CROSS-FIELD TRANSPORT OF SOLAR ENERGETIC PARTICLES IN A LARGE-SCALE FLUCTUATING MAGNETIC FIELD

In this work the IMF model originally introduced by Giacalone (1999), and further developed by Giacalone & Jokipii (2004), is used. Large-scale magnetic field fluctuations are generated by supergranular motion of the footpoints at the solar surface, with characteristic time T_c and speed V_g.

The resulting fluctuating IMF is given by

$$\begin{equation} B_{r}=\frac{B_{0}r_{0}^{2}}{r^{2}} \end{equation} \tag{ 1 }$$

$$\begin{equation} B_{\theta }=\frac{B_{0}r_{0}^{2}}{r^{2}}\frac{V_{\theta }(\theta,\phi,t_{0})}{V_{w}} \end{equation} \tag{ 2 }$$

$$\begin{equation} B_{\phi }=\frac{B_{0}r_{0}^{2}}{r^{2}}\frac{V_{\phi }(\theta,\phi,t_{0}) - r\Omega _{0}\sin {\theta }}{V_{w}} \end{equation} \tag{ 3 }$$

$$\begin{equation} t_{0}=t - \frac{r-r_{0}}{V_{w}}, \end{equation} \tag{ 4 }$$

where V_θ(θ, ϕ, t₀) and V_ϕ(θ, ϕ, t₀) are velocities associated with footpoint motion, V_w is the solar wind speed, Ω₀ is the solar rotation rate, B₀ is the magnetic field magnitude at position r₀, r is radial distance from the Sun, θ is colatitude, ϕ is longitude, t is time, and t₀ represents the time at which a fluctuation at a given radius was produced on the solar surface.

The velocity of the footpoint motions is modeled by a stream function ψ(θ, ϕ, t₀), satisfying

$$\begin{equation} V_{\theta } = \frac{1}{\sin \theta } \frac{\partial \psi }{\partial \phi } \end{equation} \tag{ 5 }$$

$$\begin{equation} V_{\phi }= - \frac{ \partial \psi }{\partial \theta }. \end{equation} \tag{ 6 }$$

The stream function is postulated to be

$$\begin{equation} \psi (\theta, \phi, t_{0}) = \sum _{n=1}^{N} \sum _{m=-n}^{n} a_{n}^{m} \exp \big(i\omega _{n}^{m} t_{0}+ i \beta _{n}^{m}\big) Y_{n}^{m}(\theta, \phi), \end{equation} \tag{ 7 }$$

where

$$\begin{equation} a_{n}^{m} = 6 V_{g}G_{n}^{m}\left(\sum _{n=1}^{N} \sum _{-n}^{n} G_{n}^{m} \right)^{-1} \end{equation} \tag{ 8 }$$

$$\begin{equation} G_{n}^{m} = \left(\left[1 + \left(\frac{n}{N_{c}}\right)^{10/3} \right] \big[1+ \big(\omega _{n}^{m} T_{c}\big)^{5/3} \big]\right)^{(-1/2)}, \end{equation} \tag{ 9 }$$

with N_c being the characteristic number of supergranular cells, defined as N_c = πr₀/V_gT_c, where r₀ is taken to be one solar radius, r_s. ω^m_n is the characteristic frequency of each mode and is taken as ω^m_n = 2πn/150T_c. The remaining parameters in the stream function are random phase angles, β^m_n, drawn from a uniform distribution, and Y^m_n(θ, ϕ), the spherical harmonic function. The values of various parameters used in these simulations are presented in Table 1.

The electric field is given by $\mathbf {E}=-\mathbf {V}_w \times \mathbf {B}/c$ since the plasma is quasineutral.

In Section 3 the Cartesian system used is defined as follows: the z-axis is the rotational axis of the Sun, while the x and y coordinates lie in the heliospheric equatorial plane with the x-axis corresponding to a longitude of ϕ = 0°.

In the simulations presented here all particles are run through a realization of this IMF model. We set t = 0, giving t₀ = −(r − r₀)/V_w, as the timescale of variation of the large-scale turbulence is large compared to the particle propagation times considered.

