Betelgeuse, also known as Alpha Orionis is the eighth brightest star in the night sky and second brightest star in the constellation of Orion, outshining its neighbour Rigel (Beta Orionis) only rarely. Distinctly reddish-tinted, it is a variable star whose apparent magnitude varies between 0.2 and 1.2, the widest range of any first magnitude star.
A red supergiant, Betelgeuse is one of the largest and most luminous stars known. However, with distance estimates in the last century that have ranged anywhere from 180 to 1,300 ly. from Earth, calculating its diameter, luminosity and mass have proven difficult.
It is believed that Betelgeuse is only 10 million years old, but has evolved rapidly because of its high mass. Currently in a late stage of stellar evolution, Betelgeuse is expected to explode as a type II supernova, possibly within the next million years.
The cool star Betelgeuse is an example of an abundantly observed late-type supergiant that displays irregular brightness variations interpreted as large-scale surface structures (e.g. Lim et al. 1998 and Gray 2000). It is one of the stars with the largest apparent sizes on the sky--corresponding to a radius in the interval 600-800. Freytag et al. (2002) performed detailed numerical 3-d radiation-hydrodynamic (RHD) simulations of the convective envelope of the star under realistic physical assumptions, while trying to determine if the star's known brightness fluctuations may be understood as convective motions within the star's atmosphere: the resulting models were largely successful in explaining the observations as a consequence of giant-cell convection on the stellar surface, very dissimilar to solar convection. Dorch & Freytag (2002) performed a kinematic dynamo analysis of the convective motions in the above model (i.e. not including the back-reaction of the Lorentz force on the flow) and found that a weak seed magnetic field could indeed be exponentially amplified by the giant-cell convection on a time-scale of about 25 years.
On the observational side of things, maser polarization is known to exist in circumstellar envelopes of AGB stars (e.g. Gray et al. , Vlemmings et al. , and recently Sivagnanam ) and X-ray emission has been observed from some cool giant stars (e.g. Hünsch et al.  and Ayres et al. ). These observations are generally taken as evidence for the existence of magnetic activity in late-type giant stars (cf. Soker & Kastner ).
There are indications from both dynamo theory and observations that some late-type giant stars such as red supergiants and asymptotic-giant-branch stars (AGB stars) may harbor magnetic fields. On the theoretical side, it has been suggested that non-spherically symmetric planetary nebulae (PNe) formed during late stages of AGB star evolution may be a result of the collimating effect of a strong magnetic field: Blackman et al. () studied interface dynamo models similar to mean field theory's solar -dynamo and found that the generated magnetic surface fields typically could be Gauss, strong enough to shape bipolar outflows, producing bipolar PNe, while also braking the stellar core thereby explaining the slow rotation of many white dwarf stars. Also using mean field dynamo theory Soker & Zoabi () propose instead an dynamo due to the slow rotation of AGB stars rendering the -effect ineffective. They find that the magnetic field may reach strengths of Gauss, significantly less than that found by Blackman et al. (). On the one hand, they believe that the large-scale field is strong enough for the formation of magnetic cool spots (see also Soker & Kastner  on AGB star flaring). These spots in turn may regulate dust formation, and hence the mass-loss rate, but the authors argue that they cannot explain the formation of non-spherical PNe (see also Soker ): on the other hand, the locally strong magnetic tension could enforce a coherent flow that may favor a maser process.
The variation of Betelgeuse: Numerical simulations
Freytag et al.
It is appropriate to discuss here also the properties of the convective flows in the model, since these ultimately supply the kinetic energy forming the basic energy reservoir for any dynamo action that might be present. It is not expected that the flows match exactly what is found in more realistic RHD simulations, but at least a qualitative agreement should be inferred since the fundamental parameters of this MHD model and the RHD model of Freytag et al. () are the same.
The velocity is initialized with a random flow with a small amplitude. Rapid large scale convection cells develop throughout the star: the giant cell convection is evident in both the thermodynamic variables, such as temperature and gas pressure, as well as in the flow field. The observational equivalent however, is the surface intensity. Since the model does not incorporate realistic radiative transfer (as opposed to the model of Freytag et al. ), only a simulated intensity can be derived.
Simulated intensity snapshots at four different instances show the typical contrast between bright and dark patches on the surface is 20-50%, and only 2-4 large cells are seen at the stellar disk at any one time corresponding to a hand full of cells covering the entire surface. The primary physical reason for the large scale of the convective cells relative to the Sun is the much larger pressure scale height in the surface layers, cf. Schwarzchild (). The simulated intensity is in qualitative agreement with the RHD models of Freytag et al. (): the surface is not composed of simply bright granules and dark intergranular lanes in the solar sense--sometimes the pattern is even the reverse of this -- e.g. in the Figure the simulated intensity snapshot at time years, the cool dark area in the center of the stellar disk is actually a region containing an upward flow.
