A&A 436, 9-16 (2005)
DOI: 10.1051/0004-6361:20040364
F. Spanier - R. Schlickeiser
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
Received 1 March 2004 / Accepted 12 February 2005
Abstract
The heating rate of the interstellar medium, especially the
warm ionized medium, is calculated considering damping of fast
magnetosonic waves. Starting with an anisotropic spectrum the energy
dissipation rate is derived from the damping rates. The results
show that the damping of fast magnetosonic waves is an
extraordinarily strong source of heat. The ion-neutral
and viscous damping contribute largely to the heating. Furthermore it is
shown that one would expect a strong parallelization of the spectrum
with exponents well above
.
Key words: magnetohydrodynamics (MHD) - plasmas - turbulence - waves - ISM: general - ISM: magnetic fields
The calculation of interstellar cosmic ray transport parameters, like the parallel and perpendicular spatial diffusion coefficients, the rate of adiabatic deceleration and the momentum diffusion coefficient of cosmic ray particles, requires the knowledge of the power spectra of magnetic and electric fluctuations in the interstellar medium at frequencies below the non-relativistic proton gyrofrequency . Adopting the plasma wave viewpoint, in which one models the turbulent electromagnetic fields as a superposition of low-frequency magnetohydrodynamic plasma modes (shear Alfvén waves, fast magnetosonic waves and slow magnetosonic waves), the electric power spectrum is related to the magnetic power spectrum through the Faraday induction law via the dispersion relation of the individual plasma modes.
A problem arises because interstellar magnetic power spectra cannot be measured directly. Only the power spectrum of interstellar electron density fluctuations is accessible via radio scintillation and dispersion measure studies (e.g. Armstrong et al. 1995). Therefore, the relation between magnetic field fluctuations and electron density fluctuations has to be investigated. According to classical MHD theory, shear Alfven waves are incompressible whereas fast magnetosonic waves for low-beta plasmas exhibit a direct correspondence between electron density fluctuations and the fluctuations in the parallel magnetic field. In a full kinetic description of the turbulence (Schlickeiser & Lerche 2002), shear Alfvén waves become compressible with the wavenumber-dependent relation .
An additional constraint on interstellar magnetic power spectra is
provided by considering the heating of the interstellar medium by
plasma wave damping. In the first paper of this series, Lerche &
Schlickeiser (2001 - hereafter referred to as Paper I)
have calculated the heating rate of the ISM by collisionless
Landau damping of fast magnetosonic waves. They demonstrated that
the energy loss rate for this process agrees well with the cooling
rate of the diffuse interstellar medium if one considers an
anisotropic power spectrum and large spatial scales, especially
without strong spatial inhomogeneities so that to a first
approximation a simple balance of heating and cooling rates
As before, we distinguish between three different phases of the interstellar medium coexisting in pressure equilibrium: cold clouds, a warm intercloud medium (also: warm ionized medium) and hot coronal gas, which is generated by supernova explosions. The temperature of the warm medium can be determined by 21 cm radio studies. These indicate a temperature range from 6000 to 10^{4} K. Also, from radio studies we can infer the mean HII-density, which is about 0.08 cm^{-3}, and the neutral (HI) hydrogen density that ranges between 0.1-0.2 cm^{-3}. The warm intercloud medium provides the dominant contribution to interstellar electron density fluctuations because its filling factor and its degree of ionization are much higher than that of cold clouds, while the coronal gas, having approximately the same filling factor and degree of ionisation, has a much lower electron density.
As in Paper I we adopt a power spectrum of electron density fluctuation in the form
(2) |
In this work we use a wavenumber-independent anisotropy factor , while Goldreich & Sridhar (1995) predict it to be wavenumber dependent. This approximation is used for two reasons: first, it allows a direct comparison of our calculations with the earlier results of Paper I for fast magnetosonic waves, which were based on the power spectrum (3), and secondly, the mathematically simple form of (3) allows an exact analytical calculation of the wave dissipation rates, which is not possible for more complicated power spectra.
