Solar wind speed theory and the nonextensivity of solar corona

Transcription

1 AXiv: Sola wind speed theoy and the nonextensivity of sola oona Du Jiulin *, Song Yeli Depatment of Physis, Shool of Siene, Tianjin Univesity, Tianjin 30007, China Abstat. The sola oona is a omplex system, with nonisothemal plasma and being in the self-gavitating field of the Sun. So the oona plasma is not only a noneuilibium system but also a nonextensive one. We estimate the paamete of desibing the degee of nonextensivity of the oona plasma and study the genealization of the sola wind speed theoy in the famewok of nonextensive statistial mehanis. It is found that, when use Chapman s oona model (1957) as the adial distibution of the tempeatue in the oona, the nonextensivity edues the gas pessue outwad and thus leads a signifiant deeleation effet on the adial speed of the sola wind. Key wods: Sola wind theoy; Nonextensive statistis. PACS Ci; m * Addess: [email protected] 1

2 1. Intodution Sola oona is a omplex system. The sola wind flow (oonal outflow) is due to the huge diffeene in the pessue between the sola oona and the intestella spae. This pessue diffeene dives plasma outwad despite the estaining influene of the sola gavitation. Taditionally, the gas of oona plasma is assumed natually to follow fom the kineti theoy in the Maxwellian sense, whee the pessue is taken to be the one of an ideal gas (Hundhausen, 197). Howeve, due to the long-ange natue of the gavitating inteations, the gas unde the field of sola gavitation behaves nonextensively, whih would lead to the patiles not stitly to follow Maxwellian distibution but, in some situations, to be powe-law one (Canme, 1998; Lavagno and Quaati, 006; Du, 004a, 006a, 007). Let us now study the nonextensive effet of the oona plasma in the gavitational field of the Sun on the sola wind speed. We will onside this system in tems of the theoy of astophysial fluid dynamis and intodue the nonextensive effet into the sola wind theoy. In the onventional theoy fo the sola wind speed, the veloity distibution funtion fo the patiles has been usually assumed to be the Maxwellian one, whih has been inanated in the theoy by the use of the euation of ideal gas as well as the Boltzmann-Gibbs entopy sine the basi theoy of sola wind was poposed by Pake in Howeve, obsevations of spae plasmas seldom indiate the pesene of ideal Maxwellian veloity distibutions. Spaeaft measuements of plasma veloity distibutions in the sola wind have evealed that non-maxwellian distibutions ae uite ommon (see, e.g., Canme 1998 and the efeenes in). In many situations the distibutions appea easonably Maxwellian at low enegies but have a supathemal powe-law tail at high enegies. This has been well modeled by the κ -distibution (Olbet 1969; Collie et al.1996), one statistial distibution euivalent to the -distibution pesented now in the famewok of nonextensive statistial mehanis (Leubne, 00; Leubne and Voos, 004; Du, 004a, 004b, 006a, 006b, 006; Lima et al, 000; Silva et al, 1998) by the powe-law fom, 1 1 mv f ( v) = B 1 (1 ), (1) kt

3 It has been found that the nonextensivity of the systems with slef-gavitating long-ang inteations, in etain situations, an be desibed by the above powe-law -distibution. In the sola oona, theκ -like distibutions have been poposed to aise fom stong noneuilinium themodynami gadients, Femi aeleation at upwelling onvetion-zone waves o shoks, and eleton-ion unaway in a Deieode eleti field (see, e.g., Canme 1998 and the efeenes in). We want to know suh a κ -like distibution will podue elatively to the Maxwellian one what effets on the speed of the sola wind athe than to explain the phenomena obseved. Fo this pupose, we fist in Se. simply eview the onventional theoy fo the sola wind speed, and then in Se.3 we study the degee of nonextensivity of the sola oona plasma. In Se.4, we genealize the theoy in the famewok of nonextensive statistis and in Se.5 we investigate the nonextensive effet on the sola wind speed. Finally in Se.6, we give the onlusive emaks.. The onventional theoy fo the sola wind speed As we know that the sola wind plasma is believed to be a noneulibium system and the atmosphee of the Sun is assumed to be steady and spheially symmeti. In suh situations, the basi fluid dynamial euations in the sola wind theoy ae given ( Zhang, 199; Hundhausen, 197) by the mass euation (the ontinuity euation), 4π ρ v = onstant, () the momentum euation, dv dp GM ρ v = ρ + ρd, (3) d d and the enegy euation, ds Q ρ v =, (4) d T whee is the distane fom the ente of the Sun; ρ,t, P and s ae the density, the tempeatue, the pessue and the speifi entopy, espetively; v is the speed of the fluid, Q is the ate of the intenal enegy geneated by the nonadiative poesses (inluding the heat tansfe), and D is the ate of the momentum obtained by the ways exept the heat pessue gadient and the gavity. Usually, the pessue of the 3

