Compact Objects & Cosmology

Every star you see is a nuclear furnace holding back its own gravity with the outward push of fusion pressure. When the fuel runs out, gravity wins — but how it wins depends on one number: mass.

Below about 8 M☉, a dying star sheds its outer layers as a planetary nebula and settles into a white dwarf — an Earth-sized ember of carbon and oxygen held up by electron degeneracy pressure, a quantum rule that says you can't pack electrons any tighter than their wavefunctions allow. Sirius B, the companion to the brightest star in our sky, is one. So is Procyon B. These are the low-budget endings.

But electron degeneracy has a limit. Subrahmanyan Chandrasekhar showed in 1930 — aboard a steamer from Madras to England, at age 19 — that if a white dwarf's mass exceeds 1.4 M☉, its electrons are forced to move near the speed of light, the pressure they provide grows more slowly than the crushing inward weight, and the star collapses. The whole of stellar-death physics pivots on this single number. For the furnace side of the story — how the star got here and which reactions fired along the way — see Nuclear Fusion in Stars.

Above the Chandrasekhar limit, the star has two remaining options. Compress into a neutron star — a city-sized ball of neutron-degenerate matter, density 10¹⁴ g/cm³ — or, if it's heavier still, keep collapsing all the way down to a black hole.

The upper cap on neutron stars — roughly 2.3 M☉, known as the Tolman–Oppenheimer–Volkoff (TOV) limit — is fuzzier than Chandrasekhar's, because we don't fully understand matter at that density. The densest known neutron star weighs in at about 2.14 M☉. The first neutron-star collision observed by LIGO in 2017 probably produced a black hole after a brief hypermassive-neutron-star intermediary.

For any mass M, there is a radius at which the escape velocity equals the speed of light. Squeeze the mass inside that radius and nothing — not even light — can climb out.

That radius is the Schwarzschild radius, named for Karl Schwarzschild, who derived the first exact solution to Einstein's field equations in 1916 while serving on the Russian front (he died of disease a few months later). The formula is elegant:

R_s = 2GM / c²

Two G, one M, one c squared. For a solar mass, the answer is 2.95 km. For Earth, 8.9 mm. For the 4.3-million-solar-mass black hole at the centre of the Milky Way (Sagittarius A*), R_s is about 12.7 million kilometres — roughly 0.085 AU, or a fifth of Mercury's orbit. For M87's central black hole at 6.5 billion solar masses, R_s is 130 astronomical units — wider than Pluto's orbit.

The Schwarzschild radius is not an observable surface — inside it, the equations say a singular point forms at the centre; outside, general relativity is well-tested. The radius defines the event horizon, the one-way membrane across which information can cross only inward.

For 103 years we had the equations but no picture. That changed on 10 April 2019, when the Event Horizon Telescope — a planet-sized radio interferometer synthesized from dishes on five continents — released the first direct image of an event horizon: the dark silhouette of M87* against its orbiting disk of plasma. The bright ring they captured was about 2.5 times R_s wide, exactly as general relativity predicted.

A black hole by itself emits nothing. But wrap it in a disk of infalling gas and the gas shines — furiously. The disk heats by friction, radiates X-rays and ultraviolet, and that radiation pushes back on the still-infalling gas.

For every mass, there is a luminosity at which the outward radiation pressure on a free electron–proton pair exactly balances the inward gravitational pull. Push brighter than that and you blow the accreting gas away before it can reach the compact object.

This ceiling is the Eddington luminosity, after Sir Arthur Eddington, who first wrote it down in 1916 while studying radiation pressure inside stars. It scales cleanly with mass:

L_Edd ≈ 3.2 × 10⁴ (M / M☉) L☉

A one-solar-mass accretor can't steadily outshine 32,000 Suns. A 10-million-solar-mass quasar can't outshine 3 × 10¹¹ Suns — still far brighter than any normal galaxy in starlight, which is why quasars were first mistaken for stars at cosmic distances when Maarten Schmidt cracked their spectra in 1963.

Reverse the argument and the Eddington limit becomes a weighing scale: a quasar's observed brightness tells you a lower limit on the mass of its central engine. If it shines at 10¹³ L☉ and respects Eddington, the black hole must be at least 3 × 10⁸ M☉. The most luminous quasars we see require SMBHs of 10⁹ M☉ and up — exactly what reverberation mapping and stellar dynamics find when we measure the masses directly.

