2.1 General scalings

In addition to the gyro-cooling examined in Paper I, 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, Reference Güdel, Cargill and Vlahos2009, e.g. their section 8.4.3). For potentially relativistic electrons with speed $v= {\unicode{x03B2}} c$ near the speed of light c, thus with Lorentz factor ${\unicode{x03B3}} \equiv 1/\sqrt{1-{\unicode{x03B2}}^2}$ and kinetic energy E= $({\unicode{x03B3}} -1)m_\mathrm{e}c^2$ ( $m_\mathrm{e}$ is the mass of an electron), the cooling time for collisions with thermal electrons of number density $n_\mathrm{e}$ is (Leach & Petrosian, Reference Leach and Petrosian1981, see their Equation 1)

(1) \begin{equation}t_\mathrm{c}\equiv \frac{E}{dE/dt}= \frac{{\unicode{x03B2}} ({\unicode{x03B3}}-1)}{n_\mathrm{e} 4{\unicode{x03C0}} r_\mathrm{o}^2 \ln \Lambda \, c }= 1.7 \times 10^{12} \, \textrm{s} \, \frac{{\unicode{x03B2}} ({\unicode{x03B3}}-1)}{ \, n_\mathrm{e}\ \textrm{cm}^3} \, ,\end{equation}

where for electron charge $q_\mathrm{e}$ and mass $m_\mathrm{e}$ , $r_\mathrm{o} = q_\mathrm{e}^2/m_\mathrm{e} c^2$ is the classical electron radius, and the latter evaluation assumes a characteristic value $\ln \Lambda \approx 20$ for the Coulomb logarithm (which accounts for the cumulative effect of many small-angle scatterings, $\Lambda^{-1}$ represents the minimum angle of deflection in the Coulomb integral, Leach & Petrosian, Reference Leach and Petrosian1981).

Assuming an isotropic pitch angle distribution with $\langle \sin^2 {\unicode{x03B1}} \rangle = 2/3$ Footnote ^b , (where ${\unicode{x03B1}}$ is the pitch angle), comparison with Equation 2 of Paper I shows that the ratio of the Coulomb to gyro-synchrotron cooling times $t_\mathrm{e}$ is given by

(2) \begin{equation}\frac{t_\mathrm{c}}{t_\mathrm{e}} \approx 2.2 \times10^3 \, \frac{({\unicode{x03B3}}^2 -1) {\unicode{x03B2}} B^2}{ n_\mathrm{e}} \, ,\end{equation}

where B is the magnetic field strength. Both B and $n_\mathrm{e}$ are taken to be in CGS units.

2.2 Cooling in the dense CM layer

To proceed, we need to specify a model for the density distribution of thermal electrons. For CMs of hot stars, the strongest collisional cooling in the extended CM should be in the dense, compressed layer near the tops of magnetic loops. For the simple axisymmetric case of a rotationally aligned dipole field, the CBO analysis by (Owocki et al., Reference Owocki and Shultz2020, see their Equation 7) provides a scaling for the radial variation of mass column density in this CM layer. Through the mean mass per electron ${\unicode{x03BC}}_\mathrm{e} = 1.16 m_\mathrm{p}$ , where $m_\mathrm{p}$ is the mass of a proton, this can be used to derive an associated electron column density of this thin, dense CM layer,

(3) \begin{align} N_\mathrm{e} &\approx 0.3\frac{B_\textrm{K}^2}{4 {\unicode{x03C0}} {\unicode{x03BC}}_\mathrm{e} g_{\textrm{K}}} \,\left ( \frac{r_\mathrm{a}}{R_\textrm{K}} \right )^{-p} \end{align}

(4) \begin{align} &\approx 4.85 \times 10^{22} \textrm{cm}^{-2} \, \frac{B_\mathrm{kG}^2}{g_4}\left ( \frac{R_\textrm{K}}{R} \right )^{-4}\left ( \frac{r_\mathrm{a}}{R_\textrm{K}} \right )^{-5},\end{align}

where $B_\textrm{K} $ and $g_\textrm{K}$ are the magnetic field strength and stellar gravity at the Kepler radius $R_\mathrm{K}$ , and the power index p allows for generalisations from the original $p=6$ assumed in Equation 7 of Owocki et al. (Reference Owocki and Shultz2020); $r_\mathrm{a}$ is the apex radius of the magnetic field loop under consideration.

Following Berry et al. (Reference Berry, Owocki and Shultz2022) and ud-Doula et al. (2023), the latter equality instead takes $p=5$ , and provides numerical scalings in terms of the scaled surface gravity $g_4 \equiv g_\ast/(10^4 \,$ cm/s $^2$ ), and the polar surface field in kG, $B_\mathrm{kG}\equiv B_\mathrm{p\ast }/$ kG. For a star with critical rotation fraction W at its equatorial surface, $R_{\textrm{K}}/R = W^{-2/3}$ .

