Trajectory Determination for Coronal Ejecta Observed by WISPR/Parker Solar Probe

Abstract

The Wide-field Imager for Solar Probe (WISPR) onboard the Parker Solar Probe (PSP), observing in white light, has a fixed angular field of view, extending from 13.5^∘ to 108^∘ from the Sun and approximately 50^∘ in the transverse direction. Because of the highly elliptical orbit of PSP, the physical extent of the imaged coronal region varies directly as the distance from the Sun, requiring new techniques for analysis of the motions of observed density features. Here, we present a technique for determining the 3D trajectory of CMEs and other coronal ejecta moving radially at a constant velocity by first tracking the motion in a sequence of images and then applying a curve-fitting procedure to determine the trajectory parameters (distance vs. time, velocity, longitude, and latitude). To validate the technique, we have determined the trajectory of two CMEs observed by WISPR that were also observed by another white-light imager, either the Solar and Heliophysics Observatory (SOHO) / Large Angle and Spectrometric COronagraph (LASCO)-C3 or the Solar Terrestrial Relations Observatory (STEREO)-A/HI1. The second viewpoint was used to verify the trajectory results from this new technique and help determine its uncertainty.

Appendices

Appendix A: Transformation of Coordinates

We transform a feature’s position in a heliocentric frame (the $x$–$y$–$z$-frame), in which the $x$–$y$-plane is the solar equatorial plane, to its position in a heliocentric frame (the $x'$–$y'$–$z'$-frame), in which the $x'$–$y'$-plane is PSP’s orbit plane and the $z'$-axis is the direction of the orbital angular momentum (Figure 2). For the transformation, we define the $x$-axis in the $x$–$y$–$z$-frame to be the ascending node of PSP’s orbit relative to the solar equatorial plane, and the $x'$-axis pointing from the Sun to PSP. The inclination of PSP’s orbit $\epsilon$ is the amount of rotation of PSP’s orbital plane about the $x$-axis. The spacecraft’s motion in its own orbital plane can be described by its distance to the Sun [$r_{1}$] and the angle [$\phi _{1}$] measured from the $x$-axis. In other words, $\phi _{1}$ is the amount of rotation about the $z'$-axis. The feature’s position in the $x$–$y$–$z$-frame is denoted $[x_{2}, y_{2}, z_{2}] \equiv [r_{2}\cos \delta _{2}\cos \phi _{2}, r_{2} \cos \delta _{2}\sin \phi _{2}, r_{2}\sin \delta _{2}]$, where $r_{2}$ is the distance to the Sun, $\phi _{2}$ is the angle in the $x$–$y$-plane, measured from the $x$-axis, or the ascending node of PSP’s orbit, and $\delta _{2}$ is the angle with the $x$–$y$-plane. Note that $\phi _{2}$ in this frame is offset from the HCI longitude by a constant, which is the HCI longitude of the ascending node of PSP’s orbit relative to the solar equatorial plane, and this constant is determined from the ephemeris. The feature’s position in the $x'$–$y'$–$z'$-frame is denoted $[x'_{2}, y'_{2}, z'_{2}] \equiv [r_{2}\cos \delta '_{2}\cos \phi '_{2}, r_{2}\cos \delta '_{2}\sin \phi '_{2}, r_{2}\sin \delta '_{2}]$ (see Figure 2), and the transformation from $(x_{2}, y_{2}, z_{2})$ to $(x'_{2}, y'_{2}, z'_{2})$ is given by

