Jan¿ 1906. Dr. Böberts, Variable Sta/r. 123

On a Method of Determining the Absolute Dimensions of an Algol Variable . By Alex. W. Roberts, D.Sc.

I. Introduction.

The present paper deals with an extension of the problem of Algol variation in the direction of a determination of the absolute dimensions of a close . Although the more definite and more accurate consideration of the dimensions of such systems falls more properly within the area of spectroscopic research, yet, theoretically at least, the light-curve of any variable exhibits data which when properly discussed yield a determination of the absolute size of the system. The theory that underlies this important determination is the simple one that light takes an appreciable interval of time to traverse the orbit of a binary star. A moment’s reflexion will make it evident that this circum- stance must make itself manifest as an acceleration in the apparent occurrence of both the primary and secondary maximum phases. The time of passing the primary and secondary maxima will, however, remain unchanged ; that is, the approach and recession of the component relative to the Earth as they revolve round one another will be translated, owing to the measurable velocity of light, into a corresponding hastening and retardation of the successive phenomena of eclipse. It will be clear, therefore, that if we had the means of ascer- taining the light-curve of an Algol variable with perfect precision and completeness, and if all the phenomena of eclipse were capable of geometrical explanation and exposition, then, at all times, the light-curve of the system under consideration would provide sufficient data for a definite determination of the absolute dimensions—size, mass, and density of the eclipsing stars. It will require little acquaintance with the many perplexities of curves to assure one that, while in theory the problem of determining the dimensions of a binary star from an examination of its light changes has the merit of simplicity, in actual practice the solution is one beset with difficulties and obscured by uncertainties. (1) It is not possible to determine the light-curve of a close binary star with the accuracy and refinement necessary for a numerical solution of the problem, except in cases where the variable completes its full period in a few hours. In the latter circumstance the intervals between the four cardinal phases of variation, principal minimum, principal maxi- mum, secondary minimum, secondary maximum, can be deter- mined to within a minute of time. This is a quantity sufficiently refined to indicate, at least, a major limit to the size of the star. (2) If all the phenomena of variation are to be included in a L

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 1906MNRAS..66..123R personal orinstrumental, influencingtheobservationsisremote. in purpose. and inartthatthepossibilityofacommon systematicerror, inquiry regardingtheabsolutedimensionsof the system. from thismassofdetailbeingsufficientlyrefined towarrantan presents averyfavourableobjectfortheinvestigation wehave its light-curvehasbeendeterminedbyProfessor Wendell, been securedofRRCentauri,themeanlight-curve resulting weaken theweightofsolution,thenwemusttravelbeyond consideration oftheproblem,andanysystemexclusionmust solution. due toeccentricityoforbit,aswelltheirrhythmicfluctuations conditions tofindacompleteexplanationoftheiroriginandan rapid