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The influence of small scale magnetic field on the p olar cap X-ray luminosity of old radio pulsars
Tsygan A.I.1 , Barsukov D.P.1
1

,2

Ioffe Institute

2

SPbPU

The influence of small-scale magnetic field on the polar cap heating by reverse positrons is considered. We use the polar cap model with steady space charge limited electron flow. To calculate the electron-positron pairs production rate we take into account only the curvature radiation of primary electrons and its absorption in magnetic field. The reverse positron current is calculated in the framework of two models: rapid [1] and gradually screening [2, 3]. It is shown that some pulsars are better described by the rapid screening model and some other pulsars have better agreement with calculation by the gradually screening model.
m
4.

free electron emission from neutron star surface
small surface magnetic field Bsu r f < 1 0 1 3 G hot polar caps T (1 - 3) · 106K Z.Medin, D.Lai (2007) · m > 0, = 2 P is angular velocity of star

Returning current from altitude zf

The p olar cap luminosity
10
31

Lpc, erg/s

5.

no vacuum gaps, no sparks
steady space charge limited flow W.M.Fawley, J.Arons, E.T.Scharlemann (1977)

positrons, n
GJ

m

=

B 2 ce

7 · 1010cm-

3

1s P

B 1012 G

5

Lpc, erg/s

Gradual screening mo d e l
2 1
zc
Bn
ondip

1 + (GJ (zf ) - GJ (zc)) ~ ~ ~ 2
where n
+

B1929+10
5

B0823+26
10
31

gradual rapid

=n

~ GJ +

­ number density of returning
10

2

10
30

30

10

29

r
1.

We suppose zf (3 - 15)r

ns
10

2

LC

2. 3.

Old isolated radiopulsars Bd ip 1 0 1 1 - 1 0 1 2 G P 100ms - 1s = P /(2P ) 106 years Goldreich-Julian model inner gaps

6. 7.

stationary case only curvature radiation
the inverse compton scattering and synchrotron emission do not taken into account

1. zf < zrad (5 - 50)rns at large z plasma waves affect on pair dynamics 2. zf < zmax (1 - 5)rns where zmax is maximum of GJ (z ) ~ at z zmax the solution satisfied both conditions exists E|| = 0 and (B · )E|| = 0

29

10

28

5

gradual rapid Bsc / Bdi
p

10

27

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bsc / Bdip

8.

only photon absorption in magnetic field
no photon splitting, photon scattering

Yu.E. Lyubarskii A&A 261 544 (1992)

Bdip = 1.0 · 1012G, P = 3 · 106 years, = Lpc from [18] is shown from [12] is shown by

= 0.23s Bdip = 1.9 · 1012G, P = 0.53s, 45 = 4.9 · 106 years, = 58 by orange area. Lpc range Upper limit from [11] is shown by orange black dashed area. area.

The p olar cap luminosity Small scale magnetic field
m



m

The reverse p ositron current for pulsar J2043+2740
Lpc, erg/s

2
5

B0943+10

B1133+16

gradual rapid

10
2 29

30 5

10

2

Lpc, erg/s
10

5

10

29 5

Bnondip
m1





m
10

2

2 28

2 28 5

-1

10

5



5
2 27

Bsc

gradual rapid Bsc / Bdip

2

10

27 5


+

10
2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bsc

A

Bsc / Bdip



10

-2

3r (r · m) - mr2 3 ( · m1) - m12 + B= 5 r 5
= r - (rns - )ez , m = mez , m1 = B 0 = Bdsicp 1 = 110 rns
3 rns

5

mex 2



rapid: + 10-2 ~ ~ gradual + rt GJ ~ z rapid -1 gradual: + 10 ~ 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 ~ ~ + 2 (GJ (zf ) - GJ (zc)) ~ Bnondip / Bdip zf - zc rns rt Bdip = 7.1 · 1011G, P = 96ms, = 1.2 · 106 years, = 55
2 -3

Bdip = 3. = 4.98 Lpc from Lbol from
0.9 1.0

96 · 1012G, P = · 106 years, = [19] is shown by [5] is shown by

1.097s 21 orange area. solid green line.

Bdip = 4.26 · 1012G, P = 5.04 · 106 years, Lpc from [20] is shown Lpc range from [12] is area.