The simulations presented in this paper use a full-orbit test-particle method, which involves numerically integrating the equations of motion of one particle at a time, repeating the process for a variety of initial conditions to assess the propagation of a population.

The code is fully relativistic and is a modified version of one previously used to study particle acceleration during magnetic reconnection (Dalla & Browning 2005). The numerical technique used for integration is a Bulirsch–Stoer method (Press et al. 1993). The Bulirsch–Stoer routine is driven by a function that adaptively controls the stepsize of integration so as to produce results to a prescribed accuracy with minimum computational effort. The solution's accuracy is determined by a user-specified tolerance value, which sets the fractional error allowed when integrating from t_n to t_{n + 1}.

Extensive testing was carried out to find an acceptable tolerance level for the solver; from these tests it was concluded that particle trajectories in the fluctuating IMF outlined in Section 2.1 converged when the tolerance was less than 10⁻¹¹, hence the simulations were carried out at a tolerance of 10⁻¹².

The presence of small-scale turbulence in the solar wind affects the trajectories of particles. Rather than providing a full description of the interaction of particles with turbulence in these simulations, its effect is described as a series of scattering events, each causing an instantaneous change in a particle's velocity. This "ad hoc" scattering has been used by many others in the literature (e.g., Pei et al. 2006).

A mean free path, λ, is fixed and the mean scattering time t_scat is obtained by t_scat = λ/v, where v is the particle's velocity in a frame of reference which is stationary with respect to the solar wind. A series of scattering events are then numerically generated for each particle, with mean time t_scat, in such a way as to form a Poisson distribution. A scattering event involves first transforming the particle's velocity vector into the solar wind frame, then the particle's velocity vector is reassigned by drawing a velocity vector at random from an isotropic distribution, changing both the particle's pitch and phase angles. The particle's new velocity is then converted back into the inertial frame of reference, and the integration of the particle's equations of motion continues.

Initial simulations were performed without ad hoc scattering in order to investigate the cross-field transport of SEPs subjected only to large-scale fluctuations, and contrast their propagation in such a field with that in a Parker spiral field.

Figure 1 shows the spatial distributions of 10 MeV protons 30 and 60 minutes after injection. The left panel shows an x–y projection (along the heliospheric equatorial plane) and the right panel an x–z projection. Red dots are particles in the fluctuating IMF configuration and blue dots those in the Parker spiral (long dashed line). Field lines associated with the fluctuating IMF are indicated by short dashed lines. It can be seen that the longitudinal and latitudinal spread of SEPs is much greater in the fluctuating IMF model than in the Parker spiral.

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The smallest scale of the fluctuations is 2πr_s/50 = 87 Mm, this is large compared to the gyroradii of SEPs propagating close to 1 AU, hence the cross-field transport observed in the simulation results is simply due to the field-line meandering and drifts. The particles propagating in the fluctuating IMF model focus at the same rate as those in the Parker spiral, as can be seen in Figure 2, showing the evolution of the mean pitch angle of the populations with time.

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To investigate any dependence of SEP propagation characteristics in the fluctuating IMF on particle energy, proton energies were varied from 500 KeV to 1 GeV. Results showed that SEP populations followed the same path through a single realization of the fluctuating IMF regardless of energy, and hence cross-field transport in this configuration is due solely to field-line meandering.

The scatter-free simulation described in Section 3.1 showed that in such conditions cross-field transport in the fluctuating IMF is due solely to field-line meandering. In this section the effect of ad hoc pitch-angle scattering, incorporated into the model as outlined in Section 2.2, is studied.

Figure 3 shows populations of 100 MeV protons propagating in the fluctuating IMF. Each column shows a particle population at three different times for a given propagation condition. The initial conditions are the same in the three cases. The central and right columns are affected by pitch-angle scattering, with mean free paths of 2.0 AU and 0.3 AU, respectively, and the left column shows scatter-free propagation. In the scattered populations there are a number of particles close to their injection point at late times, in addition to a number of outlying particles that have strayed a large distance across the field compared to the scatter-free case.

