TO THE STATISTICS OF DOUBLE STARS

Transcription

1 TO THE STATISTICS OF DOUBLE STARS It was indicated by a nube of authos, that the study of distibution law of eleents of double stas obits, as well as of othe statistical inteelations fo these objects, can give inteesting esults fo cosogony in geneal and fo the age poble of ou sta syste in paticula. Howeve, as indicated by the autho in the peliinay note [1], wong conclusions ae often ade fo obsevational data. The ai of this study is to show the eoneous natue of soe old conclusions widely spead in the liteatue [2]. We also indicate soe new iplications fo the obsevational ateial concening double stas. Contay to Jeans the obseved distibution of eccenticities aong the double stas with known obits is fa fo giving a poof of equipatition of enegies. A diect consideation of enegies of double stas ( o lage sei-axes of thei obits) poves, that the equipatition of enegy does not occu even aong wide pais. This cicustance, togethe with the absence of dissociative equilibiu between double and single stas, leads to an uppe age bound of yeas fo the enseble of double stas. 1. Distibution of eccenticities of the double sta obits The distibution of eccenticities of double sta obits was discussed often enough. Naely it was established, that aong the double stas whose obits have been deteined, the nube of pais with eccenticities less than ε is popotional to ε 2. On the othe hand Jeans has shown that unde statistical equilibiu (Boltzan distibution ) the sae dependence should be obseved. Fo this conclusions ae ade, that we aleady deal with the ost pobable distibution, and futhe about the long tie scale. Accoding to Jeans oe caefull foulation, given in his esponse [3] to authos peliinay note, equipatition exists at least fo soe paaetes. Fist of all one should ealize that the distibution of eccenticities now at hand can diffe to a lage degee fo the eal one on account of selectivity of obsevational ateial. Fo the tie being we know only the obits of pais with copaatively shot peiods. On the othe hand, as obsevations show, the aveage eccenticity cetainly inceases with peiod. Theefoe the tue elative nube of pais with lage eccenticities excedes the elative nube of such pais aong the double stas whose obits have been deteined. To ake things clea fo theoetical point of view, we conside the distibution of states of a satellite in the phase space. Fo coodinates in the phase space we take thee coponents of satellite position and thee coponents of its ipulse with espect to the ain sta. Unde statistical equilibiu, the nube of satellites dn in the volue eleent dx dy dz dp x dp y dp z of the phase space is: 34

2 ( dn = C exp E(x,y,z,p ) x,p y,p z ) dx dy dz dp x dp y dp z, (1) θ whee E = 1 2 (p2 x + p 2 y + p 2 γm z) (2) x2 + y 2 + z 2 is the enegy of satellite, M and ae asses of the ain sta and the satellite, γ is the gavitational constant and θ is the Boltzan distibution odule. We will conside a boade class of distibutions. Suppose that in the phase space the density is an abitay function f(e) of the enegy E, athe than has a special fo C exp E θ. Then dn = f(e) dx dy dz dp x dp y dp z. Now let us ake canonical tansfoations in the phase space going ove fo the vaiables x,y,z,p x,p y,p z to the vaiables L,G,H,l,g, and h of the Delonais luna theoy [4]. The fist thee of these quantities ae in the following way expessed in tes of usual eleents of the elliptic otion, which ae the lage sei-axis a, the inclination i and the eccenticity ε: L = γm a 1/2, G = γm a 1/2 (1 ε 2 ) 1/2, H = γm a 1/2 (1 ε 2 ) 1/2 cos i. The angula coodinates l, g and h epesent the aveage anoaly, the distance of peiaston fo the node and the longitude of ascending node espectively. It is known that unde the canonical tansfoation of the phase space volues eain intact (the Jacobian is equal 1). In othe wods dx dy dz dp x dp y dp z = dl dg dh dl dg dh. On the othe hand E = γ2 M 2 3 2L 2 = 1 2 γm. a i.e. the enegy depends solely on L. Hence, in ou case the density in the phase space also depends on L only, and we can wite: dn = f(l) dl dg dh dl dg dh. This iplies that the nube of pais with L between L and L + dl, and G G 0 is equal to L 8π 3 f(l) dl dg G 0 G since l, g and h independently vay within (0, 2π). 0 dh =4π 3 f(l)(l 2 G 2 0) dl, 35

