J/A+A/712/A76ivo://CDS.VizieR/J/A+A/712/A76CDSQi S.Song H.Meynet G.Maeder A.Ban Z.-L.Han Z.Li Q.-L.He Y.,Qu X.-Y.Zhong W.-L.Zhan Q.2026-08-06T17:00:54Z2026-08-06T17:52:16ZCDS support team

CDS, Observatoire de Strasbourg, 11 rue de l'Universite, F-67000 Strasbourg, France

cds-question@unistra.frmultiple-starsstellar-radiiastronomical-modelsAccurate Roche-lobe radii are vital for mass transfer and binary evolution modeling. While the Eggleton (1983ApJ...268..368E) formula is widely used for its simplicity and 1% accuracy, observations reveal systematic underestimations in certain mass-ratio ranges - a discrepancy expected within the standard Roche framework. Retaining these standard assumptions, we refined the numerical accuracy using Eggleton's formula as a baseline. Crucially, because Eggleton's model only constrains the inner critical surface (L1), it cannot accommodate over-contact binaries that require outer boundary modeling. By extending numerical integrations to the outer critical surface, we derived a more precise formula for the outer Roche-lobe radii. This provides a more realistic physical representation and is a key improvement in this work. Beyond determining the inner and outer Roche-lobe radii, we also established more precise fitting formulae for the positions of L1, L2, and L3. This study aimed to construct volume-equivalent radii formulae for the equipotential surfaces passing through these three Lagrangian points, providing a more comprehensive tool for binary evolution modeling. Furthermore, observational data of contact binaries covering mass ratios q=M2/M1 from 0 to 20 were utilized for validation. In this work, we defined a spherical coordinate system centered on the primary star (M1) and derived analytical approximations for the positions of the three collinear Lagrangian points (L1, L2, and L3). Furthermore, we proposed a new method to calculate the volume-equivalent radii for L2 and L3. By using the lowest points along the L2 and L3 equipotential surfaces as boundaries, this approach divides the peanut-shaped structure into two components, thereby yielding the corresponding volume-equivalent radii R_L2_ and R_L3_. The improved Roche-lobe radii formulae in this study achieve errors <=0.5% across all mass ratios q. The research results provide a more precise theoretical framework for simulating mass overflow and mass loss through L1, L2 and L3. Determining the positions of the Lagrangian points aims to calculate the Roche potential and investigate mass transfer and loss. These new formulae are applicable to any q with higher precision, and L2, and L3 facilitates the analysis of mass loss through the outer Lagrangian points.2026A&A...712A..76Qhttps://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/712/A76CatalogResearchIsServedByTAP VizieR generic serviceIsServedByConesearch servicehttps://cds.unistra.fr/vizier-org/licences_vizier.htmlhttps://vizier.cds.unistra.fr/viz-bin/VizieR-2?-source=J/A+A/712/A76https://vizier.iucaa.in/viz-bin/VizieR-2?-source=J/A+A/712/A76http://vizieridia.saao.ac.za/viz-bin/VizieR-2?-source=J/A+A/712/A76https://vizier.cds.unistra.fr/viz-bin/votable?-source=J/A+A/712/A76https://vizier.iucaa.in/viz-bin/votable?-source=J/A+A/712/A76http://vizieridia.saao.ac.za/viz-bin/votable?-source=J/A+A/712/A76GETtext/xml+votablehttps://tapvizier.cds.unistra.fr/TAPVizieR/tapCone search capability for table J/A+A/712/A76/tablea1 (Comparison of equal-volume radii approximations)https://vizier.cds.unistra.fr/viz-bin/conesearch/J/A+A/712/A76/tablea1?https://vizier.iucaa.in/viz-bin/conesearch/J/A+A/712/A76/tablea1?http://vizieridia.saao.ac.za/viz-bin/conesearch/J/A+A/712/A76/tablea1?GETtext/xml+votable180.050000true7.637945657.72980440.0055555555555555566/49 88 147 186 227 263 400 420 454 474 678 689 759-761 780 811 891 931 1072 1118 1266 1288 1302 1314 1329 1384 1397 1432 1441 1485 1570 1599 1621 1679 1691 1773 1796 1800 1820 1882 1916 1933 1945 2332 2436 2452 2462 2483 2488 2499 2511 2566 2581 2615 2619 2623 2702 2704 2726 2789 2809 2907 2910 2914 2929 2952 2990 3040 3042 3064 3201 3219 3284 3425 3444 3559 3597 