periodandbecauseofthefulnessaccuracy withwhich tion oftheproblemweareconsideringifatthisstageits leading toafalseresult,IaskedProfessorPickeringpermit the simplefactsofeclipsetoaregionphysicalcausesand me tomakeuseoftheveryfineseriesobservations sequences, mightcontainwithinthemsomesystematicerror varying stars—theproblemadmittedofadefiniteandreliable in brightness,consequentonalterationssurface-pressureand into accounttheever-changingfigureofcomponentstars, development wesimplydealwithtwosuitablestars,oneinthe northern hemisphereandoneinthesouthernhemisphere. character toconvincemethatincertaincases—i.e.rapidly than Ihadexpected.Theywere,indeed,sufficientlyfinalin the materialsofmyinvestigation. ness ProfessorPickeringatoncesentmethewholeof with theHarvardMeridianPhotometer.Withhisusualkind- rapidly varyingstarUPegasimadebyProfessorO.C.Wendell problem, takingtheLovedaleobservationsofAlgolvariablesas of theuncertaintywhichmustsurroundasolutionsoconditioned to bemaderegardingtheformandsurfacebrightnessof tension. interpretation oftheircharacter.Wemust,forinstance,take ingly, earlyin1905,workedoutagraphicalsolutionofthe component stars. as thatalreadyindicatedintheprecedingparagraphs.Iaccord- 124 Dr.Boberts,OnDeterminingtheAbsolutelxvi.3, Wendell’s observationsofUPegasi. me worthwhileinstitutinganinquiryastotheextentandnature the questionofdimensionsanAlgolvariable,itseemedto Then amongsouthernstarsover10,000observations have The modeofobservingbothstarsissodifferent inprinciple The northernvariablestarTJPegasi¡bothbecauseofits I thinkitwillberegardedasasufficientlycompleteexposi- Lest myownobservations,obtainedbythemethodof To mysurprise,theresultsobtainedweremuchmoredefinite Notwithstanding thesehindrancestoahopefuldealingwith (3) Anumberofuntenableassumptionshavealwaysto © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 1906MNRAS..66..123R the velocityoflightandcertainportionsunsymmetric variation ofbothUPegasiandRRCentaurias duetotheeclipse sideration, aswellinalllight-curvesofthe sametype. of onestarbyitsfellow. flow ofbrightnessrepresentedinthelight-curves undercon- circumstances whichoperateinproducingtheregularebband may bereadilydeducedtoexhibittherelation that existsbetween orbit, tobeofregularandunchangingfigure, thenexpressions distinguish anddefine,asfarwecan,themoreimportant Jan. 1906.DimensionsofanAlgolVariableStar,125 variation exhibitedintheselight-curvesitwillbenecessaryto the symmetricalportion ofanAlgolstar’svariationandthechief V PegasiandRRCentaurirespectively. elements ofthesystem. If weconsiderthecomponentbodiestomove inaclosed (1) Itseemsreasonabletoregardthemajor portionofthe In thetwofollowingfiguresaregivenlight-curvesof Before dealingdirectlywiththerelationwhichexistsbetween h o i2345678910ii12s h o I23456789 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System II. StatementoftheProblem. L 2 1906MNRAS..66..123R is thecorrectioninmagnitudetobeappliedmeanplaces, IŒ Uentauri,givenabove, indicatesthateclipsealonewillnot account forallthephenomena ofvariation. places. and themeananomalyatanytimeTifeccentricitybe expresses thedifference(inminutes)betweentrueanomaly In thecasethereforeofeccentricityand the positionof computed onthebasisofacircularorbit,toreducethemtrue a meansofdeterminingtheirvalues. expression (2)becomes apsidal linebeingunknown,theaboveexpression (3)willafford small. Then itisevidentthat is necessarytointroduceintotheequationsdefiningvariation