Einstein's theory of general relativity — published in November 1915, while artillery rolled past his Berlin window — said that mass curves the geometry of spacetime and that light follows that geometry. A star passing near the Sun's limb would appear displaced by 1.75 arcseconds, twice the Newtonian prediction.

On 29 May 1919, during a solar eclipse visible from Sobral in Brazil and the island of Principe off West Africa, Arthur Eddington's two teams measured exactly that displacement. General relativity beat Newton. The story made the front page of The Times.

Any mass will do this to any light. Today we use the effect as a natural telescope:

• Strong lensing — a foreground galaxy cluster bends the light of a background galaxy into multiple images, arcs, or a complete Einstein ring. Abell 2218 and Abell 1689 are the gold standard, with dozens of lensed background galaxies in a single Hubble image.
• Weak lensing — distant galaxy images are slightly stretched by foreground mass. Statistical analysis of millions of galaxy shapes maps the invisible mass between us and the sources.
• Microlensing — a foreground star briefly magnifies a background star as it passes across the line of sight. The OGLE and MOA surveys use this effect to hunt exoplanets and probe the dark halo for compact objects.

In 1929, Edwin Hubble — working with Milton Humason's redshift measurements at Mount Wilson — published a striking correlation. The more distant a galaxy, the faster it recedes from us. Plot velocity against distance and the data fit a line through the origin.

v = H₀ d

The slope H₀ is the Hubble constant. Hubble's original value was roughly 500 km/s/Mpc, but his distance calibration was badly wrong. Today, we measure H₀ at about 70 km/s/Mpc — a galaxy at 100 Mpc (about 326 million light-years) recedes at 7000 km/s.

Hubble's Law is not galaxies flying away from us through static space — it's space itself expanding between every pair of galaxies. There's no centre. From any galaxy in the observable universe, every other galaxy is receding, faster the farther you look. It's a common mistake to picture an explosion in a pre-existing void; picture a raisin cake rising in an oven instead — every raisin sees every other raisin getting farther away, and no raisin is at the "centre".

There is currently a real tension in the field: the CMB-based method gives H₀ ≈ 67 km/s/Mpc, while the local distance-ladder method (Cepheids → Type Ia supernovae) gives H₀ ≈ 73 km/s/Mpc. This Hubble tension is either a systematic error someone hasn't found yet, or a hint that our cosmological model is missing something. It is one of the hottest open problems in cosmology.

Run Hubble's Law backward. If everything is flying apart today, everything was squeezed together yesterday — and further back, the universe was hot.

When the universe was about 380,000 years old, it had cooled to roughly 3000 K, cool enough for free protons and electrons to combine into neutral hydrogen. The fog lifted. Photons that had been scattering off free electrons every few femtoseconds suddenly streamed free. They've been travelling through an expanding universe ever since, stretching with the geometry, cooling from 3000 K to 2.725 K today.

This is the cosmic microwave background, the universe's baby picture — redshifted by a factor of about 1090 over 13.8 billion years.

Arno Penzias and Robert Wilson stumbled onto the CMB in 1964 at Bell Labs, mistaking it for antenna noise and famously chasing pigeons out of their horn antenna before they accepted the signal was real. It came in isotropically, from every direction. They asked Princeton's Robert Dicke for help; Dicke — who had been planning his own CMB search — listened to their description, hung up the phone, turned to his team and said, "Well boys, we've been scooped." Penzias and Wilson got the Nobel in 1978.

Since then, COBE (1992) showed the CMB is a perfect blackbody to 10⁻⁴ precision — no astrophysical process can produce a spectrum that clean; it has to be thermal equilibrium. COBE also found the tiny (10⁻⁵) temperature anisotropies that seeded every galaxy, galaxy cluster, and supercluster we see today. WMAP (2001) and Planck (2013) measured those anisotropies down to arcminute resolution, and pulled out the composition of the universe from their statistical pattern.

In 1933, Fritz Zwicky measured the velocities of galaxies in the Coma cluster and found them moving so fast that the cluster's visible mass could not possibly be holding them in. He coined the term dunkle Materie — dark matter — and was roundly ignored for forty years.

Vera Rubin revived the idea in the 1970s from an entirely different angle: galaxy rotation curves. Her measurements of the rotational speed of stars and gas out to the edge of Andromeda, the Milky Way, and dozens of other spirals all showed the same shocking pattern. Past the visible disk, the orbital speed didn't fall off the way Keplerian orbits require — it stayed essentially flat.