$$\begin{aligned} \left [ \textstyle\begin{array}{c} x'_{2} \\ y'_{2} \\ z'_{2} \end{array}\displaystyle \right ] = \begin{bmatrix} \cos \phi _{1} & \sin \phi _{1} & 0 \\ -\sin \phi _{1} & \cos \phi _{1} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \epsilon & \sin \epsilon \\ 0 & -\sin \epsilon & \cos \epsilon \end{bmatrix} \left [ \textstyle\begin{array}{c} x_{2} \\ y_{2} \\ z_{2} \end{array}\displaystyle \right ] & \\ = \left [ \textstyle\begin{array}{c} r_{2}(\cos \delta _{2}\cos \phi _{2}\cos \phi _{1} + \cos \delta _{2} \sin \phi _{2}\sin \phi _{1}\cos \epsilon + \sin \delta _{2}\sin \phi _{1}\sin \epsilon ) \\ r_{2}(-\cos \delta _{2}\cos \phi _{2}\sin \phi _{1} + \cos \delta _{2} \sin \phi _{2}\cos \phi _{1}\cos \epsilon + \sin \delta _{2}\cos \phi _{1}\sin \epsilon ) \\ r_{2}(-\cos \delta _{2}\sin \phi _{2}\sin \epsilon + \sin \delta _{2} \cos \epsilon ) \end{array}\displaystyle \right ] . \end{aligned}$$

(5)

We can find the relationship between the $\gamma$–$\beta$-coordinates and the feature’s position $[r_{2}, \phi _{2}, \delta _{2}]$ in the $x$–$y$–$z$-frame,

$$\begin{array}{rll}\frac{tan\beta }{sin\gamma }& =\frac{z_2^{\prime }}{y_2^{\prime }}& \\ & =\frac{-cos{\delta }_{2}sin{\varphi }_{2}sinϵ+sin{\delta }_{2}cosϵ}{-cos{\delta }_{2}cos{\varphi }_{2}sin{\varphi }_1+cos{\delta }_{2}sin{\varphi }_{2}cos{\varphi }_{1}cosϵ+sin{\delta }_{2}cos{\varphi }_{1}sinϵ}& \end{array}$$

(6)

and

$$\begin{array}{rll}cot\gamma & =\frac{r_1-x_2^{\prime }}{y_2^{\prime }}& \\ & =\frac{r_1-r_2(cos{\delta }_{2}cos{\varphi }_{2}cos{\varphi }_1+cos{\delta }_{2}sin{\varphi }_{2}sin{\varphi }_{1}cosϵ+sin{\delta }_{2}sin{\varphi }_{1}sinϵ)}{r_2(-cos{\delta }_{2}cos{\varphi }_{2}sin{\varphi }_1+cos{\delta }_{2}sin{\varphi }_{2}cos{\varphi }_{1}cosϵ+sin{\delta }_{2}cos{\varphi }_{1}sinϵ)}.& \end{array}$$

(7)

In the above equations, $\gamma$, $\beta$, $\phi _{1}$, $r_{1}$, and $r_{2}$ are time dependent, and $\phi _{2}$, $\delta _{2}$, and $\epsilon$ are constant. For convenience, we omit the $t$-dependence in the expressions of relevant properties. For a small angle $\epsilon$, we expand Equations 6 and 7 and keep the first-order terms of $\epsilon$ to arrive at

$$\frac{tan\beta }{sin\gamma }=\frac{tan\delta }{sin(\varphi -{\varphi }_1)}(1-Fsinϵ)$$

(8)

(same as Equation 3 in Section 2.1), and

$$cot\gamma =\frac{r_1-r_{2}cos{\delta }_{2}cos({\varphi }_2-{\varphi }_1)}{r_{2}cos{\delta }_{2}sin({\varphi }_2-{\varphi }_1)}(1-Gsin{\delta }_{2}sinϵ)$$

(9)

(same as Equation 4 in Section 2.1) where the coefficients of the first-order terms are given by

$$F({\varphi }_2,{\delta }_2,{\varphi }_1)\equiv \frac{sin{\varphi }_2}{tan{\delta }_2}+\frac{tan{\delta }_{2}cos{\varphi }_1}{sin({\varphi }_2-{\varphi }_1)}$$

(10)

and

$$G(r_2,{\varphi }_2,{\delta }_2,r_2,{\varphi }_1)\equiv \frac{sin{\varphi }_1}{r_1/r_2-cos{\delta }_{2}cos({\varphi }_2-{\varphi }_1)}+\frac{cos{\varphi }_1}{cos{\delta }_{2}sin({\varphi }_2-{\varphi }_1)}.$$

(11)