For our assumed standard model with $W=1/2$ (and thus $R_K=1.59 \, R$ ), Figure 1 illustrates the spatial variation of electron density to an outer radius $r=12 \; R$ , with yellow curves showing dipole field lines.

Figure 1.

For our assumed model of a centrifugal magnetosphere (CM) limited by centrifugal breakout (CBO), the colour plot shows the log of electron density normalised 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.

For high-energy electrons with speed v along a path s through this CM layer at the loop apex, application of Equation (1) implies that the fractional loss of energy is set by the collision depth,

(5) \begin{align} {\unicode{x03C4}}_\mathrm{c} &= \int \frac{ds}{v \, t_\mathrm{c}} = \frac{N_\mathrm{e} 4 {\unicode{x03C0}} r_\mathrm{o}^2 \ln \Lambda }{{\unicode{x03B2}}^2 ({\unicode{x03B3}} -1)} \end{align}

(6) \begin{align} &\approx 0.96 \, \frac{B_\mathrm{kG}^2}{g_4 {\unicode{x03B2}}^2 ({\unicode{x03B3}}-1)} (2W)^{8/3} \left ( \frac{r_\mathrm{a}}{R_\textrm{K}} \right )^{-5}\, . \end{align}

For each pass through the CM layer, the fractional energy loss is $1-e^{-{\unicode{x03C4}}_\mathrm{c}}$ .

2.3 Reduction of gyro-emission

Let us next consider how such cooling from Coulomb collision competes with the gyro-cooling over a given mirror cycle. Equation 11 of Paper I defined a constant k to characterise the energy loss to gyro-cooling during each mirror cycle. Noting that the Thomson cross section $\sigma_\mathrm{T} = (8/3) {\unicode{x03C0}} r_\mathrm{o}^2$ and casting this in terms of these Kepler-radius parameter scalings, we have

(7) \begin{align} k &=\frac{4 B_\textrm{K}^2 R_\textrm{K} r_\mathrm{o}^2}{3 m_\mathrm{e} c^2 {\unicode{x03B2}}_0} \,\left ( \frac{r_\mathrm{a}}{R_\textrm{K}} \right )^{-5} \end{align}

(8) \begin{align} &= 0.0032 \, \frac{ B_\mathrm{kG}^2 R_{12} }{{\unicode{x03B2}}_0} (2W)^{10/3}\left ( \frac{r_\mathrm{a}}{R_\mathrm{K}} \right )^{-5},\end{align}

where $R_{12} \equiv R/(10^{12}$ cm) and ${\unicode{x03B2}}_0 \equiv v_0/c$ .

Comparison of Equation (8) with (5) shows that, quite remarkably, both gyro-cooling along magnetic loops and Coulomb cooling in the CM layer have the same dependence on the magnitude and radial variation of magnetic field, implying then that their ratio is a global parameter that is fully independent of the field,

(9) \begin{equation}\boxed{\frac{k}{{\unicode{x03C4}}_\mathrm{c}}\approx C_\mathrm{k}\, ({\unicode{x03B2}}^2/{\unicode{x03B2}}_0) ({\unicode{x03B3}} -1) \,(2W)^{2/3}\,{\mathcal M}/{\mathcal R}}\,\end{equation}

where ${\mathcal M}/{\mathcal R}$ is the star’s ratio of mass to radius in solar units. Combining (3), (5), and (7), we find the coefficient $C_{k}$ has a very small value, set by the scaling

(10) \begin{equation}C_\mathrm{k} \equiv\frac{2.8}{\ln \Lambda} \,\frac{G M_\odot}{R_\odot c^2} \, \frac{{\unicode{x03BC}}_\mathrm{e}}{m_\mathrm{e}}\approx 6.3\times 10^{-4}\, ,\end{equation}

where the latter evaluation again uses $\ln \Lambda = 20$ and ${\unicode{x03BC}}_\mathrm{e} = 1.16 \, m_\mathrm{p}$ .

For typical scaled parameters of order unity, equations (9) and (10) indicate that collisional cooling should dominate over gyro-cooling, at least for electrons with large pitch angle that mirror near the loop apex.

However, for lower pitch angles that mirror closer to the loop footpoints, where the much stronger field implies a much stronger gyro-emission, quantifying the net reduction requires integration over the full loop, as discussed next.

