CAN SLOW ROTATING NEUTRON STAR BE RADIO PULSAR?

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1 CAN SLOW ROTATING NEUTRON STAR BE RADIO PULSAR? Ya.N. Istomin, A.P. Smirnov, D.A. Pak LEBEDEV PHYSICAL INSTITUTE, MOSCOW, RUSSIA Abstrat It is shown that the urvature radius of magneti field lines in the polar region of rotating magnetized neutron star an be signifiantly less than usual urvature radius of dipole magneti field. Magneti field in polar ap is distorted by toroidal eletri urrents flowing in the neutron star rust. These urrents lose up the magnetospheri urrents driven by generation proess of eletron-positron plasma in pulsar magntosphere. Due to the urvature radius derease eletronpositron plasma generation beomes possible even for slow rotating neutron stars, P B 2/3 12 < 10se. P is the period of star rotation, B 12 is the magnitude of the magneti field on the star surfae B in the units of Gauss, B 12 = B/10 12 G. 1 Introdution Disovery of radio pulsar PSR J , having the period P = 8.51se [1], was a surprise for physis ommunity dealing with radio pulsars. Radio pulsars with suh long periods have never been observed before. Aording to observations, pulsars lie bellow the line P B 8/15 12 = 1se on the diagram P B 12. This line is known as death line for radio pulsars. The death line means that slow rotating neutron stars annot be radio soures. We an interpret observation of the soure with order of the period P 10se as a neutron star with a super strong magneti field B Suh magneti field exeeds so alled ritial magneti field B > B h = m 2 e 3 /e h = Gauss. For B = B h the eletron ylotron energy hω = heb/m e beomes equal to the eletron rest energy m e 2. Neutron stars with strong magneti field B > B h are alled magnetars [2], but they are not soures of radio emission. The reason for that is the suppression of eletron-positron generation by strong magneti field in pulsar magnetosphere. Besides, the estimation of the value of magneti field B for pulsar PSR J , obtained from the value of spin down of its rotation P, B 12 (P P 15 ) 1/2, P 15 = P/10 15 se/se, points 1

2 out the normal value of its magneti field, B However, this estimation follows from the assumption that pulsar rotation energy losses are the magnetodipole losses, when the energy of a magnetized rotating neutron star radiates in the form of vauum eletromagneti wave having a frequeny equal to star rotation frequeny, Ω = 2π/P. Here it is also assumed that suh unknown parameters as neutron star radius R, its momentum of inertia I and its inlination angle of magneti dipole axis to the rotation axis χ are equal to R = 10 6 m, I = g m 2, χ = π/2. Real values of R, I and χ an differ from these values, but the differene is not big enough to reate a magneti field with magnitude B two order times exeeding the normal value B The death line defined by the relation P B 8/15 12 = 1se seams to be inorret, but reflets observation seletion. Indeed, pulsar PSR J is the weakest among observable soures. Its small spin down luminosity erg/se and little distane to it 180p [3,4] makes impossible observation of suh soures at Galati distanes. In radio pulsar theories the death line is defined by the onditions of eletronpositron generation in the polar region of the neutron star magnetosphere. The mehanism of radio emission in the pulsar magnetosphere is not established now, but there is no doubt that radio emission is related to existene of dense plasma in pulsar magnetosphere. Dense plasma means that eletron onentration n e exeeds signifiantly the so alled Goldreih-Julian density n GJ = ΩB/2πe, whih for a usual pulsar (P 1se, B 12 1) near its surfae is n GJ m 3. Therefore, for the radio emission to be generated it is neessary to produe plasma, whih under the ondition n e n GJ have to arise in pulsar magnetosphere. The elementary at of birth of an eletron-positron pair e, e + is a one-photon generation of a pair by a gamma quantum in strong magneti field, γ + B e + e + + B. For suh a proess to take plae, the transverse photon momentum (perpendiular to the magneti field) must exeed the threshold value 2m e. This means that the energeti gamma quanta with energies larger than 1MeV have to be generated in the magnetosphere. The soure of gamma photons are eletrons and positrons thierself, but they must be energeti enough to radiate gamma quanta. Thus, there must be regions in the pulsar magnetosphere with strong eletri field, where eletrons and positrons are aelerated to large energies. Charged partiles move along magneti field lines in strong magneti field. They an either radiate photons if magneti field lines have a urvature, in this ase they are alled urvature photons, or satter thermal photons, radiating from a star surfae, by Compton proess. In both ases gamma quanta move initially almost along the trajetory of radiated partile, i.e. tangential to magneti field lines. For the asade generation of e, e + pairs to take plae, i.e. for eah primary aelerated harge partile to produe many seondary partiles, radiated gamma photons must bear a large number of e, e + pairs. For that a gamma quantum has to obtain the threshold value of its transverse momentum. That is possible only if magneti field lines have a finite value of radius of their urvature ρ. Then the distane travelled by the photon before the birth of e, e + pair, i.e. its free path l, will be proportional 2