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To quantify the cross-field transport within the simulations, the parallel and perpendicular displacements of the particles' location from the Parker spiral field line originating from their initial positions were calculated, for runs of 10,000 protons of 50 MeV energy. The first step consists of deriving the so-called target point, i.e., the location on the Parker spiral field line starting at the particle's initial position, located nearest to the actual final position (Tautz et al. 2011; see Appendix A.2 and Figure 11). If $\mathbf {r}$ indicates the location of a particle at a given time and $\mathbf {r}_t$ is the location of the target point, the displacement of the particle with respect to its target location is $\Delta \mathbf {s} = \mathbf {r}-\mathbf {r}_t$. Next, a local Parker spiral coordinate system, centered in the target location, is introduced, with an axis pointing outward along the spiral (the "parallel" direction), a second axis in the direction of $\mathbf {e}_{\theta ^{\prime }}=-\mathbf {e}_{\theta }$ with $\mathbf {e}_{\theta }$ the standard spherical coordinate system unit vector, and a third axis $\mathbf {e}_{\phi ^{\prime }}$ completing the orthogonal system (see Appendix A.1). The components of $\Delta \mathbf {s}_{\perp }$ in this system are $\Delta s_{\phi ^{\prime }}$ and $\Delta s_{\theta ^{\prime }}$. The distance traveled along the Parker spiral between the initial location and the target point is indicated as l_||.

The distributions of displacements for the three propagation conditions of Figure 3 are examined first after a fixed time from injection (when the average radial distance from the Sun in the population will be different in the three cases) and second at the times when the average distance from the Sun is the same for the three populations and set to 1 AU.

Figure 4 shows the distributions of the displacement's components for the scatter-free, λ = 2 AU, and λ = 0.3 AU conditions, for 50 MeV protons, 3 hr after injection. The top panel, displaying the distribution of l_||, shows that the population with λ = 0.3 AU has traveled the shortest distance along the field, followed by the λ = 2.0 AU case, and that the unscattered case has traveled the farthest, as one would expect. This difference in average field-parallel displacement affects the magnitude of perpendicular displacement, as the spread of the field lines in the perpendicular direction increases with distance along the field. This leads to the distributions in the middle and bottom panels of Figure 4, where the scatter-free population has wide distributions since all these particles have been able to travel a greater distance along the field.

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In an attempt to remove the effect of field-line expansion on Δs_⊥ plots, $\Delta s_{\theta ^{\prime }}$ and $\Delta s_{\phi ^{\prime }}$ values were normalized by the scale lengths of a Parker spiral flux tube at the corresponding distance from the Sun, as defined in Appendix A.3. Figure 5 shows the normalized distributions. In the θ' component, the unscattered population still has the widest distribution, while in the ϕ' component the widest distribution is in the λ = 2 AU case. It should be noted that the normalization introduced only corrects for the expansion of the Parker spiral field lines: the superimposed large-scale turbulence may be characterized by a different kind of expansion which is not accounted for here.

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Figure 6 compares $\Delta \mathbf {s}$ distributions at the time when each population has an average radial distance from the Sun of 1 AU. For the scatter-free, λ = 2 AU, and λ = 0.3 AU cases, this is 29, 33, and 91 minutes after injection, respectively. Particles are grouped in this way in an attempt to find the most appropriate time to compare populations with different scattering conditions. Populations with little scattering will reach greater radial distances in a given time than populations with strong scattering, and at larger radial distances the field-line expansion due to magnetic field fluctuations will be larger, artificially increasing the Δs_⊥ value assigned to a particle.

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Grouping particles this way results in the λ = 0.3 AU population having the broadest distribution in both perpendicular directions; however, that population has traveled the farthest along the field (see the l_|| distribution in the top panel of Figure 6) and this produces the pronounced "wings" in the middle and bottom panels of Figure 6. This effect is similar to that seen in Figure 4 and is due to field-line expansion.

Figure 7 shows the Δs_⊥ components normalized by the Parker spiral flux-tube scale lengths. The λ = 0.3 AU population has the broadest distribution in both the ϕ' and θ' populations. The distributions for the scattered populations are broader than for the scatter-free one.