3.1 Energy loss rate from Coulomb cooling

Using Equation (1), the rate of energy loss from Coulomb collisions is

(11) \begin{equation}\frac{dE}{dt} = - \frac{E}{t_\mathrm{c}} = - \frac{m_\mathrm{e} c^2 n_\mathrm{e} c 4 {\unicode{x03C0}} r_\mathrm{o}^2 \ln \Lambda }{{\unicode{x03B2}} }\, .\end{equation}

Using $E=({\unicode{x03B3}}-1)m_\mathrm{e} c^2$ and $e=({\unicode{x03B3}}-1)/({\unicode{x03B3}}_0-1)$ , for initial Lorentz factor ${\unicode{x03B3}}_0$ , we have

(12) \begin{equation}\frac{de}{dt} = - \frac{n_\mathrm{e} c 4 {\unicode{x03C0}} r_\mathrm{o}^2 \ln \Lambda }{{\unicode{x03B2}} ({\unicode{x03B3}}_0-1)}\end{equation}

Scaling the time in units of the initial apex crossing time $t_\mathrm{a}=r_\mathrm{a}/v_0$ , we find

(13) \begin{align} {\dot e}_\mathrm{c} \equiv \frac{de}{dt} t_\mathrm{a} &= - \frac{n_\mathrm{e} c 4 {\unicode{x03C0}} r_\mathrm{o}^2 \ln \Lambda}{{\unicode{x03B2}} ({\unicode{x03B3}}_0-1)}\frac{r_\mathrm{a}}{v_0}\end{align}

(14) \begin{align} &= - \frac{{\unicode{x03C4}}_\mathrm{c} {\unicode{x03B2}} ({\unicode{x03B3}}-1)}{{\unicode{x03B2}}_0 ({\unicode{x03B3}}_0-1)} \, \frac{\textrm{e}^{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2}}{\sqrt{{\unicode{x03C0}}} {\unicode{x03BC}}_\mathrm{H}},\end{align}

where the second expression uses (5) and invokes the gaussian density stratification of the CM layer over a scale height given by (see Owocki & Cranmer, Reference Owocki and Cranmer2018, their Equation 4)

(15) \begin{equation}H = \frac{\sqrt{2} c_\mathrm{s}/\Omega}{\sqrt{3-2(R_\mathrm{K}/r_\mathrm{a})^3}}\, ,\end{equation}

with $\Omega$ the stellar rotation frequency. For $r_\mathrm{a} \ge R_\mathrm{K}$ , the corresponding scale in co-latitude ${\unicode{x03BC}}$ is

(16) \begin{equation}{\unicode{x03BC}}_\mathrm{H} \equiv \frac{H}{r_\mathrm{a}} = \frac{0.063}{ \sqrt{{\mathcal M}/{\mathcal R}}\sqrt{3W^2-2(R/r_\mathrm{a})^3}}\, \frac{R}{r_\mathrm{a}}\, ,\end{equation}

where the latter evaluation assumes an isothermal sound speed $c_\mathrm{s} = 20$ km/s. For standard parameters $2W = {\mathcal M}/{\mathcal R} = 1$ , we find ${\unicode{x03BC}}_\mathrm{H} = 0.026 $ for $r_\mathrm{a}=3 R$ and ${\unicode{x03BC}}_\mathrm{H}=0.012 $ for $r_\mathrm{a}=6 R$ .

Finally, using (9) to eliminate ${\unicode{x03C4}}_\mathrm{c}$ in favour of the gyro-cooling parameter k, we find

(17) \begin{equation}\boxed{{\dot e_\mathrm{c}}= -\frac{1600 \, k}{{\unicode{x03B2}} ({\unicode{x03B3}}_0-1) (2W)^{2/3} {\mathcal M}/{\mathcal R} }\, \frac{\textrm{e}^{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2}}{\sqrt{{\unicode{x03C0}}} {\unicode{x03BC}}_\mathrm{H}}}\, .\end{equation}

3.2 Quantitive scaling of collisional cooling vs. gyro-cooling

By comparison, Equation (A5) of Paper I gives the relativistic form for the competing gyro-cooling emission term. Noting that the relativistic correction factor in curly brackets just becomes $({\unicode{x03B3}} +1)/2$ , their ratio can be characterised by

(18) \begin{equation}\frac{\langle {\dot e}_\mathrm{e} \rangle} {\langle {\dot e}_\mathrm{c}\rangle } = 3.1 \times 10^{-4} {\unicode{x03B2}} ({\unicode{x03B3}}_0-1)({\unicode{x03B3}}+1)\langle pb^3 \rangle\, (2W)^{2/3} {\mathcal{M}}/{\mathcal{R}}\, ,\end{equation}

where the angle brackets denote the cumulative time average (see Equation 21 below) from the apex ${\unicode{x03BC}}=0$ to the mirror point ${\unicode{x03BC}}={\unicode{x03BC}}_\mathrm{m}$ .

In the common simple case that both energy losses are small across a single mirror loop crossing, implying then that ${\unicode{x03B3}} \approx {\unicode{x03B3}}_0$ and ${\unicode{x03B2}} \approx {\unicode{x03B2}}_0$ , we can approximate the ratio of gyro-emission to Coulomb loss as

(19) \begin{equation}\boxed{\frac{\delta e_\mathrm{g}}{\delta e_\mathrm{c}} \approx3.1 \times 10^{-4} {{\unicode{x03B2}}_0^3}{{\unicode{x03B3}}_0^2} \, \langle pb^3 \rangle\, (2W)^{2/3} {\mathcal M}/{\mathcal R}} \, .\end{equation}

Note that for non-relativistic electrons with ${\unicode{x03B2}}_0 {\lt} 1$ , this ratio declines as $\delta e_\mathrm{g}/\delta e_\mathrm{c} \sim {\unicode{x03B2}}_0^3$ , implying strong dominance of Coulomb cooling.

