The effect of collisional cooling of energetic electrons on radio emission from the centrifugal magnetospheres of magnetic hot stars

TL;DR

This paper analyzes how Coulomb collisions with ambient thermal electrons within the centrifugal magnetosphere (CM) of rapidly rotating magnetic hot stars modify the energetics and spatial distribution of non-thermal electrons responsible for gyrosynchrotron radio emission. By comparing collisional cooling to gyro-cooling in an aligned-dipole geometry, it derives that the two loss channels share similar magnetic-field scalings, outcome quantified by a near-field-independent ratio, with collisional cooling often dominating for high-pitch-angle electrons near loop apex. Numerical simulations show that CM collisions suppress equatorial emission and shift residual gyrosynchrotron radiation toward magnetic poles, a result largely insensitive to deposition geometry and electron energy (within moderate ranges). The findings have implications for interpreting radio maps of hot stars, for applications to magnetospheric emission in ultra-cool dwarfs, and for planning VLBI imaging to test the CM cooling framework. Overall, the work provides a quantitative framework linking ambient plasma density, particle pitch-angle, and magnetospheric geometry to the observed radio morphology and spectra of magnetic hot stars.

Abstract

This paper extends our previous study of gyro-emission by energetic electrons in the magnetospheres of rapidly rotating, magnetic massive stars, through a quantitative analysis of the role of Coulomb collisions with thermal electrons from stellar wind material trapped within the centrifugal magnetosphere (CM). For a dipolar field with aligned magnetic and rotational axes, we show that both gyro-cooling along magnetic loops and Coulomb cooling in the CM layer have nearly the same dependence on the magnitude and radial variation of magnetic field, implying that their ratio is a global parameter that is largely independent of the field. Analytic analysis shows that, for electrons introduced near the CM layer around a magnetic loop apex, collisional cooling is more important for electrons with high pitch angle, while more field-aligned electrons cool by gyro-emission near their mirror point close to the loop base. Numerical models that assume a gyrotropic initial deposition with a gaussian distribution in both radius and loop co-latitude show the residual gyro-emission is generally strongest near the loop base, with highly relativistic electrons suffering much lower collisional losses than lower-energy electrons that are only mildly relativistic. Finally, we briefly discuss the potential applicability of this formalism to magnetic ultracool dwarfs, for which VLBI observations indicate incoherent radio emission to be concentrated around the magnetic equator, in contrast to our predictions here for magnetic hot stars. We suggest that this difference could be attributed to either a lower ambient density of thermal electrons, or more highly relativistic non-thermal electrons, both of which would reduce the relative importance of the collisional cooling explored here.

Figures (11)

• Figure 1: For our assumed model of a centrifugal magnetosphere (CM) limited by centrifugal breakout (CBO), the color plot shows the log of electron density normalized to its maximum value in the equator at the Kepler radius, which for this model with critical rotation fraction $W=1/2$, occurs at $R_K/R=W^{-2/3}= 1.59$ (denoted here by the white circle). The yellow contours show magnetic field lines extending to an outer radius $r=12 R$, with spacing set to follow the field strength.
• Figure 2: Top: On a log-log scale, the black curve plots the mirror-cycle time-average of $\left < p b^3 \right >$ computed from (\ref{['eq:pb3int']} vs. sine of apex pitch angle $\sin \alpha_\mathrm{a}$. The red and blue dashed lines show that $(\sin \alpha)^{-3.3}$ is a much better fit to $\left < p b^3 \right >$ than the simple estimate $(\sin \alpha)^{-4}$. The vertical dotted lines mark the minimum apex pitch angle for the mirror radius $r_\mathrm{m}$ to remain above the stellar radius $R$ for the labeled values of apex radius $r_\mathrm{a}$. The black dots thus mark the maximum possible value of $\left < p b^3 \right >$ for loops with these apex radii. Bottom: The black curve now shows this $\left < p b^3 \right >_\mathrm{max}$ plotted vs. $r_\mathrm{a}/R$. The red dashed curve shows that this maximum is quite well fit by $(r_\mathrm{a}/R)^5$.
• Figure 3: For a mildly relativistic electron with initial Lorentz factor $\gamma_0 = 1.5$ and initial pitch angle $\alpha_\mathrm{a} = 30^o$, the variation of magnetic moment $p$, energy $e$, and latitudinal cosine $\mu$ plotted vs. the dimensionless time $t/t_{\rm a} = v_0 t/r_{\rm a}$ (left columns), latitude $\mu$ (middle columns), and magnetic moment $p$ (right columns), for apex radii $r_\mathrm{a}=6R$ (left) and $r_\mathrm{a}=10R$ (right). The weaker field strength and gyro-cooling in the right set leads to many more mirror cycles on the right vs. left. But the net relative importance of collisional vs. gyro-synchrotron cooling across the CM layer -- shown by the drops in $e$ and $p$ across $\mu=0$ -- are the same in the left vs. right cases.
• Figure 4: Top: Comparison between numerically computed values of $\langle pb^3\rangle$ (as defined in §\ref{['subsec:alpha_dep_analytical']}) with the analytically predicted ones, plotted as a function of initial pitch angle for a fixed apex radius of $10\,R$. Middle: The ratio between final value of $p$ to its initial values over the time ranges considered for the top panel. Bottom: Ratio between predicted mirroring cosine to the 'actual' (numerical) mirroring cosine.
• Figure 5: Numerically computed $\langle pb^3\rangle_\mathrm{max}$ as a function of $r_\mathrm{a}/R$.