= 1.19s = 55 by orange area. shown by black dashed

The p olar cap luminosity
2

B0950+08
2 30

B1451-68

10

30

Lpc, erg/s

2

Lpc, erg/s

In the reference frame rotating with the star all values do not depend on time. = -4 ( - GJ ), E = -

Charge density
m

The p olar cap luminosity for pulsar J2043+2740
m
Lpc, erg/s

5

10

5

2 29

10

32

J2043+2740
5

10

29

10

5

5

GJ ­ Goldreich-Julian density B B = and GJ = - ~ GJ ~ 2 c 2 c = 2 /P is angular velocity of neutron star, B is magnetic field strenght Particles move along field lines v B with relativistic velocity v c

2
2

2

10

31

gradual rapid

10

28

gradual rapid

5

10

28

5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bsc / Bdi
2

p

Bsc / Bdip



10

30

5

Bsc


10

2 29



gradual rapid Bsc / B

Bdip = 4.9 · 1011G, P = 0.25s = 17.5 · 106 years, = 30 Bdip = 3.2 · 1011G, P = 0.26s Lpc from [11] is shown by orange area = 42.5 · 106 years, = 50 and black lines. Upper limits from [10] are Lpc from [23] is shown by orange area shown by green lines, solid when we see one cap, dashed when we see both caps.

div (v ) = 0 => (B · without frame dragging GJ (z ) cos ~ ~

) = 0 ~

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
dip

Lpc =

e|z

=zc

B 2 c e

+ d S ~
z =0

Conclusion
For some pulsars the gradual screening model predicts the polar cap heating which is larger than the observed polar cap luminosity. Possible explanations: 1. Surface magnetic field Bsu no free charge emission vacuum gaps, sparks [24] 3. Large redshift r < 2r
rf

is the angle between B and ~

Bdip = 7.1 · 1011G, P Upper limits of polar cap, dashed when we Emission of star surfa

= 96ms, = 1.2 · 106 years, = 55 cap emission from [10] are shown by green lines, solid when we see one see both caps. ce taken from [11] is shown by black line.

> 101 4 G

Rapid screening mo d e l
m

1. 0 < z < zc acceleration region no pairs production, no pair plasma large E|| = (E · B )/B 2. zc < z < zr partial screening area pair plasma, small E|| positrons return to the polar cap 3. z > zr full screening area pair plasma, E|| = 0 no positrons return
ondip

The p olar cap luminosity
10
32 5

2. Inner gaps occupy only small part of pulsar tube [25]
B0656+14
10
32

B1055-52 gradual rapid

5

ns

g

2

2

10

31

10

31

Lpc, erg/s

Lpc, erg/s

3 2
zr

5

5

2

10

30 5

gradual rapid

2

10

30

4. Viscous forces at z rt [26] Backflowing radiation [27, 28, 29] Radiation locked inside inner gaps [30, 31, 32] sound waves from neutron star interior [33]
We sincerely thank O.A. Goglichidze, D.N. Sobyanin, I.F.Malov and V.A.Urpin for help, comments and usefull discussions, I.F. Malov, E.B.Nikitina for provided values of inclination angles, G.G.Pavlov for provided observational data, A.I. Chugunov, M.E.Gusakov, E.M.Kantor, Yu.A. Shibanov, D.A. Zyuzin, A.A. Danilenko, A.Yu.Kirichenko, M.A.Garasyov, V.M.Kontorovich, A.B.Flanchik, D.Mitra and D.M.Sedrakian for usefull discussions. The work has been partly supported by the RFBR (pro ject 13-02-00112) and by the State Program "Leading Scientific Schools of the Russian Federation"(grant NSh-294.2014.2).

1

zc
Bn

5 2

Condition (a) E|| z=zr = 0 electric field is continous ( b ) (B · )E
||

10

29 5

2

10

29

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bsc / B

dip

Bsc / Bdip

J.Arons, E.T.Scharlemann ApJ 231 854 (1979)

charge density is continous

z =zr

=0

Bdip = 9. = 1.1 · Lpc from Lpc from Lpc from

3 · 1012G, P = 0.385s 105 years, = 23 [5] is shown by black line. [13] is shown by green line. [12] is shown by orange area

Bdip = 2. = 5.4 · Lpc from Lpc from

2 · 1012G, P = 0.197s 105 years, = 50 [5] is shown by black line. [12] is shown by orange area

References
[1] Arons J., Fawley W.M., Scharlemann E.T. // ApJ, V.231 p.854 (1979) [2] Harding A.K., Muslimov A.G. // ApJ, V.556 p.987 (2001) [3] Yu.E. Lyubarskii // A&A V.261 p.544 (1992)

Rapid screening mo d e l
m

Lpc, erg/s

2

2
zr

1

zc
Bn
ondip

GJ ~ + r t ~ z

F
z =zc

zc rt

10

29

gradual rapid

Lpc, erg/s

3

pairs are generated by curvature radiation zr - zc rt, zc a t rt at the central line the reverse positron current density may be estimated as

The p olar cap luminosity
2

[4] J.A.Gil, G.I.Melikidze, D. Mitra // A&A, V. 388, p. 235 (2002) [5] A. Szary // arXiv:1304.4203

B2224+65
10
32

J1741-2054
gradual rapid
10
31

[6] R.N.Manchester et al // Astron. J., V. 129, p. 1993 (2005) http://www.atnf.csiro.au/research/pulsar/psrcat [7] I.F.Malov, E.B.Nikitina // Astronomy Reports, V. 55, p.19 (2011) [8] J.M. Rankin // ApJSS, V. 85, p. 146 (1993) [9] A.Noutsos et al // ApJ, V. 728, p.77 (2011)