3 to the urvature radius of the magneti field line l ρ (2m e 2 /ε ph ) (ε ph is the photon energy). Magneti field strength in the pulsar magnetosphere dereases rapidly with the distane from the star enter r, B (r/r) 3. Therefore, large free paths of photons with respet to e, e + pair prodution do not result in effetive eletron-positron plasma prodution in the polar region of the pulsar magnetosphere. It is lear that the urvature radius ρ of magneti field lines in the polar region of the pulsar magnetosphere, from whih the radio emission esapes, is an important harateristi of asade proess of plasma generation. For small urvature radius plasma prodution beomes more effetive. The polar region of magnetosphere is the projetion of the light surfae, i.e. the outer boundary of the magnetosphere, where the orotation veloity of magnetospheri plasma approahes the speed of light, to the star surfae along the magneti field lines. That is the region around the axis of the magneti dipole with the dimension of r 0, r 0 R(ΩR/) 1/2 R. There the urvature of magneti field, onsidering it as dipole, is small. On the dipole axis the urvature radius is equal to infinity, ρ. On the edge of polar ap r = r 0 the urvature radius is finite, but large (ρ ) dip = 4 3 R ( ΩR ) 1/2 R. For a normal pulsar (P 1se, R 10 6 m) ρ is of the order of 10 8 m. This value of the urvature radius fixes the death line on the diagram P B 12, P B12 α < 1se. Here α is the power index depending on the onrete model of plasma generation. In order to expand the region of radio pulsar existene on the plane P B 12 one needs first of all to diminish the value of urvature radius of magneti field lines ρ in the polar region of the magnetosphere. For that Arons [5] suggested that the dipole enter an be displaed with respet to the star enter at some distane. It is lear that approahing the dipole enter to the star surfae we derease the urvature radius. In order to explain the phenomena of pulsar PSR J Arons shifted the dipole on the distane omparable with the star radius. However, the reason for suh asymmetry for the distribution of the star matter is not lear. Here we suggest that the magneti field lines in the polar region are strongly distorted in more natural manner. That is for the sak of eletri urrents flowing in the star rust. They lose the magnetospheri eletri urrents arising in the magnetosphere at proess of eletron- positron plasma generation. Due to the fat that Hall ondutivity signifiantly exeeds Pedersen ondutivity in the neutron star rust, the eletri urrent falling onto the star surfae from the magnetosphere is amplified in toroidal diretion distorting the magneti field under the surfae. Therefore, the magneti field urvature hanges namely in the region of plasma generation. 3