On the other hand, in the relativistic regime ${\unicode{x03B3}}_0 {\gt} 1$ , the ratio increases as $\delta e_\mathrm{g}/\delta e_\mathrm{c} \sim {\unicode{x03B3}}_0^2$ , implying a much reduced importance of Coulomb cooling for such relativistic electrons.

For electrons with large pitch angles, which mirror near the loop top, $\langle pb^3 \rangle$ remains of order unity, showing again that, unless the electrons are highly relativistic ${\unicode{x03B3}}_0 \gg 1$ , losses by Coulomb collisions should dominate over gyro-cooling.

But this factor $\langle pb^3 \rangle$ can become large for field-aligned pitch angles, for which the mirror radius $r_\mathrm{m} \ll r_\mathrm{a}$ , implying an increase field strength that scales as $b \sim 1/r_\mathrm{m}^3$ , allowing a potentially significant gyro-cooling even for the mildly relativistic case, ${\unicode{x03B3}}_0 \,{\gtrsim}\, 1$ .

3.3 Pitch angle dependence

To quantify this pitch angle dependence, let us again assume we can neglect the energy losses for a single mirror cycle. By conservation of the scaled magnetic moment $p = \sin^2 {\unicode{x03B1}}_\mathrm{a}$ , the apex-scaled field strength at the mirror point then just varies as

(20) \begin{equation}b_\mathrm{m} \equiv \frac{B_\mathrm{m} }{B_\mathrm{a}} = \frac{1}{\sin^2 {\unicode{x03B1}}_\mathrm{a}} = \frac{1}{p}\, .\end{equation}

Since an electron mirroring from the loop apex spends the greatest time near its mirror radius, one might initially estimate that $\langle p b^3 \rangle\approx b_\mathrm{m}^2 \approx (\sin {\unicode{x03B1}}_\mathrm{a} )^{-4}\, .$

But using $p = \sin^2 {\unicode{x03B1}}_\mathrm{a}$ , the full average over the mirror time $t_\mathrm{m}$ can be readily computed from numerical integration over a mirror cycle,

(21) \begin{equation}\langle p b^3 \rangle = \frac{\sin^2 {\unicode{x03B1}}_\mathrm{a}}{t_\mathrm{m}} \int_0^{t_\mathrm{m}} b(t)^3 \, dt \, .\end{equation}

In the upper panel of Figure 2, the black curve plots the resulting $\langle p b^3 \rangle$ vs. $\sin {\unicode{x03B1}}_\mathrm{a}$ on a log-log scale. Comparison with the red and blue dashed lines show that, instead of the above simple estimate $(\sin {\unicode{x03B1}})^{-4}$ , a much better fit to the full integral (21) is given by

(22) \begin{equation}\langle p b^3 \rangle \approx 0.4(\sin {\unicode{x03B1}})^{-3.3}\, .\end{equation}

Figure 2.

Top: On a log-log scale, the black curve plots the mirror-cycle time-average of $\langle p b^3 \rangle$ computed from (21 vs. sine of apex pitch angle $\sin {\unicode{x03B1}}_\mathrm{a}$ . The red and blue dashed lines show that $(\sin {\unicode{x03B1}})^{-3.3}$ is a much better fit to $\langle p b^3 \rangle$ than the simple estimate $(\sin {\unicode{x03B1}})^{-4}$ . 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 labelled values of apex radius $r_\mathrm{a}$ . The black dots thus mark the maximum possible value of $\langle p b^3 \rangle$ for loops with these apex radii. Bottom: The black curve now shows this $\langle p b^3 \rangle_\mathrm{max}$ plotted vs. $r_\mathrm{a}/R$ . The red dashed curve shows that this maximum is quite well fit by $(r_\mathrm{a}/R)^5$ .

The vertical dotted lines in Figure 2 mark the minimum apex pitch angle for the mirror radius $r_\mathrm{m}$ to remain above the stellar radius R for the labelled values of apex radius $r_\mathrm{a}$ . The black dots thus mark the maximum possible value of $\langle p b^3 \rangle$ for loops with these apex radii.

The black curve in the lower panel of Figure 2 then plots this maximum vs. $r_\mathrm{a}/R$ . Comparison with the red dashed line shows that, quite remarkably, the maximum has the approximate simple scalingFootnote ^c

(23) \begin{equation}\langle p b^3 \rangle_\mathrm{max} \approx \left ( \frac{r_\mathrm{a}}{R} \right )^5\, .\end{equation}

This indicates that in high-lying loops with large ratio of apex to stellar radius $r_\mathrm{a}/R$ , electrons with low pitch angle ${\unicode{x03B1}}_\mathrm{a}$ that mirror just above the stellar radius can have $\langle pb^3 \rangle$ that approach the large maximum set by Equation (23); this can thus compensate for the small numerical factor in Equation (19).

