ASTROPHYSICAL APL - DIAMONDS IN THE SKY

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In this paper we will outline the development of anastrophysical model using APL. To illustrate the flexibilityand lucidity of the notation, we attempted to identify thoseconditions which are often stated by APL opponents as beingadverse to the creation of an APL model, and then fmding anastrophysically interesting situation for which thoseconditions applied. In essence, we looked for things whichwe, and others, have conjectured that APL does not handlewell, and forced their incorporation into our model.Specifically, we demanded that the modeling of the physicalprocess under evaluation must be fundamentally iterative innature, forcing branching and re-execution of the linguisticalconstructs to be employed. Additionally, since APL naturallyhandles the manipulation of extensive multidimensionalarrays with ease (Metzger, 1981), in an adverse test the data tobe processed were to be of low rank (scalars, vectors, andmatrices) containing, in general, very small numbers ofelements. Since we were testing a numerical model, we feltthat the process should inherently have been computationallyextensive. Finally, for this illustrative example to bemeaningful to a broad spectrum of potential readers, we wishedto find a physical situation which was intrinsicallyinteresting, and not too narrowly channeled. To the latterpoint, the techniques used in the model would have to beroutinely used in diverse astronomical and physicaldisciplines.We felt that the classical problem of developing a stellarstructure model for white dwarf stars fit all of the abovecriteria. The numerical integrations of the structure equationsfrom the center of the star to the surface are inherently iterativeand computationally intensive. Yet, the physical situation isparametrized by only a few variables, and the number ofnumerical elements involved in each iteration step is small. Indeveloping the APL model, the method of approach to thesolution is as important as, and indeed is part of, the solutionitself. The actual “code” (a term we use hesitantly) isdeveloped in a bottom-up fashion, following the structuring ofthe physical principles employed in the model. There isnothing unique, ground breaking, or particularly inventive inthe implementation of the model which we present. Theunderlying physics is well understood. Indeed, this was anadditional requirement we imposed upon ourselves in selectinga situation to model for illustrative purposes.STRUCTURE MODELS - GENERAL CONCEPTSFor the purposes of developing a model of a stellar interior,the star may be thought of as being comprised of manyinfintesmally thin, spherically concentric shells. A structuremodel seeks to determine how the physical properties of thestar change at varying depths within the interior. Thisamounts to determining the “run of physical variables” fromthe center of the star to the surface (or vice-versa), througheach of the spherical shells. Some of the physical variableswhich are of interest include pressure and density at varyingdepths (i.e., within each shell), and the mass interior to eachshell.The process of solving the interior model involves expressingthe physical relationships between the structure variables asfunctions of the radial distance from the center of the star, andthen determining the numerical values of each of thosevariables throughout the interior. The form of the classicaldifferential equations of stellar structure, as described bySchwarzschild (1958), are discussed here as a background todeveloping the APL model which we will present.In the interior of a star two forces are in constant competition.The inward acting force of gravity, due to the star’s own mass,seeks to pull the outer layers of the star downward. Theinternal pressure of the gas, which comprises the stellarmaterial, acts to check this gravitational collapse. For thevast majority of a star’s life these two forces are in balance andthe star remains stable. This condition of hydrostaticequilibrium is expressed, for a region in the stellar interior, byequation [ 11.dPldr = -Gmrp ,/r* 111The pressure gradient (d Prldr ) across the regionspherical shell) in hydrostatic equilibriumdepends upon:(i.e.,r = the radial distance from the center of the starmr= the total mass interior to the regionpr= the density of the regionThe distribution of mass as a function of the radial distancefrom the center of the star is expressed by the equation of masscontirmity, written in differential form in equation [2]. As canbe seen, the differential equations of hydrostatic equilibriumand mass continuity both require a knowledge of the density ofthe region before they can be integrated.dmjdr = 4m-* p, VIThe gas pressure is related to the temperature and densitythrough an equation of state. The form of the equation of statedepends upon the underlying physics which governs thebehavior of the gas. The “perfect gas law”, which most of uslearned in high school chemistry, comes quite close to realityin physical situations of relatively low gas density, such as inthe Earth’s atmosphere and for most of the interior regions ofmain sequence stars. The perfect gas law assumes that thecollisional statistics between the particles in the gas arestrictly Maxwellian; all collisions are perfectly elastic, andboth external and inter-particle forces (such as Van der Wallsforces) are ignored (Sears, 1952). Under higher gas density, asoccur in the case of white dwarf stars, such simple assumptions’will not suffice.The third equation which describes the structure of the stellarinterior is the luminosity gradient equation, expressed in themost general form in equation [3]. This equation states thatthe radially directed luminosity gradient (i.e., the the netenergy loss), across the region is exactly compensated for bythe energy generated within the region. The energy generatedper unit mass, E, must include all energy sources. For mainsequence stars only nuclear energy sources are important,During the pre-main sequence phases of stellar evolution,when the star is undergoing gravitational contraction, boththermal and gravitational energy must also be considered.dLrldr = Edmjdr [31Equation [4], which has two forms, describes the radialtemperature gradient (d Tr/d r ) across a spherical shell in thestellar interior.APL QUOTE QUADSchneider, Paluui and Webb
