S tockm an et al. (1977) noted th a t plasm a in stab ilities will play a significant role in th e th re a d in g of th e accretion stream and th e subsequent behav io u r of th e flow once it is th read ed . L iebert & Stockm an (1985) provide a q u a lita tiv e description of th e process; ad d ition al details are provided by Lam b (1985), H am eury et al. (1986), Lam b (1988), W arner (1995) and Li (1998).
It is believed th a t th e m agnetic field will begin to affect th e shape and density of th e flow when r ~ Rth, where Rth is th e rad iu s a t which th e m agnetic pressure balances th e th erm al pressure of th e plasm a. T he th re a d in g of th e stream begins (b u t isn ’t neccessarily com pleted) in th e region R ^ < r < R th, w here th e m agnetic pressure increases faster th a n th e stream m a terial can a d ju st therm ally. Since the stream is denser a t its core th a n in its envelope, th e effect of th e m agnetic field will be significant initially in th e ou ter p a rts of th e stream . As th e flow approaches the th re a d in g region, stream m otion p erp en d icu lar to th e field begins to be resisted. T his is because th e stream m aterial, being p a rtia lly ionized, is a conducting fluid, and w herever a conducting fluid moves across field lines, cu rren ts are induced th a t
C H A P T E R 2. T H E A C C R E T IO N F L O W S IN P O L A R S 63
g en erate Lorentz forces opposing the cross-field m otion.
By th e tim e th e flow reaches th e th read in g region, th e kinetic energy of the stre am is significant, ab o u t 1-5 keV per nucleon. Unless th e th re a d in g process is very gentle, a su b sta n tia l fraction of this energy will be released in th e th read in g region. T h e stre am m ay produce a cavity in th e m agnetosphere (th e ‘stag n a tio n region’) in w hich p lasm a accum ulates. If all th e stre a m ’s kinetic energy were released in one or two hydrom agnetic shocks in th e stag n atio n region, th e energy w ould be rad iated as b rem sstrah lu n g w ith a tem p eratu re ~few keV. Since th is ra d ia tio n is n o t observed, it is likely th a t th e stream adjusts to th e field by a series of m uch weaker shocks (L iebert k Stockm an 1985; H am eury et al. 1986).
A m echanism for the production of TeV g am m a rays in th e th read in g region v ia th e diffusive shock acceleration process (B landford k Eichler 1987) has been proposed by K aul, K aul k B h at (1993). T h e shocked m a te ria l in th e th read in g region m ay accelerate protons to very high energies (th is is possible because of th e large sp atial extent of th e th read in g region). TeV g am m a rays are produced when th e p article beam im pacts on th e shock near th e w hite d w arf surface.
T h e p lasm a in th e threading region, su p p o rted by th e m ag n etic field, is subject to a wide variety of plasm a instabilities. Several of these resu lt in th e fragm enta tio n of th e stream into a m ixture of diam agnetic blobs an d finely-divided droplets. R ayleigh-Taylor instabilities arise where a heavy fluid (in th is case th e accretion stream ) is su p p o rted against gravity by a lighter fluid (here th e m agnetic field and tra p p e d low-density plasm a), and result in the frag m en tatio n of th e stream into large diam ag n etic blobs. A second instability, th e K elvin-H elm holtz instability, is caused by stro n g shear in a fluid w ith a density gradient; th is results in th e shredding of th e stre am plasm a into finely-divided droplets. T h e K elvin-H elm holtz in stab ility pro b ab ly strip s m aterial from th e stream along its entire tra je cto ry , not ju s t w ithin
Hu-
A th ird in stab ility th a t can result in th e form ation of blobs in th e th read ing region is th a t due to rad iatio n cooling in th e post-shock flow near R T he cooling
of th e gas in th e stag n atio n region will ten d to form condensations whose in itial size depends on th e cooling tim e and th e sound speed of th e h o t gas.
To describe the subsequent behaviour of th e blobs an d th e finely-divided m a terial, one can com pare th e tim e scales on which th e m ain physical processes are occurring. These include th e tim e scale on which th e blobs are m oving (the dyn am ical tim e scale yn = \J r 3 /(G M \)), th e tim e scale on which th e blobs are eroded by th e K elvin-H elm holtz in stab ility (rKH) , the tim e scale on which th e m otion of th e blobs p erp en d icu lar to th e field is resisted by th e field (Tdrag), and th e tim e scale on which th e blobs can be p e n etrate d by th e field (rpen) (H am eury et al. 1986; K ing 1993).
