On the spatial distribution of electron energy loss due to gyro-cooling in hot-star magnetospheres

For a potentially relativistic electron with pitch-angle $\alpha$ in gyration about a local magnetic field, the associated synchrotron power emitted is given by Condon & Ransom (2016, their eq. (5.37)),

where $U_{B}\equiv B^{2}/8\pi$ is the (cgs) energy density of the magnetic field $B$; $\sigma_{T}=0.67\times 10^{-24}$ cm² is the Thomson cross section for electron scattering; $\beta\equiv v/c$ is the ratio of electron speed $v$ to the speed of light $c$; and $\gamma\equiv 1/\sqrt{1-\beta^{2}}$ is the associated relativistic energy factor.

For electron mass $m_{e}$, the associated electron kinetic energy is KE$=(\gamma-1)m_{e}c^{2}$. Averaging the power loss over a given pitch-angle distribution $\left<P\right>\sim\left<\sin^{2}\alpha\right>$, we can then define an electron energy-loss time as $t_{e}\equiv$ KE/$\left<P\right>$, which, after some manipulation, can be shown to scale as

For the simple case of a gyrotropic distribution (for which $\left<\sin^{2}\alpha\right>=2/3$) and a canonical field $B=1$G, the latter relation in (2) shows that the loss time is more than a decade (recalling that 1 yr $\approx 3\times 10^{7}$ s).

However, in the context of massive-star magnetospheres, this loss timescale can be much shorter. Analyses by Shultz et al. (2022) and Owocki et al. (2020) have shown that H-alpha emission requires a field at the Kepler radius $B_{K}\approx 100$G, implying a cooling time for non-relativistic electrons, $t_{e}\approx 5.1\times 10^{4}{\rm s}\approx 0.6$ d. The implication is then that the observed quasi-steady circularly polarized gyro-synchrotron radio emission must be replenished by many small centrifugal breakouts that occur on such a timescale of order a day or less.

On the other hand, this loss time is generally much longer than the typical propagation time of such energetic electrons across loops. As an example, for 25 keV electrons with speed $v\approx c/3$, the characteristic advection time across of loop of apex radius $r_{a}$ is

Comparison with eqn. (2) shows that for a loop with $r_{a}\approx 10^{12}$cm and $B_{a}\approx 100$G, $t_{a}\ll t_{e}$. As detailed below, this implies that electrons will generally mirror many times across a dipole loop as they gradually lose energy.

An electron with energy $E$ and pitch angle $\alpha$ gyrating around field of local strength $B$ has a magnetic moment given by

Under the common assumption of fixed energy $E$, this magnetic moment is also conserved. For electrons with pitch angle $\alpha_{a}$ at the apex radius $r_{a}$ of a loop with apex field strength $B_{a}$, the pitch angle will thus become perpendicular ($\sin\alpha=1$) at a mirror radius $r_{m}$, with field strength

For a simple dipole field with spatial scaling $B\sim\sqrt{1+3\mu^{2}}/r^{3}$ in radius $r$ and colatitude cosine $\mu=\sqrt{1-r/r_{a}}$, this leads to a sixth-order polynomial for $r_{m}$,

The simple approximation $r_{m}/r_{a}\approx\sin^{2/3}\alpha_{a}$ is within a few percent of the full solution except for the small pitch angle regime. For the analysis presented in the subsequent sections, we have used the full solution.

More generally, the time variation of energy $dE/dt$ associated with the gyro-synchrotron radiation leads to an associated change in the magnetic moment,

For an electron with speed $v$, energy $E=m_{e}v^{2}/2$, and pitch angle $\alpha$, the gyro-emission power $P$ from eqn. (1) gives for the time change of energy,

where the second equality applies for the non-relativistic case $\gamma\approx 1$, and the final equality uses the definition of magnetic moment $p_{m}=E\sin^{2}\alpha/B$ to eliminate $\alpha$ in favor of $B/E$.

The initial analysis here focuses on this sub-relativistic case because then the relative cooling becomes independent of the electron energy, depending only on the magnetic field strength $B$ and electron pitch angle $\alpha$. Appendix A derives the generalized equations for relativistic electrons.

