Lecture 13: Radiation Pressure and the Eddington Luminosity

Context and Why This Matters

Every gravity-only argument so far in this course (hydrostatic equilibrium, virial theorem, orbital mechanics) has ignored radiation pressure. For an ordinary star like the Sun, that is an excellent approximation; for a compact object accreting matter (a key endpoint discussed in ASTR 230), radiation pressure can become the dominant force and impose a hard ceiling on how fast material can be accreted or how luminous a source can shine.

Learning Goals

1. Derive the radiation-pressure force on free electrons (via Thomson scattering) in a spherically symmetric radiation field.
2. Derive the Eddington luminosity as the luminosity at which outward radiation pressure exactly balances inward gravity.
3. Explain why the Eddington limit is irrelevant for normal stars like the Sun but crucial for compact accreting objects.

Opening Phenomenon

The most luminous accreting X-ray binaries and active galactic nuclei are observed to cluster just below a maximum luminosity that scales linearly with the accreting object’s mass, exactly as predicted by treating radiation pressure and gravity as competing outward and inward forces on the infalling gas.

Terms Developed in Context

These terms form the reasoning chain from observation to model, derivation, and interpretation. Each entry expands the corresponding term-explorer panel and should be read alongside the worked example.

radiation pressure

For accretion onto a compact object, exceeding L_Edd means radiation pressure exceeds gravity, halting or reversing further infall (or driving a wind); this makes L_Edd a natural, physically motivated ceiling on sustained accretion luminosity, independent of the details of the accretion flow.

Use and relationship: Sustained super-Eddington accretion (luminosities exceeding the naive Eddington limit) is observed in some systems, but only accompanied by strong outflows/winds, consistent with radiation pressure successfully driving away excess infalling material rather than simply being "overcome." The lecture's quantitative connection is \( L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

Thomson (electron) scattering

A radiation field of luminosity L exerts an outward force on free electrons (the dominant opacity source in fully ionized, hot gas) via Thomson scattering, with force per electron equal to (flux) × (Thomson cross-section)/c; because protons are tied to electrons by electrostatic attraction (charge neutrality), this force effectively acts on the ionized gas as a whole.

Use and relationship: Derive the radiation-pressure force on free electrons (via Thomson scattering) in a spherically symmetric radiation field. The lecture's quantitative connection is \( L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

Eddington luminosity

Setting this outward radiation force per unit mass equal to the inward gravitational force per unit mass (using the proton mass as the effective mass per scattering electron, since protons carry essentially all the mass in ionized hydrogen) and solving for the luminosity at which they exactly balance gives the Eddington luminosity, L_Edd = 4πGMm_p c/σ_T.

Use and relationship: Ordinary main-sequence stars like the Sun operate enormously below their own Eddington luminosity, confirming that radiation pressure is negligible compared to gas pressure in the hydrostatic balance of normal, non-accreting stars (consistent with Lecture 05’s ideal-gas treatment being adequate there). The lecture's quantitative connection is \( L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

super-Eddington accretion

Sustained super-Eddington accretion (luminosities exceeding the naive Eddington limit) is observed in some systems, but only accompanied by strong outflows/winds, consistent with radiation pressure successfully driving away excess infalling material rather than simply being "overcome."

Use and relationship: It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s. The lecture's quantitative connection is \( L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

opacity (electron-scattering)

A radiation field of luminosity L exerts an outward force on free electrons (the dominant opacity source in fully ionized, hot gas) via Thomson scattering, with force per electron equal to (flux) × (Thomson cross-section)/c; because protons are tied to electrons by electrostatic attraction (charge neutrality), this force effectively acts on the ionized gas as a whole.

Use and relationship: Setting this outward radiation force per unit mass equal to the inward gravitational force per unit mass (using the proton mass as the effective mass per scattering electron, since protons carry essentially all the mass in ionized hydrogen) and solving for the luminosity at which they exactly balance gives the Eddington luminosity, L_Edd = 4πGMm_p c/σ_T. The lecture's quantitative connection is \( L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

accretion flow

For accretion onto a compact object, exceeding L_Edd means radiation pressure exceeds gravity, halting or reversing further infall (or driving a wind); this makes L_Edd a natural, physically motivated ceiling on sustained accretion luminosity, independent of the details of the accretion flow.