terms dependingontheeccentricity. presume thattheeccentricityofsystemiszero. investigation arereferredtothisarticle. Those curiousregardingthepreliminaryportionofpresent sion oftheproblemthansimpledeterminationele- that afairlyfullexpositionoftherelationconnectingmovement and variationisgiveninMonthlyNotices^vol.Ixiii.pp.531-534. ments oforbitalmovement,Imayherejustrefertothefact 126 Dr.Boberts,OnDeterminingtheAbsolutelxvi.3, Putting (3) Anexaminationofthelight-curves U Pegasiand Consequently Let In ordertocorrectanyerrorarisingfromthisassumptionit (2) TheformulaegiveninMonthlyNotices,vol.Ixiii.p.534, As thepurposeofthispaperistodealwithawiderexten- Am =ChangeinmagnitudeperminuteattimeT. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System P =Periodofvariableinminutes. X =Longitudeofperiastronreckonedfromlinesight* 6 =Trueangulardistanceofeclipsingstarfromline £ =Eccentricityoforbit. ^ eosöjp+i—-—smdjq (3) sight attimeT. • sin(0+X)(2) e ?-^-sin(0 +X)...(1) 7T 7T £ sin\=p £ cosX==q 1906MNRAS..66..123R the dimensionsof*theorbitareforthwithalso known. cos 0beingalwayspositive. the starthenasimpleexpressionoftype variation duetoeclipsesolely,itisclearthat theresiduals simple exposition as welladefiniteandrhythmicalterationintheirform. the assumedtimeofpassingprincipalminimum phase. Then correctionforlightequationbecomes time ofthevariousphenomenavariation. to bemadewhenthecomponentstarsofabinarysystem a measurablevariationintheintrinsicbrightnessofstars, component starsproducestidalperturbations,andconsequent Algol oreclipselight-curves,duetotheapparent accelerationin this furthercorrectionbecomes due tophysicalcauses. mutually eclipseoneanother.Thecorrectionadmitsofvery change offigure. readily ontheassumptionthateccentricmovementof light toconsider.Theoreticallythiscorrectionalwaysremains Jan. 1906.DimensionsofanAlgolVariableStar.-127 may betakenascoveringalltheoutstandingperiodicvariation U Pegasi&nàRECentaurioftwounequalmaxima. . Let This expression(5)givestheaberrationinbrightness inall It isevidentthatiftheamountoferrancy be known,then If nowwehavecomputedtheamount of symmetrical If weregardthesechangestobecompletedintheperiodof Physical movementsofthisnaturemustnecessityproduce Putting For example,wehavedistinctevidenceinthecaseofboth (4) Westillhavetheeffectofmeasurablevelocity This, aswellcertainotherirregularities,canbeexplained (5) Itremainsonlytointroduceaterm,,A¿,as acorrectionto © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System Am =Changeinmagnitudeperminute. 0 =Distancefromlineofsight. Z =Time(inminutes)thatlighttakestocrossthe semi-orbit. rcosö+ssinö ...(4) Am .Icos0...(5) g sin(0+Z) g cosZ=s g sin7i—r (O-O) 1906MNRAS..66..123R used inthisreductionbeing fully withthemeasuresofthisstarthanIwouldotherwise. where p,q,r,s,l,andMareunknownquantities. representing forthemostpartunsymmetricalvariation,mustto a considerableextentbeaccountedforbytheexpressionsset forth inparagraphs(2),(3),(4)>and(5).,*„. while col.(3)givesthemagnitudescorrespondingtodatesin reduced tothemeanlight-curveof1900January1,period emerge outofthepresentinvestigation,impelmetodealmore U Pegasi,themanyquestionsofextremeinterestthataroseout as wellthedesiretoobtainconfidenceinanyresultsthatmay of eventheseeminglyunimportantfeaturesAlgolvariation, 128 Br.Roberts,OnDeterminingtheAbsolutelxvi.3, typical equation: col. (1). U Pegasi. No. Date. 12 30*553 11 *599 10 *576 1 4 ‘SOI 9 *556 2 *481 8 *544 6 *520 5 *5o8 3 *494 7 '53* 1 Nov.16-464 The highaccuracyoftheHarvardseriesobservations I accordinglygiveinfulldetailtheHarvardobservationsof Col. (1)givesthedateofobservation;col.