3 We have L 2 G 2 0 = 2 γ 2 M 2 aε 2 0 = L 2 ε 2 0, whee ε 0 is the eccenticity, coesponding to an obit L, G 0. Thus the nube of stas with ε<ε 0 (i.e. G>G 0 ) and L between L and L + dl is 4π 3 f(l)l 2 ε 2 0 dl, which iplies that the nube of obits fo which ε<ε 0 is We have the following theoe: N(ε 0 )=4π 3 ε f(l)l 2 dl. (3) If the density in the phase space is a function of L, i.e. of total enegy and solely of this quantity, then fo abitay density function f(l) the nube of stas, with eccenticities less than ε 0, is popotional to ε 2 0. It follows, that even if we assue that the obseved N(ε 0 ) is popotional to ε 2 0 (we have ou doubts about this because of selectivity of the ateial) this does not iply, that the phase density is popotional to exp ( E/θ), o in othe wods, that equipatition of enegy holds. Actually fo any distibution of enegy unde single condition that the phase density does not depend on othe eleents, we have N(ε) ε 2 0. Thus, even if we assue that in fact N(ε) ε 2 0, we can not conclude about equipatition of enegy, let alone the life peiod of a sta syste. Howeve we note the following cicustance. Accoding to the above, in the case whee the phase density depends on L only (equivalently on E o on the lage sei-axes), fo each inteval dl we have, that the nube of obits with eccenticities less than ε is also popotional to ε 2 0. In othe wods, the nube of obits with eccenticities between ε and ε+dε, should be popotional to εdε egadless of a. Theefoe the aveage eccenticity fo each inteval of values of lage sei-axes equals ε = 1 0 ε2 dε 1 0 εdε = 2 3 (4) independent of a. The obsevational ateial is in countadiction with this, as deonstated by the following table, obtained by Aitken [5] 36

4 P ε n 16.8 yeas Table 1. The table contains aveage eccenticities fo stas gouped accoding to peiods. The fist colun gives the aveage peiod fo each goup while the last the nube of stas in a goup. If we add to this the statistical esult by Russel, which says that fo stas with peiods about 5000 yeas the aveage eccenticity is 0.76, we should conclude, that ε depends on P. It is known that P L 3. Theefoe ε depends on L. It becoes evident that the ain assuption ade above should be false and the phase density does not depend on lage sei-axes alone. This eans that the phase density is by no eans popotional to exp ( E/θ). Even the assuption that it depends on enegy alone is false. Thee ae indications that the given dependence of ε on P is subject to stong obsevational selection [10, 11, 12]. Possibly fo distant coponents (P > 100 yeas) the vaiation of ε is sall and phase density depends only on E. It could be inteesting to study the dependence of phase density on E basing upon obsevations and to easue its deviation fo Boltzan dependence. Deivation of the phase density fo obsevational data In this paagaph we assue that the phase density depends solely on E. We will ty to get the fo of this dependence fo the epiical ateial. We saw that at least fo salle values of L the phase density pobably depends on othe eleents too. Hence ou esult should be teated athe as an aveage with espect to othe eleents. Even in this fo ou conclusion has soe value, all the oe, fo distant coponents ou assuption is pobably valid. Let the phase density again be f(l). This eans that in the volue eleent dx dy dz dp x dp y dp z the nube of stas is f γm 2Mγ p2 dx dy dz dp x dp y dp z, whee = x 2 + y 2 + z 2, p = p 2 x + p 2 y + p 2 z. 37

5 Theefoe the distibution density in the space is ρ = f γm 2Mγ =4π 0 2Mγ f γm p2 2Mγ dp x dp y dp z = p2 p2 dp. By the spatial distibution of satellites we ean the distibution obtained afte paallel shifts of the pais in a way binging the ain stas to a single point. The uppe bound in the last integal is obtained fo the condition, that ou satellites, ove along eliptical obits, i.e. ou syste have negative total enegy. Now a change of vaiable in the last integal yields L = γm 2Mγ p2 ρ() = γm 2 1/2 f(l) 2M2 γ γ2 M 2 4 L 2 4π2 γ 2 M 2 4 dl L 3. Denote then o ρ() =4π 2 γ 3 M 3 6 ρ(k) =C γm K = 2 1/2, K K 1 f(l) K 1 dl 2 L 2 L, 3 f(l) L 2 K 2 dl KL 4. (5) Using this integal equation we can find the phase density f(l) in tes of given ρ. Epic in [6] has shown that the obsevational ateial at hand, afte coection fo obsevational selectivity gives ρ 1. (6) 3 38