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An alternative way to remove the effect of field-line expansion on Δs_⊥ values involves normalizing Δs_⊥ by l_|| for each particle in the population at a fixed time 3 hr after injection, giving the cross-field displacement per unit distance along the field. The results of this normalization are presented in Figure 8, which shows that the population with λ = 2.0 AU has the broadest distribution of $\Delta s_{\phi ^{\prime }}/l_{||}$ and has many outliers in the plot of $\Delta s_{\theta ^{\prime }}/l_{||}$. This result ties in with Figure 3, where the λ = 2.0 AU population qualitatively appears to have outliers that have strayed the largest distance across the field.

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Particle transport can be characterized by diffusion coefficients, under the assumption that it is of diffusive character. The values of diffusion coefficients κ_||, $\kappa _{\phi ^{\prime }}$, and $\kappa _{\theta ^{\prime }}$ are calculated from the equation

$$\begin{equation} \kappa _{||, \phi ^{\prime }, \theta ^{\prime } } = \frac{ \langle (\Delta s_{||, \phi ^{\prime }, \theta ^{\prime }})^{2}\rangle }{2t}. \end{equation} \tag{ 10 }$$

In the expanding magnetic field the propagation parallel to the field is a combination of diffusion and focusing-driven streaming. This can be represented as the advection of a diffusively spreading particle population, in the strong scattering case (Earl 1976). In line with this concept, the parallel displacement Δs_|| is defined relative to the mean position of the particle distribution as

$$\begin{equation} \Delta s_{||}=l_{||}-\langle l_{||} \rangle, \end{equation} \tag{ 11 }$$

where the average is taken over the particle population.

This definition of diffusion coefficient includes the contribution of displacements in the particle's trajectory due to large-scale fluctuations in the magnetic field, and as such may not be suitable to directly compare with the diffusion coefficient used in the Parker transport equation.

Figure 9 shows the diffusion coefficients calculated from Equation (10) for the three propagation conditions as a function of time. For diffusive behavior it is expected that the diffusion coefficient would reach a constant value. The top panel shows κ_|| for the three populations, although the κ_|| values of the unscattered population are included only for reference, as this case is not diffusive since the particles are rapidly focused and travel as a beam. The increase in κ_|| with time for the scatter-free case is due to field-line meandering.

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The slopes of plots shown in Figure 9 can be characterized by the relative differences between values at the midpoint and final time point in the simulations, as given in Table 2. In the λ = 2.0 AU case Δκ_||/κ_|| = 0.28, whereas for λ = 0.3 AU, Δκ_||/κ_|| = −0.16, so neither curve is constant. For the perpendicular components of the diffusion coefficient an approximately constant value is reached only in the λ = 0.3 AU case.

Table 2. The Difference between the Midpoint and Endpoint Values of the Diffusion Coefficients, Normalized by the Final Value, for Different Propagation Conditions

	λ = 2.0 AU	λ = 0.3 AU
Δκ||/κ||	0.284	−0.163
$\Delta \kappa _{\phi ^{\prime }}/\kappa _{\phi ^{\prime }}$	−0.506	−0.080
$\Delta \kappa _{\theta ^{\prime }}/\kappa _{\theta ^{\prime }}$	0.454	−0.083

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The ratios of various diffusion coefficients from each population are presented in Table 3. It can be seen that in both scattering conditions the value of $\kappa _{\theta ^{\prime }}/\kappa _{||}$ is about 0.01, while $\kappa _{\phi ^{\prime }}/\kappa _{||}$ is approximately 0.08 for λ = 0.3 AU and 0.04 for λ = 2.0 AU. The ratio of the two perpendicular components of the diffusion coefficient is not one, as $\kappa _{\theta ^{\prime }}/\kappa _{\phi ^{\prime }}\sim 0.2$. This difference may be due to the fact that electric field drift aids diffusion in the $\mathbf {e_{\phi ^{\prime }}}$ direction (Burns & Halpern 1968) or may be a property of the magnetic field meandering.

