Ghanbari, J

Khesali, A R

IACS Institutional Repository

Indian! Phys 75B (2). 79-84 (2001)U P  Ban international journalDynamical structure of planetary nebulae in three dimensions :a numerical solutionJ G h an b a riDepartment of Physics, University ofl-crdowsi, Mashhad, IranandA R K hesali*Department of Physics, University of Mazandaran, Babolsar, Iran F-mail khcsali((i^umcc ac irR eceived  20  Ju ly  2000, accep ted  1 F ebruary 2001Abstract The interacting two winds model and a nonspherical density function in three dimensions is introduced to study the dynamical structure of planetary nebulae A fast wind with a mechanical energy interacts with a super wind with mass-loss rate of 2 x 1 0 " '- ^  and a velocity of 10 km/s As a result, it produces a dense and luminous mediumKeywoids Planetary nebulae, ISM dynamics, numerical solutionP \( 'S  Nos. 98 38 Am, 98 38 Ly1. I n t r o d u c t io nThe idea th a t th e  p la n e ta ry  n e b u la e  (P N e ) o rig in a te  from  th e  o u ter  a tm o sp h e re  o f  red  g ia n ts , g o e s  b ack  to  S h k lo v sk i [ I ] .  Later on G o ld re ic h  a n d  A b e ll [2] a rg u e d  c o n v in c in g ly  th a t red g ian ts  a re  th e  m o s t p ro b a b le  p ro g in a to rs  o f  P N e an d  th a t th e y  are  th e re fo re , sh e lls  o f  g a se o u s  m a te r ia l th a t h av e  been  lo st in the  re c e n t p a s t  b y  th e ir  c e n tra l s ta rs . T h e ir  o ld  cen tra l stars are h o tte r  th a n  g a la c tic  O  s ta rs , 5 x 1 0 "* K® -  3 x 1 0  ^ K°, and less lu m in o u s , My =  - 3  to  5. T h e  e x p a n d in g  sh e lls  h av e  a typical v e lo c ity  o f  2 5  k m /s . A s a  c o n se q u e n c e  o f  th e  sh e ll expansion , th e  sh e ll d e n s i ty  d e c re a s e s  w h ic h  c a u se s  th e  neb u lar e m iss io n  to  d e c re a s e  as  w e ll an d  th e re fo re , th ey  rapid ly  b e c o m e  u n o b se rv a b le . O b se rv e d  P N e  h a v e  d e n s itie s  in the ran g e  o f  5 x  1 0  ^ -  1 0 ^ c m  a n d  ty p ic a l m a sse s  in th e  order o f  0.1 -  \.0M q . T h e  s tu d y  o f  P N e  is im p o rta n t in several d if fe re n t f ie ld s  in a s tro n o m y . A s lo w  d e n s ity  o b jec ts , generally  tr a n s p a re n t to  ra d ia tio n  e x te n d in g  lo n g w a rd s  from  the L y m an  lim it to  ra d io  w a v e le n g th s , th e y  fu rn ish  e x c e lle n t exam ples fo r  th e  s tu d y  o f  p h y s ic a l p ro c e s se s  in  d ilu te  g a se s  siubjected to  p h o to io n iz a t io n  b y  a  c e n tr a l  s ta r  so u rce . S in ce  their to rm a tio n  m u s t b e  a  s ig n if ic a n t e v e n t in  th e  ev o lu tio no f  a  la rg e  n u m b e r o f  sta rs, a  s tu d y  o f  P N e an d  a sso c ia te d  s ta rs  can  g iv e  im p o rtan t in fo rm a tio n  on  so m e  a sp e c ts  o f  the  late  s tag e  o f  s te lla r ev o lu tio n . T h e  a im  o f  th e  th e o re tic a l w o rk  is to  ex p la in  th e  d y n am ica l s tru c tu re s  o f  P N e. P ik e ln e r  [3], K w o k  [4], an d  K ahn  [5] a m o n g  o th e rs , p re se n te d  o n e  d im e