6 o ρ 1 K. 6 It is evident that if ρ has this special fo, then the function f(l) 1 L 3. (7) satisfies the equation (5). Let us copae this obseved density in the phase space with that unde statistical equilibiu whee we have ( f(l) =C exp E ) = C exp γ2 M 2 3 θ L 2. (8) θ It eains to deteine the value of θ. If the double stas enseble eached statistical equibliu as a esult of inteaction with othe stas, then θ by ode should equal two thids of aveage kinetic enegy of tanslational oveent of suounding stas. Suppose that the aveage speed of tanslational oveent of stas is 25 k/sec by ode, then aleady fo a>20 A.U. the exponent on the ight-hand side of (8) becoes uch less then 1. Theefoe fo lage values of L (as well as of a) f(l) = const (9) is a satisfactoy appoxiation. Meanwhile the esult obtained by Epic efes just to distant coponents. Hence (7) efes to lage values of L. We see that the obseved phase density (7) is quite diffeent fo that of statistical equilibiu (9). One can show that the distibution in the space in the case of statistical equilibiu also diffes stongly fo one obseved. Indeed fo (9) and (5) follows that unde statistical equilibiu ρ 1 3/2 (10) in contadiction with the obseved distibution (6). The diffeence between (10) and (6) is so geat that leaves no doubt that in fact (10) is unsatisfactoy even as a ough appoxiation. The foula (3) was established by Epic fo distant coponents,up to A.U.. We conclude that even fo such distant coponents the encountes has not yet led to statistical equilibiu (that is to the ost pobable distibution ) in the sizes of geat sei-axes,i.e the enegies. We will see that this stongly educes the uppe bound fo the life peiod of a sta syste. 3. Testing Epic law of invese cubes by new obsevational ateial In the pesent paagaph we conside a vey siple testing ethod fo the law (6), obtained by Epic fo the distibution of coponents in the space. We will see that this new ethod of analysis confis an appoxiate coectness of foula (6). 39

7 As a atte of fact, if coponents ae distibuted aound cental stas by the law 1 / n, whee n is abitay, then the pojection of the distibution density onto the celestial sphee will be 1 / n 1. Assue we have double stas aggegate govened by this distibution within a volue. Fo any shifts of the volue, the distibution of appaent distances obviously will continue to satisfy the law 1 / n 1. A suation of this distibutions fo volue eleents both along a diection and in vaious diections also leads to 1 / n 1. Theefoe fo abitay lage pat of the sky, fo appaent (pojected) distances we obtain the sae law. In paticula unde the altenative assuptions fo distibution densities in pojections we obtain ρ 1 3 and ρ 1 3/2 ρ 1 2 and ρ 1 1/2 which iplies that the nube of stas with appaent distances between 2 and 1 is popotional to ln 2 and 3/2 2 3/2 1. (11) 1 Fo Aitken s catalogue [7] we took all stas with appaent agnitude up to 9.0 lying in the nothen heisphee (4640 stas altogethe). In this bounds Aitken s catalogue sees to be sufficiently hoogeneous, since all stas with agnitude up to 9.0 wee tested by Aitken at Lick. The following table gives the nube of pais with distances anging fo 0.5 to 8. The second and the thid lines give nubes popotional to ln 2 1 and to 3/2 2 3/2 1 espectively. The popotionality coefficient C was chosen to have the sae total nube in each line. Inteval T otal The obseved nube of pais C ln ( 1 ) C 3/2 2 3/ It is seen fo this copaison that C ln 2 indeed yields an appoxiation (with about 10% ( 1 ) pecision) to the obseved nubes, while C 3/2 2 3/2 1 can not be justified. The deviations fo C ln 2 will definitely becoe salle if optical pais ae excluded. 1 Thus Epic s law ρ 1 3 is confied in the fist appoxiation. Once again this shows that enegies of stella pais ae not distibuted by Boltzan law. 40

