Simultion of Eclipsing Binry Str Systems Boris Yim 1, Kenny Chn 1, Rphel Hui 1 Wh Yn College Kowloon Diocesn Boys School Abstrct This report briefly introduces the informtion on eclipsing binry str systems. We mde progrm which simultes the orbitl motion nd perform light curve plotting. A detiled description on our project is provided. We lso performed simultion on fmous eclipsing binry, Algol, nd compred it with the ctul experimentl result. 1. Introduction [1] Hlf or more of ll strs in the universe re in orbit round nother str or strs. In most of these multiple-str systems, there is type of system which consists of two strs only, known s binry str system, whose components my be seprted by lrge frction of light yer, or they my be lmost touching. In binries, individul strs orbit in ellipticl orbits round common center of mss. The more mssive component, which is not necessrily the brighter of the two strs, hs the smller orbit; the reltive size of ech str's orbit is inversely proportionl to its mss. One type of binry system is known s eclipsing binries, in which the eclipse of one str by nother is the key to identify its binry nture. In such systems, n eclipse occurs becuse the strs re firly close to ech other nd their orbits re seen more or less edgewise. Thus their periodic motion cuses first one str nd then the other to pss between its compnion nd us, temporrily cutting off ll or prt of the eclipsed str's light. Consequently there will be decrese in the pprent brightness of the system ech time n eclipse occurs. The resultnt light curve of the system depends on the brightness nd size of both strs which provides very useful informtion for deriving model of the system. The ims of our project re to simulte the orbitl motion nd chnges in luminosity of n eclipsing binry system with computer progrm, nd try to study the reltionship between the light curve nd the chrcteristics of binry strs with the id of the progrm. By using rel exmple, we will compre the output results of the progrm with ctul experimentl results. 1
. The Physics of Binry Strs.1. Bsic Assumptions In order to simplify the clcultions involved in the progrm, we ssume tht ll strs in the binry systems involved in the progrm re sphericl nd re blckbodies. Also, the limb drkening effect, mteril trnsfer phenomenon between the strs nd distortion of strs re neglected... Period of the system The period of binry system τ is given by 3 d τ = (1) M A + M B where d is the verge seprtion of the strs in stronomicl units, M A nd M B re respectively the msses of the two strs, nmely A nd B, of the binry system, mesured in units of solr msses.[].3. Position of the strs t prticulr time The orbit of the binry strs cn be computed using Newtonin mechnics [3]. Firstly, for given time t, the eccentric nomly ψ of the true reltive orbit cn be found by the eqution ωt = Ψ ε sin Ψ. () This cn be solved by using the following itertion formul derived from Newton s Method. ωt + ε sin Ψn 1 Ψn 1 Ψ = Ψ n n 1 (3) ε cosψn 1 1 where ε is the eccentricity of the orbit, ω is the ngulr velocity derived from τ, nd t vries from 0 to π. [3] Then the rel distnce r is clculted, from the definition of eccentric nomly, by the following formul. r = d( 1 ε cosψ) (4) After tht, the polr ngle θ is found by the ellipticl orbit eqution. cosψ ε θ = cos 1 (5) 1 ε cosψ The coordintes of str A in the orbitl plne (x A, y A ) nd tht of str B (x B, y B ) re then respectively given by
