Stars as Laboratories for Fundamental Physics - MPP Theory Group

236 Chapter 6
that one may employ the simple analytic form Eq. (6.92) of the emission
rate with Q n = 1. The core of HB stars consists at first of helium, later
also of carbon and oxygen, for all of which Y e = 0.5. Then,
⎧
5.0 e ⎪⎨
2 14 Millicharge,
ϵ x = 1 erg g −1 s −1 × T8 3 × 0.098 µ 2 12 ρ 4 Dipole Moment,
⎪⎩
0.0127 ρ 2 4 Standard Model,
(6.95)
where T 8 = T/10 8 K and ρ 4 = ρ/10 4 g cm −3 . The core averages for a
typical HB star are ⟨T 3 8 ⟩ = 0.44, ⟨T 3 8 ρ 4 ⟩ = 0.47 and ⟨T 3 8 ρ 2 4⟩ = 0.57. The
requirement ⟨ϵ x ⟩ < 10 erg g −1 s −1 then gives the limits
e ν ∼ < 2×10 −14 e and µ ν ∼ < 14×10 −12 µ B . (6.96)
Of course, for such large dipole moments the core would grow far beyond
its standard value before helium ignites, causing an additional
acceleration of the HB lifetime. In fact, this indirect impact on the HB
lifetime would be the dominant effect as shown, for example, by the
numerical calculations of Raffelt, Dearborn, and Silk (1989).
From Fig. 6.14 it is clear that the dipole-induced emission rate is
larger for the conditions of the second criterion, based on the heliumignition
argument where ⟨ρ⟩ ≈ 2×10 5 g cm −3 . According to Eq. (D.12)
the relevant plasma frequency is ω P = 8.6 keV so that Q charge /Q SM ≈
1.2 e 2 14 and Q dipole /Q SM ≈ 0.4 µ 2 12 in Eq. (6.94). The average total emission
rate is then given by the standard rate times F ν = 1 + Q j /Q SM
where j stands for “charge” or “dipole.” In order to prevent the core
mass at helium ignition from exceeding its standard value by more
than 5% one must require F ν < 3. Then one finds e ν ∼ < 1.3×10 −14 e
and µ ν ∼ < 2×10 −12 µ B . For the dipole case, a detailed numerical implementation
yielded µ ν ∼ < 3×10 −12 µ B (Sect. 2.5.2), nearly identical with
this simple analytic estimate. The limit on the charge could also be
slightly degraded and so I adopt
e ν ∼ < 2×10 −14 e and µ ν ∼ < 3×10 −12 µ B (6.97)
as the final globular-cluster limits.