Interpret that curve with Newton and gravity alone and you are forced to the conclusion: there is more mass at large radii, distributed in a roughly spherical halo extending far beyond the visible stars. It doesn't emit light. It doesn't absorb light. It doesn't scatter light. It has mass and it has gravity and that's all we can see of it — but every spiral galaxy is embedded in such a halo, weighing five to ten times the stars.

This is not a fringe claim. The evidence stacks from independent directions:

• Rotation curves of thousands of galaxies.
• Velocity dispersions of galaxy clusters (Zwicky's original argument).
• Gravitational lensing by galaxy clusters — the mass inferred from lensing exceeds the visible mass by the same factor of 5–10.
• The Bullet Cluster — two galaxy clusters caught in the act of collision. The gas (visible in X-ray) got slowed by friction; the dark matter (inferred from lensing) sailed through untouched and is now spatially offset from the gas. You can see the two populations separated on the sky.
• The CMB power spectrum — the distribution of temperature fluctuations at different angular scales encodes the matter content of the early universe. It nails ordinary matter at 5% and total matter at ~32% of the cosmic energy budget. The missing 27% has to be something that doesn't couple to photons.

What is it? We don't know. The main candidates — weakly interacting massive particles (WIMPs), axions, and primordial black holes — have been increasingly constrained by decades of direct-detection experiments. The boring answer (ordinary compact halo objects — free-floating planets, brown dwarfs, small black holes) has been ruled out by microlensing surveys. Modified gravity theories like MOND fit galaxy rotation curves handsomely but fail on cluster scales, where the Bullet Cluster is their nemesis.

Five percent of the universe is ordinary atoms. Of that five percent, only a small fraction shines as starlight. When you look at the night sky you are seeing a thin luminous veneer on the ordinary-matter slice of a universe whose majority composition — dark matter and dark energy — you cannot see at all, by any technology we have.

But the veneer is enough to work with. Every photon that reaches your eye tonight:

• was emitted by a star whose mass is below the TOV limit (otherwise it would have collapsed before shining at you),
• travelled through a spacetime curved by dark-matter haloes at every scale along its path,
• was redshifted by a small amount by cosmic expansion if the source was extragalactic,
• was deflected — very slightly — by the gravity of the Sun, the Galactic centre, and any foreground mass along the way,
• is now one more photon in a night sky bathed in 400 CMB photons per cubic centimetre, 13.8 billion years old.

The night sky is not a backdrop. It is the physics, acting on you.

Ten observational hooks into the big physics, from naked eye to long-term project:

• Naked eye — Go outside, face the zenith. The sky you see looks dim, but every cubic centimetre of it holds roughly 400 CMB photons from the epoch of recombination. You are immersed in the afterglow.
• Naked eye — Our motion through the CMB rest frame points toward Leo at about 370 km/s. That's the "CMB dipole" — one hemisphere of sky is 3.4 mK warmer than the other because of our velocity.
• Binoculars — Sweep the Coma Berenices / Virgo border. You're looking at the Coma and Virgo galaxy clusters, the same clusters Zwicky weighed in 1933 and got dark matter as the price.
• 4-inch scope — M1, the Crab Nebula in Taurus. At its heart is a pulsar — a neutron star spinning 33 times per second, a testbed for matter at 10¹⁴ g/cm³.
• 8-inch scope — M87 in Virgo. You can see the core; from its 6.5-billion-solar-mass black hole, general relativity curves the spacetime of a region larger than Pluto's orbit.
• 8-inch scope — M31, the Andromeda Galaxy. Its rotation curve — Vera Rubin's 1970s work — was the cleanest early evidence for dark matter. You're looking at a dark-matter halo five times more massive than everything you can see.
• 14-inch scope, CCD — the Einstein Cross in Pegasus. Four images of one quasar, split by a foreground galaxy 400 million light-years away.
• Astrophotography project — image a rich cluster like Abell 2218. The giant arcs in your frames are background galaxies smeared around the cluster's gravity well.
• Photometry project — time a Type Ia supernova in a nearby galaxy for AAVSO. Every clean light curve is a data point on the Hubble diagram.
• Patient long-term — revisit M87 each year and remember that the Event Horizon Telescope image is now fully corroborated. The dark disk at its heart is 2.6 times the Schwarzschild radius wide, exactly as Einstein's equations predicted in 1916.