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Pulsar Magnetic Field *

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Problem: Estimate the magnetic field strength of the Vela pulsar. (The worksheet draws on data from Bates, Bailes, Barsdell, et al. (2012),  Carroll and Ostlie (2007), CSIRO (n.d.), and Lyne and Graham-Smith (2012).)

 

Hints:

All pulsars have masses near the Chandasekhar limit: 1.4 solar masses.

All pulsars have radii near 10,000 m.

 

Data:

For the Vela pulsar (PSR J0835-4510):

R := 10^4*Unit('m')

10000*Units:-Unit('m')

(1)

M := (1.4*1.989)*10^30*Unit('kg')

0.2784600000e31*Units:-Unit('kg')

(2)

II := (2/5)*M*R^2

0.1113840000e39*Units:-Unit('kg')*Units:-Unit('m')^2

(3)

theta := (78*((1/180)*Pi))*Unit('rad')

(13/30)*Pi*Units:-Unit('rad')

(4)

mu[0] := 4*Pi*10^(-7)*Unit('N')/Unit('A')^2

(1/2500000)*Pi*Units:-Unit('N')/Units:-Unit('A')^2

(5)

NULL

c := 2.998*10^8*Unit('m')/Unit('s')

299800000.0*Units:-Unit('m')/Units:-Unit('s')

(6)

P := 0.8933e-1*Unit('s')

0.8933e-1*Units:-Unit('s')

(7)

NULL

PP := 1.25008*10^(-13)

0.1250080000e-12

(8)

``

Useful Equations:

B[su] = 3.2*10^19*(P*PP)^(1/2)

B[po] = (3*mu[0]*c^3*II*P*PP/(2*Pi))^(1/2)/(2*Pi*R^3*sin(theta))

 

 

Solution: Substitute these values into the two equations:

 

This formula gives a result in gauss.

NULL

B[su] = 3.2*10^19*(P*PP)^(1/2)*Unit('G')/sqrt(Unit('s'))

B[su] = 0.3381563514e13*Units:-Unit('G')

(9)

NULL

This is very close to the currently accepted figure: 3.38 * 108 T or 3.38 * 1012 G for the magnetic field at the surface of the pulsar (CSIRO n.d.). The value at the poles , in tesla, may be estimated with

``

NULL

B[po] = simplify((3*mu[0]*c^3*II*P*PP/(2*Pi))^(1/2)/(2*Pi*R^3*sin(theta)))

B[po] = 729655367.2*Units:-Unit('T')

(10)

NULL

NULL

This value, approximately 7.3 * 108 T., is a little more than twice the surface value and lies within the range given by Lyne and Graham-Smith (2012), who state that, for pulsars, the polar value is approximately twice the surface value.

 

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References

 

Bates, S., Bailes, M., Barsdell, B. et al. (2012). The High Time Resolution Universe Pulsar Survey – VI. An artificial neural network and timing of 75 pulsars. MNRAS, 427(2): 1052-1065.

 

Carroll, B. and Ostlie, D. (2007). An Introduction to Modern Astrophysics (2nd ed.). San Francisco: Pearson / Addison-Wesley.

 

CSIRO. ATNF Pulsar Catalog. http://www.atnf.csiro.au/research/pulsar/psrcat/ (Accessed 2016-04-02).

 

Lyne, A. and Graham-Smith, F. (2012). Pulsar Astronomy (4th ed.). Cambridge: Cambridge University Press.

 

 

 

 

``