## 2. Semi-analytical formulaeAn electron in a magnetic field can emit radiation only by satisfying the resonance condition: where is the cyclotron frequency,
is the emission frequency, The important factor in Eq. 1 is the refraction index for the Ordinary Mode (OM or `+'), and the eXtra-Ordinary mode (XO or `-'). The refraction index is a function of the ambient density, of the cyclotron frequency, of the emission frequency, and of the angle to the magnetic field, and in general is quite complicated. The resonance condition defines an ellipse on the electron velocity
plane for a given
For the fully relativistic case it is more convenient to consider the plane , and the resonance condition in the form: This equation has two possible forms of a curve. One for , where as approaches 1 from above, the solution goes to minus infinity. The second form is for , where the solution goes to positive infinity as the kinetic energy approaches zero. Both curves go asymptotically to as goes to infinity. For the fully relativistic case the equivalent curve to the special resonance circle is a curve of the second form which is tangent to the line . The equation which results for the fully relativistic case, giving the frequency at which ECM occurs is: This new approximation is identical with the Melrose and Dulk approximation (Eq. 2) for the case . It is, however, more general and is a good approximation for the frequencies of negative absorption, even when is large. It is important to note that even though Eq. 4 is a better
approximation, it still assumes a single harmonic Our equation includes the refraction index which is a function of the frequency , and also of the ratio between the plasma frequency and the cyclotron frequency . Our approximation is therefore only semi-analytical, and some numerical computation must be made. The computation, however, is very simple and can be performed very quickly. Some general properties of the solution can be deduced, and in summary they are for the OM and XO mode: -
For the OM the frequency should be derived from a harmonic *s*which has . -
For the XO mode the frequency should be derived from a harmonic *s*which has . -
For both modes the frequency is smaller than a limiting frequency
The harmonic number Since it is reasonable to assume that the smallest harmonic has the largest contribution, it is sufficient to consider it alone for the purposes of the approximation. There is another magneto-ionic mode, the Z-mode. The Z-mode is the lower branch of the XO mode, for frequencies smaller than Emission in the Z-mode can not emerge from the plasma, and therefore can not be observed. Since our purpose is to derive estimates for the observable frequencies, approximate solutions for the Z-mode are not of great interest. However, the Z-mode may be the dominant mode and quench the maser before the other modes are amplified. It is therefore of interest to determine where possible negative frequencies can appear. The condition for negative absorption in the Z-mode is either , or and not too small. Empirically "not too small" translates as . We compared the Dulk-Melrose approximation, our new approximation, and a numerical calculation of the absorption coefficients from which the frequencies where the absorption is negative was extracted. The results are that for small the Dulk-Melrose approximation(Eq. 2) is very similar to our new approximation. However, for larger , the new approximation is much better. The new approximation is identical with the result of the full numerical computation for most of the range where there is negative absorption. © European Southern Observatory (ESO) 1999 Online publication: July 16, 1999 |