Stars as Laboratories for Fundamental Physics - MPP Theory Group

186 Chapter 5
The off-diagonal term in the mixing matrix Eq. (5.27) is ∆ aγ =
1
g 2 aγB = 0.98×10 −9 eV g 10 B 12 with g 10 = g aγ /(10 −10 GeV −1 ). Ignoring
m a the oscillation length is 2π (∆ 2 ∥ + ∆2 aγ) −1/2 ≈ 1.3 cm B12 −2 ωkeV −1 while
the geometric dimension of the dipole field is of order the stellar radius,
i.e. of order 10 km. Therefore, many oscillations occur within the magnetosphere,
and the average transition probability between photons and
axions is θ 2 with the mixing angle θ ≈ ∆ aγ /∆ ∥ = 1.1×10 −5 g 10 B12 −1 ω −1
keV .
Therefore, the transition rate is very small.
The vacuum refractive index is larger than unity, the plasma contribution
less than unity, and so near the stellar surface a crossover must
occur where axions and photons are degenerate. However, the length
scales do not work out to have a resonant MSW-type transition (Raffelt
and Stodolsky 1988; Yoshimura 1988).
At a pulsar, reducing B increases the mixing angle and thus the
transition rate while the oscillation length becomes larger. When l osc
far exceeds the geometric dimension R of the stellar magnetosphere,
and when the mixing angle is small, one can expand the sine functions
in Eq. (5.29) so that the transition probability is (θ∆ osc R) 2 ≈ (∆ aγ R) 2 .
This transition rate scales with B 2 as expected and becomes smaller for
smaller B. Therefore, the optimal situation is when the magnetic field
strength and geometric dimensions are matched such that R ≈ l osc .
This condition is approximately met in magnetic white dwarfs with,
say, B = 10 9 G, ω = 10 eV, and R = 10 3 km. In these systems one may
even expect a resonant level crossing if they have a dilute atmosphere
with an appropriate scale height (Raffelt and Stodolsky 1988; Gnedin
and Krasnikov 1992). However, a fortuitous combination of particle and
white-dwarf parameters is required, and, even then, observable effects
apparently have not been proposed.
Most recently, Carlson and Tseng (1995) have performed a study of
the conversion of very low-mass pseudoscalars in the magnetic field of
sunspots. They find that for certain parameters the x-ray flux from the
conversion process could be observable in solar x-ray telescopes such as
SXT and Yohkoh.
5.5.2 Birefringence in a Pulsar Magnetosphere
In the previous section it was shown that near a pulsar the QED vacuum
Cotton-Mouton effect (Fig. 5.7a) induces a sizeable amount of
birefringence between the photon states which are linearly polarized
parallel or perpendicular to the transverse component of the magnetic
field. Recently, Mohanty and Nayak (1993) showed that in addition