4 atmosphee of the Sun takes the fom of the euation of state of ideal gas, P = ρ kt / m based on the standad statistial mehanis, with m the mass of the poton and k the Boltzmann onstant and the speifi entopy takes the fom, 3 k 5 / s = ln( Pρ ) + onst. (5) m In the liteatues that disuss the sola wind theoy, one usually makes use of E.(5) and ombine Es.(3) and (4) to deive the diffeential fom of the euations, d d 1 kt GM Q v 5 + = + D. (6) m ρv This fom an be used to explain why the enegy flux bought by the steady onvetion of the plasma may be hanged by the loal heating to podue the heat pessue gadient o by dietly ineasing the momentum, suh as the wave pessue gadient. If delete ρ fom Es.() and (3), we an obtain the so-alled ontinuity momentum euation, s dv vs dvs GM v = v d d + D, (7) whee v s = kt / m is the sound speed. E.(7) tells us that the oona plasma might be aeleated by the ombination effets of those tems on the ight hand side of E.(7). Hee we will intodue the nonextensivity into the above theoy. It might be one possible aeleation effet on the sola wind speed. 3. Tempeatue, pessue and nonextensivity of the oona The sola oona is a plasma system fa fom euilibium. Aoding to Chapman s oona model (1957), the oona tempeatue vaies dependent on the adial distane fom the ente of the Sun (Lin, 000; Zhang, 199) as follows, T / 7 0 ( ) T0, 0 =, (8) whee 0 is a efeene distane fom the ente of the Sun. Fo example, it may be taken as 0 = 1. 05R, R is the adius of the Sun. T 0 is the tempeatue at = 0. The dependene of pessue on the adial distane is expessed by 4

5 GMµ P ) = P R 0 exp ' d' T ( ' ( 0 ), (9) whee P 0 is the pessue at = 0 ; G, M, R and µ ae the gavitational onstant, the mass of the Sun, the gas onstant and the mean moleula weight, espetively. We now onside the nonextensivity of the sola oona. The nonextensivity is one non-additive popety of the noneulibium system being an extenal field with the long-ang inte-patile inteations. We have aleady known examples of this in the many-body teatment of self-gavitating systems and plasma systems. The degee of the nonextensivity fo a noneuilibium system with the long-ange inte-patile inteations suh as the gavitational foe an be estimated by using the nonextensive paamete defined in nonextensive statistial mehanis (NSM) in tems of its deviation fom unity. The fomulation of the paamete fo the self-gavitating system an be given (Du, 004a; 006a) by k dt GM 1 =, (10) µ m d H whee k is the Boltzmann onstant and m H is the mass of hydogen atom. The deviation of fom unity is thought to desibe the degee of nonextensivity. If substituting Es.(8) into E.(10), we have lealy undestood that the sola oona is not only a noneuilibium system but also a nonextensive system. Theefoe, we have to genealize the sola wind speed theoy in the famewok of NSM so as to take the nonextensive effet into onsideation, though the onventional theoy has made vey extensive appliations. Fom E.(8) we an find the tempeatue gadient in the oona, / 7 dt T 1 0 = 0, (11) d 7 and then, make use of (10), / 7 kt0 0 GM 1 =, (1) 7m whee m is the mass of poton. When we onside the ase of 0 =1.05R and 5

6 T 0 = K (Zhang, 199), the tempeatue gadients and the values of 1- at the diffeent distanes fom the sun an be obtained fom Es.(11) and (1). They ae shown in Table I. We find that the value of 1- (the degee of nonextensivity) in the oona ises with the inease of the adial distane fom the Sun, though the tempeatue gadient slopes moe and moe gently. The fathe the adial distane is fom the Sun, the highe the degee of nonextensivity is, whih, as we will see, with the distanes fa away fom the Sun, would have a moe signifiant deeleation effet on the sola wind s speed. Table I. The values of 1- in the sola oona / 0 T/ T 0 dt/d (K /km) A genealization of the sola wind speed theoy in NSM The nonextensive statistial mehanis based on Tsallis entopy an be defined by the so-alled -logaithmi funtion, ln f, and -exponential funtion, exp f, (fo example, Tsallis et al, 1998; Lima et al, 001) ln 1 1 f = (1 ) ( f 1), f >0; (13) 1/1 exp f = [1 + (1 ) f ], (14) 6

