A calculator for Roche lobe properties
- Denis A Leahy^{1}Email author and
- Janet C Leahy^{2}
https://doi.org/10.1186/s40668-015-0008-8
© Leahy and Leahy; licensee Springer. 2015
Received: 9 January 2015
Accepted: 24 March 2015
Published: 6 May 2015
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 \(R(\theta,\phi)\), 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
1 Background
The Roche potential is the potential energy per unit test mass which is orbiting the center-of-mass of a binary star system at the same rate as the two stars. A circular orbit is assumed for the binary system. The Roche potential includes both gravitational and centripetal energy, such that its derivative gives the force (and thus acceleration) on the test mass.
Many works have studied and calculated properties of the Roche potential and the Roche lobe, starting with the pioneering work of Kopal (1959). Plavec and Kratochvil (1964) present tables of distances for several points on the equipotential surfaces which pass through the L1 and L2 Lagrange points. This was done for a range of mass ratios from 0.1 to 1.0. Eggleton (1983) calculated volumes, V, of the Roche lobe for a range of mass ratios to obtain the equivalent volume radius \(R_{eq}\) (defined by \(V=4 \pi R_{eq}^{3}/3\)). He then presented a simple fitting formula for \(R_{eq}\) as a function of q, accurate to better than 1%, which is often used and now known as the Eggleton formula. Mochnacki (1984) presented tables of potentials and positions of the L1 and L2 Lagrange points as a function of mass ratio. He defined a fill-out factor F in terms of the potential of an equipotential surface relative to the potential of the surfaces through the L1 and L2 points. F was defined so that \(F<1\) corresponds to surfaces inside the Roche lobe, and \(1< F<2\) corresponds to surfaces between the Roche lobe and the equipotential through the L2 point. He gave tables of volume radius, area, average gravity and average inverse gravity as a function of mass ratio and F. Pathania and Medupe (2012) present series expansions for the radius of Roche equipotential surfaces, and compare various orders of the expansion to numerically calculated values.
The purpose of the present work is to present an freely available software tool, written in two versions (Java and Fortran), which calculates radii of the Roche lobe. It does this for any specified direction, and gives some other commonly used quantities (such as the Lagrange points and values of the potential). The calculator is designed to be accurate for any mass ratio q between 0.01 and 100, for dimensionless rotation rates between 0.01 and 2, and fill-out factors between 0.1 and 1.0. The calculator may work for parameters outside these limits but has not been tested for accuracy or errors outside these limits.
The coordinates are Cartesian (x, y, z) and spherical-polar (r, θ, ϕ), centred on star 1 (with mass \(M_{1}\)), with the x-axis (\(\theta=\pi/2\), \(\phi=0\)) pointing towards star 2 (with mass \(M_{2}\)). The z-axis is perpendicular to the orbital plane. The mass ratio is defined as \(q=M_{2}/M_{1}\) and the binary separation is a.
For a physical system, one knows the actual values of the masses \(M_{1}\) and \(M_{2}\). Either one specifies the binary separation a, or calculates it from the orbital period P, using Kepler’s third law, \(4 \pi^{2} a^{3}= G (M_{1}+M_{2}) P^{2}\). Dimensionless distances are converted into physical units by multiplying by a.
2 Implementation
The purpose of the calculator is to allow one to calculate the precise size of the Roche lobe along the main coordinate axes or in any specified direction (θ, ϕ). Sizes are given along the x-axis, from the center of the star to the L1 Lagrange point, toward star 2 (positive x, called \(R_{L1}\)), and on the back side away from star 2 (negative x, called \(R_{bk}\)). They are given along the y-axis (called \(R_{y}\)) and along the z-axis (called \(R_{z}\)). The Roche lobe has size \(R_{y}\) along the positive and negative y-axes and size \(R_{z}\) along positive and negative z-axes. Additional values are given for the x-coordinate of the L2 and L3 points along the x-axis (called \(X_{2}\) and \(X_{3}\)), and the values of the potential at the L1 point (\(\Omega_{1}\)), at the L2 point (\(\Omega_{2}\)) and at the L3 point (\(\Omega_{3}\)). The area, A, and volume, V, of the Roche lobe are given (in units of \(a^{2}\) and \(a^{3}\), respectively), and the equivalent volume radius \(R_{eq}\), defined by \(V=4 \pi R_{eq}^{3}/3\) and area radius \(R_{a}\), defined by \(A=4 \pi R_{a}^{2}\). For the case of \(F<1\), i.e. star 1 inside its Roche lobe, the values of \(R_{bk}\), \(R_{y}\), \(R_{z}\), A, V, \(R_{a}\) and \(R_{eq}\) are for the stellar surface. Additionally the front radius, \(R_{fr}\) of the star (along the positive x-axis) is given because it is not equal to \(R_{L1}\) for \(F<1\).