Table 3. Diffusion Coefficient Ratios in Different Propagation Conditions

	λ = 2.0 AU	λ = 0.3 AU
$\kappa _{\theta ^{\prime }}/\kappa _{||}$	0.011	0.0139
$\kappa _{\phi ^{\prime }}/\kappa _{||}$	0.043	0.083
$\kappa _{\theta ^{\prime }}/\kappa _{\phi ^{\prime }}$	0.259	0.169

Note. The values of each diffusion coefficient at the final time were used to calculate these ratios.

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The increased value of $\kappa _{\phi ^{\prime }}$ compared with $\kappa _{\theta ^{\prime }}$ found in these simulations differs from the conclusion of Jokipii et al. (1995), based on analysis of magnetic field fluctuations in Ulysses data. However, this difference may be explained by the fact that the Ulysses measurements were made at a much larger radial distance from the Sun.

The diffusion coefficients were also calculated by means of values of $\Delta s_{\theta ^{\prime }}$ and $\Delta s_{\phi ^{\prime } }$ normalized by the width of a Parker spiral flux tube, as outlined in Appendix A.3; the plots of the resulting diffusion coefficients are shown in Figure 10. The observed decrease in the diffusion coefficients with time can be understood by noting that in a simple radially expanding field without scattering, field-line expansion would generate a dependence like t⁻¹ for the normalized coefficients.

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The simulations were also run with two new sets of random phase angles (β^m_n) in the stream function defined in Equation (7), and it was found that this does not produce a significant qualitative change in the results. Plots corresponding to Figures 6–10 for the two additional realizations display the same trends as the figures shown in this paper. Actual values of the diffusion coefficients are broadly consistent. However, in one of the additional realizations the ratio κ'_θ/κ_ϕ' at the final time was equal to 1.6 (while in the other additional realization this value was 0.1). This appears to be related to a large deviation of the magnetic field lines, from the Parker spiral, in the θ' direction.

If the value of N is increased from N = 50 to N = 75 it is found that the particles' trajectories are very similar to those in the N = 50 case, with some small differences due to the fact that there is less power in the low N modes.

A new coordinate system with origin at a point (x_t, y_t, z_t) in space and one axis coinciding with the Parker spiral magnetic field direction at the point is introduced. The $\mathbf {e}_{l}$ axis points outward along the local direction of the Parker spiral magnetic field, a second axis is in the direction of $\mathbf {e}_{\theta ^{\prime }}=-\mathbf {e}_{\theta }$, with $\mathbf {e}_{\theta }$ the standard spherical coordinate system unit vector, and a third axis $\mathbf {e}_{\phi ^{\prime }}$ completes the orthogonal right-handed system. The latter axis does not coincide with the standard spherical coordinate system unit vector $\mathbf {e}_{\phi }$.

The angle between the radial direction and the direction of the magnetic field at the point is indicated by β and is given by

$$\begin{equation} \beta = \tan ^{-1} \left(\frac{-B_{\phi }}{B_r} \right) =\tan ^{-1}\left(\frac{r\,\Omega _0 \, \sin {\theta }}{V_w}\right), \end{equation} \tag{ A1 }$$

where the terms B_r and B_ϕ are the radial and longitudinal components of the Parker spiral magnetic field and can be found by setting V_θ = V_ϕ = 0 in Equations (1) and (3).

If a vector has components (v_x, v_y, v_z) in the Cartesian coordinate system introduced in Section 1, its components $(v_l, v_{\phi ^{\prime }}, v_{\theta ^{\prime }})$ in the new Parker spiral coordinate system are given by

$$\begin{eqnarray} &&\left[\begin{array}{@{}c@{}}v_l \\ v_{\phi ^{\prime }} \\ v_{\theta ^{\prime }}\end{array} \right] = \left[\begin{array}{@{}ccc@{}}\sin {\theta } \cos {\phi } \cos {\beta } +\sin {\beta } \sin {\phi } & \sin {\theta } \sin {\phi } \cos {\beta } -\sin {\beta } \cos {\phi } & \cos {\beta } \cos {\theta } \\ -\cos {\beta } \sin {\phi } +\sin {\theta } \sin {\beta } \cos {\phi } &\cos {\beta } \cos {\phi } +\sin {\theta } \sin {\beta } \sin {\phi } &\cos {\theta } \sin {\beta } \\ -\cos {\theta } \cos {\phi } &- \cos {\theta } \sin {\phi } &\sin {\theta }\end{array} \right] \left[\begin{array}{@{}c@{}}v_x \\ v_y \\ v_z\end{array} \right]. \end{eqnarray} \tag{ A2 }$$