More quantitatively, a kinetic power-spectrum illustrates that there is much more power on large scales than on the small scales of the velocity field: below a wavenumber of 20 (based on the box size in units of the star's radius) power is decreasing fast, but at larger scales the power is proportional to corresponding to normal Kolmogorov scaling, the inertial range spans however only roughly one order of magnitude. In conclusion the large-scale convective patterns are then typically larger than 15-30% of the radius, and are actually often on the order of the radius in size. The corresponding radial velocities range between 1-10 km/s in both up and down flowing regions. There are at least three different evolutionary phases of convection in the simulations, depending on the level of the total kinetic energy E of the convection motions: initially there is a transient of about 30 years after which the RMS velocity field reaches a level where it fluctuates around a value of about 800 m/s (this corresponds to the kinematic phase of the dynamo, where the flow is unaffected by the presence of the still weak magnetic field, see below). During the rest of the simulation after about 290 years, the RMS speed measured in the entire box decreases to 500 m/s (when the energy in the magnetic field becomes comparable to the kinetic energy density). During the stretch of the simulation however, the maximum speed in the computational box fluctuates around a constant value of about 90 km/s. The flows are not particularly helical and the mean kinetic helicity is on the order of m/s. Mean field -type solar dynamos do not produce large-scale fields if the kinetic helicity is less than a certain value (cf. Maron & Blackman ) and hence we cannot expect a large-scale toroidal field in the solar sense to be generated.
Magnetism of Betelgeuse: A dynamo
Dorch et al.
In an early kinematic study of Betelgeuse using a completely different numerical approach (Freytag et al.  and Dorch & Freytag ), dynamo action was obtained when the specified value of Re was larger than approximately 500 and at lower values of Re the total magnetic energy decayed. In the present case Re is of the same order of magnitude and we find an initial clear exponential growth over several turn-over times, and many orders of magnitude in energy. Figure shows the evolution of E as a function of time, for the first 225 years (in Betelgeusian time): once the giant cell convection has properly begun the magnetic field is amplified and we enter a linear regime of exponential growth. There are two modes of amplification in the linear regime; the initial mode with a growth rate of about 4 years, which in the end gives way to a mode with a smaller growth rate corresponding to a time-scale of 25 years. This is a slightly unusual situation, since normally modes with smaller growth rates are overtaken by modes with larger growth rates (cf. Dorch 2000); the explanation is that while both modes are growing modes, only the one with the largest growth rate is a purely kinematic mode--while the exponential growth of the second mode is linear it is not kinematic--the presence of the magnetic field is felt by the fluid through the back-reaction of the Lorentz force becoming important. This quenches the growth slightly and henceforth one can refer to the second mode as a ``pseudo-linear" mode.
No exponential growth can go on forever and eventually the magnetic energy amplification must come to a halt: the question is then whether the magnetic field retains a more or less constant saturation value, or if it dissipates. The latter is only possible if the non-linear mode is a decaying mode that corresponds to a negative growth rate. In case of saturation the typical field strength is expected to be on the order of the equipartition value corresponding to equal magnetic and kinetic energy densities. Figure shows the RMS magnetic field strength B within the entire model star as a function of time for Betelgeusian years: the pseudo-linear mode as well as the mode in the non-linear regime are shown. The RMS magnetic field saturates at a value slightly above the RMS equipartition field strength B 90-100 Gauss, corresponding to a value of about 120-130 Gauss. In terms of total energy this means that the magnetic energy E is above equipartition with the kinetic energy E by approximately a factor of two. Hence the field cannot be said to be extremely strong, but it is not particularly weak in most parts of the star either. What may be interesting from an observational point of view is the strength of the field at the surface. In the non-linear regime the field strength at the sphere with radius can be up to Gauss, while in the interior going downwards in the star the field strength rises and can be as high as a few kG: the strongest intermittent magnetic structures almost completely quenches the velocity field in these regions that are small-scale compared to the scale of the convection; i.e. the local field can be far above equipartition. The downward increase of the field strength is analogue to the flux pumping effect that has been found in the solar context (cf. Dorch & Nordlund 2001).
It can be interesting to examine the geometry of the magnetic field that the saturating non-linear dynamo generates since this could be relevant for the influence of the field on e.g. asymmetric dust and wind formation. Qualitatively speaking the field becomes concentrated into elongated structures much thinner than the scale of the giant convection cells, but perhaps due to the very irregular nature of the convective flows, no ``intergranular network is formed in the solar sense. On the one hand, at times magnetic structures coincide with downflows, but not as a general rule. On the other hand, strong fields are seldomly located within the general upflow regions.
Figure XX shows the PDF of the magnetic field: the distribution is a typical signature of highly intermittent structures, i.e. only a very small fraction of the volume carries the strongest structures and the probability of finding a vanishing field strength at a random point in space, is far greater than finding strong fields.