For example, applying (23) into (19), we can solve for the apex radius ratio that would give $\delta e_\mathrm{g}/\delta e_\mathrm{c} =1$ ,

(24) \begin{equation}\boxed{\left ( \frac{r_\mathrm{a}}{R} \right )_1 \approx5 \left ({{\unicode{x03B2}}_0^3}{{\unicode{x03B3}}_0^2} \,(2W)^{2/3} {\mathcal M}/{\mathcal R}\right )^{-1/5}}\, .\end{equation}

The upshot here is that for even mildly relativistic electrons ( ${\unicode{x03B2}}_0 \approx 0.76$ , ${\unicode{x03B3}}_0 \approx 1.5$ ) for which ${\unicode{x03B2}}_0^3 {\unicode{x03B3}}_0^2 \approx 1$ , loops with an apex radius near $r_\mathrm{a} \approx 5 R$ can have gyro-cooling that competes with collisional cooling, despite the small numerical value of the factor ( $C_\mathrm{k}/2$ ) in (19).

4.1 Single electron evolution

In analogy with Figure 1 of Paper I, Figure 3 here illustrates the evolution of the energy e, magnetic moment p, and latitudinal cosine ${\unicode{x03BC}}$ for an electron introduced at loop apex radius $r_\mathrm{a}$ with initial pitch angle ${\unicode{x03B1}}_\mathrm{a} = 30^o$ , and with an initially relativistic Lorentz factor ${\unicode{x03B3}}_\mathrm{0}=1.5$ . The left and right panels contrast results for $r_\mathrm{a}=6 R$ vs. $r_\mathrm{a}=10R$ . For the higher apex radius, the much lower value of the magnetic field, and thus of the gyro-cooling constant k, leads to many more mirror cycles than for the lower apex radius $r_\mathrm{a}= 6R$ .

Figure 3.

For a mildly relativistic electron with initial Lorentz factor ${\unicode{x03B3}}_0 = 1.5$ and initial pitch angle ${\unicode{x03B1}}_\mathrm{a} = 30^o$ , the variation of magnetic moment p, energy e, and latitudinal cosine ${\unicode{x03BC}}$ plotted vs. the dimensionless time $t/t_\textrm{a} = v_0 t/r_\textrm{a}$ (left columns), latitude ${\unicode{x03BC}}$ (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 ${\unicode{x03BC}}=0$ – are the same in the left vs. right cases.

One key difference from the case without collisional cooling is that here the time evolution of the particle is governed by the evolution of p, since once p becomes zero (i.e. the particle’s velocity becomes parallel to the magnetic field), it will no longer lose energy by radiation. Thus, any further loss of energy occurs only by collisions, which do not produce observable radio emission. For the initial pitch angle of $30^\circ$ , collisional cooling always dominates over gyrocooling, which is consistent with the analytical analysis in ${\S}$ 3.3 with $\langle pb^3\rangle \approx 4$ following Eq. 22.

For the case with $r_\mathrm{a}=10\,R$ , the top panel of Figure 4 compares the numerical values $\langle pb^3 \rangle$ with the analytic predictions from Eq. 22. For the relatively large values of initial pitch angle, the two agree remarkably well; but for the smaller values of initial pitch angles, the numerically obtained values are higher than predicted. The reasons for this are illustrated in the middle panel, which shows that the final to initial value of p decreases sharply with lower initial pitch angle $\sin {\unicode{x03B1}}_\mathrm{a}$ , demonstrating then that the analytic assumption that p does not change significantly over a mirror cycle is simply not valid for such low initial pitch angles. The bottom panel shows that this enhances electron penetration into the lower magnetosphere, thus increasing values of $\langle p b^3 \rangle$ from what’s predicted from the analytic analysis.

Figure 4.

Top: Comparison between numerically computed values of $\langle pb^3\rangle$ (as defined in ${\S}$ 3.3) 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.

Another important difference between the analytical and numerical results is that, while the former shows the minimum value of $\sin{\unicode{x03B1}}_\mathrm{a}$ for $r_\mathrm{a}=10\,R$ to be $\approx 0.024$ , in the numerical simulation, electrons with such low pitch angle completely lose their magnetic moment p by the time they come out of the dense plasma at the apex, and so no longer contribute to radiation.