7 with 1+(1-)f >0 and exp f = 0 othewise, whee is the nonextensive paamete diffeent fom unity. In othe wods, the deviation of fom unity desibes the nonextensive degee of the system unde onsideation. When 1 all the above expessions epodue those veified by the usual elementay funtions, and Tsallis entopy funtion, S = k f ln f dω, edues to the standad Boltzmann-Gibbs logaithmi one, S = k f ln f dω. In this new statistial famewok, the pessue of the atmosphee of the Sun an be witten (Du, 006a, 006) as P = C ρ kt m (15) / with the -oeffiient = /(5 3), 0<<5/3. And natually, the speifi entopy C an be witten by 3 k 5 / s = ln ( ρ P ) + onstant, (16) m Afte the -logaithmi funtion is eplaed by E.(13), one has 3 k 1 1 5(1 ) / s = [ P ρ 1]+ onstant. (17) m 1 We use Es.(17) and (15) instead of E.(5) and the euation of state of ideal gas, espetively, then in the new famewok, E.(6) and E.(7) beome d d 1 v 5C + kt m GM = C 1 kt m (5 3) / Q ρ + D, (18) v and s dv vs dvs GM v C = C C v d d + 7 D. (19) Thus the nonextensivity of the sola oona plasma has been intodued into the theoy of the sola wind. These new euations tell us that the nonextensive paamete diffeent fom unity will play a ole in the aeleation of the sola wind speed. 5. The nonextensive effet on the sola wind speed In Pake s theoy of sola wind speed (1958), the sola oona is assumed appoximately to be isothemal one and the nonextensive effet is negleted beause the tempeatue gadient is thought to be zeo. Thus, the fluid dynamial euations ae

8 the euation of the onsevation of mass, E.(), and the euation of the onsevation of momentum, E.(3), while the enegy euation, E.(4), is disaded. In suh a ase, the dynamial euations fo the sola wind speed beome dietly E.(7). Namely, s dv vs GM v =. (0) v d We now onside the nonextensive effets expessed by E.(10) on the Pake s speed theoy. In the pesent ase that the nonextensive effet is taken into onsideation, the dynamial euation fo the sola wind speed should be eplaed by the genealized one, E.(19). Namely, s dv vs dvs GM v C = C C. (1) v d d If 1 it edues to the onventional fom, E.(0). In the new dynamial euation, the new itial speed fo the sola wind is no longe the sound speed but is v ( ) = v C = C kt m. The new itial distane () an be detemined by s / the euation T T (1) ( ) ( ) + = 0, () dt / d dt / d C whee = = is the old itial distane in the Pake theoy ( 1) ( = 1) GM / vs when the nonextensive effet is not onsideed. When is vey lage, we onside dt/d 0, then we get and E.(1) beomes GMm (1) ( ) = v v ( ) =. (3) 4CkT C dv (1) d 1 = 1 d lnvs, (4) v C Complete the integal of above euation, we find v ( ) 4 (1) ln v = 4ln + 4ln v C s + onst. (5) whee the integal onstant an be detemined by using = ( ), v v ( ). Thus 8 =

9 E.(5) beomes v v ( ) ln v v ( ) 4 (1) 1 1 = 4ln ( ) ( ), (6) C It detemines vaiations of the adial speed dependent on the adial distane fom the Sun. It is lea that all the euations depend expliitly on the nonextensive paamete. When 1, E.(6) eoves pefetly the speed euation of Pake s sola wind theoy unde the assumption of isothemal oona (Zhang, 199), (1) v ln (1) v 4 (1) = 4ln + 3, (7) (1) whee v ( 1) = v = kt m. s / The adial dependene of the sola wind speed on the distane fom the Sun 3 v(1) v() Speed v( 10²km/s) /o Fig.I. The adial vaiation of the sola wind speed dependent on the distane fom the ente of the Sun. The dashed line denotes the speed in the ase that the nonextensive effets ae taken into aount. The solid line denotes the Pake s speed but onsideing the adial vaiation of the tempeatue. By the numeial alulations, the adial dependene of the solutions of E.(6) and E.(7) ae obtained, espetively. They ae shown in Fig.I The dashed line denotes the solution of E.(6), being the sola wind speed in the ase that the 9

10 nonextensive effets ae taken into aount. The solid line denotes the solution of E.(7), being the Pake s sola wind speed but onsideing the adial vaiation of the tempeatue. These esults lealy show that the nonextensivity of the sola oona has a signifiant deeleation effet on the adial speed of sola wind. And this effet would be enhaned with the distane fa away fom the Sun. Fom E.(6) we an deive the speed fomula of the sola wind at the distanes fa away fom the Sun. Fo instane, when is vey lage, we have 4 (1) / C 0 and, by using E.(3), 4 (1) / C ( ) = 4. Then ln( v / v ( ) ) small as ompaed with the tems, ( v / v ( ) ) and 4ln( / ( ) ) negleted. Thus E.(6) an be edued to and -3 ae vey, and so may be v ( ) 4v ( ) ln. (8) ( ) It is lea that when 1 the speed fomula of Pake s sola wind theoy fo to be vey lage, 1) v [ ln / (1) ] 1/ v (Zhang, 199; Hundhausen, 197), an also be ( s eoveed oetly fom E.(8). Substitute the elated paametes into E.(8), we find v ( ) = 4C v s ln ( ) 8kT = v (1) + ln. (9) 5 3 m 5 3 By using this fomula we an alulate the suae of speed of the sola wind fa away fom the Sun in the ase that the nonextensive effet is taken into onsideation. Fo example, the adial speed of sola wind at = an be witten as 14 1/ ( ) 0.7 (1) 1 (1) = = v v v m s -1. (30) with the nonextensive paamete =0.37. If take Pake s speed as v (1) =63 km s -1, fo example, then we find the new speed to beome v ( = 0.37) =157 km s -1. This 10

