Galaxies: the Wells and the Stars Around Them

Where do the stars of a universe gather, and why do the gatherings take the shapes they do? In ours, nearly every star lives in a galaxy, a system of millions to trillions of stars held by the gravity of its dark matter, with a well at its heart. Galaxies come in a small number of shapes, and what decides the shape is known: spin, collisions, feedback and surroundings. This part sets out the types, the causes, the wells at their centres and the brightest things in the universe, the quasars, and reads each in Binary Pulse Theory.

The Hubble Sequence

Hubble sorted the galaxies he could resolve into a sequence that still frames the subject (Hubble, 1926; Hubble, 1936), which de Vaucouleurs extended (de Vaucouleurs, 1959):

• Ellipticals (E0 to E7): smooth, round to elongated, of old red stars with little gas; their stars orbit every way at once rather than in a disk. The giants sit at the hearts of clusters of galaxies.
• Lenticulars (S0): a disk with no spiral arms and little gas, between the ellipticals and the spirals.
• Spirals (Sa to Sd): a turning disk with arms, from Sa, with a large central bulge and tight arms, to Sd, with almost no bulge and loose arms. About two thirds carry a bar of stars across their centre (SB, and SAB for a weak bar); our own galaxy is a barred spiral.
• Irregulars (Irr, Im): no settled shape, rich in gas and forming stars, often small or disturbed.

Beyond the Sequence

• Dwarfs, by far the most numerous: dwarf ellipticals, dwarf spheroidals, the faintest of all and almost entirely dark matter, dwarf irregulars and blue compact dwarfs.
• Ultra-diffuse galaxies: as wide as ours but a hundred times fainter (van Dokkum et al., 2015).
• Starbursts: forming stars a hundred times faster than usual, most often after a collision.
• Ring and peculiar galaxies: shaped by encounters, collected in Arp's atlas (Arp, 1966); a ring forms where a small galaxy has passed straight through a disk.
• Active galaxies: galaxies whose central well is feeding, below.

What Sets the Type

Spin. Every clump of a young universe is spun up by the tidal pull of its neighbours (Part 3.12). Its spin is measured by a pure number, the spin parameter, which simulations and observations find spread about 0.035 (Peebles, 1969; Bullock et al., 2001).

Halo Spin Parameter G

λ = J · E^(1/2) / (G · M^(5/2))

How fast a clump turns, against how fast it would have to turn to hold itself up

Where:

• λ [∅] — the spin parameter
• J [𝕄·𝕃²·𝕋⁻¹] — the clump's angular momentum
• E [𝕄·𝕃²·𝕋⁻²] — the size of its total energy, which for a bound clump is negative
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the gravitational constant
• M [𝕄] — its mass

➢ A clump of λ = 1 would be held up by its turning alone; real ones turn at a few hundredths of that. It is small, but it decides everything after, because gas that cools keeps its spin.

Gas that cools in a spinning clump sinks until its spin holds it up, and settles into a disk whose size its spin sets (Fall & Efstathiou, 1980; Mo, Mao & White, 1998).

Disk Size G

R_d = λ · R_h / √2

Dimensional analysis:
Where (from the book): R_d [𝕃], λ [∅], R_h [𝕃]
[𝕃]
[∅] × [𝕃] ÷ √[∅] = [𝕃] ÷ √[∅] = [𝕃] ÷ [∅] = [𝕃]
[𝕃] = [𝕃] ✓ Valid PulseCore Verified ✓

A disk is as wide as its halo's spin allows

Where:

• R_d [𝕃] — the scale radius of the disk
• λ [∅] — the halo's spin parameter
• R_h [𝕃] — the radius of the halo it settles in

➢ With λ ≈ 0.035 a disk is a few hundredths of its halo across, which is why disks are thin and small inside their great dark halos.

Collisions. When two disks of similar mass merge, their orderly orbits are scrambled and the result is an elliptical (Toomre & Toomre, 1972); a small galaxy swallowed by a large one thickens its disk and grows its bulge. Feedback. Supernovae and a feeding central well heat and blow out gas: a galaxy that keeps its gas goes on forming young blue stars, one that loses it turns red and dead. Surroundings. Crowded clusters are full of ellipticals and lenticulars and the open field of spirals, the morphology–density relation (Dressler, 1980).

How the Type Is Read

The shape of a galaxy is the share of its stars' motion that is orderly rotation. A disk turns together; an elliptical's stars swarm. The kinematic measure is the fraction of the stars' kinetic energy in rotation about the axis (Sales et al., 2012).

