this is not a proof of stable hydrostatic structure or a strict observed luminosity ceiling.

Real stars are significantly more complex. This simulation assumes a constant mass opacity (k) and spherical isotropic luminosity. It does not include a real stellar opacity database, and ignores effects like convection, rotation, or anisotropic accretion. Positive masses, radii, and opacities are assumed, while a luminosity of 0 is permitted as a boundary case.

References

For a detailed exploration of stellar astrophysics, radiation pressure, and the Eddington limit, consult the primary course notes used to inform this model:

Notes on Stellar Astrophysics by Edward F. Brown. These notes provide the theoretical derivation of the radiative momentum and the critical luminosity limit (G=c=1 reduced units).