An energy spectrum reveals that the magnetic structures are well resolved with little power at the Nyquist wavenumber , and that the power at the largest wavenumbers is two-orders of magnitude smaller than that at the largest scales (see Figure XX). Maximum power is obtained on the largest scales corresponding to wavenumbers of a few, while there is a dip at corresponding to the scale of the radius, where the power is minimum. The power on scales 10-20 is flat leaning towards being proportional to corresponding to Kolmogorov scaling, and at small-scales power steeply drops: magnetic structures in the non-linear regime are then large by solar standards, but smaller than the giant convection cells that show increasing power towards large scales.
Blackman  argues that in turbulent dynamos the magnetic filling factor would be of the order ( being the ratio of gas pressure to magnetic pressure) so that for relatively weak fields the filling factor would be small. This is partly the case in the present model where the average field is weak field above the stellar surface where it has a low filling factor : However, the filling factor scales very well with the average field strength.
Figure XX is a map of the spherical surface at R in terms of magnetic energy in the non-linear regime: there are both dark patches of strong magnetic field, e.g. a 500 Gauss region at longitude around and latitude north of the equator, and large areas with a vanishing field (e.g. at longitude on the equator). On this map the areal magnetic filling factor is 55% for Gauss, while it is only 0.6% for Gauss. However, this mapping does not represent any physical surface of the star. Due to the fact that the actual upper boundary consists of a few large cells the surface cannot be captured by a simple sphere with radius R; this is illustrated by a volume rendering of the isosurface at the cooling temperature. E.g. in the upper left corner of Figure there is a ``hill in the temperature isosurface and further large ``slopes can be seen across the star in this illustration. In the latter figure, the magnetic field lines illustrate that there is a slight trend inside the star, looking through the partly transparent surface, towards a radial orientation of the magnetic field, while the strong fields near the surface of the star are predominantly horizontally aligned. This was also observed by Dorch & Freytag () and may be a generic trade of giant-cell fully convective slowly rotating stars.
In summary three different modes of dynamo action are recognized:
A relatively fast growing linear mode with an exponential growth time of years. A relatively slowly growing pseudo-linear mode with an exponential growth of years. A saturated non-linear mode operating a factor of two above equipartition (through-out the star). More modes may of course exist but these must then have very low growth rates and/or very small initial amplitudes since they have not appeared in the simulations. It is worth noting that in case 2) of the pseudo-linear mode, the same value of the growth time (around 25 years) was found in the previous purely kinematic dynamo models Dorch & Freytag () although they employed a different computational method. This may in fact not be so strange, since the growth rate in a kinematic dynamo is set in part by the convergence of the flow across the field lines and if the flows are similar so should the growth rates be.
Based on the numerical results, it is not possible to state conclusively if Betelgeuse actually has a magnetic field. However, one may conclude that it seems possible that late-type giant stars such as Betelgeuse can indeed have presently undetected magnetic fields. These magnetic fields are likely to be close to or stronger than equipartition yielding surface strengths on the order of 500 Gauss at maximum; this may be difficult to detect directly, due to the relatively small filling factors of the strong fields, but even the moderately strong fields may have influence on their immediate surroundings through altered dust, wind and mass-loss properties. The formation of dust in the presence of a magnetic field will be the subject of a subsequent paper along the lines presented here: the ``Pencil Code has recently been augmented with modules for radiation and dust modeling.
Soker et al.
Aurière et al. (2010)
Vlemmings et al. (masers)
The dynamo of the late-type giant studied here may be characterized as belonging to the class called ``local small-scale dynamos another example of which is the proposed dynamo action in the solar photosphere that is sometimes claimed to be responsible for the formation of small-scale flux tubes (cf. Cattaneo ). However, in the case of Betelgeuse this designation is less meaningful since the generated magnetic field is both global and large-scale, but because of the slow and non-differential rotation, no large-scale solar-like toroidal field is formed although the situation might be different in more rapidly rotating AGB stars.
It is interesting to note that very recently Lobel et al. () published spatially resolved spectra of the upper chromosphere and dust envelope of Betelgeuse. Based on various emission lines they provide evidence for the presence of warm chromospheric plasma away from the star at around 40 R. The spectra reveal that Betelgeuse's upper chromosphere extends far beyond the circumstellar envelope. They compute that temperatures of the warm chromospheric gas exceed 2600 K. The presence of a hot chromosphere lead this author to speculate on the possible connection to coronal heating in the Sun, which is likely to be magnetic in origin and caused by flux braiding motions in the solar photosphere (cf. Gudiksen & Nordlund ): it remains to be proven whether a similar process could be operating in late-type giant stars.
- Nordlund, Å., Galsgaard, K. & Stein, R. F. (1994) Magnetoconvection and magnetoturbulence in NATO Advanced Science Institutes (ASI), (Dordrecht, Kluwer), p.471
- Parker, E.N. (1979) Cosmical Magnetic Fields - Their Origin and Their Activity, Clarendon Press, Oxford