Rotation Fraction G

κ_rot = K_φ / K

Dimensional analysis:
Where (from the book): κ_rot [∅], K_φ [𝕄·𝕃²·𝕋⁻²], K [𝕄·𝕃²·𝕋⁻²]
[∅]
[𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻²] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

A disk turns together; a spheroid swarms

Where:

• κ_rot [∅] — the fraction of the stars' kinetic energy in ordered rotation
• K_φ [𝕄·𝕃²·𝕋⁻²] — the kinetic energy of their motion round the axis
• K [𝕄·𝕃²·𝕋⁻²] — their whole kinetic energy

➢ Above about one half a galaxy is disk-dominated, a spiral or a lenticular; below it, spheroid-dominated, an elliptical. The game reads each galaxy's type this way, from the stars round its well: κ_rot and the share of young blue stars together give spiral (rotating, forming stars), lenticular (rotating, quiet), elliptical (swarming), irregular (few stars, no settled motion), dwarf (small) and starburst (forming stars fast).

A Well at Every Heart

Every large galaxy has a well at its centre, millions to billions of Suns, and the well's mass follows the swarm of the bulge around it so closely that the two must have grown together (Ferrarese & Merritt, 2000; Gebhardt et al., 2000; Kormendy & Ho, 2013).

Well–Bulge Relation G

M∅ = M◇ · (σ / σ◇)^(438 / 100)

Dimensional analysis:
Where (from the book): M◇ [𝕄], σ [𝕃·𝕋⁻¹], σ◇ [𝕃·𝕋⁻¹]
[𝕄]
[𝕄] × ([𝕃·𝕋⁻¹] ÷ [𝕃·𝕋⁻¹])^([∅] ÷ [∅]) = [𝕄] × ([𝕃·𝕋⁻¹] ÷ [𝕃·𝕋⁻¹])^([∅]) = [𝕄] × [∅]^([∅]) = [𝕄] × [∅] = [𝕄]
[𝕄] = [𝕄] ✓ Valid PulseCore Verified ✓

The central well grows with the swarm of stars round it

Where:

• M∅ [𝕄] — the mass of the central well
• M◇ [𝕄] — the reference mass, 3.1 × 10⁸ Suns
• σ [𝕃·𝕋⁻¹] — the spread of speeds of the bulge's stars
• σ◇ [𝕃·𝕋⁻¹] — the reference spread, 200 km/s

➢ Kormendy & Ho (2013). In Binary Pulse Theory the galaxy is the basin of its central well, the stars it gathers and the Data Matter that holds them (Part 3.13), so the well and its galaxy are one structure, grown together by the same infall.

Quasars and Active Galaxies

When the central well feeds fast, the gas spiralling in heats until it outshines the whole galaxy: a quasar, the brightest steady light in the universe, first recognised in 1963 (Schmidt, 1963) and explained as a feeding well (Salpeter, 1964; Lynden-Bell, 1969). Gas falling in turns about a tenth of its rest energy into light (Shakura & Sunyaev, 1973), and a spinning well throws two jets out along its spin axis (Blandford & Znajek, 1977).

Accretion Luminosity G

L = η · M′ · c²

Dimensional analysis:
Where (from the book): L [𝕄·𝕃²·𝕋⁻³], η [∅], M′ [𝕄·𝕋⁻¹], c [𝕃·𝕋⁻¹]
[𝕄·𝕃²·𝕋⁻³]
[∅] × [𝕄·𝕋⁻¹] × [𝕃²·𝕋⁻²] = [𝕄·𝕋⁻¹] × [𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻³]
[𝕄·𝕃²·𝕋⁻³] = [𝕄·𝕃²·𝕋⁻³] ✓ Valid PulseCore Verified ✓

A feeding well turns a tenth of what falls in into light

Where:

• L [𝕄·𝕃²·𝕋⁻³] — the light given off
• η [∅] — the share of rest energy radiated, about 0.1
• M′ [𝕄·𝕋⁻¹] — the mass falling in each unit of time
• c [𝕃·𝕋⁻¹] — light speed

➢ Ten times more efficient than nuclear fusion. A quasar swallowing a few Suns a year outshines a trillion stars.