3628 3669 3707 3814 3874 4078 4146 4298 4422 4425 4427 4488 4558 4595 4618 4666 4693 4698 4703 4711 4713 4724 4772 4795 4820 4947 5110 5122 5212 5414 5427 5541 5588 5656 5724 5799 5836 5868 5947 5962 5979 6248 6252 6312 6354 6408 6471 6585 6594 6609 6628 6652 6671 6704 6740 6742 6786 6921 6978 7006 7055 7092 7166 7225 7233 7247 7327-7328 7580 7682 7770 7875 8067 8129 8144 8357 8450 8480 8551 8579 8656 8663 8708 8716 8802 8871 8880 9008 9107 9165 9167 9244 9263 9277 9306 9387 9404 9415 9450 9492 9514 9578 9582 9625 9628 9689 9737 9748 9778 9780 9890 9911 9922 9951 9969-9970 10002 10023 10105 10160 10183 10295 10324 10326 10335 10351-10352 10355 10393 10462 10487 10601 10648 10706 10775 10809 10817 10855 10902 11031 11100 11152 11162 11181 11194 11256 11278 11290 11312 11327 11337 11340 11395 11576 11642 11650 11785 11796 11824 11830 12184 12223 12355 12587 12676 12701 12729 12745 12792 12885 12943 13079 13159 13168 13181 13306 13419 13437-13438 13499 13509 13618 13696 13699 13772 13811 13859 13964-13965 14087 14131 14141 14187 14219 14227 14286 14484 14528 14572 14580 14584 14600 14619 14643 14666 14698 14751 14760 14770 14779 14792 14794 14819 14825 14827 14909 14916 14918 14923 14930 14934 14938 14940-14941 14959 14966-14967 14993 15053 15108 15116 15119-15122 15130 15140 15155-15156 15168 15179 15187 15329 15396 15488 15501 15581 15900 15952 16058 16071 16186 16272 16337 16409 16411 16435 16872 17158 17328 17508 18006 18016 18234 18500 19075 19448 19468 19650 19698 19914 19934 19937 19960 19998 20046 20080 20085 20131 20229 20255 20270 20318 20397 20470 20476 21027 21193 21718-21719 21742 21749 21828 21907 21962 21969 22079 22117 22138 22180 22207 22241 22334 22460 22680 22696 22786 22902 22978 23261 23760 23772 23974 23993 24026 24091 24164 24232 24266 24276 24343 24378 24391 24522 24616 24983 25408 25553 25835 25949 26097 26169 26391 26441 26495 26611 26746 26850 26974 27017 27045 27116 27250 27481 27603 27605 27619 27639 27708 27770 27852 27855 27954 28129 28141 28143 28149 28244 28296 28432 28457 28778 28839 29671 29999 30190 30220 30491 30565 30609 30935 31135 31148 31263 31499 31506 31546 31604 31781 31812 31843 31886 31944 32109-32110 32307 32315 32385 32444 32713 32864 32999 33102 33288 33294 33317 33321 33418 33450 33591 33837 33880 34326 34611 34947 34958 35208 35331 35422 35641 36065 36494 36650 36805 36872 37212 37890 38674 39035 39933 40030 40294 40427 40656 40939 40970 41440 41515 41641 41705 41864 41983 42060 42076 42955 43191 43455 44772 44853 44872 44981 45010 45241 45287 45597 46084 46722 46763 47131 47243 47825 48035 48452https://cdsarc.cds.unistra.fr/viz-bin/moc/J/A+A/712/A76?format=asciidefaultJ/A+A/712/A76/tablea1Comparison of equal-volume radii approximationsSimbadNameSimbad column added by the CDSmeta.idcharnullableSystemBinary system namemeta.id;meta.maincharrecnoRecord number assigned by the VizieR team. Should Not be used for identification.meta.recordintBibcodeADS Bibcode for the system data sourcemeta.bib.bibcodecharqMass ratio (M1/M2)phys.massfloat1/qInverse mass ratio (M2/M1)stat.fit.paramfloataSemi-major axissolRadphys.angSize.smajAxisdoubleM1Mass of the primary componentsolMassphys.massdoubleM2Mass of the secondary componentsolMassphys.massdoubleR1Observed/Model radius of the primarysolRadphys.size.radiusdoubleR2Observed/Model radius of the secondarysolRadphys.size.radiusdoubleR1EggEqual-volume radius R1 from Eggleton formulasolRadphys.size.radiusdoubleR2EggEqual-volume radius R2 from Eggleton formulasolRadphys.size.radiusdoublee_R1Egg[] Percentage error of Eggleton R1 approximation%stat.errordoublee_R2Egg[] Percentage error of Eggleton R2 approximation%stat.errordoubleR1ThisEqual-volume radius R1 from this worksolRadphys.size.radiusdoubleR2ThisEqual-volume radius R2 from this worksolRadphys.size.radiusdoublee_R1This[] Percentage error of R1 from this work%stat.errordoublee_R2This[] Percentage error of R2 from this work%stat.errordouble_RARight Ascension (J2000) from SIMBAD (not part of the original data)degpos.eq.ra;meta.main_DEDeclination (J2000) from SIMBAD (not part of the original data)degpos.eq.dec;meta.main