(2)thesedates Grouping theseexpressionstogetherwehavethefollowing © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 1895. + sin0,s(Am.cos0)?+AmA*=(O—C)...(6) Jan. i-ioo960 Red. DateMagni- H.M.T. tude. ? cosd'jp-tsin0^+6,r 1900. 8 III. SizeofUPegasi. •156 -26 •130 -44 •117 *40 •321 -30 •144 "23 •23s 'S •139 36 •212 -65 •I92 *50 •180 -48 Observations ofUPegasi. 167 -33 gh m418-34 59 Table I. No. Date. 24 '690 15 'SSs 14 '572 23 -682 21 ’öjS 20 -647 17 -615 16 594 13 Nov.30-563 22 ‘667 18 -631 19 -639 1895. Jan. 1-331937 Bed. BateMag- H.M.T. nitode. 1900. •362 -37 •024 *50 -oo8 58 •340 *20 •351 -24 •083 -57 •032 -47 •060 *6^ •051 -8o •040 -82 •07s -58 1906MNRAS..66..123R 1 Jan. 1906.DimensionsofanAlgolVariableStar, No. 60 61 55 45 44 62 57 53 46 43 40 28 -510 27 Feb.25*499 59 58 56 54 52 49 48 47 50 42 4 38 36 30 25-521 29 SIS 51 37 39 35 34 26 Nov.30715 33 32 31 Oct.18-522 25 Nov.30706 i8 ^ 97- 1895. Date. 1896. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System ^578 744 •700 ’690 753 749 740 734 724 •715 •706 •634 '626 •618 *559 •660 •570 •729 •685 •678 •668 •645 •600 •593 •587 •552 •653 •607 *537 ’544 •531 Jan. 1-099 Red. Date H.M.T. 1900. •108 •024 *355 •333 •325 •318 •310 •299 •243 •224 •350 *343 •291 •258 •252 •217 •034 •019 •014 •030 •005 •371 •265 *209 •325 •320 •309 •283 •202 *196 •272 *331 ■187 365 235 043 039 Magni- 9-50 9*50 tude. •30 '53 '38 •25 •26 •90 •85 •36 •40 ♦30 '32 •38 •40 •38 •86 •94 •30 •62 •56 '42 •40 •48 •63 76 •28 •33 •64 •70 '22 •66 74 •69 •51 •60 •58 •72 100 No. 63 Oct.18-569 93 92 91 86 85 84 65 64 96 94 90 87 77 67 97 95 89 88 83 76 69 68 66 75 74 98 82 Dec.28-435 73 80 79 72 70 81 Oct.18-684 78 99 71 1897. Date. '595 '455 •589 '583 '497 •462 •440 •659 •528 •486 •480 •467 •446 •665 '653 •647 •601 '578 *535 '523 •518 •492 '639 •608 •560 '553 •672 •631 •626 •619 •567 •678 •613 '583 ■576 Jan. 1-367 Ref. Date H.M.T. i goo. •001 •006 •018 •012 •294 •076 •024 '324 '307 •082 *031 •012 •005 •362 '355 '350 '345 '313 •289 •282 '273 •267 •070 •019 '319 •088 •062 •042 •107 •101 •095 •049 •036 •262 •054 ■028 '035 Magni- 9'36 tude. *49 •48 '45 •36 •62 '38 •50 '38 •50 '58 '65 '48 .32 •29 '35 '34 •58 *45 '34 •30 •32 '45 '58 •72 *58 '54 •36 '50 •70 '71 '74 '77 •65 •42 •69 •62 '35 129 1906MNRAS..66..123R 138 137 136 134 135 131 133 132 130 129 125 Jan.1-531 128 I23 127 126 124 Dec.30-668 122 121 120 119 115 II7 lió US 114 US 112 111 no 109 108 107 106 105 104 103 102 IOI Dec.28*596 i3° Dt.Roberts,OnDeterminingtheAbsolutelxvi.3, No. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 1898. Date. 1898. 30-608 28-687 •626 •575 •622 •603 •591 *571 •585 •580 •5S8 •SSI •544 •662 ■537 •644 •655 '635 598 •628 •618 •612 •681 *669 •676 •665 •648 •644 *640 •635 •630 •618 •613 •607 *600 Jaü. 1-0489-80 Ref. DateMagni H.M.T. tude. 1900. •330 -31 •279 •295 -42 •326 -28 •307 -32 •302 -37 •289 *40 •284 -46 •235 •275 •262 -62 •248 •240 •246 *90 *255 73 •241 •222 •213 •206 •196 •190 -47 •186 -44 •139 -36 •133 *40 •121 233 •117 •100 •096 -44 •092 *44 •087 -50 •082 *58 •070 *67 •065 77 128 •059 -77 •052 *79 *44 •48 •94 •96 •88 •90 •86 •64 •60 •82 •50 *39 •37 •36 *44 176 175 1 174 173 172 171 170 169 1)11110 168 167 163 162 161 159 166 165 164 160 158 157 156 155 154 153 152 5 150 149 147 148 146 145 144 No. DateMagni- 143 142 141 140 139 Jan.1-6301-3349‘34 3 898.1900. 