Astronomy & Celestial Mechanics
the sky, computed
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Eddington limit — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The main sequence against its own ceiling. Luminosity against mass, in solar units, with the Eddington limit drawn on the same axes. The limit is exactly linear in mass. The main sequence is not — until it is: above about 55 solar masses the relation flattens to a straight proportionality as well, and the two lines then run parallel at a fixed ratio of 83 per cent. They never cross on this range, which is the more interesting outcome: the most massive stars do not approach their limit gradually, they are built at a fixed fraction of it. The Sun, by contrast, radiates 0.0026 per cent of its own ceiling.

    A brightness that would blow the star apart

    Light pushes. For a star of a given mass there is a luminosity at which the outward push on its own outer layers equals the inward pull of gravity, and it depends on nothing but the mass and the opacity — not on the star's structure, its composition or its age.

    part 1 · stars
  2. A thousand times the supply, and eight times the light. The luminosity of an accretion disc, in units of its Eddington luminosity, against the rate at which gas is supplied to it, in units of the rate that would make a thin disc shine at exactly that limit — both on logarithmic axes. Below the limit the luminosity is proportional to the supply, the dashed line, with a tenth of the rest-mass energy radiated. Above it the disc cannot radiate the excess: the luminosity grows only as 1 + ln ṁ, so ten times the supply gives 3.3 times the limit, a hundred times gives 5.6, and a thousand times gives 7.9. The limit is a ceiling that bends rather than breaks — and whatever the supply is doing above it, the light does not say, because the same curve follows whether the surplus is blown away in a wind or carried into the hole.

    A ceiling on the light and not on the mass

    Pour gas onto a black hole a thousand times faster than the Eddington rate and it shines at only about eight times its limit. The rest of the energy is either blown away or swallowed, and those two fates give the same luminosity while differing by a factor of the supply in how fast the object grows — so the light a super-Eddington source emits cannot say which one it is.

    part 2 · stars
  3. A field that makes electrons transparent. The electron-scattering cross-section for photons polarised across a strong magnetic field, as a share of the free-electron value, against photon energy, for fields of 10¹², 10¹³, 10¹⁴ gauss, on logarithmic axes. Below the electron cyclotron energy — 12, 116, 1160 keV for these fields — an electron bound to the field can barely respond to a wave whose electric field is perpendicular to it, and the cross-section falls as the square of the ratio of energies. At the 10 keV of these sources' X-rays, a 10¹⁴ gauss field reduces it 13,456-fold. The Eddington limit is written with the free-electron cross-section; in such a field it is the wrong number.

    A limit written for free electrons

    The Eddington limit says a star cannot shine brighter than the point where its light pushes its own gas away, and the push is calculated for free electrons scattering light. In a magnetic field of a hundred trillion gauss an electron is not free. It is bound to the field, it can barely respond to light polarised across it, and the limit calculated for it is the wrong number by a factor of hundreds — which is one of two ways a neutron star can shine five hundred times brighter than it is supposed to.

    part 3 · stars
  4. The nebula's line, if the source is beamed. The luminosity of the He II 4686 Å line a surrounding nebula would emit, against the source's beaming fraction b, for sources that appear at 10⁴⁰ and 10⁴¹ erg/s to an observer in the beam, with 10% of their true output in photons able to ionise He⁺. The nebula sees the true luminosity, b times the apparent one, so its line falls in proportion to b: a source apparently at 10⁴¹ that is beamed into a hundredth of the sky lights its nebula like an unbeamed source at 10³⁹. Measured He II luminosities of nebulae round ultraluminous sources are 10³⁵ to 10³⁷ erg/s (shaded). On this conversion a source apparently at 10⁴⁰ whose nebula shows a few 10³⁶ needs no beaming at all; one apparently at 10⁴¹ with the same line is beamed into a tenth of the sky or less.

    A nebula that cannot be beamed at

    An X-ray source a hundred times brighter than a neutron star should be allowed to shine may be lying about its luminosity: if its light leaves through a narrow funnel pointed at the Earth, it looks as bright as a source a hundred times more powerful. A single observer cannot tell. The gas around it can. A nebula surrounding the source is lit from every direction at once and records what the source emits in all of them, and a bubble blown by its wind records the power it has put out over a million years. Each is a calorimeter no beam can fool.

    part 4 · stars

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