The L1, L2 and L3 Lagrange points are along the x-axis (\(\mathrm{y}=\mathrm{z}=0\)). A root-finding routine is used to find the x-value where the derivative of \(\Omega(r,\theta,\phi)\) along the x-axis is zero. The root is found to an accuracy of 1 part in 10^{8} (Java version) and 1 part in 10^{9} (Fortran version) using Brent’s method (Brent 1973). This method combines the bisection method, the secant method and inverse quadratic interpolation. It has been shown previously that the L4 and L5 points lie in the x-y plane, and either of these when taken with the mass centers of \(M_{1}\) and \(M_{2}\) form an equilateral triangle, so are at fixed positions independent of mass ratio. For further discussion of the Lagrange points see the textbook Rosswog and Brüggen (2007).
Values of sizes of the Roche lobe along the major axes, positions of the Lagrange points and the potentials are calculated to accuracy which is limited by the root-finding algorithm (the accuracy of the root finding algorithm can be changed by changing tolerance parameter in the root-finding subroutine of the source code). The Roche lobe volume V and area A, and the radii \(R_{eq}\) and \(R_{a}\), are obtained by using a Romberg integration method in the Fortran program and have been tested to be accurate to 8 digits (see Section 4 below). A set of results of these integrated quantities were calculated using the Fortran code for ≃150 different values of q between 0.01 and 100, nearly equally spaced in \(\log(q)\). These were used as input to the Java code, within which V, A, \(R_{eq}\) and \(R_{a}\) are calculated by interpolation. As a result the integrated quantities from the Java calculator are accurate to 4 digits or better in the range of q from 0.01 to 100.
Example input ( q , θ , ϕ ) and output ( \(\pmb{R(\theta,\phi)}\) )
q | θ (degrees) | ϕ (degrees) | R(θ,ϕ) |
---|---|---|---|
1 | 90 | 0 | 0.5000000 |
1 | 90 | 10 | 0.4551819 |
1 | 90 | 20 | 0.4256989 |
1 | 90 | 30 | 0.4055942 |
1 | 90 | 40 | 0.3918260 |
1 | 90 | 50 | 0.3826327 |
1 | 90 | 60 | 0.3769055 |
1 | 90 | 70 | 0.3738997 |
1 | 90 | 80 | 0.3730828 |
1 | 90 | 90 | 0.3740461 |
3 Results
3.1 Roche lobe radii dependence on q, p and F
To carry out the calculations of the Roche Lobe properties, a Java version with a graphic interface (see Additional file 1), and a Fortran 90 version, with a command line interface (see Additional file 2), were created.
3.2 Fits to Roche lobe radii
The Eggleton formula has been widely used to calculate volume equivalent radius of the Roche lobe for the case of synchronous rotation \(p=1\) (here we note that the q we use is equivalent to \(1/q\) used by Eggleton (1983)). It has accuracy better than 1%. However, no similarly accurate formula for the other radii (\(R_{L1}\), \(R_{bk}\), \(R_{y}\), \(R_{z}\)) has been given. Here, we check the accuracy of the Eggleton formula and provide a somewhat improved version. We also show that using a cubic spline fit yields accurate approximations of the various radii.
4 Discussion and conclusions
Here we present a freely available calculator for various quantities related to the Roche lobe of a star in a binary system. The calculator is provided in two versions: a Java version which is easily installed and run on any computer system with Java, and a Fortran 90 version, which can be easily used and edited by anyone familiar with Fortran 90. The calculator provides two general types of output. The first type yields values of \(R_{L1}\), \(\Omega_{1}\), \(X_{2}\), \(\Omega_{2}\), \(X_{3}\), \(\Omega_{3}\), \(R_{bk}\), \(R_{y}\), \(R_{z}\), A, V and \(R_{eq}\), where the various symbols are defined above, given input values of mass ratio, q, stellar rotation parameter, p, and stellar fill-out factor, F. The second type gives the Roche radius in any direction (θ, ϕ) specified by the user. Both types allows a single input parameter set, or multiple sets of input parameters in the form of a text file, so that a large number of values of Roche radius can be calculated quickly and conveniently.
The present calculator can be used to calculate the instantaneous Roche lobe for elliptical orbits, for synchronous or non-synchronous stars, by using the value of \(\frac{p (1+e)^{2}}{(1+e \cos(\nu))^{3/2}}\) instead of p as an input, as explained in Section 1 above using equations (2) and (3). The calculator has only been tested for the range \(0.1 \le p \le 2\), which restricts the corresponding range of p, e and ν for the elliptical orbit case.