3-671 5*525 •636 •618 •614 •609 3*549 •585 •603 •581 •565 1658 •576 *549 •665 •570 *543 •531 •658 •642 •633 *537 •638 •629 •624 •619 •601 *597 *592 •585 •580 •562 *574 •556 •647 •642 653 H.M.T. tude. •217 -72 •199 *52 •195 -48 •162 -35 •190 *42 •I66 *36 •184 -45 •156 •Ï12 -36 •151 ’34 •146 -32 •130 -36 ■124 -38 •106 -40 •127 •121 •094 *46 •i 18-37 •089 -52 •114 •098 •085 •54. •080 *59 ‘053 *76 •041 *76 •036 -69 •048 -79 •030 *65 •018 *50 •012 *45 075 -64 057 -8o •005 -44 •362 *28 *357 *32 •351 *32 *346 -32 •40 •30 *44 •32 1906MNRAS..66..123R for seconddifferences. from theforegoing204observations. 190 *513*220-8o other measuresthatnogoodpurposewouldbeservedbyretaining 189 *508-215-67 187 -494-201-52 Nos. iand19.Thesearesomanifestlyoutofaccordwiththe forty meanplacessetforthinTableIII. 188 *501-208-59 been allowedfor:— 186 *490‘197-50 them. 183 -469-176-36 Jan. 185 -485-192*46 179 7-442-149-35 178 5-649-23084 182 -462*169*32 177 Jan.56421-2239-86 184 -476-183-42 181 -457-164-31 180 *450-157*35 No. Date.H.M.T.tude. i^r -nRef.DateMagni- ofri The meanmagnitudesinTableIII.aretherefore corrected We nowproceedtoobtainthemeanlight-curveofUPegasi In TableII.wehaveindetailtheoperationoffinding It willbeobservedthattwoobservationsareexcluded—viz. Also thefollowingcorrectionsforseconddifferenceshave Table III.Date. Number Mean 1906. DimensionsöfanAlgolVariâblèStar. 27 25 26 8 7 6 Jan.io59 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System Z898. I9OO. Ref. Date d hm i 555 5 42 5 28 i ii i 21 Magnitude. SecondDiff. 9-878 9-756 Mean CorrectionforCorrected •906 •seo •736 •778 No. 204 203 202 201 200 199 198 196 192 197 195 194 193 191 Jan. 7*523Jan1-2309*85 Date. 1898. 1900. + 0-004 + 0-002 •010 •OO4 •002 •OO4 7'633 •626 •62I •6I3 •590 •53« •607 •598 '574 •558 •547 •583 •567 Ref. DateMagni- H.M.T. tude. 9-882 9-758 •916 738 •782 •864 1340 9’35 *333 *38 •328 *40 •254 -78 •24s '87 •320 "38 •265 *66 •314 *38 •306 *37 *297 ‘37 •290 -46 •274 -54 •281 -50 13t 1906MNRAS..66..123R 132 Jan. 1001 (•032 •012 •014 •012 •008 •024 •024 •019 •018 •018 •0116 •0044 •006 •024 •0196 •012 •035 •028 •019 •005 •0344 •036 •036 •034 •005 •040 •005 •041 •031 •0272 •030 •030 •0410 •042 •039 043 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System Dr. Roberts,OnDeterminingtheAbsolutelxvi,3, Determination ofMeanPlacesUPegasi. 9*36 •50 •58 *49 *45 •48 •62 •550 •62 •504 •48 •58 *44 •62 •56 •50 •53 •424 *45 •42 •47) •69 •69 *45 •69 •610 •66 •65 •65 •82 •698 •72 •756 74 •76 74 77 Jan. 1-048 Table II. •054 •070 •070 •053 •0496 778 •052 •0684 -672 •075 •065 •057 •059 •051 •049 •048 •087 •062 •0566 736 •060 •080 •076 •075 •089 •088 •095 •0864 -526 •085 •0S3 •082 •082 •094 •092 •096 •0790 •0950 •098 979 72 76 79 •64 •58 77 •80 •67 70 77 •80 •63 •58 *8o •50 •50 •58 *52 •54 •58 •59 •57 •50 •42 •44 *44 •44 •46 •566 •440 71 Jan. 1-099 (•100 •107 *1026 •106 •1222 •124 •121 •121 •118 •101 •1136 ■17 •117 •114 •112 •108 •130 •100 •151 •146 •130 •149 •128 •156 •1458 •144 •139 •1320 •133 •139 •157 •156 •1590 •162 •164 127 9-50 •38 •410 *35 .32 •36 '36 •40 •30 •39 •37 •36 •60) •40 •40 •368 •36 •34 *35 *44 •320 •36 •36 *37 •44 *32 •386 •40 •23 •36 •26 *36 •326 *35 •31 360 35 1906MNRAS..66..123R Jan. 1906.DimensionsofanAlgolVariableStar.133 Jan. I*1669*36 'i 84*45 •216 ‘67 •217 72 •195 *48 •192 *50 •i860 *446 •2184 762 •222 -86 •220 *8o •217 76 •2096 *636 •213 -64 •209 70 •1990 -544 •201 *52 •199 *52 •196 *58 •1930 *482 •196 *50 •190 -47 •180 -48 •167 -33 •212 65 •208 *59 •202 *6o •197 -50 •187 -50 •183 *42 •1716 -370 •176 *36 •206 -6o •192 *46 •190 *42 •186 *44 •169 *32 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System Jan. 1-2239*86 •246 *90 •245 'S? •2372 *906 •241 ’88 *265 *64 •2468 ^ •248 *82 •224 *85 •279 '44 *274 *54 •2684 *586 •2782 *492 •28I *50 ‘273 ‘54 •265 -66 •243 *94 •235 *94 •233 ‘9° •230 -85 •230 -84 •267 *58 •2582 *682 •2Ó2 *65 •2Ó2 *62 •258 -63 •254 78 •252 -86 •240 •235 ’»s •2280 *86o •282 *50 •272 *51 •235 *90 •255 73 275 '48 Jan. 1-2839’48 1 - -2 •307 '34 •289 -40 •330 ‘3 •284 *46 •306 -37 •287O *450 •29O *46 •289 -45 •3280 -322 •331 '37 •326 '28 •3220 3io •325 ~28 •3148 -322 •318 '26 •3062 -346 •307 -32 •29I -38 •302 -37 •294 -38 •328 ‘40 •325 5 •320 -30 •314 '38 •3 «3'35 •310 -30 •309 '33 •2952 -390 •299 ’40 •295 -42 •324 '29 •321 -30 •320 '38 •297 ‘37 319 32 1906MNRAS..66..123R 20 21 17 No. ir 18 16 14 19* 15 12 13 10 134 Dr-BobertS,OnDeterminingtheAbsolute S 6 4 9 2- 7 5 3 i - r ' 1900.d Jan. 1*0044 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System *1860 •0950 •1026 ,*0496 '0116 •1930 *1590 •1320 •1222 •1136 •0864 •0410 •1458 •0684 •0344 •0272 •0790 •0566 •0196 •1716 Jan. 1*331*9*22 •350 -30 •345 -32 •343 -32 •3448 *292 •346 *32 •340 *20 '3342 ‘334 •340 -35 •334 '34 •333 -38 •333 'S« Date. Mean Light-curveofUPegati. Jan. 1900. d I h m Table III. o 17 4 38 O 6 2 28 O 59 o 50 4 28 O 28 o 39 3 49 2 56 2 44 3 lo 4 7 3 30 39 17 21 ii 54 5 Jan. 1*3509*30 Observed 9*424 m Mag. •504 •440 •550 •482 •672 •698 •360 •410 •526 •446 •326 •782 •610 •320 758 •566 •386 738 •368 •370 1*3648 9*365 1371 9'4° •367 *36 •365 *40 •362 *38 •362 *37 •362 *28 •3532 -317 ~357 '32 •355 '34 •355 -38 •351 *32 •351 *24 Computed 9*447 m Mag. •500 •782 *443 •562 •330 •500 •704 •481 •431 •320 *323 *337 •400 •660 •764 •756 •361 •358 •562 •633 LXVI. 3* -OO23 - *003 + *004 — *004 - *026 + *031 - *006 — *012 + *001 + *063 + *002 + *002 + *015 + *OIO ~ *023 + *026 + *012 + *004 + *009 (0-0.) m •000 •000 1906MNRAS..66..123R 37 24 23 40 39 lío. 38 29 26 25 22 28 36 51 50 27 the lastcolumnofsametable. 33 32 Jan. 1906.DimensionsofanAlgolVariableStar. elements aregivenincol.5ofTableIII.,and theresidualsin 35 34 Notices, 1903June),weobtainthefollowingelements: acceleration ofthetwominima,dueto light-equation,be fluctuation inbrightnessduetotidaldeformation, and(3)ifthe sufficiently greattobediscernible, thenthecomputedmagnitudes revolve roundoneanother,incontact,acircular orbit. The componentsareconsideredtobeequal insize,andto require afurthercorrection forthesesecondarycausesofvariation. indicated inmypaperontheorbitofBECentauri(Monthly TJ Pegasibeeccentric,(2)iftherearhythmic andphysical Now, asalreadystated,itisevidentthat(1) iftheorbitof Dealing withtheforegoingmeanmagnitudesinmanner Computed magnitudesbasedontheforegoing groupof Inclination oforbit Brightness ofstar Epoch ofminimum Prolateness ofstars Brightness ofstar Period ofUPegasi (l) (2) IQOO. Jan. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 1*3648 1*1990 d. •2184 •3448 •3342 •2782 •2096 •3532 •3280 •2952 •2870 •2372 •2280 •3062 •2684 •3220 •2468 •2582 •3148 Date. Jan. 1900. 