A first order estimate of the minimum $\sin{\unicode{x03B1}}_\mathrm{a}$ for a given value of k and $r_\mathrm{a}$ can be obtained as follows. For electrons close to the apex radius, gyrocooling is negligible compared to collisional cooling. Thus, we can write:

\begin{align*} \frac{dp}{dt}&\approx \frac{1}{b}\left(\frac{de}{dt}\right)_\mathrm{col}\end{align*}

Changing the variable from t to ${\unicode{x03BC}}$ , we get:

\begin{align*} \frac{dp}{d{\unicode{x03BC}}}&\approx -\frac{1}{b}\frac{\sqrt{1+3{\unicode{x03BC}}^2}}{\sqrt{e-pb}}\frac{1600 k}{{\unicode{x03B2}}({\unicode{x03B3}}_0-1)}\frac{\exp{\{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2\}}}{\sqrt{{\unicode{x03C0}}}{\unicode{x03BC}}_\mathrm{H}}\\[4pt] &\approx -\frac{1600 k}{\sqrt{e-p}{\unicode{x03B2}}({\unicode{x03B3}}_0-1)\sqrt{{\unicode{x03C0}}}{\unicode{x03BC}}_\mathrm{H}}\exp{\{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2\}}\end{align*}

where we have set $b=1$ and $\sqrt{1+3{\unicode{x03BC}}^2}\approx 1$ for regions close to the apex. Also, under the assumptions:

(25) \begin{align} \dot{p}&=\dot{e} \\[-7pt] & \nonumber \end{align}

(26) \begin{align} \Rightarrow p&=p_0+\dot{e}dt\approx p_0+e-1 \\[-7pt] & \nonumber \end{align}

(27) \begin{align} \Rightarrow e-p& \approx 1-p_0 \\[-7pt] & \nonumber \end{align}

where $p_0$ is the initial value of p. Thus, we can write,

\begin{align*} p &= p_0-\frac{1600 k}{({\unicode{x03B3}}_0-1)\sqrt{1-p_0}}\int_0^{{\unicode{x03BC}}}\frac{\exp{\{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2\}}}{{\unicode{x03B2}}\sqrt{{\unicode{x03C0}}}{\unicode{x03BC}}_\mathrm{H}} d{\unicode{x03BC}}\end{align*}

Here ${\unicode{x03BC}} \gg {\unicode{x03BC}}_\mathrm{H}$ represents a value just outside the dense plasma layer. Now p will be greater than zero, if

\begin{align*} p_0&{\,\gt\,}\frac{1600 k}{({\unicode{x03B3}}_0-1)\sqrt{1-p_0}}\int_0^{{\unicode{x03BC}}}\frac{\exp{\{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2\}}}{{\unicode{x03B2}}\sqrt{{\unicode{x03C0}}}{\unicode{x03BC}}_\mathrm{H}} d{\unicode{x03BC}} \\ &\,\geq\, \frac{1600 k}{{\unicode{x03B2}}_0({\unicode{x03B3}}_0-1)\sqrt{1-p_0}}\int_0^{{\unicode{x03BC}}}\frac{\exp{\{-({\unicode{x03BC}}/{\unicode{x03BC}}_\mathrm{H})^2\}}}{\sqrt{{\unicode{x03C0}}}{\unicode{x03BC}}_\mathrm{H}} d{\unicode{x03BC}}\end{align*}

For ${\unicode{x03BC}} \gg {\unicode{x03BC}}_\mathrm{H}$ , the integral equals $1/2$ . Also, since we are considering the cases where the initial $p_0$ is already small, we can set $1-p_0\approx 1$ . Thus, we get:

(28) \begin{align} p_0&{\,\gt\,}\frac{800 k}{{\unicode{x03B2}}_0({\unicode{x03B3}}_0-1)} \nonumber \\ |\sin{\unicode{x03B1}}_\mathrm{a}|&{\,\gt\,} {\left( \frac{800 k}{{\unicode{x03B2}}_0({\unicode{x03B3}}_0-1)} \right)}^{1/2}\end{align}

For our numerical simulation, we have used $B_*=3.8$ kG, $R=2\, R_\odot$ and $W=0.5$ (parameters similar to CU Vir), which gives $|\sin{\unicode{x03B1}}_\mathrm{a}|{\gt} 0.043$ for $r_\mathrm{a}=10\, R$ . The minimum value of $\sin{\unicode{x03B1}}_\mathrm{a}$ in Figure 4 is 0.055.

Because the actual minimum value of pitch angle is larger than that considered in the analytical analysis, we also find that the value of $\langle pb^3\rangle_\mathrm{max}$ is smaller than that predicted by Eq. 23, which can also be seen from Figure 5. The consequence is that the value of $r_\mathrm{a}$ beyond which gyrocooling becomes significant is larger than that estimated from Eq. 24. For example, the numerical values can be reasonably approximated as $0.8\times(r_\mathrm{a}/R)^5$ , in which case, the minimum $r_\mathrm{a}$ will be $6.25\,R$ (instead of 5 R, ${\S}$ 3.3). Note, however, that beyond this radius, gyrocooling will dominate over collisional cooling only for the nearly field aligned electrons. Thus, for an isotropic distribution, collisional cooling will still dominate unless we consider very high values of the Lorentz factor. The effect of increasing the initial electron energy ( ${\unicode{x03B3}}_0$ ) is demonstrated in Figure 6.