Eddington Luminosity G

L⥣ = 4 · π · G · M · m_p · c / σ_T

Dimensional analysis:
Where (from the book): L⥣ [𝕄·𝕃²·𝕋⁻³], G [𝕄⁻¹·𝕃³·𝕋⁻²], M [𝕄], c [𝕃·𝕋⁻¹], σ_T [𝕃²]
[𝕄·𝕃²·𝕋⁻³]
[∅] × [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [∅] × [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕄⁻¹·𝕃³·𝕋⁻²] × [𝕄] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕃³·𝕋⁻²] × [𝕄] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕄·𝕃³·𝕋⁻²] × [𝕃·𝕋⁻¹] ÷ [𝕃²] = [𝕄·𝕃⁴·𝕋⁻³] ÷ [𝕃²] = [𝕄·𝕃²·𝕋⁻³]
[𝕄·𝕃²·𝕋⁻³] = [𝕄·𝕃²·𝕋⁻³] ✓ Valid PulseCore Verified ✓

The most a well can shine before its light holds back its food

Where:

• L⥣ [𝕄·𝕃²·𝕋⁻³] — the Eddington luminosity
• G [𝕄⁻¹·𝕃³·𝕋⁻²] — the gravitational constant
• M [𝕄] — the well's mass
• m_p [𝕄] — the proton mass
• c [𝕃·𝕋⁻¹] — light speed
• σ_T [𝕃²] — the Thomson cross-section of an electron

➢ Eddington (1926). A well feeding near this limit is a quasar; one feeding at a small part of it is a Seyfert galaxy (Seyfert, 1943), a bright core in an ordinary spiral; one whose jets reach far beyond its stars into great lobes is a radio galaxy. Collisions pour gas inward, so mergers light quasars, and the light and jets that follow are feedback that ends the burst.

Feedback: What Keeps Galaxies Blue

Without feedback a galaxy forms its stars too fast, too early, and too many: gas cools, falls in and turns into stars until none is left, and every galaxy ends red and dead. Early simulations made exactly this mistake, the overcooling problem (White & Frenk, 1991). Two sources of energy push back, and with them the galaxies come out as observed.

Supernovae. Each massive star ends by throwing 10⁵¹ erg into the gas around it. Most of that energy radiates away, but the shell it drives carries a nearly fixed momentum out through the gas, heating it, stirring it, and in a small galaxy blowing much of it out altogether (Dekel & Silk, 1986).

Supernova Momentum G

p_sn = p◇ · (n_H / n◇)^(−2 / 17)

Dimensional analysis:
Where (from the book): p_sn [𝕄·𝕃·𝕋⁻¹], p◇ [𝕄·𝕃·𝕋⁻¹], n_H [𝕃⁻³], n◇ [𝕃⁻³]
[𝕄·𝕃·𝕋⁻¹]
[𝕄·𝕃·𝕋⁻¹] × ([𝕃⁻³] ÷ [𝕃⁻³])^([∅] ÷ [∅]) = [𝕄·𝕃·𝕋⁻¹] × ([𝕃⁻³] ÷ [𝕃⁻³])^([∅]) = [𝕄·𝕃·𝕋⁻¹] × [∅]^([∅]) = [𝕄·𝕃·𝕋⁻¹] × [∅] = [𝕄·𝕃·𝕋⁻¹]
[𝕄·𝕃·𝕋⁻¹] = [𝕄·𝕃·𝕋⁻¹] ✓ Valid PulseCore Verified ✓

The momentum one supernova drives into the gas around it

Where:

• p_sn [𝕄·𝕃·𝕋⁻¹] — the momentum the remnant's shell carries when it stops sweeping up gas
• p◇ [𝕄·𝕃·𝕋⁻¹] — the reference momentum, about 2.8 × 10⁵ Suns at 1 km/s
• n_H [𝕃⁻³] — the density of the gas it explodes in
• n◇ [𝕃⁻³] — the reference density, one atom per cubic centimetre

➢ Kim & Ostriker (2015), from simulations of single remnants: the momentum hardly depends on the gas, falling only as the −2/17 power of its density, so every supernova delivers about the same push. It is enough to move a hundred thousand Suns of gas at a few kilometres a second, and a galaxy's supernovae together hold its gas up against its own gravity.

Quasars. A feeding well gives off a tenth of its food's rest energy as light (Accretion Luminosity, above), and a few per cent of that couples to the galaxy's gas as winds and heat. When the well is fed fast, in a quasar, the wind clears the gas from the galaxy's centre and cuts off its own food; when it is fed slowly, its jets keep the gas round a massive galaxy too hot to fall in and form stars. This is what turns the largest galaxies red and keeps them so (Di Matteo, Springel & Hernquist, 2005; Croton et al., 2006; Bower et al., 2006), and it is why a well's mass follows its bulge: the well grows until its own feedback stops the gas that feeds both (Silk & Rees, 1998).