The target point is the location on the Parker spiral field line starting at the particle's initial position with the shortest distance to the particle's actual final position (Tautz et al. 2011; see Figure 11).

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The target point is determined by means of an iterative procedure that differs from the one used by Tautz et al. (2011). As an initial guess for the distance from the Sun of the target location, r_t, the value of the actual radial distance for the particle is used. For a given r_t, the target longitude can then be found as ϕ(r_t) = ϕ₀ − (Ω₀(r_t − r₀)/V_w) given the particle's initial colatitude θ₀, longitude ϕ₀, and initial radius r₀. The target colatitude is θ₀. If the Cartesian components of this initial target location are indicated as x_t, y_t, and z_t, the Cartesian components of the $\Delta \mathbf {s}$ vector with respect to this target can be obtained. Using the transform defined in Equation (A.1), this can be converted into its components in the local Parker spiral coordinate system defined in Appendix A.1.

The optimal target location will have a zero component along the $\mathbf {e}_l$ axis and can therefore be found by solving the equation

$$\begin{equation} (\sin {\theta } \cos {\phi } \cos {\beta } +\sin {\beta } \sin {\phi })[x_{e} - x_{t}] +(\sin {\theta } \sin {\phi } \cos {\beta } -\sin {\beta } \cos {\phi })[y_{e} - y_{t}] + (\cos {\beta } \cos {\theta })[z_{e} - z_{t}]=0, \end{equation} \tag{ A3 }$$

where (x_e, y_e, z_e) is the particle's actual location. The equation is solved by means of Brent's method (Press et al. 1993). This ensures that the particle's displacement with respect to the optimal target location is perpendicular to the Parker spiral.

Care must be taken when specifying the bracketing region to ensure that only one root of Equation (A3) lies within the region specified, and that this root corresponds to the closest target point to the particle end point.

The cross-section of a Parker spiral flux tube increases with radial distance from the Sun. In this section, the scale lengths characterizing the cross-section in the ϕ and θ directions are determined, indicated as L_ϕ and L_θ, respectively. These quantities are used to normalize perpendicular particle displacements so as to remove the effect of Parker spiral field-line expansion.

The borders of the particle injection region, located at a distance r₁ from the Sun and with angular extent Δϕ and Δθ in longitude and latitude, respectively, define a Parker spiral flux tube with cross-section approximated by σ₁ = r²₁ Δϕ Δθ. If the cross-section of the same flux tube at a distance r from the Sun is indicated as σ, the conservation of magnetic flux requires that σ = σ₁B₁/B, where B₁ is the magnetic field magnitude at r₁ and B its value at r.

In the Parker spiral,

$$\begin{equation} B=\frac{B_{0}r_{0}^2}{r^{2}}\sqrt{1+\frac{r^2}{a^2}}, \end{equation} \tag{ A4 }$$

where a = v_sw/(Ω₀sin θ). Hence,

$$\begin{equation} \sigma =r^2 \, \Delta \phi \,\Delta \theta \, \sqrt{\frac{r_1^2+a^2}{r^2+a^2}}, \end{equation} \tag{ A5 }$$

giving the following expressions for the cross-sectional scale lengths:

$$\begin{eqnarray} L_{\phi } &=& r \, \Delta \phi \, \sqrt{\frac{r_1^2+a^2}{r^2+a^2}} \end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray} L_{\theta }&=& r \, \Delta \theta. \qquad\ \ \ \ \ \end{eqnarray} \tag{ A7 }$$

L_ϕ and L_θ are calculated using r₁ = 21.5 r_s and Δϕ = Δθ = π/60.