3.2. Structures and Timescales of the Macroworld

Essentially all astrophysical structures, sizes and timescales are controlled by one dimensionless ratio, sometimes called the ``gravitational coupling constant,''

(2)

where m_Planck = sqrt( c/G) 1.22 x 10¹⁹ GeV is the Planck mass and G = m_Planck^-2 is Newton's gravitational constant. ⁽⁶⁾ Although the exact value of this ratio is not critical - variations of (say) less than a few percent would not lead to major qualitative changes in the world - neither do structures scale with exact homology, since other scales of physics are involved in many different contexts (Carr and Rees 1979).

The maximum number of atoms in any kind of star is given to order of magnitude by the large number

(3)

Many kinds of equilibria are possible below M_* but they are all destabilized above M_* (times a numerical coefficient depending on the structure and composition of the star under consideration). The reason is that above M_* the particles providing pressure support against gravity, whatever they are, become relativistic and develop a soft equation of state which no longer resists collapse; far above M_* the only stable compact structures are black holes.

A star in hydrostatic equilibrium has a size R / R_S m_proton/E where the particle energy E may be thermal or from degeneracy. Both R and E vary enormously, for example in main sequence stars thermonuclear burning regulates the temperature at E 10^-6 m_proton, in white dwarfs the degeneracy energy can be as large as E_deg m_e and in neutron stars, E_deg 0.1 m_proton.

For example, the Chandrasekhar (1935) mass, the maximum stable mass of an electron-degeneracy supported dwarf, occurs when the electrons become relativistic, at E m_e,

(4)

where Z and A are the average charge and mass of the ions; typically Z/A 0.5 and M_C = 1.4 M, where M = 1.988 x 10³³ g 0.5 M_* is the mass of the Sun.

For main-sequence stars undergoing nuclear burning, the size is fixed by equating the gravitational binding energy (the typical thermal particle energy in hydrostatic equilibrium) to the temperature at which nuclear burning occurs at a sufficient rate to maintain the outward energy flux. The rate for nuclear reactions is determined by quantum tunneling through a Coulomb barrier by particles on the tail of a thermal distribution; the rate at temperature T is a thermal particle rate times exp[-(T₀/T)^1/3] where T₀ = (3/2)³(2 Z )² Am_proton. Equating this with a stellar lifetime (see below) yields

(5)

note that the steep dependence of rate on temperature means that the gravitational binding energy per particle, GM/R, is almost the same for all main-sequence stars, typically about 10^-6 m_proton. The radius of a star is larger than its Schwarzschild radius R_S by the same factor. Since M/R is fixed, the matter pressure M/R³ M^-2 and at large masses (many times M_*) is less than the radiation pressure, leading to instability.

There is a minimum mass for hydrogen-burning stars because electron degeneracy supports a cold star in equilibrium with a particle energy E = m_e(M/M_C)^4/3. Below about 0.08 M the hydrogen never ignites and one has a large planet or brown dwarf. The maximum radius of a cold planet (above which atoms are gradually crushed by gravity) occurs where the gravitational binding per atom is about 1 Rydberg, hence M = M_C ^3/2 - about the mass of Jupiter.

The same scale governs the formation of stars. Stars form from interstellar gas clouds in a complex interplay of many scales coupled by radiation and magnetic fields, controlled by transport of radiation and angular momentum. Roughly speaking (Rees 1976) the clouds break up into small pieces until their radiation is trapped, when the total binding energy GM² / R divided by the gravitational collapse time (GM/R³)^-1/2 is equal to the rate of radiation (say x times the maximum blackbody rate) at T/m_proton GM/R, giving a characteristic mass of order x^1/2(T/m_proton)^1/4 M_*, controlled by the same large number.

Similarly we can estimate lifetimes of stars. Massive stars as well as many quasars radiate close to the Eddington luminosity per mass L_E/M = 3G m_proton/2r_e² = 1.25 x 10³⁸ (M / M) erg/sec (at which momentum transfer by electrons scattering outward radiation flux balances gravity on protons), yielding a minimum stellar lifetime (that is, lower-mass stars radiate less and last longer than this). The resulting characteristic ``Salpeter time'' is

(6)

The energy efficiency 0.007 for hydrogen-burning stars and 0.1 for black-hole-powered systems such as quasars. The minimum timescale of astronomical variability is the Schwarzschild time at M_*,

(7)

The ratio of the two times, t_* and t_min, which is _G^-1/2, gives the dynamic range of astrophysical phenomena in time, the ratio of a stellar evolution time to the collapse time of a stellar-mass black hole.

A ``neutrino Eddington limit'' can be estimated by replacing the Thomson cross section by the cross section for neutrinos at temperature T,

(8)

In a gamma-ray burst fireball or a core collapse supernova, a collapsing neutron star releases its binding energy 0.1 m_proton 100 MeV per nucleon, and the neutrino luminosity L_E 10⁵⁴ erg/sec liberates the binding energy in a matter of seconds. This is a rare example of a situation where weak interactions and second-generation fermions play a controlling role in macroscopic dynamics, since the energy deposited in the outer layers by neutrinos is important to the explosion mechanism (as well as nucleosynthesis) in core-collapse supernovae. The neutrino luminosity of a core-collapse supernova briefly exceeds the light output of all the stars of universe, each burst involving _G^1/2 of the baryonic mass and lasting a little more than _G^1/2 of the time.

Note that there is a purely relativistic Schwarzschild luminosity limit, c⁵ / 2G = m_Planck²/2 = 1.81 x 10⁵⁹ erg/sec, corresponding to a mass divided by its Schwarzschild radius. Neither Planck's constant nor the proton mass enter here, only gravitational physics. The luminosity is achieved in a sense by the Big Bang (dividing radiation in a Hubble volume by a Hubble time any time during the radiation era), by gravitational radiation during the final stages of comparable-mass black hole mergers, and continuously by the PdV work done by the negative pressure of the cosmological constant in a Hubble volume as the universe expands. The brightest individual sources of light, gamma ray bursts, fall four or five orders of magnitude short of this limit, as does the sum of all astrophysical sources of energy (radiation and neutrinos) in the observable universe.

Using cosmological dynamics - the Friedmann equation H² = 8 m_Planck² relating the expansion rate H and mean density - one can show that the same number N_* gives the number of stars within a Hubble volume H^-3, or that the optical depth of the universe to Thomson scattering is of the order of Ht_*. The cosmological connection between density and time played prominently in Dicke's rebuttal of Dirac. Dicke's point is that the large size and age of the universe - the reason it is much bigger than the proton and longer-lived than a nuclear collision - stem from the large numbers M_* / m_proton and t_* / t_proton, which in turn derive from the large ratio of the Planck mass to the proton mass. But where does that large ratio come from? Is there an explanation that might have satisfied Dirac?

⁶ The Planck time t_Planck = / m_Planckc² = m_Planck^-1 = 0.54 x 10^-43 sec is the quantum of time, 10¹⁹ times smaller than the nuclear timescale t_proton = / m_protonc² = m_proton^-1 (translating to the preferred system of units where = c = 1). The Schwarzschild radius for mass M is R_S = 2M / m_Planck²; for the Sun it is 2.95 km. Back.