T h e tim e scale r KH on which th e blobs are eroded by K elvin-H elm holtz in stab il ities is
Tk„ = (2.1)
T ) C s V h
(A rons & Lea 1980), where 77 ~ 0.1 is th e K elvin-H elm holtz efficiency, /b is th e blob length, Vb ~ Vff is the infall velocity of th e blob, cs is th e sound speed in th e blobs, and ca = B / (An p\)1/ 2 is th e Alfven speed in th e inter-blob plasm a. T he density pi
of th e inter-b lo b plasm a can be estim ated by assum ing th a t th e inter-blob m aterial is th e p lasm a whose density is too low near th e shocks a t RM to have condensed by th e tim e it reaches th e w hite dw arf surface. T his gives p\ ~ 2 x 10-11 g s -1 (H am eury et al. 1986).
T h e tim e scale r pen on which th e m agnetic field can p e n e tra te th e blobs (via ohm ic diffusion) is
r pen = 1 0 -7/gT43/2s (2.2)
(H am eury et al. 1986), where T4 is th e blob te m p e ra tu re in u n its of 104 K. T he blobs are optically thick, so T4 is a t least as high as th e local rad ia tio n tem p e ratu re. T he blob te m p e ra tu re is probably higher th a t this, since com pression by converging field lines will h e at th e entrained m aterial.
C H A P T E R 2. T H E A C C R E T IO N F L O W S IN P O L A R S 65
p erp en d icu lar to th e field, currents are induced on th e blob surfaces. T h e surface cu rren ts will generate a Lorentz force which acts as a drag force, opposing th e blobs’ cross-field m otion. T he blobs lose energy by exciting Alfven waves in th e surrounding field, since th e field is d isto rted by th e m otion of th e blobs p erp en d icu lar to th e field. T h e tim e scale for energy loss is
_ CAm b 0\
Tdrag — £ 2 ^ 2
(D rell, Foley & R uderm an 1965), where mb is th e m ass of a blob. K ing (1993) and W y n n & K ing (1995) have quantified th is effect by m odelling th e d rag as a velocity- d ep en d en t force th a t is p ro po rtio n al to th e square of th e local field stren g th .
T he subsequent behaviour of th e dense blobs and th e finely-divided m aterial depends on th e relative m agnitudes of Tpen, Tdrag> Tkh, and Tdyn. Since th e finely- divided m aterial has r pen Tdyn, it will be th re a d ed alm ost im m ediately and forced to co -ro tate w ith th e field (i.e. a t th e angular velocity of th e b in ary ). T h e blobs, however, have r pen 7dyn, and proceed to plough th e ir way th ro u g h th e field via a series of m agnetic reconnections (Li 1999). A t each reconnection event, Kelvin- H elm holtz instabilities erode th e surface of th e blob. T h e efficiency of th e in stab ility a n d th e d rag tim e scale will determ ine w hether a blob is eroded com pletely before it reaches th e w hite dw arf or w hether th e blob can reach a region in which Tpen < rdyI1, where the blob as a whole can be th read ed . T he fine rain produced a t each reconnection event is p en etrated by th e field and th re a d ed (this process is illu strated in Fig. 2.6). T he th read ed droplets c o n stitu te a cross-w ind th a t can assist th e Kelvin- H elm holtz in stab ility in eroding th e u n th read ed blobs.
T he precise m anner in which th read ed m a te ria l is lifted ou t of th e o rb ital plane is a difficult problem which has only recently been addressed (Li 1998). T h e accel e ra tio n of th e particles a t each reconnection event m ay be significant in lifting th e m a te ria l o u t of th e o rb ital plane and onto th e field lines.
W ith in th e m agnetosphere, th e flow consists of th re a d ed droplets, th read ed blobs t h a t have survived th e shredding process, and p erh ap s some denser blobs th a t are
Magnetic Fields
Curtain matter
Curtain width