View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Leahy and Leahy Computational Astrophysics and Cosmology (2015)2:4 DOI 10.1186/s40668-015-0008-8 S O F T WA R E Open Access A calculator for Roche lobe properties Denis A Leahy1* and Janet C Leahy2 Abstract Background: The Roche lobe geometry is important to understand and study the properties of the mass-losing component in a semi-detached binary system. However it is not easy to calculate accurately, and existing tables usually do not include the parameters of the binary system under study, nor do they allow for non-synchronous rotation. Results: A calculator for properties of the Roche lobe is presented in two formats. An easy-to-use Java version has a graphic interface, and a Fortran 90 version has a command line interface. The Fortran version allows for easy modifications by the user. Both versions have two basic output options: one provides values of a set of various quantities (such as the Lagrange points along the binary axis, and area and volume of the Roche lobe); the second provides R(θ, φ), the distance from the stellar center to the stellar surface for any specified polar angle. A single set of input parameters can be entered directly or a large set of input parameters can be specified in a text file. The calculator includes the options to have non-synchronous rotation of the star, or to have the star underfill its Roche lobe. It can be used to calculate Roche lobe properties for the case of elliptical orbits, with some restrictions. Conclusions: We present a convenient software tool for quickly and accurately calculating Roche lobe properties for mass ratio in the range 0.01 to 100, for Roche lobe fill-out factor in the range 0.1 to 1.0, and for dimensionless rotation rate of the star in the range 0.1 to 2.0. This will allow anyone working with a binary star system to obtain the Roche lobe or stellar surface geometry for their system. Keywords: stars; binaries 1 Background has the potential equal to the L potential. The Roche lobe The Roche potential is the potential energy per unit test has the property that any material outside this surface is mass which is orbiting the center-of-mass of a binary star unbound from star and will be lost. system at the same rate as the two stars. A circular orbit Many works have studied and calculated properties of is assumed for the binary system. The Roche potential in- the Roche potential and the Roche lobe, starting with the cludes both gravitational and centripetal energy, such that pioneering work of Kopal (). Plavec and Kratochvil its derivative gives the force (and thus acceleration) on the () present tables of distances for several points on the test mass. equipotential surfaces which pass through the L and L The Roche potential has many uses, including finding Lagrange points. This was done for a range of mass ra- the locations in the binary system where the acceleration tios from . to .. Eggleton () calculated volumes, V , is zero- the Lagrange points L, L, L, L and L. In the oftheRochelobeforarangeofmassratiostoobtainthe case of a binary star system where one of the stars (star ) equivalent volume radius Req (defined by V =πReq/). He increases its size, star will reach a limiting surface. This then presented a simple fitting formula for Req as a function surface is called the Roche lobe, which is sketched in Fig- of q,accuratetobetterthan%,whichisoftenusedand ure . The Roche lobe is also defined by the surface that now known as the Eggleton formula. Mochnacki () presented tables of potentials and positions of the L and L Lagrange points as a function of mass ratio. He defined *Correspondence: [email protected] a fill-out factor F in terms of the potential of an equipoten- 1Department of Physics and Astronomy, University of Calgary, 2500 University Dr. NW, Calgary, Canada tial surface relative to the potential of the surfaces through Full list of author information is available at the end of the article the L and L points. F was defined so that F <corre- © 2015 Leahy and Leahy; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduc- tion in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Leahy and Leahy Computational Astrophysics and Cosmology (2015)2:4 Page 2 of 6 Figure 1 Roche lobe geometry. Illustration of the Roche lobe for star 1, with x, y, z axes and the various Roche radii (RL1, Rbk, Ry, Rz and Req). sponds to surfaces inside the Roche lobe, and < F < tential (r, θ, φ) in the case of synchronous rotation is: corresponds to surfaces between the Roche lobe and the equipotential through the L point. He gave tables of vol- (r, θ, φ)= + q ume radius, area, average gravity and average inverse grav- r –r sin θ cos φ + r ity as a function of mass ratio and F. Pathania and Medupe q + – r sin θ cos φ + r sin θ .