Figure 5.

Numerically computed $\langle pb^3\rangle_\mathrm{max}$ as a function of $r_\mathrm{a}/R$ .

Figure 6.

Time evolution of energy lost by radiation and collision for an electron with initial pitch angle of $30^\circ$ , injected at the magnetic equator at an apex radius of $10\,R$ for three different values of initial Lorentz factor ${\unicode{x03B3}}_0$ . Note that the times here refer to the dimensionaless times (scaled by the respective $t_\mathrm{a}, \S$ 3.1).

Finally, for the case of a gyrotropic distribution of electrons injected at an apex radius of $10\,R$ , Figure 7 shows that, except for the ultra-relativistic case shown in green, most of the energy deposition occurs close to the magnetic poles.

Figure 7.

Distribution of energy lost by radiation over co-latitude for a gyrotropic electron distributions with three different values of initial Lorentz factor. The apex radius for all three cases is 10 R.

4.2 Spatial distribution of emission from Gaussian deposition

To complete comparisons with Paper I, let us next consider a model in which the initial energy deposition has a gaussian spread in both field co-latitude ${\unicode{x03BC}}=\sqrt{1-r/r_\mathrm{a}}$ and apex radius $r_\mathrm{a} = r/(1-{\unicode{x03BC}}^2)$ , centred on a peak radius $r_p$ with radial dispersion $\sigma_{r}$ (cf. Equation 16 of Paper I),

(29) \begin{equation}e_{{\unicode{x03BC}},r_\mathrm{a}} =C_{{\unicode{x03BC}},r_\mathrm{a}}\exp\left[ -\frac{{(r_\mathrm{a}-r_p)}^2}{2\sigma_{r}^2}\right] \exp\left(-\frac{{\unicode{x03BC}}^2}{2\sigma_{\unicode{x03BC}}^2}\right)\, ,\end{equation}

where $C_{{\unicode{x03BC}},r_\mathrm{a}}$ is a normalisation factor such that

\begin{align*} \int_{R}^{\infty} \int_{-{\unicode{x03BC}}_\ast}^{+{\unicode{x03BC}}_\ast} e_{{\unicode{x03BC}},r_\mathrm{a}}d{\unicode{x03BC}}\,dr_\mathrm{a}&=1 \, ,\end{align*}

with ${\unicode{x03BC}}_\ast \equiv \sqrt{1-R/r_\mathrm{a}}$ .

Figure 8 shows the distribution of radiative energy loss for mildly relativistic electrons ( ${\unicode{x03B3}}_0=1.5$ ) with $\sigma_{\unicode{x03BC}}=0.1$ and $\sigma_r=0.5$ deposited around three different values of the apex radius: $3\,R$ (left), $5\,R$ (middle), and $10\,R$ (right). The top panel shows the input energy distribution, the middle panel shows the resulting radiative cooling taking into account of collisional cooling (this work) and the bottom panel shows the corresponding distribution in the absence of thermal plasma causing collisional cooling (Paper I). Note that the net emission is scaled by the respective maxima, which differ significantly, since, the total radiative energy lost is significantly smaller (by $\approx 2$ orders of magnitude) when collision is considered for this mildly relativistic case (e.g. see the top panel of Figure 6). In terms of spatial distribution of emission, the key difference is that, in the absence of collisional cooling, radiative cooling predominantly occurs close to the magnetic equator, but when collisional cooling is taken into account, the high-density plasma at the magnetic equator suppresses radiative loss, especially for smaller values of mean $r_\mathrm{a}$ , and the electrons lose their energy via radiation predominantly close to the magnetic poles only. Also, for the pure radiative cooling case (Paper I), the spatial distribution remains qualitatively the same for the different values of the mean deposition radius. However, in the presence of collisional cooling, the radiative energy loss close to the magnetic equator gradually increases with increasing apex radius. This is because the thermal plasma density falls sharply away from the Kepler radius (taken to be $1.59\,R$ , Figure 1).

Figure 8.

Top: Spatial distribution of input energy (Equation 29) for three different values of mean $r_\mathrm{a}$ . The mean Lorentz factor and the parameter $\sigma_{\unicode{x03BC}}$ (see ${\S}$ 4.2) are fixed at $1.5$ and $0.1$ , respectively. The Kepler radius is $1.59\,R$ . Middle: The corresponding distribution of energy lost via radiation. The white lines represent contours of magnetic field strength B, spaced logarithmically by $-0.1$ dex from the stellar surface value at the magnetic equator. For comparison, the corresponding distributions of radiative energy lost in the absence of collisional cooling are also shown in the bottom panels. Note that, since we are interested in the distribution only, we have normalised the energy values by dividing them by the respective maxima (which vary significantly between the cases with and without collisional cooling, but are comparable for the different values of mean $r_\mathrm{a}$ ).