Defining apex-scaled variables for energy $e\equiv E/E_{a}$ and magnetic field $b\equiv B/B_{a}$, along with scaled magnetic moment $p\equiv p_{m}B_{a}/E_{a}$, eqns. (8) and (7) can be recast as a coupled system of two first-order, dimensionless ODE’s,

and

where the dimensionless constant $k$ is given by

In the latter equality, $t_{a}\equiv r_{a}/v_{a}$ is a characteristic propagation time, used now to scale the time variable, $t\rightarrow t/t_{a}$. The cooling constant $k$ is thereby cast as the ratio of this to a characteristic cooling time $t_{c}$, given by the inverse of the square bracket term ²²2This is closely related to the electron energy loss time for a specific pitch angle, as given by eq. (2)..

For typical values $r_{a}=10^{12}$ cm, $B_{a}=100$ G, $E_{a}=25$ keV, and $v_{a}=c/3$, we find this cooling constant is quite small, $k\approx 0.002$. For initial pitch angles that are not too small, with thus mirroring not too far below the loop apex, only a small fraction of the particles energy is lost per mirror cycle. But the cubic scaling of cooling with magnetic field, and its steep increase inward, means that lower pitch angles can lose a significant fraction of their energy near their mirror point.

The apex-scaled dipole field $b$ can be written as a function of the co-latitude cosine $\mu$,

A key to proceeding thus regards the time variation of this co-latitude variable, $d\mu/dt$. For electron speed $v$, movement along the field line coordinate $s$ is given by

where $b_{\theta}=\sqrt{1-\mu^{2}}/\sqrt{1+3\mu^{2}}$ is latitudinal projection of the unit vector along the dipole field, for which also $r=r_{a}(1-\mu^{2})$. Using eqn. (11) and the above definitions of scaled energy $e$ and scaled magnetic moment $p$, we can write a third dimensionless time equation that must be solved,

where the sign function allows one to keep track the directionality of propagation as the electron mirrors across the loop.

In this approach, one solves the coupled system of 3 ODE’s (14), (9), and (10) for the associated dependent variables $\mu$, $e$, and $p$ as a function of the independent variable, the scaled time $t$. Appendix A gives the correponding relativistic forms, (25, (27 and (28), wherein the curly bracket factors represent the corrections from the sub-relativistic expressions here.

In our basic scenario here, we assume electrons are introduced at a dipole loop apex with fixed energy and pitch angle $\alpha_{a}$, giving then initial conditions $\mu(0)=0$, $e(0)=1$, and $p(0)=\sin^{2}\alpha_{a}$. From this inital condition, we use the odeint function from scipy.integrate (Virtanen et al., 2020) to evolve the system of ODE’s (9), (10), and (14) in time until the energy retained by the electron drops below 0.1% of its initial energy.

Figure 1 shows solutions for a representative case with $k=0.002$ and $\alpha_{a}=60^{o}$. The upper two panels of the left column plot the temporal decrease in the magnetic moment and energy, while the bottom panel shows the temporal variation of the co-latitude (due to magnetic mirroring). Note how the non-conservation of magnetic moment leads to a systematic increase in the mirror co-latitude, from its initial $|\mu_{min}|\approx 0.24$ to a maximum $\mu_{max}\approx 0.55$, corresponding to a minimum radius $r_{min}/r_{a}\approx 1-\mu_{max}^{2}\approx 0.7$. For any loop with apex $r_{a}/R_{\ast}>1/0.7\approx 1.4$, electrons with this initial pitch angle $\alpha_{a}=60^{o}$ will thus have their full energy dissipated without interaction with the underlying star. We thus defer consideration of such stellar interaction dissipation to the fuller models below.

The central column of Figure 1 shows the associated evolution of magnetic moment and energy with co-latitude. The latter in particular forms the basis for deriving the spatial distribution of energy loss along the loop. Figure 2 plots the energy emission $\epsilon$ versus co-latitude $\mu$, color coded by the deposition time.

By binning the energy loss over a grid in co-latitude, Figure 3 plots the total distribution of emission $\epsilon_{\mu}$ in co-latitude $\mu$, comparing now results for the various labeled values of the cooling parameter $k$. Note that results for the smallest $k$ are all very similar, since they all represent the cumulative effects of energy loss over many mirror cycles; in contrast, for the case with cooling constant $k=0.2$ approaching unity, the relatively stronger energy loss within any given mirror cycle gives a much more irregular distribution.