8 In his answe to the autho s peliinay note Jeans acknowledged [3], that equipatition in enegies does not exist, but he added that in cetain espects thee is a toleably good appoxiation to equipatition. Howeve, in view of the above coollaies fo Epic s law thee can be no wod on any appoxiation to equipatition at all. Relaxation tie fo a double stas enseble Let us conside the tie equied fo a double stas enseble in a sta syste to each the statistical equilibiu with suounding stas. In statistical equilibiu we usually have two pocesses acting in opposite diections: on one hand, the destuction of physical pais in cosequence of inteaction with stas of the field and, on the othe hand, foation of pais when thee initially independent stas coe togethe. In the latte case the thid body caies away the excess enegy. Below we show that, because of absence of statistical equilibiu in ou sta syste, coplete utual copensation of these pocess does not take place: the nube of pais foed is negligibly sall in copaison with the nube of pais destoyed. Along with the destuction of pais by appoaching stas, sall enegy vaiations ay accuulate to cause a destuction. These pocesses lead to a statistical equilibiu in the sence of Boltzan distibution. It will be evident that the aveage tie of destuction of a sta pai, which we calculate below, is quite enough to each Boltzan distibution. Boltzan distibution is eached by eans of enegy vaiations salle than those needed fo destuction. Theefoe the tie equied fo this is not geate than the aveage lifetie of a pai. Thus the aveage life necessay fo destuction of a pai gives the ode of elaxation tie of a double sta syste. Ou coputations will efe to distant pais with a distance between coponents lage than 100 A. U. and thousands A. U. in aveage by ode. A passage of the thid sta nea a stella pai can be of two types: 1) the inial distance of the passing body fo the cente of gavity of the pai is lage as copaed with the geat sei-axes of the obit; 2) the inial distance of the thid body fo one of coponents is sall in copaison with the lage sei-axes of the pai. The coesponding types we call distant and close passages. Passages of inteediate type also ay occu, but we will not dwell on the, since thei ole is seconday. Consideing passages of Coulob paticles nea an ato, Boh has shown [8,9] that the ole of distant passages is negligably sall as copaed with the ole of close passages. Theefoe we will conside close passages only. The contibution of distant passages soewhat shotens the elaxation tie but leaves its ode intact. Fo pais of the type we now conside, velocities of obital otions aound the gavity cente has the ode of one o at ost 2 o 3 k/sec. Meanwhile the elative velocities in the sta syste 41

9 ae about 30 k/sec. Theefoe in the coodinate syste attached to the gavity cente, the satellite can pactically be consideed to be otionless. Due to the entioned velocity atio a closely passing sta will delive ost of its influence on the satellite duing a sall faction of the otation peiod of a pai. The satellite will acquie exta kinetic enegy while its potential enegy will eain unchanged. As a esult, we will have eithe gowth of geate seiaxis o coplete destuction of the pai. A contay patten of inteaction equies salle kinetic enegy of the passing sta as copaed with that of the satellite. Howeve, the pobability of such an event is too sall. The above eaks iply that to influence the satellite, the cental sta needs a tie peiod uch geate than the duation of an encounte. In this way we coe to the poble of evaluation of the change of kinetic enegy of a satellite unde influence of a passing sta, in the coodinate syste attached to pai s cente of asses. A siple calculation gives the enegy incease due to a distant passage to be E = v2 2 1, (12) 1+ p2 v γ 2 assuing the asses of the passing sta and the satellite ae equal. Hee p is the encounte paaete, i.e. the distance fo the satellite fo the line along which the passing sta oved befoe encounte. The nube of encountes fo which p lies between p and p + dp, the velocity of the passing sta between v and v + dv, and which occu within tie inteval dt equals 2πp dp v dt dn, whee dn is the nube of stas within unit volue possesing velocities between v and v + dv. Theefoe the enegy incease duing tie t will be πt v 3 dn pdp 1+ p2 v γ 2 Integation in p is ove values coesponding to a close encounte, i.e. ove p<a, a is the geat seiaxis. Theefoe lg (1+ a2 v 4 ) E =2πt 3 γ γ 2 dn, v o E =2πt 3 γ 2 n ( v lg 1+ a2 v 4 ), (13) 4 2 γ 2 whee n is the total nube of stas in unit volue (the stella density), v is the ean velocity. The tie equied fo E to each the total enegy of the syste which is γ2 2a is ; 42

