Albert P. Linnell

The volume and surface area of a uniformly rotating polytrope

This paper develops algebraic expressions for the volume and surface area of a uniformly rotating polytrope. The expressions depend on an analytic theory for boundary shape developed in previous papers. A comparison with the calculations of James indicates the present theory improves on the original Chandrasekhar theory by a factor 10 or more.

Rotational distortion of polytropes by analytic approximation, II

Chandrasekhars (1933) paper on rotational distortion of polytropes contained a perturbation term in the potential which was linear inv, the rotation parameter. The same paper, and subsequent papers by various authors, developed an analytic expression for the boundary also linear inv. The latter expression is equivalent to a two term Taylor series about the unperturbed boundary, and is in error by 12% near critical rotation, for a polytropic index 3.0. The boundary can be located directly from the functions representing density, potential, and the potential gradient. The boundary error by this procedure is 0.2% near critical rotation.

A boundary equation for uniformly rotating polytropes

A linear approximating equation exists for the boundary of a uniformly rotating polytrope. The equation in η=(ξ1−ξ)/ξ1 permits rapid calculation of the polytrope radius for any latitude, and is accurate for angular velocities of rotation nearly to critical rotation. Data in this paper apply to a polytrope indexn=3.0.

Light Synthesis Modeling of Close Binary Stars

Recent years have seen substantial progress in understanding residual departures of observed light curves, for close binary stars without degenerate components, from the canonical Roche model. Examples of observed departures include the O’Connell effect and ”anomalies” in W Ursae Majoris systems.

A computer study of EE Pegasi and CM Lacertae

Computer routines permit the solution of eclipsing binary light curves on the Russell Model. With additional automatic point plot routines, the operator has available all necessary supervisory control for an optimum solution. A solution of a synthetic light curve, whose parameters simulate those calculated for the physical system, is an important adjunct to test convergence properties of the physical system solution.Application to EE Peg determines values ofxxandxxeven though secondary minimum is only 0.08 mag. deep. First order perturbation theory is used with the Russell Model to calculate a final triaxial ellipsoid model.Solution of the CM Lac light curve shows that the data require an occultation eclipse at primary minimum, in contrast to the available nomographic solution. The point plot routines demonstrate a substantial improvement effected by the computer solution and show that the latter technique can determine limb darkening coefficients in this partially-eclipsing system.

Approximation functions to the emden and associated emden functions near the first zero, II

Analytic expressions are presented which approximate the Emden function θ, and the associated Emden functions,ψo andψ2, and their derivatives, near the first zero of θ,ξo. The range of accurate representation extends toξ= 1.5ξ1. This range is sufficient to encompass the boundary of a critically rotating polytrope.

W Ursae Majoris star models: Observational constraints

W Ursae Majorls stars can be understood as contact binary stars with a common envelope (Lucy 1968). They subdivide into two types: The A-type are earlier in spectral class than about F5, are believed to have radiative envelopes, and associate primary (deeper) eclipse minimum with transit eclipse. The W-type have spectral classes later than F5, are believed to have convective envelopes, and associate primary minimum with occultation eclipse. Controversy has surrounded the explanation of W-type light curves. Four distinct models have been introduced to describe the envelopes or photospheres of W UMa stars. (I) The Ruclnskl hot secondary model directly explains W-type light curves on a postulatlonal basis. Since 70%-90% of the emitted radiation from the secondary (less massive) component is believed to reach the secondary via circulation currents from the primary, there is an apparent thermodynamic mystery why the secondary should be hotter. (2) The Lucy Thermal Relaxation Oscillation (TRO) model argues that the secondary component is perpetually out of thermal equilibrium and that the components are in contact only during part of a given TRO cycle. During contact the photosphere is supposed to be barotroplc. In this case primary minimum always associates with transit eclipse, in disagreement with observation for W-type systems. (3) The Shu et al. thermal discontinuity (DSC) model als0 argues for a barotropic photosphere but differs from Lucy on the gravity brightening exponent. The changes are insufficient to produce W-type light curves. (4) Webblnk (1977), and, separately, Narlal (1976), argue for a barocllnlc envelope. If the barocllnlclty extends to the photosphere there is a possibility that W-type light curves could be explained. In particular, the Webblnk scenario produces a hot secondary.

The Physical Status of VW Cephei

The decreasing period of VW Cep, interpreted on conservative case mass transfer, implies mass flow from primary to secondary. The TRO theory then predicts semi-detached status. The system b-v color is the same at the two minima, to within observational error. This is in marginal disagreement with a hot secondary. It also implies close thermal contact, in disagreement with the postulated semi-detached status. Unanimous agreement is lacking among research workers concerning the evolutionary status of W UMa stars and the interpretation of their light curves. The evolutionary status protagonists include the supporters of thermal relaxation oscillations (TRO) in one group, led by Lucy (1976), Flannery (1976), and Robertson and Eggleton (1977). The other group supports a thermal equilibrium model, made possible by an entropy discontinuity at the inner contact surface. This group is led by Shu, Lubow, and Anderson (1976). In the study of W UMa light curves, the usual interpretation of the W-subclass (Binnendijk, 1970) systems has required a secondary (less massive) component whose photosphere is hotter than that of the primary. This accounts for the observed association of primary minimum with the eclipse of the less massive component. Rucinski (1974) used the hot secondary model in his study of W UMa light curves, as have several other investigators. The W-subclass systems were a problem for the Lucy (1973) light curve study, which otherwise was successful in representing the A-subclass systems. The Lucy common convective envelope model asserts a gravity brightening coefficient of 0.08 for convective envelopes, and incorporates a Rucinski (1969) bolometric albedo factor of 0.5. The most extensively developed light curve synthesis program is that of Wilson and Devinney (1971). Wilson and Devinney (1973) were able to fit the W-type light curve of RX Com successfully, but required a value of gravity brightening much larger than the Lucy coefficient.

An Automated Photometric Telescope

A photometric system providing automated telescope control and data gathering recently has been completed at Michigan State University. At present it is possible to do sequential 3-, 4-, or 5-color photometry.

Computer calculation of eclipse functions

Abstract The direct comparison of individual observed light losses in a binary system and theoretical expressions for the same quantities on the Russell Model can be performed on a digital computer provided with routines for the various alpha functions. A broader use of these routines requires a constant fractional accuracy for them. This objective greatly complicates the programming because of the properties of the generating equations.