r cosθ M A 1+ x A ' M B = y A' r sin θ M A 1+ M B (5) r cosθ M B 1+ xb ' M A = yb ' r sin θ M B 1+ M A (5b) Finlly, they re trnsformed into the coordintes of str A nd str B in the true orbit, i.e. (x A, y A, z A ) nd (x B, y B, z B ) respectively, s follows x cosγ cosβ sinγ y = cosγ + cosβ sinγ z sinβ sinγ xb cosγ cosβ sinγ yb = cosγ + cosβ sinγ zb sinβsinγ sinγ cosβ cosγ sinγ + cosβ cosγ sinβ cosγ sinγ cosβ cosγ sinγ + cosβ cosγ sinβcosγ sinβ x ' sinβ y' cosβ 0 sinβ xb' sinβ yb ' cosβ 0 where α is the ngle of inclintion with z-xis s the xis-of rottion, β is the ngle of inclintion with y-xis s the xis-of rottion, nd γ is the ngle of inclintion with x-xis s the xis of rottion. (Refer to Fig. 1) (6) (6b) y z x Fig. 1 - The directions of the xes of rottion. The y-xis is pointed into the pper..4. Observble power [4] To discuss the vrious cses of eclipses, we let d = the distnce between the centers of the strs, projected onto the plne trnsverse to the line of sight. Also, = the rdius of the lrger str, b = the rdius of the smller str, F = light flux from the surfce of the lrger str nd F b = light flux from the surfce of the smller str. 3
Cse 1: d > + b (No eclipse) This is the full phse. The entire surfces of both strs re not blocked. The observble power P is P = Fπ + Fbπb (7) Cse : b < d < + b (Eclipse occurs) In this cse, only the projected re not being shded by the front str contributes to the observed light. Assuming tht the smller str is eclipsing the lrger str, the unshded re of the eclipsed str is given by = ( π α) + sin α b β + b ( sin α + π α) + b cos β sin β (cosβ sin β β ) where α nd β (Refer to Fig. ) cn be obtined by the cosine lw. 1 + d b α = cos (8b) d 1 b + d β = cos (8c) bd Therefore, the observed power from the eclipsed str (in this cse, P ) is P = F [ ( sin α + π α ) + b (cosβ sin β β )]. (9) And the totl observble power P is the observble power from the eclipsed str nd its full-phse prtner, i.e. [ ( sin α + π α ) + b (cosβ sin β β )] + F πb (8) P = F (10) b b α d β Fig. - Digrm of the binry str during prtil eclipse Cse 3: d < b (The entire smller str is eclipsing the lrger one) The totl observble light is contributed by the whole smller str nd the non-eclipsed 4
prt of the lrger str. Therefore the observble power P is P = Fπ ( b ) + Fbπb (11) Cse 4: d < b (The smller str is completely eclipsed by the lrger one) The totl observble light is solely contributed by the whole lrger str. Therefore the observble power P is P = F π (1) 3. Study of the properties of binry systems 3.1. Reltionship between str rdii nd resultnt light curve In n eclipsing binry system, if both strs hve similr rdii, minim with shrp bottom re observed. (Refer to Fig. 3) If the difference of rdii of the two strs is significnt, the minim re flt t the bottom. (Refer to Fig. 3b) It is worth noticing tht the durtion of the flt bottom is the time for the smller str to trvel out from the bck of the lrger str. Hence, the rtio of the flt-bottom durtion to the eclipse durtion revels the rtio of the rdii of the two strs. () 5
(b) Fig. 3 - Light curves of binry system with strs of () similr msses nd (b) msses with significnt difference 3.. Reltionship between inclintion of orbit nd resultnt light curve If the orbit of binry system hs no or very smll inclintion, totl eclipse occurs nd flt bottom minim will be observed. (Refer to Fig. 4) If the inclintion increses, the flt bottom will dispper, nd shrp bottom is observed, becuse the smller str is no longer fully eclipsed. (Refer to Fig. 4b) If the inclintion continues to increse, no eclipse will occur nd hence no periodic chnge of the light curve is observed. (Refer to Fig. 4c) 6
() (b) 7
(c) Fig. 4 - Light curves of binry system with orbits of () no inclintion, (b) smll inclintion nd (c) lrge inclintion. In this exmple, both strs hve msses with significnt difference. 3.3. Reltionship between msses nd resultnt orbit If both strs hve similr msses, their orbits will be the sme with their rdii identicl, no mtter wht their volumes re. (Refer to Fig. 5) However, if their msses differ lot, the rdii of their orbits will lso be different. The orbit of the more mssive str will hve shorter rdius thn tht of the less mssive one. For exmple, if the mss of str is twice of its compnion, the rdius of its orbit is hlf of its compnion. (Refer to Fig. 5b) () 8