For the standard cooling parameter $k=0.002$, Figure 4 overplots the energy emission $\epsilon_{\mu}$ versus co-latitude $\mu$ for a range of initial pitch angles ranging from $\cos(\alpha_{a})=0.1$ to 0.9. The peak energy emission again occurs near the mirror point for each initial pitch angle, with a narrow spread. But these mirror points range from near the loop apex for oblique pitch angles (e.g., $\cos\alpha_{a}=0.1$) to much further down the loop for more field-aligned cases (e.g., $\cos\alpha_{a}=0.9$).

Figure 5 plots the total energy emission for a gyrotropic distribution of initial pitch angles, assuming the labeled small values for the cooling parameter $k$. The close overlap for all 3 cases shows that this spatial distribution of energy is not sensitive to the exact cooling parameter, as long as it is significantly below unity, and so allows for gradual cumulative energy loss over many mirror cycles.

The strong peak around $\mu=0$ reflects the strong energy emission near the loop apex from electrons with nearly orthogonal initial pitch angles $\alpha_{a}\approx 90^{o}$, which, due to the relatively large solid angle, are relatively more numerous. But more field-aligned pitch-angles contribute to significant energy emission further down the loop, here extending to a maximum colatitude $|\mu_{max}|$ that depends on just how small the cooling parameter $k$ is.

We first consider the case in which the initial energy deposition along a given loop follows a gaussian form that is still centered on the loop apex $\mu=0$, but with co-latitude dispersion $\sigma_{\mu}$,

where for a loop with apex radius $r_{a}$, $C_{\mu}$ is a normalization constant that ensures that the total energy ($\int e_{\mu}d\mu$) is unity ($e=1$) when integrated from the loop apex ($\mu=0$) to the stellar base ($\mu=\pm\mu_{\ast}\equiv\pm\sqrt{1-(R_{\ast}/r_{a})}$).

For a loop with $r_{a}=3R_{\ast}$, the top panel of Figure 6 compares the associated cumulative energy emission $\epsilon_{\mu}$ vs. co-latitude $\mu$ for cases with $\sigma_{\mu}=0.1$, 0.3, and 0.5. For the small dispersion case $\sigma_{\mu}=0.1$, the cumulative energy loss still has its peak at the loop apex, though with a broader range than found for models with energy deposition confined to just the loop apex.

For intermediate dispersion case $\sigma_{\mu}=0.3$, the central emission is flatter, with even a modest dip around $\mu=0$.

Indeed, for the largest dispersion case $\sigma_{\mu}=0.5$, the central emission shows an even deeper local minimum, even though that is still where the deposition is greatest. Instead there are two distinct peaks located at $|\mu|\approx 0.5$, distinctly below the loop apex, where the energy deposition is highest.

To help understand this rather unexpected result, the light green curves in the bottom panel plot the emission distribution for a subset with deposition peaks separated by $\Delta\mu=0.1$, showing then these generally have much stronger emission away from the apex, due the higher magnetic field strength there. Thus, even though the black dots show these represent a somewhat lower number in the gaussian distribution, their higher emission leads to an overall peak in the cumulative emission at co-latitudes $|\mu|\approx 0.5=\sigma_{\mu}$.

These results have important implications for both the spatial and spectral distribution of radio emission for more realistic models in which the energy deposition has a broader radial distribution, as we discuss next.

Let us now consider a model in which the initial energy deposition has a gaussian spread in both field co-latitude $\mu=\sqrt{1-r/r_{a}}$ and apex radius $r_{a}=r/(1-\mu^{2})$, centered on a peak radius $r_{p}$ with radial dispersion $\sigma_{r}$,

where $C_{a}$ is a normalization factor such that

where $\mu_{\ast}\equiv\sqrt{1-R_{\ast}/r_{a}}$.