The power spectrum itself results from the balance of all wave damping and driving processes, although a detailed theory and explanation is not available at present. Therefore, our calculations are limited in the sense that we assume a given and fixed power spectrum to determine the heating rate of the interstellar medium, but we do not self-consistently investigate the effect of this energy loss rate on the form of the power spectrum, nor do we investigate the formation of real turbulence spectra in wavenumber space. Although highly needed, such a self-consistent theory lies beyond the scope of the present investigation.
For our calculations we describe the warm intercloud medium with the
following set of parameters:
(5) | |||
(6) | |||
(7) | |||
(8) | |||
(9) | |||
(10) |
(11) | |||
(12) |
(13) |
(14) |
= | |||
(15) |
= | |||
(16) |
= | |||
(17) |
As we can see from Schlickeiser & Lerche (2002), we have the
following relation between electron density (P_{nn}) and magnetic
fluctuation (
as the z-component of the magnetic
field is dominant) spectra for fast magnetosonic waves in a low
-plasma
(18) |
For waves damping at a rate
the energy loss
rate
is conventionally written in the form (Spangler
1991)
As already described in Paper I the wave damping rate for
collisionless Landau damping may be written as
Lerche & Schlickeiser (2001) found an energy
dissipation rate
(27) | |||
(28) | |||
(29) |
There exist approximations for the anisotropy term in the limit
and
.
(30) |
(31) | |||
(32) |
Tsap (2000) gives the following damping rate for Joule
dissipation
(33) |
(34) |
The Joule energy dissipation is then calculated as
= | |||
= | |||
= | |||
(36) |
(37) |
For our given parameters we have the numerical values
(38) | |||
(39) |
From Braginskii (1965) we know that viscosity dissipation is
given by
(40) |
(42) | |||
(43) | |||
(44) |
This damping rate is now multiplied again with the power spectrum
in order to get the energy dissipation rate
= | |||
(46) | |||
= | |||
(47) | |||
= | |||
(48) |
= | (49) | ||
Kulsrud & Pearce (1969) have given the following damping
rate for ion-neutral collisions
= | (53) | ||
= | (54) | ||
= | |||
(55) |
(56) |
The dissipation rates for all damping processes are shown in Fig. 1.
From Minter & Spangler (1997) we know that the
cooling rate of the diffuse interstellar medium is
Figure 1: Anisotropy dependence of different energy dissipation processes. | |
Open with DEXTER |
Within the current hypothesis of a steady-state configuration of heating and cooling, in order to achieve agreement of the wave turbulence spectrum with the constraints from the cooling rate we have two options: we can either modify
As the ion-neutral damping is only determined by the collision
frequency between ions and neutrals and the fluctuating magnetic power
,
the heaiting rate by ion-neutral friction can be used to fit the fluctuating magnetic power as it is the only parameter here. We attribute 50% of the heating to ion-neutral damping,
i.e.
,
so that Eqs. (58) and (59) yield
(60) | |||
(61) |
As mentioned above we have fixed the value of
by demanding
(62) |
(63) |
The first method of fixing
and varying s yields from
The second modification of fixing s=5/3 and varying yields from Eq. (64) 10^{-12} cm^{-1}, which is five orders of magnitude smaller than the values of Spangler.
Varying both the spectral index s and
yields the relation
(66) |
Figure 2: Relation between cutoff wavenumber and spectral index s for isotropic turbulence. The dotted line shows Spangler's while the dashed shows his . | |
Open with DEXTER |
As Fig. 1 shows, a small value of does not modify the viscous damping rate. However, large values of have a strong influence on the damping. Therefore, only for values of giving a fully parallel spectrum are we able to justify s=5/3.
We have calculated the main damping processes of fast magnetosonic waves in the interstellar medium. Using numerical parameters derived from observations we found that for isotropic turbulence ion viscosity and ion-neutral damping in particular lead to a much higher energy dissipation than the cooling rate allows in a steady state case. This leads to two conclusions:
We will initially analyze the energy conservation in the ISM turbulence to justify our model. Our basic assumption was the steady state model
(67) | |||
(68) | |||
(69) |
As a simplified model we assume a cylindrical galaxy (r=50 000 Ly, h=3000 Ly) completely filled with diffuse ionized gas, which is heated by supernovae (the typical energy output for supernovae is E=10^{51} erg and the mean time between two supernovae is approx. s =30 years).