4 2 Eletri urrents While generating an eletron-positron plasma in the polar region of the neutron star magnetosphere, there arises an eletri urrent j, flowing along open field lines of the magneti field. The open field lines are the lines of magneti field, whih are not losed inside of the magnetosphere, but ross the light surfae R L = /Ω and esape to infinity. The projetion of open magneti field lines on the neutron star surfae is the polar ap of the size of r 0 R (ΩR/) 1/2. The eletri urrent j streams to one diretion in the enter of polar ap, and to opposite diretion near its edge. The total urrent j ds is equal to zero. This urrent realizes the spin down of a star, flowing in the rust aross the magneti field. The magnetospheri plasma at n e n GJ totally sreens the magnetodipole radiation. At j = 0 the flux of eletromagneti energy does not ross the light surfae. Only the eletri urrent j generates in the pulsar magnetosphere toroidal magneti field B ϕ, whih reates the radial flux of the eletromagneti energy from the magnetosphere. The value of j is of the order of the harateristi Goldrieh-Julian urrent j GJ = en GJ. Let us introdue a demensionless value of the eletri urrent i = j/j GJ. Aording to the paper [6] the value of i is less than unit, i 1. On the outer edge of the magnetosphere, near the light surfae, the eletri urrent j is losed. Also it must be losed in the neutron star rust, hitting to it from the magnetosphere. Under strong magnetization of the matter of the rust ω τ e 1 (τ e is the eletron relaxation time), the eletri ondutivity of the rust is strongly anisotropi. The biggest omponent of the rust ondutivity is its longitudinal part, along the magneti field, σ. The value of the ondutivity, whih is perpendiular to the magneti field, along the eletri field, σ, is (ω τ e ) 2 times less than the longitudinal one, σ = σ (ω τ e ) 2. Besides, Hall ondutivity σ exists also, whih results in the eletri urrent oriented orthoganal to both magneti B and eletri E fields, j H = σ [EB]/B. The Hall ondutivity σ is less than the longitudinal one σ, but is muh larger than the perpendiular one σ, σ = σ /(ω τ e ) = σ (ω τ e ). The eletri urrent E appears in the rust when the urrent j inroads into it from the magnetosphere. The field is potential E = Ψ, beause we onsider all quantities in the frame rotating with the star, there everything is stationary. The eletri urrent flowing in the rust j = ˆσ Ψ has a large toroidal omponent j ϕ j(σ /σ ) j. This is due to the fat that the eletri field whih is orthogonal to the magneti field indues first of all the Hall eletri urrent. The toroidal urrent j ϕ will be an additional soure of the poloidal magneti field. Struture of the eletri urrents arising in the rust is drawn on the figure 1. Now we will disuss qualitatively how the magneti field in polar ap is disturbed by eletri urrents flowing in the neutron star rust. The toroidal urrent j ϕ = j(σ /σ ) = i(σ /σ )(BΩ/2π) streams in the rust near the star surfae and oupies the region with size of r 0 = R (ΩR/) 1/2. The poloidal magneti field δb generated by this urrent is δb = ib(σ /σ )(ΩR/) 3/2. The disturbed magneti field in the magnetosphere under the star surfae an be 4

5 Å Ò ØÓ Ô Ö j j B j B Æ ÙØÖÓÒ Ø Ö ÖÙ Ø Figure 1: Eletri urrents in the rust 5

6 onsidered approximately as a field of magneti dipole with dipole momentum δµ δbr0 3 = i σ ( ) 3 ΩR BR 3 = i σ ( ) 3 ΩR µ σ σ Here µ is the magneti momentum of the basi dipole plaed in the neutron star ore, µ = BR Gauss m 3. Despite the large parameter σ /σ 1, the disturbed magneti momentum is small, δµ µ, beause of small value of ΩR/ The ratio σ /σ = ω τ e is [7] ω τ e = B 12 T 1 6 ; ω = 1, B 12 se 1, τ e = 2, T 1 6 se. The value T 6 is the star surfae temperature in the units of 10 6 K, T 6 = T/10 6 K. Thus, we an neglet the disturbed magneti field at large distanes from the polar ap, r r 0. However, there are two fators whih do not permit us to neglet the disturbed magneti field near the star surfae r r 0. First, the urvature of the magneti field δb is large, ρ r 0, and seond, the transverse omponent of the basi magneti field in the polar region is muh less than its longitudinal omponent. This is the reason of large urvature radius of the dipole magneti field near its axis, ρ R. The magneti momenta µ and δµ have one axis, therefore vertial B z and radial B ρ omponents of the magneti field are B z = µ (3 z2 r 5 1 ) r 3 + δµ (3 δz2 δr 5 1 ) δr 3 ; B ρ = 3µ ρz r 5 + 3δµρδz δr 5. (1) The quantity δz is the oordinate along the axis of the dipole µ, measured from the star surfae, δz = z R, ρ is the radial oordinate, it is the distane from axis z, the quantity δr is δr = ( ρ 2 + δz 2) 1/2. The eletri urrent streaming in the rust hanges a bit the vertial magneti field B z. From the first equation of Expression (1) it follows that δb z /B z (δµ/µ)(r/r 0 ) 3 = i (σ /σ )(r 0 /R) 3 1. The radial magneti field may be disturbed signifiantly, δb ρ /B ρ (δµ/µ)(r/r 0 ) 4 = i (σ /σ )(r 0 /R) 2 = i (σ /σ ) (ΩR/). So, for the estimation of the urvature of magneti field in the polar region we an onsider that the vertial omponent of the magneti field is not disturbed, B z = µ ( 2z 2 ρ 2) /r 5 2µ/R 3. The expression for the urvature radius ρ 1 = ( 1 + B2 ρ B 2 z ) 3/2 [ z + B ρ B z ρ ]( Bρ ontains the magneti field line inlination B ρ /B z. Substituting Expression (1) into (2) we get ρ = 4 R 2 [ i σ ] 1 r 0. 3 r 0 σ R 6 B z ) (2)