Given such a distribution in energy deposition $e_{\mu,ra}$, one can, for each $r_{a}$, use the techniques described above to derive an associated emission distribution $\epsilon_{\mu,r_{a}}$ along the loop. For differential intervals in loop colatitude $d\mu$ and apex radius $dr_{a}$, the incremental contribution to radio luminosity is given by

Assuming for simplicity that the local emission is isotropic over the full $4\pi$ steradians, the associated increment in intensity (energy/time/area/solid angle) about an azimuthal interval $d\phi$ centered on the plane of the sky is

where the latter equality uses the relations $d\mu=\sqrt{1-\mu^{2}}d\theta$ and $dr_{a}=dr/(1-\mu^{2})$.

For a representative, standard case with $r_{p}=3R_{\ast}$ and $\sigma_{r}=0.5R_{\ast}$, the lower row of Figure 7 shows the spatial variation of this emitted intensity $dI/d\phi$ for the 3 distinct values of latitudinal dispersion $\sigma_{\mu}=0.1$, 0.3, and 0.5. The white lines show contours of the field strength $B$, spaced logarithmically in increments of $-0.1$ dex from the value at the equatorial surface.

The upper row of Figure 7 compares the corresponding emission for a simple “on-the-spot” model, in which the local emission is just set by the local energy deposition, $\epsilon_{\mu,r_{a}}=e_{\mu,r_{a}}$. Comparison with the middle row shows that the net effect of the electron propagation and mirroring within the closed loops is to shift the emission closer to loop footpoints, where the field strength is higher. This shift becomes more pronounced with increasing latitudinal dispersion $\sigma_{\mu}$.

Using the generalized relativistic forms for gyro-cooling derived in Appendix A, the lower row shows a similar spatial distribution for the relativistic case with initial Lorentz factor $\gamma_{a}=11$ at the loop apex. The close similarity to the middle panel shows that the relativistic corrections have only a modest effect on the overall spatial distribution of gyro-cooling emission.

The results derived in this subsection (including Figure 7) provides a visualization about the relation between the spatial distribution of energy deposited and that of the energy radiated by the non-thermal electrons. In their current forms, however, these results cannot be compared with observations, which will require a more complete treatment, including absorption effects in the magnetosphere.

In addition to the gyro-cooling examined here, non-thermal electrons can also cool by the energy exchange from coulomb collision with an ambient population of thermal electrons of much lower energy (see Güdel, 2009, e.g., their section 8.4.3). For potentially relativistic electrons with Lorentz factor $\gamma$ and thus kinetic energy E=$(\gamma-1)m_{e}c^{2}$, the coulomb collision cross section takes the form

where the last evaluation assumes a characteristic value $\ln\Lambda\approx 20$ for the coulomb logarithm (which accounts for the cumulative effect of many small-angle scatterings). Since each associated collision results in the effective loss of the kinetic energy of the non-thermal particle, the associated cooling time for collisions with thermal electrons of number density $n_{e}$ is (Leach & Petrosian, 1981, see their eqn. 1)

Comparison with eqn. (2) shows that the ratio of coulomb to gyro-synchrotron cooling (assuming an isotropic pitch-angle distribution with $\left<\sin^{2}\alpha\right>=2/3$) is given by

where $B_{kG}\equiv B/kG$ and $n_{e}$ is in ${\rm cm}^{-3}$. Setting the left side to unity, we can solve for the critical density above which cooling by coulomb collisions will dominate over gyro emission,

For highly relativistic electrons with $\gamma\gg 1$ and $\beta\rightarrow 1$, this critical density increases with $\gamma^{2}$, implying that in regions with strong kG fields, coulomb cooling will only be important in regions with quite a high density of thermal electrons.

In contrast, for non-relativistic electrons with $\beta<1$, we find $n_{e1}\sim\beta^{3}$, implying that cooling by coulomb collisions will dominate over gyro-emission for even modest densities. For example, for the mildly sub-relativistic case $\beta=1/3$, we find $n_{e1}=7.2\times 10^{7}\,{\rm cm}^{-3}\,B_{kG}^{2}$.

The overall implication is that the incoherent radio emission from magnetic hot stars most likely arises from relativistic electrons for which gyro-emission is relatively unaffected by collisional losses with thermal electrons. Further quantifying this will depend on the detailed models of the thermal electron density and its spatial distribution. We are currently examining such cooling effects in the context of hot-stars with centrifugal magnetospheres, and will report results in a follow up paper.

