ASTR 310 · Lecture 13

Radiation Pressure and the Eddington Luminosity

When starlight itself pushes back against gravity

Learning Goals

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

Reading anchor: 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.

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.

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.

First question: what here is directly observed, and what follows only once a physical law is derived and applied?

Terms for This Lecture

radiation pressure

Definition: 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.

Why it matters here: The Eddington luminosity is the direct consequence of balancing electron-scattering radiation pressure against gravity, and while it is many orders of magnitude above any normal star’s actual luminosity, it becomes a hard, physically meaningful ceiling for luminous accreting compact objects.

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."

Boundary or common 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 We Need to Explain

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.

Quantitative Tool

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

Worked Example

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☉.
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.
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.

Visual Reasoning

Study how this lecture’s figure turns the equation and worked example above into a concrete quantitative picture.

Which numbers in the figure come directly from measurement, and which are computed from this lecture’s physical law?
Where do the marked points match the worked-example values above?
What would change in the figure if a key assumption in the derivation failed?

The Eddington luminosity is the direct consequence of balancing electron-scattering radiation pressure against gravity, and while it is many orders of magnitude above any normal star’s actual luminosity, it becomes a hard, physically meaningful ceiling for luminous accreting compact objects.

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 Segment

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