Planck is expected to answer crucial questions on the early Universe , but it also provides further understanding on anomalous microwave emission .
Electric dipole emission from spinning dust grains continues to be the favored interpretation of anomalous microwave emission .
In this paper , we present a method to calculate the rotational emission from small grains of irregular shape with moments of inertia I _ { 1 } \geq I _ { 2 } \geq I _ { 3 } .
We show that a torque-free rotating irregular grain with a given angular momentum radiates at multiple frequency modes .
The resulting spinning dust spectrum has peak frequency and emissivity increasing with the degree of grain shape irregularity , which is defined by I _ { 1 } :I _ { 2 } :I _ { 3 } .
We discuss how the orientation of dipole moment ¯ in body coordinates affects the spinning dust spectrum for different regimes of internal thermal fluctuations .
We show that the spinning dust emissivity for the case of strong thermal fluctuations is less sensitive to the orientation of ¯ than in the case of weak thermal fluctuations .
We calculate spinning dust spectra for a range of gas density and dipole moment .
The effect of compressible turbulence on spinning dust emission is investigated .
We show that the emission in a turbulent medium increases by a factor from 1.2 – 1.4 relative to that in a uniform medium , as sonic Mach number M _ { { s } } increases from 2 – 7 .
Finally , spinning dust parameters are constrained by fitting our improved model to five-year Wilkinson Microwave Anisotropy Probe cross-correlation foreground spectra , for both the H \alpha -correlated and 100 \mu m-correlated emission spectra .