d Tidr = [-3KLrp/16nacr2 ~,3]~~~ PaId T/dr = [ (1 -Y-l) (frlPJ dq/d rlconv [4blThe appropriate form of equation [4] to use in the.regiondepends upon whether the mode of energy transport is radiativeor convective. To determine this, the convective form of [4]may be cast as an inequality (Chandrasekhar, 1967). If theregion is to be stable against adiabatic convection, then themagnitude of the temperature gradient (d T,/di) must be lessthan the magnitude of adiabatic temperature gradient. Here,K = the opacity of the regiona = the Stephan-Boltzman constantC = the speed of lighty = the ratio of specific heats of the gas(513 for highly ionized gas)As is apparent, the differential equations of stellar structure,[l] - [4], are ~nutually dependent and must be simultaneouslyintegrated to determine the run of the physical variables. Theboundary values for the integrations must come from anunderstanding of the physical properties at the stellar surfaceand/or center.THE CASE FOR WHITE DWARF STARSA star spends most of its life creating energy by nuclear fusiondeep in its interior, converting Hydrogen to Helium throughthe “proton-proton” or “C-N-O” cycles. More massive,evolved stars may also fuse heavier elements, such as Heliuminto Carbon via the “triple alpha” process. In either case theenergy generated in the core provides a sufficient radiantenergy flow to support the weight of the outer lying layers, asexpressed by the equation of hydrostatic equilibrium. At somepoint in the life cycle of a star, the available nuclear fuel in thecore will be expended. The condition of hydrostaticequilibrium will be violated and the star will collapse under itsown gravitational force. The final configuration which will bereached at the end of this rapid collapse depends upon the massof the star. In any case the outer envelope of the star will beexpelled (perhaps violently) or eventually depleted. Theremnant stellar core, internally heated and compressed to veryhigh densities by the stellar collapse, is all that will remain ofwhat once was a main sequence star. The compressed corematerial, often nearly pure Carbon for stars of the appropriateinitial mass, is crystalized during the collapse making somewhite dwarfs truly diamonds in the sky.White dwarf stars have peculiar physical properties which insome ways greatly simplifies developing numerical models oftheir interiors, and in other ways complicates mattersenormously. First, simplicity.White dwarfs represent the final stage of evolution for manystars. As nokd, at this phase in a star’s life nuclear energygeneration processes in the interior have terminated, as thefusionable material has been exhausted. Further, the star isonce again in hydrostatic equilibrium, so there is no thermalenergy generation to speak of. Therefore, the luminositygradient, expressed in equation [3] is 0, since E is 0.White dwarf stars are essentially isothermal. Since there is nophysical mechanism for energy generation within the interior,the thermal energy containedthere is a remnant of the corecollapse. This remnant thermal energy is bottled up within thestar since white dwarfs have small surface areas (typically, lessthan 10e6 that of the sun). Additionally, a very thin radiativeregion must exist at the surface. At the interior/surfaceinterface the temperature gradient is large, since the internaltemperatures of newly formed white dwarfs are on the order of afew million degrees, while the surface temperatures areobserved to be a few times lo4 K. Given a small thin radiativesurface area, with a large temperature gradient, white dwarfstars have very long thermal cooling times. Therefore, exceptfor this very thin region at the surface, the temperaturegradient in the interior is essentially zero. For these reasonswhite dwarfs are considered thermally cold, and d Tr ldr inthe interior is 0.Since both the luminosity and thermal gradients (equations [3]and [4]) are zero throughout the interiors of the white dwarfs,the physical characteristics of the interiors of these stars maybe inferred only from the differential equations of hydrostaticequilibrium and mass continuity. Hence, we mustsimultaneously integrate only the first two of the fourdifferential equations of stellar structure.Several complications arise due to the high particle densitieswhich occur in the interiors of white dwarfs. The gas does notbehave in accordance with the “perfect gas law”, and indeed isdegenerate. In a degent,ate gas both the momenta and thepositions of the particles are constrained.The Pauli exclusion principle dictates that no two electronscan have the same position, momentum and spin. TheHeisenberg uncertainty principle states that one cannot, evenin principle, determine the position and momentum of aparticle simultaneously. The location of the particle may bedescribed by its coordinates in a six dimensional phase space;the coordinates being the orthogonal components of theposition and momentum vectors. This phase space,conceptually, is comprised of individual cells (whose sidesrepresent conjugate variable pairs in position and momentum,and whose volume is h 3). As the gas density increases, thephase space cells begin to fill up, with the lowest momentumcells being occupied first. Since each cell can hold only twoelectrons, the distribution function becomes constrained, andis no longer Maxwellian. Such a gas is said to be partiallydegenerate.In this case, the particle number density and pressure can bederived utilizing Fermi-Dirac statistics. The equation of state(for a given atomic species) may then be characterized by asingle parameter representing the energy of the most energeticparticle found in the Fermi-Dirac distribution. This parameter,X, is referred to as the “relativity parameter”, and is related tothe particle momentum by the usual relativistic relation:x = po/mc [51where p. is the particle momentum.The density is related to the relativityequation [ 61.parameter, X, throughp = pMu(xmec/fi)3/37c2 [61Astrophysical APL - Diamonds in the Sky 306 APL89
Page 4 and 5: where,P = the ratio of atomic mass
Page 6 and 7: The Runge-Kutta integration is invo
Page 8: -. -. ---...APL.68000~ (1986a,b) di

ASTROPHYSICAL APL - DIAMONDS IN THE SKY

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