The incoherent, circularly polarized radio emission observed from magnetic massive stars is understood to arise from gyro-synchrotron emission by energetic electrons trapped within their magnetospheres. By accounting for the associated “gyro-cooling” loss of energy, this paper analyzes the spatial distribution of this radio emission for a simple model in which the electrons trapped within closed (assumed dipole) magnetic loops gradually cool as they repeatedly mirror across the loop. Some key results are:

• •

  Because the gyro-cooling time in hot-star magnetospheres is of order a day, explaining their observed persistent quasi-steady radio emission requires a persistent electron acceleration mechanism, perhaps from magnetic reconnection driven by repeated, small-scale centrifugal breakout (CBO) events in these rapidly rotating magnetospheres (Owocki et al., 2022).

• •

  So long as the ratio of the advective propagation time to cooling time is small, $k\equiv t_{a}/t_{c}\ll 1$, the spatial distribution of cumulative emission is insensitive to the specific value of this ratio.

• •

  For various assumptions for the spatial location of the deposition of energetic electrons centered about the apex of closed magnetic loops, the associated radio emission tends to be spread down the loop where the field is stronger (see Figure 7). The formalism here thus effectively represents an energy transport model between the deposition of energetic electrons to their ultimate gyro-synchrotron radio emission.

• •

  Generalization to the case of relativistic electrons introduces a formal, explicit dependence of gyro-cooling on the initial energy, but the overall effect on the resulting spatial distribution of emission is quite modest.

• •

  Cooling from coulomb collisions with thermal electrons in the dense CM layer should effectively quench any gyro-emission from sub-relativistic electrons, but have less effect on relativistic electrons.

While the work here thus provides an initial basis for modelling the spatial and spectral distribution of radio emission from these massive-star magnetospheres, it is grounded in several idealized assumptions and simplifications that should be examined and relaxed in future work.

• •

  To translate results for spatial distribution of emission within magnetic loops into observable spectra in frequency, future work should consider a realistic scenario of multiple harmonic emission and absorption in the stellar magnetosphere.

• •

  This should include the effect of cooling of the energetic electrons through coulomb collision with thermal electrons associated with material trapped within the centrifugal magnetosphere.

• •

  In contrast to the static, dipole field assumed here, even closed loops in centrifugal magnetospheres are likely to be dynamically distorted, due to the stretching by the centrifugal force from the trapped material and the associated breakout events. Future work should examine how this affects electron mirroring and the associated emission.

• •

  As found in planetary magnetospheres, such variations can excite MHD waves and even cascade to magnetic turbulence, leading to pitch-angle scattering of electrons (e.g. Summers et al., 2005; Kim et al., 2018) into a loss cone that allows interaction with the underlying planet or star. The associated energy loss can compete with gyro-cooling, and so lower the overall radio emission.

• •

  Both loss mechanisms lead to formation of anisotropic electron distribution. Future work should examine how this affects the associated electron cyclotron maser emission (ECME) seen from many such magnetic stars (e.g. Trigilio et al., 2000; Das et al., 2022).

• •

  Instead of the simplified assumption here of isotropic emission, future work should take into account the distinctive angular phase function for gyro-synchrotron emission.

• •

  Within the CBO-driven reconnection paradigm, there is a need for detailed modeling, e.g. using particle-in-cell (PIC) codes (Lapenta, 2012; Germaschewski et al., 2016), of the electron electron acceleration from reconnection, with a focus on the spatial deposition of non-thermal electrons.

• •

  Does such reconnection yield non-gyrotropic electron deposition, and if so, does it favor field-aligned or orthogonal pitch angles? This is crucial for mirroring and the spatial distribution of radio emission.

• •

  Finally, to test and constrain the basic gyro-cooling model and any extensions, there is a need for more extended observational programs on radio spectra and their level of variability on a dynamical timescale of order a day. While current radio observatories cannot spatially resolve the emission from hot-star magnetospheres, upcoming facilities such as the ngVLA (McKinnon et al., 2019) will have the sensitivity and angular resolution to do so for at least a subset of hot magnetic stars; this will allow direct tests of predictions from our simple model, and constrain characteristics of the underlying energy deposition processes.

Overall, despite the relatively idealized nature of its basic assumptions, the gyro-cooling analysis presented here forms a good initial basis for modelling the observed incoherent radio emission from these magnetic massive stars.