5 IS 6 41 h m 8 17 5 2 7 52 8 45 7 44 4 47 6 53 6 27 8 29 7 33 7 21 7 5 5 55 5 28 6 12 8 i 5 42 Observed 9*365 9*544 m Mag. dhm •762 •636 •322 •492 *334 •292 •346 •390 '317 •322 •882 •864 •310 •450 •586 •682 •916 1900 Jan.in 8h m418.34 S9 0 Computed I5 20' 9-538 9*375 m 0*46 o*57 Mag. o*43 •650 •767 .320 *342 •498 •320 •390 •874 *330 *347 •327 •321 *595 •721 •924 *435 867 — 'OO5 - *025 - *006 — *OIO — *014 -0*010 - *°3S — *008 - *011 — ’008 + *002 - *001 + 'OI4 - *009 - *039 + *015 + 0*006- + 'oi5, (0-C.> m •000 'iS 1906MNRAS..66..123R of thesystem,and(3)absolutedimensionssystem. eccentricity oforbit,(2)anyintrinsicchangesinthebrightness equations ofconditionwhichdetermine(1)theamount outstanding unsymmetricalportionofthelightchanges. held toexistbetweenthesecondarycausesofvariationand serviceable, wemaywriteitthus: determining thevalueandextentofthesecorrections. + (10,000Amcos0)—+(10,000Am)^=1000(0—C)...(7) and thej(0—C)quantitiesofcol.6,TableIII.,yielddatafor + (100cos6)ior+(ioosin0)10s 136 Dr.Boberts,OnDeterminingtheAbsoluteliXvi.3, (10,000 Amcos0)+(10,000sin&)^L We cannowfromTableIII.readilyformasystemof Recasting equation(6),soastorenderitmoreuniformand Equation (6)expressestherelationwhichmayreasonablybe © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System System ofEquationsCondition. I07T lOTT 1906MNRAS..66..123R 69509 _2334+ (i^) 77io(io»-)-S7844(ios)-16241^^-17959^^=:_ I20 2I9II +3i29+73(ior)_68(ios)+3i4= 4-20 -65 4-10 G^)-(^)i^)^)~ © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 31 46 63 19 87 83 56 75 which yield 75 forty equationsofconditionare: Jan. 1906.DimensionsofanAlgolVariableStar. o S o 3 8 2 1 The normalequationsresultingfromtheforegoingsystemof o — 57OI4 “3 —10 4- 6 IO637I 34 42 28 10 19 38 37 19 33 37 27 10 12 o o o 4- 6 - 4 + 54 — 1120 4- 1570 1 100 Fq 93605 39 56 67 37 26 24 80 99 90 94 95 86 99 14 77 lOTT I07T ior =4“0*oi8 , = —oio7 = -f0*000 -84 + 15 —16 100 100 92 97 97 83 93 99 44 59 74 64 50 31 35 o — 920 4- 4662 + 4730 69509 — i + 2 + 20 —10 + 65 31 83 46 56 19 63 87 75 75 o o 8 8 3 — 1411 + 5431 4- 5220 +78719 - 6 —10 + 36 4-10 + 66 91269 20 87 20 33 46 83 70 10 60 93 76 73 o o •= —10 137 = —5020 -35 - 8 —10 — i — ii - 5 + 14 - 8 - 14 4- 2 -25 — 6 -39 - 9 4- 6 + 15 + 15 0 4-22275 - 3370 - 570 4- 9 o 1906MNRAS..66..123R way fromasystemof orbitalelements.Wemaytreatthe responding computedmagnitudes,derived in theordinary twenty-nine meanmagnitudesofthisstar, asalsothecor- varying binarystar.Thisinitselfisa matterofsome importance. with thelastofforegoingvalues,viz. it followsthat From which,since the possibilityofobtainingabsolutedimensions ofarapidly of UPegasi. corrected tonoinconsiderableextent. accurate meanlight-curveistheresult,thisquantitymaybe component starsthatgotoformthesystemUPegasi 138 Dr.BoberUiOnDeterminingtheAbsolute-lxti.3, The purposeofthepresentpaperisfulfilledindealingonly This representsthetimelighttakestocrosssemi-orbit I think*however,thepresentinvestigationhas demonstrated As observationsofÜPegasimultiply,andamore Reducing thevaluetomilesweobtainasdistanceof In theMonthlyNotices,vol.Ixiii.pp.538, 539,wefind © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System IV. SizeofBPCentauri. 63,240,000 miles ios =+0*085 — =+0*585 -=-0587 Ai 10 0 Ai =-5^9 o m Z ==12° g =*009 X =270 i= +- l=z 5^.9 6 =0*006 59 g sinZ e sinX e cosX 172 g cosZ 1906MNRAS..66..123R - 5 - 6 + 20 + 29 + 17 + 16 2 2 2 r 5 9 17 24 21 5 26 22 26 3 20 ii i? sion (7)intheprecedingpartofthisdiscussion,wemayreadily form thefollowingsystem: intrinsic changesinthesurfacebrightnessofcomponent outstanding residualsisdueto(i)eccentricityoforbit,(2) residuals inthelastcolumnofpaperreferredto stars, (3)light-equation. That is,wemayendeavourtodiscoverwhatportionofthe same manneraswehavedonetheresidualsofUPegasi. Jan. 1906.DimensionsofanAlgolVariableStar. 