() () present series expansions for the radius of Roche equipotential surfaces, and compare various orders of the expansion to numerically calculated values. This is the same form used by Kopal () and Pathania The purpose of the present work is to present an freely and Medupe (). When non-synchronous rotation is available software tool, written in two versions (Java and included Limber (), with p = star/binary is the rate Fortran), which calculates radii of the Roche lobe. It does of stellar rotation divided by the binary rotation rate, the potential becomes: this for any specified direction, and gives some other com- monlyusedquantities(suchastheLagrangepointsand values of the potential). The calculator is designed to be (r, θ, φ)= + q r –r sin θ cos φ + r accurate for any mass ratio q between . and , for di- q + mensionless rotation rates between . and , and fill-out – r sin θ cos φ + pr sin θ .() factors between . and .. The calculator may work for parameters outside these limits but has not been tested for This potential can be further generalized to the case accuracy or errors outside these limits. of elliptical orbits, as discussed in Sepinsky et al. (). The coordinates are Cartesian (x, y, z) and spherical- In the quasi-static approximation, valid for dynamical θ φ polar (r, , ), centred on star (with mass M), with the timescale of the star much less than the tidal timescale, x-axis (θ = π/, φ = ) pointing towards star (with mass theinstantaneousshapeofthestarcanbeapproximated M). The z-axis is perpendicular to the orbital plane. The by the instantaneous surface of constant potential, even mass ratio is defined as q = M/M and the binary separa- though the instantaneous surface is changing with orbital tion is a. phase. As shown by Sepinsky et al. (), the instanta- We use the dimensionless form of the Roche poten- neous potential for an elliptical orbit can be written in the tial (or potential energy per unit mass) obtained by di- same form as equation ()aboveifonereplacesp by the function viding by GM/a,whereG is Newton’s gravitational con- stant, and using distances in units of the binary separa- p( + e) tion a. The equation for the resulting dimensionless po- A(p, e, ν)= () ( + e cos(ν)) Leahy and Leahy Computational Astrophysics and Cosmology (2015)2:4 Page 3 of 6 with e is the eccentricity and ν isthetrueanomalyforthe X), and the values of the potential at the L point (), position of the star in its elliptical orbit. at the L point ()andattheLpoint(). The area, For a physical system, one knows the actual values of the A,andvolume,V ,oftheRochelobearegiven(inunitsof masses M and M. Either one specifies the binary sepa- a and a , respectively), and the equivalent volume radius ration a, or calculates it from the orbital period P,using Req,definedbyV =πReq/ and area radius Ra,definedby π Kepler’s third law, a = G(M + M)P . Dimensionless A =πRa.ForthecaseofF < , i.e. star inside its Roche distances are converted into physical units by multiplying lobe, the values of Rbk , Ry, Rz, A, V , Ra and Req are for the by a. stellar surface. Additionally the front radius, Rfr of the star TheLagrangepointsarethelocationswherethepoten- (along the positive x-axis) is given because it is not equal tial (or dimensionless potential) has a maximum (or min- to RL for F <. imum) or saddle point. There are a total of Lagrange The L, L and L Lagrange points are along the x-axis points (L, L, L, L and L). The L point is the La- (y = z = ). A root-finding routine is used to find the x- grange point located between the two stars. One finds the value where the derivative of (r, θ, φ)alongthex-axisis positions of and the values of the potential at the Lagrange zero. The root is found to an accuracy of part in (Java points by solving version) and part in (Fortran version) using Brent’s method (Brent ). This method combines the bisection ∇(r, θ, φ)=. () method, the secant method and inverse quadratic inter- polation. It has been shown previously that the L and L The Roche lobe (of star ) is the surface surrounding star points lie in the x-y plane, and either of these when taken having the potential equal to the potential of the L point.