7 If the value of eletri urrent i flowing in the pulsar magnetosphere is not small, then the urvature radius of magneti field in the polar ap may differ strongly from the urvature radius of the dipole magneti field. Under the ondition 23i(σ /σ )(r 0 /R) = 23i(σ /σ )(ΩR/) 1/2 1 the urvature radius is defined by the eletri urrents only, ρ 4 R 3 69 r0 2 and may be ompatable with the neutron star radius. 3 Distortion of a dipole ( i σ ) 1, (3) σ Previously we gave the qualitative piture of a magneti field line distortion by the eletri urrents flowing in the neutron star rust. Here we desribe quantitatively the disussed phenomenon. As it was shown before the magneti field distortion under the star surfae may be large, so it is impossible to onsider the field of the urrent in rust as a disturbane of basi magneti dipole. We have the following physial problem. The ylindrially symmetri eletri urrent j 0 (r) falls down from the magnetosphere (h > 0) onto the onduting matter (h < 0). The boundary between the magnetosphere and the rust is h = 0. As the size of the polar ap is muh smaller than the star radius, r 0 << R, we will onsider the boundary as flat. Let us introdue ylindrial oordinates h, ρ, ϕ. The eletri urrents in the magnetosphere are along the magneti field lines, whih are not known yet. Total urrent is equal to zero, r0 0 j 0(ρ)ρdρ = 0. Initially we have the magneti field of the dipole onfiguration with its enter plaed on the axis ρ = 0 at the distane h = R from the surfae. The matter of the rust in the magneti field posses a strongly anisotropi eletri ondutivity ( σ, σ, σ ), and σ >> σ >> σ. The urrent in the rust is losed, i.e. flows perpendiular to the magneti field. It means that the eletri field must appear in the rust E = Ψ. Ψ(h, ρ) is the eletri potential. Due to the Hall ondutivity σ the toroidal eletri urrent j φ must arise in the rust. This urrent hanges the poloidal magneti field in the rust and in the magnetosphere under the star surfae. The losed, j = 0, ylindrially symmetri urrent is presented in the form j = [ J 1 ϕ] + J 2 ϕ. (4) Here J 1 and J 2 are the salar funtions of (h, ρ); ϕ is the azimuthal angle. The similar relation an be written for the magneti field B = [ f ϕ] + g ϕ. The quantity f(h, ρ) is the flux of the poloidal magneti field, g(h, ρ)/ρ is the toroidal magneti field. From the Maxwell equation urlb = 4πj/ we find the relations between the quantities J 1, J 2, f and g J 2 = 4π f, J 1 = 4π g; 7 = ρ ρ ( 1 ρ ρ ) + 2 h 2.