First, we calculate the average input power from supernovae for the
total Galaxy
(70) |
(71) |
The energy of the supernovae is approximately sufficient to sustain the energy balance. But still one question remains: is it likely that such a high fraction of the heating power from the supernovae is ultimately dissipated by ISM heating, probably via the formation of supernova shock waves, which are important for the generation of turbulence? Thus the input power is almost fully available as wave energy. On the other hand, the alternative energy-consuming process, which is according to Schlickeiser (2002) the acceleration of cosmic rays, only needs an average power of 10^{40} erg/s. As it seems there are no other processes in an interstellar phase at rest (no convective motions like galactic winds) which could possibly consume the available energy and as the input power and the heating fit remarkably well (though we used a very rough model) it is reasonable to argue that ISM heating is a stationary process driven by supernovae. It should be noted, however, that the given figures are only rough approximations, the supernova-frequency especially cannot be treated as a constant and a strong deviation from the average would lead to a cooling of the ISM.
One may also derive time scales from the total energy E instead of the power , but these time scales are not suitable for the dynamical thermal equilibrium, which we examine, as there is no intrinsic relation between E and . Those time scales can only give estimates for the time the turbulent field would need to build or decay when either the input or output power would be turned off.
The "decay time scale'' is the ratio of field energy and loss power
(the plasma wave heating rate).
(72) |
(73) | |||
(74) |
This could mean that fast magnetosonic waves can only be found in the vicinity of supernovae and are damped completely after a supernova occurs.
Another possibility is the nonlinear interaction of Alfvén waves as a generation mechanism for the fast magnetosonic waves (see e.g. Chin & Wentzel 1972). Particularly the production rate of fast waves by Alfvén waves could help our understanding of the energy transfer in the ISM.
Also the time scale of field creation should be analyzed. This
time scale is derived from turbulent electromagnetic energy and
input power. Again we differentiate between Alfvén and fast
magnetosonic waves. These "rise time scales'' are given by
(75) | |||
(76) | |||
(77) |
The next point concerning the physical plausibility is the possibility of a nearly parallel spectrum. For Alfvén waves this matter has been addressed by Goldreich & Sridhar (1995), but their (disputed) process is not applicable to fast magnetosonic waves. Nonetheless the ISM is marked by strong anisotropies, which may result in anisotropic spectra. Again further insight into the production processes of FMS waves would be helpful, but in contrast to the question of input power of the spectrum, the anisotropy may only be explained by the damping itself. If we imagine an isotropic input spectrum, we may see after short times that with increasing wavenumber the anisotropy increases, as the damping of perpendicular waves is dominant. This may be modelled by wavenumber dependent anisotropy factors , which for reasons of simplicity are not included in our model.
The last point to be discussed is the possibility of very steep spectra. If we stick to the Kolmogorov (1941) or Kraichnan-Iroshnikov (1965) theory we should expect a 5/3or 3/2 spectrum, but one point in the derivation of those theories seems not to be fulfilled in our case: The K41 (i.e. turbulence with an inertial range power law s=5/3) and KI (i.e. s=3/2) theories consider an inertial range, where the form of the turbulence spectrum is given by the local transport in wavenumber space, with damping being neglected in that range. But as we already pointed out, the ISM has strong damping features. This does not mean that there is no inertial range but there could be additional effects due to the dispersion range studied by Stawicki et al. (2001), an intermediate regime between the inertial and dissipation range. Thus our findings are not in contradiction to Kolmogorov theory but they seem to indicate that the inertial range of the ISM turbulence is negligible.
Acknowledgements
We thank the anonymous referee for the very constructive and helpful report. We acknowledge partial support by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 591. We would also like to thank Urs Schaefer-Rolffs for providing the figures.