The rotating dipolar magnetic field models included in this folder have been made with the Mathematica software, using the following field equations. The total dipolar magnetic field is the sum of the near zone (or static) field, intermediate zone (or induction) field and far zone (or radiation) field : Btot(t, r) = Bnear(t, r) + Binter(t, r) + B far(t, r) . (Equ. 1) In the case of a rotating magnetic dipole, we have (see Jackson, section 9.3) :

Bnear(t, r) = 35 (µ µ · r) r – 13 µ , r r

(Equ. 2)

Binter(t, r) = 3 4 (µ µ · r) r – 1 2 µ , cr cr

(Equ. 3)

B far(t, r) = 21 3 (µ µ · r) r – 12 µ ! 21 3 r × (r × µ ) , cr cr cr

(Equ. 4)

where the dot means a time derivative. The magnetic moment µ is steadily rotating around the z axis with an inclination angle " and angular velocity # . Since the fields must be evaluated at the retarded time, we write

µ (t, r) = µ cos !# t – k (r – R)" sin ", sin !# t – k (r – R)" sin ", cos " ,

(Equ. 5)

where k = # /c is the wave number and R is the pulsar radius. Although the equations above are sufficient to build the field models with Mathematica, it is useful to introduce the following three locally orthogonal unit vectors :

u(r) = cos !– k (r – R)", sin !– k (r – R)", 0 ,

(Equ. 6)

v(r) = – sin !– k (r – R)", cos !– k (r – R)", 0 ,

(Equ. 7)

w = (0, 0, 1) .

(Equ. 8)

The field can now be expressed as follows :

B(t, r, " ) = F(r) sin " cos(# t) + G(r) sin " sin(# t) + H(r) cos " , where :

(Equ. 9)

F(r) = 35 (u · r) r – 13 u + k 34 (v · r) r – 12 v – k 2 13 r × (r × u) , r r r r r

(Equ. 10)

G(r) = 35 (v · r) r – 13 v – k 34 (u · r) r – 12 u – k 2 13 r × (r × v) , r r r r r

(Equ. 11)

H(r) = 35 (w · r) r – 13 w . r r

(Equ. 12)

Notice that the vectors F(r) , G(r) and H(r) don’t form an orthogonal set. Remarkably, the time evolution of the magnetic field is such that all the field lines are globally rotating like a rigid body around the center r = 0 .