Finally, we examine the effect of varying the electron energies, as well as the distribution of the initial energy deposition. For cases with $r_\mathrm{a} = 10\,R$ and $\sigma_r=0.5\,R$ , the top row of Figure 9 shows the assumed energy deposition $e_{{\unicode{x03BC}},r_\mathrm{a}}$ for cases with $\sigma_{\unicode{x03BC}} = 0.1$ (left column) and $\sigma_{\unicode{x03BC}}=0.5$ (right column). The rows below show the resulting net emission (scaled here by their respective maxima) for cases with moderately relativistic ( ${\unicode{x03B3}}_0=1.5$ ; middle row) and highly relativistic ( ${\unicode{x03B3}}_0=11$ ; bottom row) injection energies.

Figure 9.

Top: Spatial distribution of input energy (Equation 29) for two values of $\sigma_{\unicode{x03BC}}$ . Middle and bottom: The corresponding distribution of energy lost via radiation for a mean Lorentz factor of 1.5 (middle row) and 11 (bottom row). Note that, since we are interested in the distribution only, we have normalised the values by dividing them by the respective maximum values. The maxima of the output energy distributions are higher for the case of the higher Lorentz factor by approximately an order of magnitude. For a fixed value of the Lorentz factor, more energy is lost via radiation when $\sigma_{\unicode{x03BC}}$ is higher. The white lines again represent contours of magnetic field strength B, spaced logarithmically by $-0.1$ dex from the stellar surface value at the magnetic equator.

The results show that, rendered and scaled in this way, the computed spatial distribution of emission is surprisingly independent of the assumed parameter for deposition width $\sigma_{\unicode{x03BC}}$ and even deposition energy ${\unicode{x03B3}}_0$ . In all cases, the dominant emission occurs near the loop footpoint, where the field strength is strongest, leading to strong gyro-emission, with also minimal collisional cooling, due to the large distance from the dense CM region around the loop top. The more highly relativistic case does also show a relatively greater level of such scaled emission near the dense loop top, owing to the reduced collisional cross section at for higher electron energies.

4.3 Emission distribution as a function of magnetic field strength

The great distance of massive stars makes it difficult to resolve directly the spatial distribution of radio emission from their magnetospheres. But because the electron gyro-frequency scales directly with magnetic field strength, characterising the emission distribution with field strength provides an initial basis for deriving observable gyro-emission spectra.

Let us thus define a differential emission luminosity $dL \equiv \epsilon_{{\unicode{x03BC}},r_\mathrm{a}}d{\unicode{x03BC}} dr_\mathrm{a}$ associated with a spatial differential $dr_\mathrm{a}$ of magnetic loops around a central apex $r_\mathrm{a}$ , and a co-latitude differential $d{\unicode{x03BC}}$ along the loops. For a dipole field, we can readily identify the associated differential in field strength db, scaled by the loop strength at the central loop apex $r_\mathrm{a}$ . For the same set of parameters shown in Figure 9, Figure 10 now plots the emission distributions $dL/db$ versus the scaled field b. The dashed curves show the corresponding input distributions for $\sigma_{\unicode{x03BC}} = 0.1$ (red) and $\sigma_{\unicode{x03BC}} = 0.5$ (blue). The other associated curves show results for the cases ${\unicode{x03B3}}_0=1.5$ (solid) and ${\unicode{x03B3}}_0=11$ (dot-dashed).

Figure 10.

Distribution of radiated energy as a function of magnetic field strength corresponding to the cases shown in Figure 9. The dashed lines represent the input energy distribution. The vertical line on the right represents the polar magnetic field strength.

Reflecting the much reduced collisional loss of energy for higher energy electrons, the curves for the strongly relativistic case ${\unicode{x03B3}}_0=11$ (dot-dashed curves) are about two orders of magnitude higher than those for the mildly relativist case ${\unicode{x03B3}}_0=1.5$ (solid curves).

But the similarly broad distribution of the emission for the red versus blue curves, for spatially narrow ( $\sigma_{\unicode{x03BC}}=0.1$ ) versus broad ( $\sigma_{\unicode{x03BC}}=0.5$ ) energy deposition along the loops, again demonstrates the surprising insensitivity of the resulting emission to assumptions about this spatial distribution of energy deposition.

Incoherent radio emission from magnetic hot stars is interpreted as gyrosynchrotron emission (e.g. Drake et al., Reference Drake, Abbott and Bastian1987) for which the emission occurs at the harmonics of the electron gyrofrequency that is proportional to the local magnetic field strength. Because of the complexity of the emission spectrum (for a fixed magnetic field) and the requirement to perform detailed radiative transfer modelling, quantitative derivation of observed spectra is beyond the scope of this paper, so will be deferred future work. But the overall results here suggest that such spectra should generally have a broad distribution in frequency.