In addition, the angle $\beta$ can be also independently derived from trigonometry,

$$\begin{array}{rll}tan\beta & =\frac{z_2^{\prime }}{\sqrt{x_2^{\prime 2}+y_2^{\prime 2}+r_1^2-2x_2^{\prime }{r}_1}}& \\ & =\frac{r_{2}sin{\delta }_2}{\sqrt{r_2^2{cos}^2{\delta }_2+r_1^2-2r_2{r}_{1}cos{\delta }_{2}cos({\varphi }_2-{\varphi }_1)}}(1-Hsinϵ),& \end{array}$$

(12)

where the coefficient for the first-order term is

$$H({\varphi }_2,{\delta }_2,r_2,{\varphi }_1,r_1)\equiv \frac{cos{\delta }_{2}sin{\varphi }_2}{sin{\delta }_2}+\frac{r_2^{2}cos{\delta }_{2}sin{\delta }_{2}sin{\varphi }_2-r_2{r}_{1}sin{\delta }_{2}sin{\varphi }_1}{r_2^2{cos}^2{\delta }_2+r_1^2-2r_2{r}_{1}cos{\delta }_{2}cos({\varphi }_2-{\varphi }_1)}.$$

(13)

A series of $[\gamma (t_{i}), \beta (t_{i})]$ measurements in the PSP orbital frame at times $t_{i}$ are fit to Equations 8 and 9 to determine parameters characterizing a particle’s motion in the HCI frame. Equation 12 is used to calculate $\beta (t_{i})$ from fitting parameters and compare with the observed $\beta (t_{i})$.

Appendix B: Initial Guess of Fitting Parameters

The convergence of a non-linear least-squares fit often depends on the initial input. For out-of-plane motions, we can calculate the zeroth-order solutions of a feature’s position $[r_{2}, \phi _{2}, \delta _{2}]$ and use them as the initial input for the fit. We first compute the initial guess of $\phi _{2}$ and $\delta _{2}$ using Equation 1, without considering the first-order corrections of the orbit inclination. For this purpose, we define

$$\eta (t)\equiv \frac{cos{\varphi }_1(t)tan\beta (t)}{sin\gamma (t)}=\frac{tan{\delta }_2}{sin{\varphi }_2-cos{\varphi }_{2}tan{\varphi }_1(t)}.$$

(14)

Taking measurements of $[\gamma (t), \beta (t)]$ at two times $t_{i}$ and $t_{j}$, we find

$${\varphi }_2={tan}^{-1}\left[\frac{\eta (t_i)tan{\varphi }_1(t_i)-\eta (t_j)tan{\varphi }_1(t_j)}{\eta (t_i)-\eta (t_j)}\right]$$

(15)

and

$${\delta }_2={tan}^{-1}[\eta (t_i)(sin{\varphi }_2-cos{\varphi }_{2}tan{\varphi }_1(t_i))].$$

(16)

We note that PSP usually only moves by a small angular distance during the observation; therefore, $\eta (t)$-values are very close to each other, and the in-plane angle calculated using Equation 15 with only two images is likely dominated by uncertainties in $\gamma (t)$ and $\beta (t)$ measurements. To overcome this difficulty, we employ all $\gamma (t)$ and $\beta (t)$ measurements to compute the initial guess of $\phi _{2}$ with a linear approximation. For a series of $[\gamma (t_{i}), \beta (t_{i})]$ measurements at times $t_{i}$, we define $\Delta \phi _{1}(t_{i}) \equiv \phi _{1}(t_{i}) - \bar{\phi }_{1}$, where $\bar{\phi }_{1} \equiv \phi _{1}(t_{m})$ is PSP’s angle at a reference time $t_{m}$ when $\phi _{1}$ is closest to the median of $\phi _{1}(t_{i})$ during the observation. For small $\Delta \phi _{1}$, Equation 15 can be linearized,

$$1-\frac{\overline{\eta }}{\eta (t_i)}\approx \left(\frac{{sec}^2{\overline{\varphi }}_1}{tan{\varphi }_2-tan{\overline{\varphi }}_1}\right)\mathrm{\Delta }{\varphi }_1(t_i),$$

(17)

where $\bar{\eta }$ is computed by Equation 14 at the reference time $t_{m}$. A least-squares fit to the above linear relation returns the scaling constant in the large bracket, from which $\phi _{2}$ is computed and referred to as $\bar{\phi }_{2}$. The other angle $\delta _{2}$ is then computed applying Equation 16 to $\gamma$–$\beta$ measurements at time $t_{m}$.