8 Introduing the unit vetor b along the magneti field, [ f ϕ] + g ϕ b = ρ [( f) 2 + g 2 ], 1/2 we present the eletri urrent in the form j = b(σ σ )(b Ψ) σ Ψ σ [ Ψb]. (5) Equating the Expressions (4) and (5), we obtain equations onneting the flux of the poloidal magneti field f and the toroidal magneti field g with the eletri potential Ψ 4π f = ρg(σ σ ) [( f) 2 + g 2 ] [ Ψ f] φ 4π [ g ϕ] = ρ(σ σ )[ Ψ f] φ [( f) 2 + g 2 ] ρσ ( Ψ f); [( f) 2 + g 2 1/2 ] ρgσ [ f ϕ] σ Ψ [ Ψ ϕ]. [( f) 2 + g 2 1/2 ] (6) Let us express the quantity [ Ψ f] φ from the first equation of the system (6), [ ( f) 2 + g 2] { } [ Ψ f] φ = ρgσ 4π f ρσ + ( Ψ f). [( f) 2 + g 2 1/2 ] The ondutivity along the magneti field is large, r 0 σ / >> 1. It means that the eletri field is almost parallel to the magneti field, Ψ = Ψ(f)+δΨ, δψ σ 1, δψ << Ψ(f). Substituting the expression for [ δψ f] φ to the seond equation of (6), we obtain the equation not ontaining the quantity σ { 4π g [ g ϕ] = 4π f [ + ρσ ( f) 2 + g 2] } 1/2 dψ [ f ϕ] σ g dψ df df f. (7) Here we use a useful oordinate system f, x 1, ϕ, x 1 = [ f ϕ]. The oordinate x 1 is direted along lines of the poloidal magneti field lying on the surfaes f = onst. The funtion g depends on the oordinates f, x 1, and is found from the omponent of Equation (7) direted along f, g = 4π dψ df x 1 0 ρ 2 (f, x 1 )σ (f, x 1 )dx 1 + F(f). We onsider the oordinate x 1 to be equal to zero at the level h = 0. But h = 0 is the boundary between the magnetosphere and the star rust. Then the funtion F(f) is defined by the eletri urrents flowing in the magnetosphere F(f) = 4π f 0 j 1 (f )df, 8

9 where the quantity j 1 (f) is the urrent density as a funtion of the magneti flux f, j 1 x 1 = j 0 e h. Total urrent equals to zero f 0 0 j 1(f )df = 0. The value of f 0 is the boundary magneti surfae limiting the region where the urrent exists, i.e. the boundary of the polar ap. Namely the quantity f 0 is the outer parameter of the problem, it is the flux of the magneti field passing through the light surfae. As a result the expression for g has the form [ g = 4π dψ x 1 ] f ρ 2 σ dx 1 + j 1 df. df 0 0 The toroidal magneti field must be equal to zero on the boundary f = f 0, g = 0, beause the total eletri urrent equals to zero inside the polar ap. It means that the quantity dψ/df must turn to zero on the boundary f = f 0 dψ df = onst(f f 0) α. The equation defining the poloidal magneti field is extrated from the omponent of Equation (7) direted along x 1 f + g g f + 4πρσ [ ( f) 2 + g 2] 1/2 dψ df = 0. Non-disturbed dipole magneti field is defined by the first term of this equation f = 0, the seond term desribes the field distortion by the eletri urrents flowing along the spiral magneti field lines. This effet exists in all magnetosphere in the region of open lines. However, the seond term is small near the star surfae where the magneti field strength is large [6]. And finally the third term is due to the Hall urrents flowing in the rust. The harateristi value of the Hall ondutivity σ is [7] σ = B 1 12 ρ2/3 5 se 1, where ρ 5 is the density of the rust matter in the units of 10 5 g m 3. The harateristi values in our problem are ρ r 0, h r 0, f f 0, x 1 f 0 /r 0, j 1 j 0 r 2 0 /f 0, Ψ j 0 r 0 /σ, g j 0 r 2 0 /. Beause of that the therm g g/ f is small, it is i 2 (ΩR/) 3 times smaller than the rest therms. As a result we obtain the following equation for the flux of the poloidal magneti field f + 4πρ2 σ B dψ df = 0. (8) The produt σ B depends only on the matter density in the rust whih hanges with the depth h. The real dependene of ondutivity on the rust density, ρ 5 ( h) 3, omes to the strong dependene of the Hall ondutivity σ on the depth h, σ [1 γ(h/r 0 ) 3 ] 2/3, where the parameter γ is large, γ