n s io n a l m o d e ls  to  e x p la in  th e  d y n a m ic s  o f  ro u n d  PN e, c h o o s in g  a sp h e rica l d en sity  d is tr ib u tio n  fo r th e  re d  g ian t e n v e lo p e  (R G B ). K ah n  an d  W est [6 ] g iv e s  a s im p lif ie d  tw o  d im e n s io n a l m o d e l fo r a n o n -sp h e ric a l d e n s ity  fu n c tio n . G h a n b a ri [7] g iv es  a tw o  d im e n s io n a l m o d e l to  c a lc u la te  th e  s tru c tu re  o f  th e  sh e ll. R ecen tly , B o b ro w sk y  [8 ] an d  L o p e z  [9] stu d y  th e  P N e in tw o  d im e n s io n s .2. T h e  d e n s i tyS in ce  th e  R G E  d en sity  fu n c tio n  is th e  m a in  fe a tu re  fo r  s tu d y  th e  P N e sh ap es , G h a n b a ri [7] u se d  a  n o n -u n ifo rm  d e n s ity  fu n c tio n  fo r R G E  to  c a lc u la te  th e  p h y s ic a l p a ra m e te rs  o f  n o n -sp h e ric a l P N e  in te rm s  o f  tim e. H e a s su m e d  th e  re d  g ian t m a ss-lo ss  ra te  p e r s te rad ia n  w h ic h  d e p e n d s  o n  th e  p o la r  an g le  0.= (1)H iorresponding Author O 2001 I ACS80 J  Ghanhari and A R Khesaliw h ere  B = ^ s in "  O)do), an d  1 0  k m /s) a rethe  s low  w in d  m a ss-lo ss  ra le  an d  v e lo c ity  re sp ec tiv e ly , w h ich  are  in d e p e n d e n t o f  tim e  an d  sp ace , an d  in teg e r n and  p a ra m e te r  e a re  free  q u a n ti tie s  w h ich  c h a n g e  d e n s ity  d is tr ib u tio n  fu n c tio n s . F o r th e  fo llo w in g  rea so n s , the  non- sp h e rica l d e n s ity  d is tr ib u tio n , d e p e n d s  on  b o th  p o la r  an g le  0 a n d  a z im u ta l an g le  (i)  S o m e  o f  th e  cen tra l s tra rs  o f  PN e can  be o n e  m e m b e r o f  b in a ry  s tars. T h en  th e  sh ap e  o f  the cen tra l s ta r can  be  d e fo rm e d  from  sp h e ric a l to  th e  sh ap e  o f  th e  b a ll o f  A m e ric a n  fo o tb a ll, b e c a u se  o f  m ass  tra n s fe r  [10].(ii)  I f  th e  d ire c tio n  o f  the  m a g n e tic  fie ld  o f  the  cen tra l s ta r . is n o n -u n ifo rm , th en  the  e jec tio n  o f  m a tte r  from  the m a g n e tic  p o les  o f  th e  s ta r w o u ld  be  d if fe re n t from  th a t o f  the  o th e r reg io n s. W ith  th e se  a ssu m p tio n s , w e  ch o o se  the  d e n s ity  o f  th e  red  g ia n t’s e n v e lo p e , to  h av e  the  fo llo w in g  0 an d  ^  d e p e n d e n c e sm ^ ( l  + ^ )(1  + f ' s i n ” ' <in: n ( 2 )f f (1 + i ’s in ^ '6?)(1 + ^ ' s i n "  ^ ) s i n ^ f t / ^Jo Jow h ere  t  'and n' a re  tw o  p a ra m e te rs  w h ich  a ffe c t on  th e  d is tr ib u tio n  o f  m ass  in th e  a z im u th  an g le  A s w e see  in th is case , th e  d e n s ity  h as its m in im u m  v a lu e  ai 0 = (p 0 an d  it w ill h av e  its m a x im u m  v a lu e  at e q u a to re  w h en  ^  -  tt/2.3. T h e  sh e ll d y n a m ic sIt is n o w  w id e ly  th o u g h t th a t th e  in te rac tio n  o f  th e  fast w in d  w ith  a m a ss-lo ss  ra te  - 1 0 "* MJyr an d  a v e lo c ityv,^, --10 k m /s , c an  ex p la in  th e  e v o lu tio n  o f  a P N e [11]. T h e  in te rac tio