11 modifiation intodued by the nonextensive effets is uite signifiant. 6. Conlusive emaks We have studied the degee of nonextensivity of the sola oona and have intodued the nonextensive effet into the theoy fo the sola wind speed. The nonextensivity is shown to be a possible deeleation effet on the adial speed of the sola wind. Ou esults ae onluded as follows. (a) The sola oona is not only a noneuilibium system but also a nonextensive system. The degee of nonextensivity in the system ises with the inease of the adial distane fom the Sun, though the tempeatue gadient slopes moe and moe gently. (b) The nonextensivity of the sola oona has a signifiant deeleation effet on the adial speed of sola wind. And this effet will be enhaned with the inease of the distane away fom the Sun. () In the new famewok of NSM, unlike the ase in Pake s theoy, the sola wind speed does not ise monotonously with the inease of the distane fom the Sun, but ises apidly with the distane in the egions not vey fa fom the Sun (< 0R ), being the maximum at about = 8 0. Futhe out fom the Sun, beyond the maximum, the speed deeases vey slowly with the distane away fom the Sun. (d) In Pake s speed euation E.(6), If we onside the dependene of the oona tempeatue T on the distane, as expessed by E.(7), all the alulated sola wind speeds at the diffeent plaes ae signifiantly less than those in Pake s sola wind speed theoy, namely unde the assumption of isothemal oona. Why does the nonextensivity have the deeleation effet on the sola wind speed? This uestion may be answeed fom the ole of the nonextensive effet in the gas pessue of the sola oona. We know that the nonextensive paamete is detemined by E.(1) and its values ae stongly depended on the adial distibution of the tempeatue in the sola oona. When we take E.(8) as the tempeatue model of the oona, the paamete is always less than unity. The intodution of suh a 11

12 nonextensive effet leads to a deease of the gas pessue outwad (see E.(15))and thus podues the deeleation of the sola wind speed. Aknowledgements This wok is suppoted by the National Natual Siene Foundation of China unde the gant No and the 985 Pogam of TJU of China. We deeply aknowledge H. J.Haubold fo ontinuous enouagement and fo his inteest in ou woks on astophysial appliations of statistial mehanis. We thank S. Abe, A. M. Mathai and C. Tsallis fo useful disussions. Refeenes Chapman S., 1957, Smithson Cont. Astophys., 1. Collie, M. R., Hamilton, D. C., Gloekle, G., Bohsle, P. and Sheldon, R. B. 1996, Geophys. Res. Lett., 3, Canme S.R., 1998, ApJ 508, 95. Du J.L., 004a, Euophys. Lett. 67, 893. Du J.L., 004b, Phys. Lett. A 39, 6. Du J.L., 006a, Euophys. Lett. 75, 861. Du J.L., 006b, Astophys. Spae Si. 305, 47. Du J.L., 006, New Aston. 1, 60. Du J.L., 007, Astophys. Spae Si.(to be published); ond-mat/ Hundhausen A.J., 197, Coonal Expansion and Sola Wind, Belin Heidelbeg New Yok, Spinge-Velag. Lavagno, A. and Quaati, P., 006, Astophys. Spae Si. 305, 53. Leubne, M. P., 00, Astophys. Spae Si. 8, 573. Leubne, M. P. and Voos, Z., 004, ApJ. 618, 547., Lima, J.A.S., Silva R. and Plastino, A.R., 001, Phys.Rev.Lett., 86, 938. Lima, J.A.S., Silva R. and Santos, J.,000, Phys.Rev.E 61, 360. Lin Y.Z., 000, Intodution to Sola Physis, Beijing: Siene Pess. Olbet S., 1969, in Physis of the Magnetosphee, ed. Caovillano, R. L., MClay, J. F. and Radoski (Dodeht: Reidel), 641. Pake E.N., 1958, Astophys. J. 18, 664. Silva, R., Plastino, A.R., and Lima, J.A.S. 1998, Phys. Lett. A 49, 401. Tsallis C., Mendes R.S. and Plastino A.R., 1998, Physia A 61, 534. Zhang Z.D., 199, Sola Physis, Beijing: Siene Pess. 1