572
J.-F. Fortin and K. Sinha
where a(x) = A cos[χ(x)]e −iφ a (x) and
E (x) = iA sin[χ(x)]e
−iφ (x) ,
E ⊥ (x) = A ⊥ e
−iφ ⊥ (x) .
(10)
Here the angle χ(x) determines the relative ALP–photon mixture, Δφ(x) =
φ a (x) − φ (x) is the phase difference between the ALP field and the parallel photon
field, Σφ(x) = φ a (x) + φ (x) is the sum of the ALP field and the parallel photon
field phases, while φ ⊥ (x) is the phase of the perpendicular photon field.
Since the quantities of interest in the following are the intensities
I a (x) = A
2 cos
2
[χ(x)],
I (x) = A
2 sin
2
[χ(x)],
I ⊥ (x) = A
2
⊥ ,
(11)
for the ALP field, the parallel photon field, and the perpendicular photon field,
respectively, only the evolution equations for χ(x) and Δφ(x) are relevant. The
evolution equations for Σφ(x) and φ ⊥ (x) can be discarded, greatly simplifying the
analysis.
3 Stokes Parameters
Before discussing polarization, it is important to note that the dominant ALP
production mechanism and the dominant hard X-ray photon production mechanism
are completely independent. Hence, the initial phase difference Δφ 0 = Δφ(1)
is effectively random and one should average over it, leading to the averaged
intensities
¯
I a (χ 0 , x) =
2π
0
dΔφ 0
2π
I a (χ 0 , Δφ 0 , x), ¯
I (χ 0 , x) =
2π
0
dΔφ 0
2π
I (χ 0 , Δφ 0 , x).
(12)
Therefore, at a distance xr 0 from the magnetar, the relevant Stokes parameters
are
I (χ 0 , x) = ¯
I ⊥ (x) + ¯
I (χ 0 , x),
Q(χ 0 , x) = ¯
I ⊥ (x) − ¯
I (χ 0 , x),
(13)
and depend on the initial relative ALP–photon mixture χ 0 = χ(1). Here I (χ 0 , x)
and Q(χ 0 , x) are the sum and difference between the (averaged) perpendicular
photon intensity and the averaged parallel photon intensity, respectively. By subtracting the Stokes parameters at a distance xr 0 from the magnetar (which are
ultimately observed on Earth) by their values at the surface of the magnetar (which
are determined by the astrophysics of magnetars), the surface-subtracted Stokes
parameters
ΔI (χ 0 , x) = I (χ 0 , x) − I (χ 0 , 1) = ¯
I (χ 0 , x) − A
2 sin
2 (χ 0 ),
(14)
J.-F. Fortin and K. Sinha
where a(x) = A cos[χ(x)]e −iφ a (x) and
E (x) = iA sin[χ(x)]e
−iφ (x) ,
E ⊥ (x) = A ⊥ e
−iφ ⊥ (x) .
(10)
Here the angle χ(x) determines the relative ALP–photon mixture, Δφ(x) =
φ a (x) − φ (x) is the phase difference between the ALP field and the parallel photon
field, Σφ(x) = φ a (x) + φ (x) is the sum of the ALP field and the parallel photon
field phases, while φ ⊥ (x) is the phase of the perpendicular photon field.
Since the quantities of interest in the following are the intensities
I a (x) = A
2 cos
2
[χ(x)],
I (x) = A
2 sin
2
[χ(x)],
I ⊥ (x) = A
2
⊥ ,
(11)
for the ALP field, the parallel photon field, and the perpendicular photon field,
respectively, only the evolution equations for χ(x) and Δφ(x) are relevant. The
evolution equations for Σφ(x) and φ ⊥ (x) can be discarded, greatly simplifying the
analysis.
3 Stokes Parameters
Before discussing polarization, it is important to note that the dominant ALP
production mechanism and the dominant hard X-ray photon production mechanism
are completely independent. Hence, the initial phase difference Δφ 0 = Δφ(1)
is effectively random and one should average over it, leading to the averaged
intensities
¯
I a (χ 0 , x) =
2π
0
dΔφ 0
2π
I a (χ 0 , Δφ 0 , x), ¯
I (χ 0 , x) =
2π
0
dΔφ 0
2π
I (χ 0 , Δφ 0 , x).
(12)
Therefore, at a distance xr 0 from the magnetar, the relevant Stokes parameters
are
I (χ 0 , x) = ¯
I ⊥ (x) + ¯
I (χ 0 , x),
Q(χ 0 , x) = ¯
I ⊥ (x) − ¯
I (χ 0 , x),
(13)
and depend on the initial relative ALP–photon mixture χ 0 = χ(1). Here I (χ 0 , x)
and Q(χ 0 , x) are the sum and difference between the (averaged) perpendicular
photon intensity and the averaged parallel photon intensity, respectively. By subtracting the Stokes parameters at a distance xr 0 from the magnetar (which are
ultimately observed on Earth) by their values at the surface of the magnetar (which
are determined by the astrophysics of magnetars), the surface-subtracted Stokes
parameters
ΔI (χ 0 , x) = I (χ 0 , x) − I (χ 0 , 1) = ¯
I (χ 0 , x) − A
2 sin
2 (χ 0 ),
(14)