HENYEY

In 1964, Louis Henyey, James Forbes, and Nancy Gould of the UC Berkeley Astronomy Department published the numerical method that permitted the evolution of stars to be accurately understood. Their method can still be found at the heart of nearly every modern code that tracks the evolution of stars, brown dwarfs and giant planets. This page carries a faithful reconstruction of their calculation: the difference equations and boundary conditions exactly as the paper gives them, written in period FORTRAN IV with punched-card input and line-printer output, and compiled by an equally faithful reconstruction of a 7094-era FORTRAN IV compiler, arithmetic and all. In 1964, the program ran overnight. Punch a deck and run it now.

Punch a deck ↓ Read the source

1–1The deck

The interface to a star was once eighty columns wide. Two cards specify the whole calculation: the first carries the star (mass, hydrogen, metals), the second the run (mesh, models, time step, convergence, mixing length). Edit the fields — the keypunch follows.


  

    
    machine idle
  






  
1–2The job

  
Unit 5 is the card reader; unit 6 is the printer. Watch the
     MAX CORRECT column: that is the Henyey method
     itself — Newton–Raphson corrections collapsing toward zero, model after model.

  






  
1–3The star

  
The same numbers, read the modern way. Left: the track in the
     Hertzsprung–Russell diagram. Right: the star's vital signs against age.

  

    
Fig. 1. The evolutionary track in the Hertzsprung–Russell diagram. The red point is the current model.

    
Fig. 2. Luminosity, in units of L☉.

    
Fig. 3. Radius, in units of R☉.

    
Fig. 4. Central temperature (K).

    
Fig. 5. Central hydrogen mass fraction.

  

  

    

       no structure printed yet

    

      

      

      

    

    
Fig. 6. The interior run of temperature, density, and luminosity
      against the mass fraction, at the printed models. Shaded bands mark convective regions,
      located by the Schwarzschild criterion (eq. 27 of the paper).

  

  
View the track as a table






  
1–4The method

  
The star is divided into concentric mass shells. On each shell live four
     unknowns — radius, luminosity, temperature, density — governed by four
     difference equations: hydrostatic equilibrium, mass continuity, energy
     balance, and energy transport (equations 22–26 of the paper). The trick that
     changed the field: solve all of them, at all shells,
     simultaneously, by Newton–Raphson iteration — linearize, sweep a
     block elimination from center to surface, back-substitute, correct, repeat.
     Each iteration the corrections shrink quadratically; four or five iterations
     build a star.

  
The reconstruction keeps the paper's own devices: the “artificial variables”
     p = P1/4, q = ρ1/3, the pseudoflux
     l = ξ²F, chosen in 1964 to tame a matrix that is otherwise
     ill-conditioned — a courtesy to a machine that carried only eight
     significant digits. It works in the original group's scaled units
     (mass in 10³⁰ g, length in 10¹⁰ cm), and it runs here under the
     same 27-bit single-precision arithmetic as the 7094, faithfully emulated.
     The physics is the Berkeley group's own: the opacity and
     electron-degeneracy formulae of their 1965 appendices, Saha ionization and
     Böhm-Vitense convection in the envelope, H⁻ at low temperature, the
     proton–proton chain and the CNO cycle.

  
Henyey, Forbes & Gould 1964, ApJ 139, 306 ·
     the annotated listing ·
     the repository ·
     the compiler's dialect contract






  
1–5The lineage

  
The method was born at Berkeley and Livermore in the 1950s, published
     tersely in 1959, and taught to the world largely by a talk Henyey gave at
     the 1961 IAU General Assembly in Berkeley. The 1964 paper — the one this
     reconstruction implements — gave the mature form: Method II. The production
     code's physics filled the 1965 “Studies in Stellar Evolution” papers.
     From Berkeley the code traveled: card decks carried to Santa Cruz in 1968
     seeded a line of descendants that has run continuously ever since, on
     problems from the faintest red dwarfs to the formation of Jupiter. Nearly
     every modern stellar-evolution program — whatever its language — still has
     a Henyey solver at its heart.

  
This site runs a clean-room reconstruction: a new
     implementation of the published method in period FORTRAN IV, compiled by a
     new 7094-dialect compiler, not recovered original source. The compiler
     translates the deck to C, and WebAssembly carries it into your browser —
     every visitor's machine becomes the mainframe, running at the 7094's own
     27-bit arithmetic, so the overnight job of 1964 completes in about half a
     second, entirely on your side of the wire. The full family tree, including
     a FORTRAN 77 line under active development, lives in
     the repository.