Quasar Feedback Energy G

E_fb = ε_f · L · Δt

Dimensional analysis:
Where (from the book): E_fb [𝕄·𝕃²·𝕋⁻²], ε_f [∅], L [𝕄·𝕃²·𝕋⁻³]
[𝕄·𝕃²·𝕋⁻²]
[∅] × [𝕄·𝕃²·𝕋⁻³] × [𝕋] = [𝕄·𝕃²·𝕋⁻³] × [𝕋] = [𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] ✓ Valid PulseCore Verified ✓

The share of a feeding well's light that stirs its galaxy

Where:

• E_fb [𝕄·𝕃²·𝕋⁻²] — the energy given to the galaxy's gas
• ε_f [∅] — the share of the light that couples to the gas, about 0.05
• L [𝕄·𝕃²·𝕋⁻³] — the well's accretion luminosity
• Δt [𝕋] — the time it feeds

➢ Di Matteo, Springel & Hernquist (2005) found that ε_f ≈ 0.05 reproduces the Well–Bulge Relation in simulated mergers, the quasar shining until its wind stops its feeding.

In Binary Pulse Theory feedback is the data of a star or a well's food returned to the field and pushed back out before the wells take what is left: the supernova's shell carries its star's matter and the data it holds back into the gas (Part 2.9), and the quasar's wind sends what its well could not swallow back out across its galaxy. The simulation gives each supernova an outward push on the gas round it and keeps that gas too hot to form stars for a while, and each well feeding past a hundredth of its Eddington rate a push and heating in proportion; the reach of both is scaled to what the device can compute, so their effect is seen in the universes deep enough to have stars, from about level 23 (the Star Existence Condition, Part 2.10).

Wells per Galaxy

The progeny law gives the number of wells a universe forms in its life, 2^(L/3) (Part 3.10), and most of a universe's wells are the small ones its dying stars leave (Part 2.9), scattered through its galaxies, with one great well at each heart.

Wells per Galaxy G

n∅ = 2^(L / 3) / N_g

Dimensional analysis:
Where (from the book): n∅ [∅], L [∅], N_g [∅]
[∅]
[∅]^([∅] ÷ [∅]) ÷ [∅] = [∅]^([∅]) ÷ [∅] = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The wells a galaxy holds, from the progeny law

Where:

• n∅ [∅] — the wells in a typical galaxy
• L [∅] — the level, 202 for ours
• N_g [∅] — the galaxies the universe holds

➢ At our level 2^(202/3) ≈ 1.8 × 10²⁰ wells. With 2 × 10¹² galaxies in the observable universe (Conselice et al., 2016), or the few hundred billion that measurements of the dark sky's light now favour (Lauer et al., 2021), that is 10⁸ to 10⁹ wells a galaxy, and our own galaxy is estimated to hold about a hundred million black holes from dead stars (Elbert, Bullock & Kaplinghat, 2018). The progeny law, derived for a universe's children, lands on the count of its stellar wells.

The Galaxy in Binary Pulse Theory

A galaxy is a well's basin gathering stars. Its Data Matter, laid on the scaffold before any matter (Part 3.13), is its dark halo; the Proto Well at the halo's densest place is its central well; its spin is the spin its clump gathered, carried down the lineage from the Pulsecore's own (Parts 3.12 and Z.11); its stars are the gas that cooled and kept that spin; and its type is how much of that order survived collisions and feedback. The wells it leaves as its stars die are the 10⁸ to 10⁹ of the count above, and its central well is the one whose child, at the final well, inherits the galaxy's data (the Return to Bifurcation).

The Game's Reading

The simulation draws each well at its true horizon, a dark point too small to see from afar, and shows it by the stars gathered round it; a tap anywhere in a galaxy selects its well. Stars are drawn by type: blue-white giants bright and few in young clusters, red giants and supergiants large and orange, red dwarfs small and dim, brown dwarfs fainter still, white dwarfs as small sharp points, neutron stars blinking as pulsars; supernova remnants and planetary nebulae as expanding shells. A central well that feeds fast lights as a quasar, a blazing core with two jets along its spin axis, standing up from the torus's surface, and a weaker one as a Seyfert glow. A selected galaxy is named by its type, read from its stars' rotation fraction and their colour. The field is far too coarse for spiral arms to grow in it, so, as cosmological simulations do, a galaxy's inner structure is drawn by the known laws while its place, its stars, its turning sense and its type come from the simulation: a spiral as an exponential disk with two trailing logarithmic arms of young blue-white stars, the arms the density waves of Lin & Shu (1964), round a warm bulge, its pattern turning in the galaxy's own sense; a lenticular as a smooth disk with a larger bulge; an elliptical as a round warm swarm; an irregular or a starburst as clumps.

Links to Binary Pulse Theory

• Part 2.9: the stars, their ends, and the Stellar Null Wells a galaxy fills with.
• Part 3.10: the progeny law that counts the wells; the central well's child inherits the galaxy's data.
• Part 3.12: the tidal spin every galaxy keeps, and the cooling that turns it into a disk.
• Part 3.13: Data Matter, the dark halo every galaxy sits in; the Proto Well at its heart.