n  se ts  u p  th e  u su a l tw o  sh o ck  flo w  p a tte rn s  w ith  fo u r d is tin c t re g io n s , a n d  th re e  d is tin c t b o u n d a rie s  tha t re p re se n t th e  s ite  o f  th e  in te ra c tio n  an d  th e  re g io n s  b ey o n d  it (F ig u re  1). T h e  in w ard  sh o c k  S\ d e c e le ra te s  the  s te lla r  w in d  a n d  th e  o u tw a rd  sh o c k  ^ 2  a c c e le ra te s  th e  a m b ie n t gas. T hedM  ^an d  rad ia l v e lo c ity  a t tim e  /, th e  e q u a tio n  o f  m otion  in th e  rad ia l d ire c tio n  isFigure 1. Shock flow pattern with distinct regions and boundariessh o c k e d  s te lla r  w in d  re g io n  h as  a  so u n d  sp eed  o f  th e  o rd e r  o f  1000 km /s. A s lo n g  as th e  sh e ll is e x p a n d in g  at a  v e lo c ity  w h ic h  is ty p ic a lly  - 2 0  -  50  k m /s , less th a n  th e  so u n d  sp e e d  1 0 0 0  k m /s, th e  b u b b le  p re ssu re  is u n ifo rm  in sp a c e  b u t c h a n g e s  in  tim e. I f  w e  m ak e  th e  a s su m p tio n  th a t th e  s tre a m  lin e s  a re  ra d ia l an d  th e  sh o ck  is s tro n g , th e n  th e  e q u a tio n  o f  m o tio n  o f  a  sh e ll e le m e n t fo llo w s fro m  th e  m o m e n tu m  c o n s id e ra tio n . W e a lso  n e g le c t b o th  th e  ram  p re s su re  in th e  ra d ia l d ire c tio n  o f  th e  sh e ll e lem en t an d  th e  c e n tr ifu g a l p re s su re  c o rre c tio n  te rm  [12]. F o r a  sh e ll e le m e n t o f  m ass=  P„dA, c o s a  , (3)aw h e re  dA^  is th e  su rfa c e  e le m e n t o f  th e  sh e ll a n d  angle( I cR is th e  d ire c tio n a l a n g le  o f  th e  surfacee le m e n t at p o s itio n  /?, b e tw e e n  th e  ra d iu s  a n d  th e  n o rm a l to th e  su rface  at th e  sam e  p o in t (F ig u re  2). S in ce  th e  con trib u tio no f  th e  s te lla r  w in d  to  th e  m ass  flux  is n e g lig ib le  re la tiv e  tt) th e  c o n tr ib u tio n  o f  th e  s lo w  w in d  [ 13], th e  ra te  o f  th e  inflow  m ass  to  th e  c o n e  o f  sem i an g le  P  isdM,dA^ R, = P^  co^a (5)th e  im p o rta n t p a ra m e te r  d e f in in g  th e  p ro p e r tie s  o f  th e  fast w in d  is th e  o u tp u t ra te  o f  th e  c e n tra l s ta rs  w in d  m ech a n ica le n e rg y , Ah =  . A s w e n o te d  b e fo re , th e  energyc o n te n t o f  th e  sh o c k e d  w in d  re g io n  is a lm o s t e n tire ly  in the fo rm  o f  th e rm a l e n e rg y . T h e  th e rm a l e n e rg y  p e r  u n it vo lum e3o f  a m o n o a to m ic  g as  is eq u a l to  y  tim e s  th e  g a s  p ressu re . T h e  to ta l th e rm a l e n e rg y  o f  th e  re g io n  th e re fo re , equals3— P^V, w h e re  V is th e  b u b b le  v o lu m e . C o n c e n tra tio n  o f e n e rg y  fo r  th is  re g io n  w h ic h  is in th e rm a l eq u ilib riu m , d e m a n d s  th a t(6)T h e  eq . ( 6 ) s ta te s  th a t th e  ra te  o f  c h a n g e  o f  th e rm a l energy o f  a h o t sh o c k e d  re g io n  is e q u a l to  th e  ra te  a t  w h ic h  energy  is fed  in to  th e  re g io n  b y  fa s t w in d  m in u s  th