0 8 3 6 4 o 1 3 9 i lOTT Taking, therefore,asourtypeofequationconditionexpres- Vp 1 Royal Astronomical Society•Provided bythe NASAAstrophysics Data System — 2 — 13 + 7 - 3 + 6 -19 20 19 10 15 14 14 18 19 15 16 18 19 10 11 20 17 o 6 4 o 8 7 2 2 ^ +92(ior)~39(ios)2+3(“) 9 —16 + 2 -15 + 8 + 81 100 100 2 2 45 Equations ofCondition. 90 64 5 79 40 98 96 97 99 85 95 58 87 54 75 34 73 48 9 65 + 4 — 14 + 7 -58 100 2 2 1 100 43 61 97 89 99 77 50 9 53 82 24 9 86 9 99 30 94 84 69 96 88 76 - 6 -17 4- 20 — i + i + 5 + 26 2 2 -3 2 + 3 5 9 26 17 22 26 24 5 20 13 21 ii 17 8 6 o 4 9 o 2 - 6 -17 + 7 — 2 + 5 + 3 + 3 1 2 2 2 2 3 3 26 5 30 3 20 3 30 17 15 21 28 10 28 26 12 26 11 19 6 = +10 139 M " +7 —16 - 9 — i — i —12 -15 — 5 - 8 + - ii + 18 -13 + 22 - 8 + 17 + 3 - 6 + + + + - 7 + 8 + 6 6 7 1906MNRAS..66..123R 8696— l62Ior2I068 (toa) ^(loï)9)3®)+77^+4^ © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System we obtainasthesemi-orbitofRRCentauri it followsthat From whichsince which yield i4° Dr.Boberts,AlgolVariableStar.lxvi.3, Dealing againonlywiththequantity Normal equations 5036 1 —2I35 145870 3,800,000 miles Jl?—= +o*oi6 lOTT in m lOTT — =+0-316 lOTT ior =+0*013 Z =+o-3 io« =+0-065 At Ai =+o-3 — =+0*029 o 10 10 Z =12° m> g =o*Oo6 l ^ X =90 p =esinX r —gsinZ ¿ =+o3 q =EcosX € =0-012 s —gcosZ + 270 + 124 =278 = +0-034 144555 -864 +336 + 23 8690 X — 1019 +106j2 + o + 39 13641 1 + 8 — I + 26 + IC +3 + 9 ! i i 1906MNRAS..66..123R o 2 2610. 022 125. ^ 1973- 2 1702... 2 1965. 5 434•• 2 1692... 2 1688... 022 4.. 2447 •• 2179 .. 022 II.. 2 40 022 2.. 262 .. continued wheretheyhadbeenalreadycommenced. are alreadysufficientlywellcaredfor,andsoonlymeasureswere list. SubsequentlyProfessorBurnhaminformedmethatthese hitherto, andalargenumberofthestarsnotedasdoublein Catalogues oftheAstron.G-esellschaftwereputonworking end of1905.Thestarsareallsituatedbetween+30°and + 40.Attentionhasmostlybeengiventostarsneglected accurate valuesofthedimensionssystem,inasmuchas— maximum, secondaryminimum,maximum—willyield cally circular,thefourdates—principalminimum,principal determination aspossibleofthefourcardinalpointsvariation. of thisstarwiththeexpresspurposesecuringasaccuratea Jan. 1906.Mr.Espin,MeasivresofDoubleStars.141 No. The followinglistcompletesthemeasuresobtainedupto Micrometrical MeasuresofDoubleStars.(ThirdSeries.) As theorbitofstarhasbeendemonstratedtobepracti- = fourtimesthelight-equationofradiusorbit. I maymentionthathaveinstitutedaseriesofobservations — (Durationofsecondquadrant+durationfourthquadrant) (Duration offirstquadrant+durationthirdquadrant) © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 13 427 H 55'4+3516 IS 3S'6 12 48-8 h m o, 3 37*4 o 26-2+33I I i*6 R.A.. 1900Dec. 1 427 539 51-4 44-8 4’ 47*3 31-5 29-8 36l6 39 2 36 45 36 58 38 31 38 50 38 51 35 13 33 10 38 4 38 7 38 3 36 50 By Rev.T.E.Espin. Stars of2 / o 237-5 303-1 343’9 203-3 294-9 228-0 322-6 3048 3i5'i l67'6 I73-9 1591 I59-7 83-0 851 85-3 P. 1 and 022. 71-08 5-08-5 30-44 35-23 62-77 36-11 12- 01 115 14-08 27-39 3°'38 3-29 19-81 55-67 11- 3 8 11-29 7’8-4 6*37 D. Mags. 3-84 8-08-1 8-8 9-0 7*o 7-0 7-0 8-0 9-2 71 Nights. Date. 3 05-87 2 2 2 3 2 2 2 3 2 2 04-81 3 3 I 1 05-38 0539 05-29 05-38 05-29 05-31 05-08 04-86 05-08 04-80 05- 8 8 04-88 04- 8 7