10

30

5

10

30

[10] W.Becker et al // ApJ, V. 615, p.908 (2004) [11] V.E. Zavlin, G.G. Pavlov // ApJ, V. 616, p. 452 (2004)

5

10
2

29

[12] J.Gil et al. // ApJ, V. 686, p. 497 (2008) [13] G.Pavlov et al // The Fast and the Furious: Energetic Phenomena in Isolated Neutron Stars, Pulsar Wind Nebulae and Supernova Remnants, held 22-24 May, 2013 in Madrid, Spain. (2013)

0.3

0.25

where rt is the pulsar tube radius, z is altitude above star
n n
+

10

28

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10

28

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

[14] M. Pierbattista et al. // arXiv:1403.3849 [15] A.Karpova et al. // ApJ, V. 789, id 97 (2014) [16] C.Y.Hui, W.Becker // A & A, V. 467, p.1209 (2007)

Bsc / B

dip

Bsc / Bdip

0.2

=n =

~ GJ +

­ number density of the returning
3 1s P B 1012 G

F (x)

0.15

positrons
GJ B 2 ce

0.1

7 · 1010cm- 4x 1 6 +1 5 x x at 4

0.05

F (x )
2

1 + 1. 19 x

x 1 +x 2 4 15

Bdip = 5.2 · 1012G, P = 0.68s = 1.1 · 106 years, = 16 Lpc from [16] is shown by solid green line. Upper limit from [16] is shown by dashed green line, upper limit from [11] is shown by orange area.

Bdip = 7.1 · 1011G, P = 96ms, = 1.2 · 106 years = 3 [14] Total surface luminosity Ltot from [15] is shown by orange area.

[17] C.Y.Hui et al // ApJ, V.747, p.74 (2012) [18] Z.Misanovic et al // ApJ, V.685, p.1129 (2008) [19] B. Zhang, D. Sanwal, G.G. Pavlov // ApJ, V. 624, p. L109 (2005) [20] O. Kargaltsev, G.G. Pavlov, G.P. Garmire // ApJ, V. 636, p. 406 (2006) [21] W.Becker et al. // ApJ, V.615, p.908 (2004) [22] A.T.Deller et al. // ApJ. V.701, p.1243 (2009)

0.0 10

10

-1

1

10

10

2

x

F (x ) x 1

1 , F (x )

at

The p olar cap luminosity Gradual screening mo d e l
m

[23] B.Posselt et al // ApJ, V.749, id 146 (2012) [24] Gil J, Melikidze G I and Geppert U // A&A, V.407, p.315 (2003) [25] S. Shibata // ApJ, V.378, p.239. (1991)

The assumptions: - all values do not depend on time t (stationary case)
10
5

10

30

B0628-28
5

B0834+06

gradual rapid

[26] S.Shibata et al // MNRAS, V.295, L53 (1998) [27] G.Melikidze, J.Gil // Chin. J. Astron. Astrophys., V.6, Suppl. 2, p.81 (2006) [28] J.Dyks et al // Chin. J. Astron. Astrophys., V.6, Suppl. 2, p.85 (2006) [29] D.Lomiashvili et al // arXiv:0709.2019 (2007) [30] V.M.Kontorovich, A.B.Flanchik // JETP Letters, V.85, p. 267 (2007) [31] V.M.Kontorovich, A.B.Flanchik // JETP, V. 106, p.869 (2008) [32] V.M.Kontorovich, A.B.Flanchik // Astrophysics and Space Science, V. 345, p. 169 (2013) [33] D.M.Sedrakyan // "The Mo dern Physics of Neutron Stars and Relativistic Gravity" Yerevan, Armenia, September 18-21, 2013

2 29

2

Lpc, erg/s

5

gradual rapid

Lpc, erg/s
10 10

- pairs are affected only by average electric field - GJ monotonically grows with the ~ altitude z
zc
Bn
ondip

10

29

5

2 28

2 28

2 1

10

5

5

2

2 27

Hence, conditions E|| z=zr = 0 and (B ·

10

27

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

)E

||

same No fullscreening area
can not be satisfied at the

z =zr

=0 point

Bsc / Bdip

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bsc / Bdip

A.K. Harding, A.G. There is only partial screening area Muslimov where the electric field is small and ApJ 556 987 (2001) a t z

Bdip = 6.0 · 1012G, P = 2.8 · 106 years, Lbol range from [12] area. Lbol from [5] is 52 Distance D = 332+40 -

= 1.24s = 30 is shown by black dashed shown by solid green line. pc [22].

Bdip = 6.0 · 1012G, P = 1.27s = 3.0 · 106 years, = 32 Lpc range from [5] is shown by orange area, Lbol from [5] is shown by solid green line. Lbol range from [12] is shown by black dashed area.