e  ra te  a t w hich th e  h o t g a s  d o e s  w o rk  o n  th e  in te rs te lla r  g a s . I f  w e  defineth e  b u b b le  v o lu m e  V s i n f t / 6 lc/^ a n d  a ssu m e  au n ifo rm  p re s su re  th e n  th e  e n e rg y  e q u a tio n  ( 6 ) g ivesp  ___________________ ____________________"  ^ io  * io(7)Dynamical structure o f planetary nebulae in three dimensions etc. 814. The solutionTo solve the equation of motion in order to find radius, velocity and other physical parameter of the nebular shell, we have at time t - - r -  /qi the superwind terminates and rye a rs  later, a physical change occurs and a fast wind begins. The interaction between the tw'o winds occurs at time t = 0, / e h time-scale to is given byh -  ■ (8)where Rrg is the red giant radius which is typically \00Rq. We assume that the shell motion obeys the similarity solutionof R, = Ro ^  and R, = at the early stages of theinteraction, + v ^ (r  + /©) is the shell radius at time/ 0 and A is a parameter which determines the initialv e lo c ity  of the shell and is therefore related to the input energy by the fast wind. Since vyw »  v»h^, thenRrg+^swT^R() =  ^ ----_ " which is muchlarger than the red giant radius Rrg eyen for small values of time-scale r which could be small as a few year. During these stages, the density has a spherical distribution ofp(R,,) = — . Putting the initial values of /?, and R, in eq. (5), the bubble pressure at the early stage isThere are several parameters which determine the shape of the shell, £, e' and /i, n* change the RGE density, and X affects the fast wind energy rate. Another parameter which changes the initial shell radius is the time-scale r. If the backward shock has a radius R\ in the early stages, then assuming the inner shock velocity R^  «  vy^ ,, this is called hot shock. The density in the hot shocked medium is^ 34\ n R lThis gives a temperature of( 10)( H )Putting the typical values in eq. (11), r=  4 x |0 ’ K . Such a hot gas can be treated as an adiabatic medium with a cooling time tc which is given by Kahn [14], as3P 2t =Iqp2temperature provides a large sound speed ci, -  400 km s~*and the sound corssing time r  ^  = —---- - — —-  is much lessCh Ctthan the dynamical time-scale, this region therefore has a uniform pressure. The gas aoes not cool effectively ifA»  qtc [5] or»  20 (13)i.e., tor A = 3 and the previous values of the other parameters, ic»  3.7 yr. This puts a lower limit for the time-scale r. Since this limit is very small compared with the typical dynamical time-scale of PNe, it does not have any appreciable effect on the shell structure, r may have an upper limit upto the dynamical time-scale of PNe (-10^ y r \5. The resultsNumerical calculations have been achieved by a program written in Matlab enviomment. We have used Matlab's subroutins to solve the differential equations and to calculate the integrals which have been discussed in the previous sections. For our models which are considered to be in thermal equilibrium, we adopted Msm, = 2 x 10"^  Moyr~ ,^= 10 kms“*. As was pointed out in Section 4, fo -  0 and Ro can be ignored and the energy