Use and relationship: Sustained super-Eddington accretion (luminosities exceeding the naive Eddington limit) is observed in some systems, but only accompanied by strong outflows/winds, consistent with radiation pressure successfully driving away excess infalling material rather than simply being "overcome." The lecture's quantitative connection is \( L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

Evidence

• X-ray binaries accreting onto neutron stars and stellar-mass black holes show observed peak luminosities consistent, to within a factor of a few, with the Eddington luminosity computed from their independently measured (or inferred) compact-object masses.
• Sustained super-Eddington accretion (luminosities exceeding the naive Eddington limit) is observed in some systems, but only accompanied by strong outflows/winds, consistent with radiation pressure successfully driving away excess infalling material rather than simply being "overcome."
• Ordinary main-sequence stars like the Sun operate enormously below their own Eddington luminosity, confirming that radiation pressure is negligible compared to gas pressure in the hydrostatic balance of normal, non-accreting stars (consistent with Lecture 05’s ideal-gas treatment being adequate there).

Derivation and Model

• A radiation field of luminosity L exerts an outward force on free electrons (the dominant opacity source in fully ionized, hot gas) via Thomson scattering, with force per electron equal to (flux) × (Thomson cross-section)/c; because protons are tied to electrons by electrostatic attraction (charge neutrality), this force effectively acts on the ionized gas as a whole.
• Setting this outward radiation force per unit mass equal to the inward gravitational force per unit mass (using the proton mass as the effective mass per scattering electron, since protons carry essentially all the mass in ionized hydrogen) and solving for the luminosity at which they exactly balance gives the Eddington luminosity, L_Edd = 4πGMm_p c/σ_T.
• For accretion onto a compact object, exceeding L_Edd means radiation pressure exceeds gravity, halting or reversing further infall (or driving a wind); this makes L_Edd a natural, physically motivated ceiling on sustained accretion luminosity, independent of the details of the accretion flow.

Working Equation

\[ L_{\rm Edd} = \dfrac{4\pi G M m_p c}{\sigma_T} \]

Worked Example

1. Eddington luminosity for the Sun’s own mass: L_Edd(1 M☉) = 4π × 6.674e-11 × 1.989e+30 × 1.6726e-27 × 2.998e+08 / 6.6524e-29 = 1.257e+31 W = 3.28e+04 L☉.
2. The Sun’s actual luminosity (3.828e+26 W) is only about 3.04e-03% of its own Eddington limit, confirming radiation pressure is utterly negligible in the Sun’s hydrostatic balance.
3. Eddington luminosity for a Chandrasekhar-mass (1.4 M☉) accreting compact object: L_Edd = 1.760e+31 W = 4.60e+04 L☉ — within the observed range of peak luminosities for accreting neutron stars and stellar-mass black holes in X-ray binaries, confirming the Eddington limit’s practical relevance for exactly the compact objects introduced qualitatively in ASTR 230.

Common Misconception

Assuming the Eddington luminosity is a universal maximum luminosity for any astrophysical object. It specifically describes the balance between electron-scattering radiation pressure and gravity for a spherically symmetric, hydrogen-dominated accretion flow; genuinely super-Eddington sources exist (with strong outflows), and L_Edd scales with the emitting/accreting object’s mass, so a supermassive black hole’s Eddington luminosity is far higher than a stellar-mass compact object’s.

Active-Learning Guidance

Using L_Edd ∝ M, predict by what factor the Eddington luminosity of a supermassive black hole with M = 10⁸ M☉ exceeds that of the 1.4 M☉ compact-object example above, without recomputing from scratch.

Lab Connection

Lab 07’s capstone characterization of Alpha Centauri AB includes an explicit comparison of each star’s actual luminosity to its own Eddington luminosity, quantitatively confirming that radiation pressure is negligible for both (ordinary, non-accreting) stars.

Synthesis Questions

• What was measured directly in this lecture’s worked example, and what was derived from a physical law?
• Which assumption in the derivation would most change the interpretation if it were wrong?
• How does this lecture’s technique connect to the lab and problem set that follow it?

References

• OpenStax Astronomy 2e does not derive the Eddington luminosity at the introductory level; this course develops it from the radiation-pressure/gravity force balance, connecting to the compact-object endpoints discussed descriptively in ASTR 230, Chapter 23.
• Every numeric result above is computed programmatically in materials/ASTR310/src/generate_astr310_content.py, not hand-typed.