10 v t = 4πγan lg (1+ a2 v 4 ). (14) 4 2 γ 2 This we can (and do) conside to be the elaxation tie. Hee a is soe ean value ove this peiod which is close to the initial value a 0, since fo the ain potion of inteaction tie the values of a eain less than a 0. Let us substitute in (14) v = c/sec and = the ass of Sun. The obseved values of A can each 1 /20 pasec, while n =0.1(pasec) 3. We get the value t = yeas. Fo salle values of a we get values of ode and yeas. Thus, fo double stas with distances between coponents less than A. U., Boltzan distibution is eached only afte a peiod of yeas. The Epic distibution (ρ 1 ) 3 was deived fo pais fo just this class. Hence fo the Boltzan distibution is not valid. We conclude that the age of these pais cannot exceed yeas. In othe wods, the distibution of geat axes of double stas obits favous athe definitely the shote tie scale. Ou pevious note entioned this cicustance, albeit above calculation of the elaxation tie was not pesented thee. This gave Jeans a chance to wite: I cannot see that Pof. Abatzuian s eaks in any way challenge this position, so that, it sees to e the obsevational data he entions ae not opposed to the long tie scale of yeas, but only to an infinitely long tie scale. Meanwhile we have seen, that siple calculations indicate that obsevational data in question ae in contadiction not only with the tie scale of yeas, but even with the tie scale of yeas, i.e. they favou copletely the shote tie scale. Dissociative equilibiu fo double stas Anothe ipotant data confiing that the encountes have not by now ceated a statistical equilibiu fo pais with distances about 10 4 A. U. by ode, ae deviations in the nube of such pais fo what is expected unde dissociative equilibiu. We denote by δn D the nube of pais whose satellites lie within eleent δγ of the phase space which we consideed above. Accoding to standad ules of kinetic theoy of gases, by a dissociative equilibiu we have δn D n 2 = δγ (πθ) ( exp E 3/2 θ ), (15) Hee E is again the intenal enegy of a pai with satellite in δγ, θ is the odule of the Boltzan distibution fo otion of the stas, n is the nube of single stas in unit volue. If we choose δγ in that pat of the phase space whee a>100a.u., then the facto exp E /θ can be eplaced by 1. We will have δn D n 2 = δγ, (16) 3/2 (πθ) By suation this foula extends to volues δγ which ae no longe infinitesial. The only estiction is that δγ should not include pats whee E/θ is no longe sall as copaed with 1. 43

11 We can (and do) take that pat of the phase space whee a 1 <a<a 2 fo soe bounds a 1,a 2. The coesponding bounds fo L will be L 1 = γm a 1/2 a 1/2 1 and L 2 = γm a 1/2 a 1/2 2. The coesponding phase volue is found to be L 2 δγ =8π 3 M L 1 L 0 dg G 0 dh = 4π3 3 ( ) = 4π3 3 3 (γm) 3/2 a 3/2 2 a 3/2 1. ( L 3 2 L 3 ) 1 = (17) Substituting (17) into (16) and putting thee M = we obtain δn D n 2 = 4 ( πγ ) 3/2 ( ) 3 3 a 3/2 2 a 3/2 1. (18) θ Fo a 1 =10 2 A. U., a 2 =10 4 A. U. and fo the sae nueical values of contstants as used above, we get δn D n =10 8. This is the faction of double stas which unde dissociative equilibiu will have a satellite with a between 100 and 10 4 A. U.. In eality, at least one fo evey seveal dozens has this popety, i.e. the event in question occues illions tie oe often then it should unde dissociative equilibiu. This pehaps is the ost stiking evidence, indicating that ou Galaxy is vey fa fo the statistical equilibiu state and, in conjunction with the esults of the pevious section, speaks fo the validity of the shote tie scale of yeas. Because at pesent we have in ou Galaxy an excess nube of double stas as copaed with the equilibiu state, dissolutions occu consideably (pehaps illion ties) oe often then ceations of pais. The esult of this section we oughly foulate as follows. The existence of such pais as α and Poxia Centaui o Washington poves the validity of the shote tie scale. In fact, satellite stas having a about 10 4 A. U. ae so nueous that even the sta closest to us possesses such a satellite. Conclusion Until now it was a widespead opinion, ainly due to Jeans, that statistic of double stas speaks in favou of the longe tie scale. While new facts fo othe doains of astonoy kept confiing the shote scale, double stas eained the ain aguent fo longe evolutionay scale. In the pesent pape the latte aguent is shown to be an illusion. A pope teatent of statistical data vey definitely points to the shote tie scale. 44

12 REFERENCES 1. Natue, vol. 137, 537, Natue, vol. 136, 432, Natue, vol. 137, 537, Fank und Mises, Die Diffeential und Integalgleichungen de Mechanik und Physik, Zweite Teil, SS, Baunschweig, pp , Aitken, The Binay Stas, Tatu Obsevatoy Publications, vol. 25, Aitken, The New Geneal Catalogue of Double Stas, Washington, Boh, Phil. Mag., vol. 25, 10, Boh, Phil. Mag., vol. 30, 581, H. B. Seyfet, 896, C. R. Babie, vol. 199, 930, M. N. Finsen, vol. 96, 862, Decebe 1936 Astonoical Obsevatoy of Leningad Univesity. 45