To check the accuracy and precision of the calculations done here we compared calculated values from the Fortran 90 version to the most recent published tables of values for the Roche radii, done by Pathania and Medupe (2012). For \(R_{L1}\) our values, in all cases they provide, agree to the 6 digits given by Pathania and Medupe (2012). For \(R_{bk}\) our values agree in all cases to 5 digits, and differ in the 6th digit in about half the cases. For \(R_{y}\) and \(R_{z}\), we agree to 4 digits, but differ in the 5th digit in many cases. Because we have calculated the radii using 3 different codes which agree, and the Fortran 90 version has good error control, we believe that the values we present are more precise than previously published values.
To check for accuracy to more than 5 digits we rely on tests that we have carried out. The Fortran 90 code was written with all calculations done in double precision. The Fortran version was run on three different Linux machines with two different flavours of Linux (RedHat Fedora and CentOS) and gave the same numerical results in all cases. Additionally, a MathCad version (Parametric Technology Corporation) was written on a Windows 7 computer system to calculate all quantities accurate to 8 digit accuracy. The Mathcad version was very slow to run for error tolerance 10^{−9} or less so it was not used to calculate higher accuracy results. Also the Mathcad program was written to use different integration and root-finding algorithms than the Fortran program as a check. The MathCad values agreed with the Fortran 90 values to 8 digits or better. The calculated values from the Java version and the Fortran 90 version agree to 7 digits or better for \(R_{L1}\), \(\Omega_{1}\), \(X_{2}\), \(\Omega_{2}\), \(X_{3}\), \(\Omega_{3}\), \(R_{bk}\), \(R_{y}\) and \(R_{z}\). For quantities obtained by interpolation in the Java version (A, V and \(R_{eq}\)) the accuracy is lower (but still better than 4 digits). The result is that we believe, for the above reasons, the Fortran 90 program gives results which are accurate and precise to 8 digits.
5 Availability and requirements
The above software is available in both Java and Fortran 90 versions. The Java version is called the Roche Radius Calculator, and is freely available as a zip file, with simple installation and usage instructions, from the Quark Nova Project website at the University of Calgary (at http://quarknova.ucalgary.ca/, follow the ‘software’ tab to the entry ‘RoLo’ as the link ‘Download’). The Fortran 90 version is available as Fortran 90 source code in a single file at the same website under the ‘software’ tab to the entry ‘RoLo’ as the link RocheLobe.f90.
Dr. P. Eggleton suggested to include the large q limit that \(R_{eq}\propto q^{-1/3}\) which implies that the exponent of q in the logarithm term is \((a4+1/3)\).
Declarations
Acknowledgements
DAL thanks Prof. Didier Barret and Dr. Natalie Webb for hospitality at Institut de Recherche en Astrophysique et Planétologie (CNRS/UPS/OMP), Toulouse, France, where part of this work was completed. The author thanks the referees for valuable suggestions which were implemented. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
Open Access 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 reproduction 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.
Authors’ Affiliations
References
- Brent, RP: Some efficient algorithms for solving systems of nonlinear equations. SIAM J. Numer. Anal. 10, 327-344 (1973) View ArticleADSMATHMathSciNetGoogle Scholar
- Eggleton, PP: Approximations to the radii of Roche lobes. Astrophys. J. 268, 368 (1983) View ArticleADSGoogle Scholar
- Kopal, Z: Close Binary Systems. Chapman and Hall, London (1959) Google Scholar
- Limber, DN: Surface forms and mass loss for the components of close binaries-general case of non-synchronous rotation. Astrophys. J. 138, 1112 (1963) View ArticleADSGoogle Scholar
- Mochnacki, SW: Accurate integrations of the Roche model. Astrophys. J. Suppl. Ser. 55, 551-561 (1984) View ArticleADSGoogle Scholar
- Pathania, A, Medupe, T: Radius of the Roche equipotential surfaces. Astrophys. Space Sci. 338, 127-145 (2012) View ArticleADSMATHGoogle Scholar
- Plavec, M, Kratochvil, P: Tables for the Roche model of close binaries. Bull. Astron. Inst. Czechoslov. 15, 165 (1964) ADSGoogle Scholar
- Rosswog, S, Brüggen, M: Introduction to High-Energy Astrophysics. Cambridge, Cambridge (2007) MATHGoogle Scholar
- Sepinsky, JF, Willems, B, Kalogera, V: Equipotential surfaces and Lagrangian points in nonsynchronous, eccentric binary and planetary systems. Astrophys. J. 660, 1624-1635 (2007) View ArticleADSGoogle Scholar