(b) Fig. 5 - Orbits of binry system with strs of () identicl msses but different volume, nd (b) msses with 3-time difference but sme volume. The orbit with smller rdius is the one of more mssive str. 3.4. Exmple - Algol (Persei b) Algol is fmous eclipsing binry system. In fct, it is the first eclipsing binry to be discovered. It is known tht Algol A is min sequence str of spectrl type B8 (surfce temperture=1000k), with mss of 3.59 nd rdius of.88. Algol B is subgint of spectrl type K (surfce temperture=4888k) with mss of 0.80 nd rdius of 3.54. The verge seprtion between Algol A nd Algol B is bout 15 solr rdius (1.04 10 10 m). The eccentricity is nerly zero (i.e. the orbit is circulr), nd the orbit is inclined with ngles of inclintion α=0, β=-7.69 nd γ=0. [5] After running the progrm, it is found tht the period is bout 3.1 dys. The light curve hs two minim occurred t 0.5p nd 1.5p. It shows tht the minimum t 0.5p (minimum ) is much shllower thn tht t 1.5p (minimum b). Minimum only hs decrese in brightness of %, while minimum b hs decrese in visul brightness of 71%. The difference between the mximum nd minimum mgnitude of the system is bout 1.36, which grees quite well with the observtionl result. (Refer to Fig. 6, b) However, the discrepncy between observtionl result nd result from the progrm is inevitble, becuse we hve neglected the effect of distortion nd mteril trnsfer. Algol is ctully semidetched binry system where the volume of Algol B is greter thn its boundry of the Roche Lobe. Significnt trnsfer of mterils occurs, which distorts the shpe of Algol B (Refer to Fig. 6c). 9
() (b) (c) Fig. 6 - Light curves of Algol. In (), the one simulted by the progrm, it shows tht the bsolute minimum of the light curve is t t=1.5p, with remining brightness of 8% only (i.e. 7% decrese in brightness). In (b), it shows the light curve of Algol bsed on experimentl dt (Adpted from http://www.vso.org/vstr/vsotm/0199.stm). In (c), it shows the mteril trnsfer from Algol B to Algol A. 4. Conclusion 10
In this project, we hve successfully simulted the orbitl motion nd chnges in luminosity of n eclipsing binry system with computer progrm, nd studied the reltionship between the light curve nd the chrcteristics of binry strs with the id of the progrm. The progrm hs been proved successful by compring the output results of the progrm with ctul experimentl results of Algol. We believe tht the progrm cn be further developed to simulte more thn two strs nd the ssumptions be ccounted for. Finlly, we sincerely hope the progrm cn ese the work of ll who engge in the field of stronomy. 5. Acknowledgements We would like to thnk Dr. K.Y. Michel Wong of Hong Kong University of Science nd Technology, our project supervisor, for his guidnce nd support. (nd lunch ) 6. Appendix: Derivtion of the Luminosity of Str Consider ring of width dθ on the surfce of the lrger str, where is the rdius of the str (Refer to Fig. 7). Then the re of the ring δa is given by δa = πrdθ (13) where r is the rdius of the ring. The power trnsmitted from the ring δp 0 is δp = 0 Fπrdθ (14) where F is the energy flux from the surfce of the str. However, this power is trnsmitted t n ngle of θ to the line of sight. Hence the observble power from the ring δp is δp = Fπr cosθdθ (15) Since r = sin θ, we hve dr = cosθdθ = r dθ (15b) Hence δp becomes δp = Fπrdr (15c) The totl observble power of the str is the re projected onto the plne trnsverse to the line of sight. In full phse of the str, the observble power of the str P is δ Pdr = Fπrdr = Fπ (16) 0 0 P = 11
θ θ r Fig.7 - Digrm for deriving the power of str 7. Reference [1] http://www.physics.gmu.edu/clssinfo/str103/coursenotes/ectext/ch14_txt.htm, Binries nd Str Clusters [] M. A. Seeds, Foundtions of Astronomy 6 th ed., Brooks/Scole, Pcific Grove, CA (001). [3] H. Goldstein, Clssicl Mechnics nd ed., Addison-Wesley, Reding, MA (1980). [4] K.Y. Michel Wong, Mgnitude of Binry Strs(Lecture Notes) (00). [5] http://www-stro.physics.uiow.edu/~rlm/lm-pge/reserch/lgol/. 1