To estimate the range of $\phi _{2}$-values, we apply Equation 17 to a pair of $\gamma$ and $\beta$ measurements obtained at the reference time $t_{m}$ and at any other time during the observation, which returns $N-1$ estimates of $\phi _{2}$, $N$ being the total number of $\gamma$ and $\beta$ measurements. The standard deviation of these $\phi _{2}$-values, referred to as $\Delta \phi _{2}$, gives the range of the initial input $\phi _{2}$. As an example, for a flux rope observed by PSP on 2 April 2019 (see Section 3.1 for the details of this event), $\bar{\phi }_{2}$ is found to be 75^∘ (in HCI coordinates), and the deviation $\Delta \phi _{2}$ is 42^∘. These initial guesses are used in the first-step fit to Equation 8 to determine the accurate values of $\phi _{2}$ and $\delta _{2}$ with the first-order correction of the inclination of the orbit.

Subsequently, we use Equation 2 to solve for $r_{2}(t)$,

$$r_2(t)=\frac{r_1(t)}{cos{\delta }_2[sin({\varphi }_2-{\varphi }_1(t))/tan\gamma (t)+cos({\varphi }_2-{\varphi }_1(t))]}.$$

(18)

With measurements of $\gamma$ and $\beta$ at two times $t_{i}$ and $t_{j}$, here chosen to be the start and end times of the observation, we compute $r_{2}(t)$, and solve for $r_{20}$ and $V$ to zeroth order:

$$V=\frac{r_2(t_j)-r_2(t_i)}{t_j-t_i},$$

(19)

$$r_{20}=r_2(t_i)-V(t_i-t_0).$$

(20)

These are used as initial input in the second-step fit to Equation 9 to determine $r_{20}$ and $V$ more accurately.

We conduct the Levenberg–Marquardt least-squares fit to Equations 8 and 9 multiple times, each time varying the initial guess of $\phi _{2}$ by 5^∘ in the range from $\bar{\phi }_{2} - \Delta \phi _{2}$ to $\bar{\phi }_{2} + \Delta \phi _{2}$. For each initial guess of $\phi _{2}$, the initial guesses of $\delta _{2}$, $r_{20}$, and $V$ are then derived, as described above. Constraining the initial guess using the zeroth-order estimates helps the fit to converge. In each fit, $\phi _{2}$ is allowed to vary from $-10^{\circ }$ to $+90^{\circ }$ around its initial guess, the range of $\delta _{2}$ is $\pm 30^{\circ}$ around the initial guess. In the second-step fit, in which $\phi _{2}$ and $\delta _{2}$ are fixed at the values determined from the first-step fit, the range of $r_{20}$ is between 1 and 35 solar radii, and the range of $V$ is between 10 to 2500 km s⁻¹. The uncertainties of the final fitting results primarily depend on the uncertainties in the $[\gamma (t_{i}), \beta (t_{i})]$ measurements (see Section 2.2.1).

For in-plane motions when $\beta \approx 0$, Equation 3 becomes trivial and cannot be applied to compute $\phi _{2}$. Instead, $\gamma (t_{i})$ measurements are fit to Equation 4 to determine $\phi _{2}$, $r_{20}$, and $V$ all together. We conduct the one-step fit multiple times using 10 different initial guesses of $\phi _{2}$ spanning 180^∘ starting from PSP’s position $\phi _{1}$. In the fit, $\delta _{2}$ is fixed, and the fit is conducted multiple times with $\delta _{2}$ varying from $-4^{\circ }$ to $4^{\circ }$ in 1^∘ increments (see Section 2.2.2).