driven mode is applicable at most stages of the evolution. Here, we have examined the effects of different parameters on the shape and physical quantities of the shell. As A increases, the wind mechanical energy increases and therefore the wind puts more energy into the bubble. If the other parameters are kept constant, the shell velocity increases in both polar and equatorial directions for all <t> angles. This can be seen in Figures 3 and 4. for(12)'vhere the parameter ^ = 4 x 10^2 cm  ^gm"* s"^  is given for normal cosmic abundances. On the other hand, this highFigure 3. Velocity as a function o f time at ^ -  90  ^for two different values o f 0, ^ -  /r/2); the parameters are to be ^ «  1. »  1, - 1 * 3 .n *  2, n ' = 2^ = 90°, as A changes from 3 to 6. Moreover, at the same conditions as we can see from Figures 5 and 6, changing A from 3 to 6 causes the shell radius to increase and the shell thickness to decrease for example, in the equatoral plane82 J  Ghanhari and A R Khesali(0 =  90®) an d  fo r ^ =  90°. A s a  re su lt, w e  c o n c lu d e  th a t th e  th in  she ll a p p ro x im a tio n  h as  b een  a su ita b le  a ssu m p tio n . TTiere a re  fo u r p a ra m e te rs  € ,€ \n ,n  w h ic h  a ffe c t th e  d e n s ity  c o n tra s t. A s lo n g  as  A is ta k e n  c o n s ta n t, £, £ t n, n ch a n g e  o n ly  th e  d is tr ib u tio n  o f  th e  s lo w  w in d  d e n s ity  in  th eFigure 4. Velocity as a function of time at ^ = Q0° for two different values of “ 0, 0=  ^ nri)\ the parameters are chosen to be f  *= I, e' -  /I -  6. n = 2. fi' := 2 .Figure 5. Radius and thickness as a function of time at ^ = 90'’ and ^ * 90", the parameters are chosen to be f  = 1, £■'=!, A -  3, n * 2, n' = 2120Figure 6. Radius and thickness as a function of time at ^ « 90" and # -  90". the parameters arc chosen to be f  -  I, e' = \, >l “ 6 , n -  2,m e d iu m . A s n a n d  n' a re  ta k e n  c o n s ta n t a n d  e a n d  c' c h a n g e  fro m  I to  1 0 , th e  sh a p e  o f  th e  sh e ll b e c o m e s  m ore e lo n g a te d  a lo n g  th e  p o la r  d ire c tio n  a n d  a  b u lg e  ap p ears  a ro u n d  th e  e q u a to r  (c o m p a re  F ig u re s  7 a n d  8 ). O n  th e  o ther h a n d  w h e n  ^  = 0  o r  phi ^ 180° w e  w ill h a v e  c u s p  a t the e q u a to r  (c o m p a re  F ig u re s  9  a n d  10). A s  n in c re a se s , w e  will- o - o s  --  O .  X -Figure 7. PNe cross section in the azimutal directions ^ = 90" (left) and = 270" (right), 0  changes from 0 to 180". The parameters \ aree - \ ,  ~ A -  3, n “ 2. n' = 2- o .  XO • O B  C p o >Figure 8 . PNe cross section in the azimutal directions ^ -  90" (Icit) and ^ = 270" (right), 0  changes from 0 to 180". The parameters are 10. £ ’ = \ 0, A = 3. n * 2 ,  n ' = 2( F>c= >O . 3- O  - X- 0 . 3- 0 .1  o o - X <r>«)Figure 9. PNe cross section in the azimutal directions ^ * 0® and ^ 180" (right), 0  changes from 0 to 180®. The parameters vc■ 1, » I, A -  3, n * 2, /i' = 2.Dynamical structure o f planetary nebulae in three dimensions etc. 83have more and more mass concentration at the equator and Simultaneously less and less mass concentration at the pole. Similarly, the larger value of n' means that the concentration of mass at dir^tion 90 is maximum and the concentrationat ^ ^ 0 is minimum. Figures 11 and 12 show how changes- 0 , 2- 0 , 6  - 0 . 