8.4. Definition of the micro-turbulent velocity in SPEX

As of SPEX version 3.06.00, we are using a different definition of velocity broadening for emission models. Regarding parameter name, we moved from V_{mic}, or micro-turbulent velocity, to root-mean-square velocity V_{rms}. The two definitions are related to each other as follows:

V_{mic} = \sqrt{2} V_{rms}

Warning

The change of definition means that the V_{rms} that you measure with SPEX 3.06 will be a factor of \sqrt(2) smaller than the V_{mic} that you previously measured with earlier SPEX versions.

8.4.1. Why change the definition?

The V_{rms} is defined such that it is equivalent to the Gaussian \sigma, which makes it a more intuitive quantity for most colleagues in the field. This new definition makes the emission models consistent with the velocity broadening parameters in the SPEX absorption models. In addition, this definition is also used by the APEC code in XSPEC. To avoid confusion when results from the two codes are compared, we decided to change the V_{mic} parameter to V_{rms}.

8.4.2. Microturbulent velocity in SPEX versions <=3.05.00

In previous versions of SPEX before 3.06.00 the turbulent velocity was called vmic. The micro-turbulent velocity (V_{mic}) was defined following a definition historically used often in spectroscopy. In this definition, the Doppler broadening (\Delta E_D) is defined as:

\Delta E_D = \frac{E_0}{c} \sqrt{\left( \frac{2 k T_{ion}}{A m_p} + V_{mic}^2 \right)}

As is shown in Zhuravleva et al. (2012; Section 7.2, Eq. 24), this relation can be written also in terms of Gaussian \sigma:

\Delta E_D = \frac{E_0}{c} \sqrt{2( \sigma_{therm}^2 + \sigma_{turb}^2)},

where \sigma_{therm} = \sqrt{\frac{kT}{A m_p}} is the thermal line broadening. From these equations, it is easy to see that V_{mic} = \sqrt{2} \sigma_{turb}, where we call \sigma_{turb} the root-mean square (RMS) of the line-of-sight velocity (called vrms in the current SPEX version).

8.4.3. Historic note

The definition V_{mic} = \sqrt{2} \sigma_{turb} historically originates from spectroscopy. It was chosen because it made the Maxwellian function and Gaussian function easier to calculate. It may also be related to the definition of the error function (ERF), in which the width is also a factor \sqrt{2} different from the Gaussian \sigma. Still, SPEX internally calculates the line broadening using the error function which uses the V_{mic} definition, but the number that is reported changed to V_{rms}.

In the early days of MEKAL and SPEX (from the ’70s onward), this V_{mic} definition was probably chosen to speed up the calculations. This definition is still used often in theoretical work, so be prepared to convert your measured V_{rms} to \Delta E_D or V_{mic} if you want to do a meaningful comparison.