10 Equation (8) is valid in the region h < 0 in the star rust where the eletri urrents flow, f < f 0. In other regions h > 0 and f > f 0, h < 0 the equation for the poloidal magneti field is f = 0. This equation ontains not only the dipole field but also all multipole omponents. The funtion f(ρ, h) must be ontinuous on the boundaries h = 0, f = f 0. The boundary f(ρ, h) = f 0 itself is unknown, it has to be found from the solution of the problem. Thus, the problem to find the poloidal magneti field flux f(ρ, h) in all spae an be formulated as f = Θ(f 0 f)θ( h) 4πρ2 σ B dψ df, (9) where Θ(y) is the step funtion, Θ(y) = 1, y > 0; Θ(y) = 0, y < 0. The boundary onditions for Equation (9) are 1. the limiting value of f at ρ 0 and 2. the approahing of the solution to the dipole field f ρ 2 /[ρ 2 +(h+r) 2 ] 3/2 at large distanes ρ/r 0, h/r 0 >> 1. 4 Calulations We solved Equation (9) numerially. For alulations we selet the following parameters: dψ/df (f f 0 ) 2, R/r 0 = 100, γ = 10 3, (4π/3)i(σ /σ )(P/1se) 1 = 1. The result of alulations are drawn on the figures. The figures 2 show the magneti field lines of the poloidal omponent for positive value of the eletri urrent i in the enter of the polar ap (fig. 2a) and for the negative urrent (fig. 2b). The position of the edge of the ap (ρ = r 0 ) orresponds to the value of ρ = 5 on the figures. The dashed lines are the pure dipole field lines. We see that the distortion of the magneti field in the ap is signifiant. The figures 3 show the urvature, (r 0 /ρ ) 1, of the poloidal magneti field lines on the star surfae versus the distane from the ap enter ρ (i > 0-3a, i < 0-3b). Even for small values of the eletri urrent i 10 2 for the seleted parameters the urvature radius of the magneti field lines differs notieably from the urvature radius of the dipole field (the dashed lines on the figures 3). Let us note that the quantity 23i(σ /σ )(r 0 /R) 1 for PSR J (P = 8.51, B 12 = 2) and for seleted parameters. And the urvature radius estimation obtained in setion 2 is onfirmed by the alulations. 5 Conlusions Here we disussed the reason of hange of the urvature of magneti field lines in the polar ap, where generation of eletron-positron plasma takes plae. The urvature radius defines the free path of a fast partile with respet to the pair birth. The less urvature radius is, the less the free path beomes. The less the free path is, the less the height of the polar ap is, and onsequently, the less the eletri voltage in the ap is. Aording to the paper of Ruderman & Sutherlend [8] the value of the voltage differene ψ between the star and magnetosphere in 10

11 h Figure 2: Poloidal magneti field lines in the polar ap, for positive magnetospheri urrent. Dashed lines are the dipole field 11

12 h Figure 3: Poloidal magneti field lines in the polar ap, for negative magnetospheri urrent. Dashed lines are the dipole field 12

13 r 0 /ρ 4 x Figure 4: Curvature of the poloidal magneti field lines on the star s surfae, for positive magnetospheri urrent. The dashed line is the urvature of the dipole field 13

14 r 0 /ρ 2 x Figure 5: Curvature of the poloidal magneti field lines on the star s surfae, for negative magnetospheri urrent. The dashed line is the urvature of the dipole field 14

15 the polar region is proportional to the urvature radius ρ, ψ ρ 4/7 B 1/7 12. This is the ase when an eletron-positron plasma is generated in the polar ap under the star surfae and small amount of primary eletrons is ejeted from the rust. On the other hand, the maximum value of this potential is ψ m πen GJ r0 2 [6]. For standard value of the urvature radius of the dipole field in polar region we obtain the usual riteria for the polar ap ignition P B 8/15 < 1se (ψ < ψ m ). Let us note that the power 8/15 just oinides with the slope of the death line on the diagram log P log B 12 for observed pulsars. If we use our results for the minimum value of the urvature radius of the field distorted by the rust urrent (see formula (3), i = 1, ρ PB12 1 T 6) we find another riteria P B 2/3 12 < 10T 2/9 6 se. This ondition explains well existene of the slow rotating pulsar PSR J Referenes 1. Young, M.D., Manhester, R.N., Johnston, S., Nature, 400, 848, Thomson, G., Dunan R.C., ApJ, 408, 194, D Amio, N., Stappers, B.W., Bailes, M., Martin, C.E., Bel, J.F., Lyne, A.G., Manhester, R.N., MNRAS, 297, 28, Teylor, J.H., Cordes, J.M., ApJ, 411, 674, Arons, J., Pulsar Astronomy and Beyond, 449, ASP Conferene series, Beskin, V.S., Gurevih A.V., Istomin Ya.N., Physis of Pulsar Magnetosphere, Cambridge University Press, Blandford, R.D., Applegate, J.H., Hernquist, L., MNRAS, 204, 1025, Ruderman, M.A., Sutherland, P.G., Ap.J., 196, 51,