0 5 0  0 . 0 5  (pc)Figure 10. PNe cross section in the azimutal directions ^  = ( f  (left) and ^ 180" (right), 0  changes from 0 to 180". The parameters aref  = 10, = 10, A *  3, n “  2, rt' = 2 .- 0 . 1 ?— 0 . 1  O 0 - 1  Cpo*Figure 11. PNe cross section in the azimutal directions ^ *  0" (left) and ^ -  180" (right), 0  changes from 0 to 180". The parameters are£ - 1 .  f ' - l ,  n » 2 ,  w' = 2 .o f  n fro m  2  to  1 0 , c a u se s  th e  sh a p e s  a lo n g  th e  e q u a to r  s ta r t  to  m a k e  c u sp  an d  th e  ra d iu s  a ro u n d  th e  p o le  to  re m a in  c o n s ta n t. F ig u re s  13 an d  14 sh o w  w h e n  n '  c h a n g e s  fro mFigure 13. PNe cross section at ^  -  90", ^ changes from 0 to 360", the parameters arc £ -  I .  «  l ,  A -  3. n -  2, n' -  2 .Figure 14. PNe cross section at ^ ■  90*. ^ changes from 0 to 36CT, the parameters are f  *  1, c ' »  1, A -  3. n -  2, n' <■  10.2 to  10, sh a p e s  a lo n g  ^  9 0  s ta r t to  m a k e  c u sp  a n d  ra d iu s  w ill re m a in  c o n s ta n t a ro u n d  ^ * 0. As th e  f in a l re s u h s . F ig u re s  15 an d  16 sh o w  th e  su r fa c e  c h a n g e s  in  th re e- 0 .1 - 0  1< po)Figure 15. PNe surface for f  “ 1. e' « 1, A -  3, » •  2, n' = 2.1  C p c )I'igure 12. PN, section in the izlmulul directioni ^  ■ 0" (left) ^ ’  '*0° (right), e  chw gei from 0 to I80". The parameters are I, c' = l, A - 3 ,  n -  1 0 . « ' - 1 0 ,- 0.1C p o >Figure 16. PNe surfree for -  1, -  I, A •  3, n »  10, ft’ »  10.84 J Ghanbari and A R Khesalidimensions corresponding to changes of n, n* from 2 to 10. This method of numerical analysis of dynamical structures has been proved to have high capabilities and our future work will be to study the ionization structure of planetary nebulae.Reference![1] I A  Shkiovski if jr. 33 315 (1956)[2] P Goldreich and G 0  Abell PubI Asiron. Soc Pac. 78 232 (1966)[3] S B Pikclncr Astrophyx J  U tt  15 91 (1973)[4] S Kwok J  Roy Astron Soc Can. 74 216 (1980)[5] K D  Kahn Planetary Sebulae (ed ) D  R Flower lAUSymp  No. 103 p305 (1983)[6] F D Kahn and K A West Mon. Rot. Roy. Astron. Soc. 212 g] (1985)[7] J Ghanbari Structure of Wind Nebulae (PhD Thesis) ( 198 9 )[8] M Bobrowsky Nature. 2 April (1998)[9J J A Lopez and R Vazqez Astron. J. (1999)[10] Z Kopal Close Binary Systems (Netherland : Reidel) ( 1959)[11] A Renzini in Physical Processes in Red Giants (eds.) 1 \\^  and A Renzini (D ordrecht: Reidel) p431 (1981)[12] J L Giuliani (Jr) Astrophys. J. 256 624 (1982)[13] J E Dyson Astrophys. Space Sci. 35 299 (1975)[14J F D Kahn Astron Astrophys 50 145 (1976)

Dynamical Structure of Planetary Nebulae in Three Dimensions: A Numerical Solution

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Dynamical Structure of Planetary Nebulae in Three Dimensions: A Numerical Solution

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