Coronal Photopolarimetry with the LASCO-C2 Coronagraph over 24 Years [1996 – 2019]

Abstract

We present an in-depth characterization of the polarimetric channel of the Large-Angle Spectrometric COronagraph/LASCO-C2 onboard the Solar and Heliospheric Observatory (SOHO). The polarimetric analysis of the white-light images makes use of polarized sequences composed of three images obtained though three polarizers oriented at $+60^{\circ }$, $0^{\circ }$, and $-60^{\circ }$, complemented by a neighboring unpolarized image, and relies on the formalism of Mueller. The Mueller matrix characterizing the C2 instrument was obtained through extensive ground-based calibrations of the optical components and global laboratory tests. Additional critical corrections were derived from in-flight tests relying prominently on roll sequences of SOHO, on the basis of consistency criteria (e.g. the “tangential” direction of polarization), and from several applications, notably the time-dependent tomographic reconstruction of the coronal electron density. This took several years of effort, but resulted in the quasi-uninterrupted photopolarimetric analysis of the corona over two complete Solar Cycles 23 and 24, and the comparison with a variety of eclipse data obtained at different phases of these cycles. Our final results encompass the characterization of the polarization of the white-light corona, of its polarized radiance, of the two-dimensional electron density, and of the K-corona. The agreement with the eclipse data is excellent except for slight discrepancies affecting the innermost part of the C2 field of view. The present work leaves the two unpolarized components, F-corona and stray light, entangled and their complex separation will be dealt with in a follow-on article.

Appendix: Determination of the Correction Function \(S(x,y)\)

The correction function $S(x,y)$ was derived from the polarization sequences performed on 2 and 3 September 1997 when the SOHO spacecraft, starting from its nominal position $\theta _{\mathrm{r}}=0^{\circ }$, was successively rolled to $\theta _{\mathrm{r}}=45^{\circ }$ and $90^{\circ }$ (Table 3). Each sequence yielded a triplet of polarized images $I_{1}$, $I_{2}$, $I_{3}$ leading to a polarized radiance $\mathrm{p}_{\mathrm{c}}I_{\mathrm{c}} = p_{\mathrm{cp}}I_{\mathrm{cp}}$ according to our notations introduced in Section 3. For simplification in the present discussion, we adopt the standard notation $pB$. The three sequences therefore produced three $\mathrm{pB}$ images denoted as follows:

• (i) $\theta _{\mathrm{r}}=0^{\circ }\rightarrow {\mathrm{pB}}_{\mathrm{0}}$,

• (ii) $\theta _{\mathrm{r}}=45^{\circ }\rightarrow {\mathrm{pB}}_{\mathrm{45}}$,

• (iii) $\theta _{\mathrm{r}}=90^{\circ }\rightarrow {\mathrm{pB}}_{\mathrm{90}}$.

Comparing those images requires a rotation by $-\theta _{\mathrm{r}}$ and this operation will be denoted by $[\Im ]_{-\theta _{\mathrm{r}}}$ where $[\Im ]$ is any image. $S(x,y)$ is determined by imposing the requirement that the three $\mathrm{pB}$ images, once corrected by $S(x,y)$ and properly rotated, are identical. This procedure requires that the corona remained unchanged during the 20 hours of the roll sequence. This is certainly true for the F-corona as well as for the large-scale K-corona with no coronal mass ejection detected during this period of low solar activity. However, small scale variations may present a difficulty, but will be ignored by considering that $S(x,y)$ is intended to correct for large-scale variations only. The very validity of our procedure, i.e. the existence of an ad hoc function $S(x,y)$ can only be established by finding a relevant solution as demonstrated in Section 7.

Let us form the following ratios of the polarized radiances:

$$\begin{aligned} R_{0/45} =& {\mathrm{pB}}_{0}/[{\mathrm{pB}}_{45}]_{-45} , \end{aligned}$$

(34)

$$\begin{aligned} R_{45/90} =& {\mathrm{pB}}_{45}/[{\mathrm{pB}}_{90}]_{-45} . \end{aligned}$$

(35)

The condition that the corrected polarized radiances be identical translates into the two following constraints for $S(x,y)$:

$$\begin{aligned} R_{0/45} =& \frac{S}{[S]_{-45}}(1+\epsilon _{1}) , \end{aligned}$$

(36)

$$\begin{aligned} R_{45/90} =& \frac{S}{[S]_{-45}}(1+\epsilon _{2}) , \end{aligned}$$

(37)

where $\epsilon _{1}$ and $\epsilon _{2}$ account for some uncertainty as the radiances cannot obviously be identical because of small changes in the corona during the time interval elapsed between successive observations and because of the intrinsic noise in the images. In order to combine the above two equations in a single constraint, we introduce a mask $M=M(x,y) \in [0,1]$ intended to mitigate the effect of the variations of the inner streamer belt. We then obtain

$$ R = \frac{M\cdot R_{0/45}+[M]_{45}\cdot R_{45/90}}{M+[M]_{45}} .$$

(38)

$S(x,y)$ is now determined by imposing the requirement that it is positive, smooth, and that it satisfies the relationship

$$ S = R[S]_{-45} .$$

(39)

To solve the above equation, we switch from Cartesian $(x,y)$ to polar $(r,\phi )$ coordinates and we change the intensity scale from linear to logarithmic (homomorphic transform). We then obtain

$$\begin{aligned} \log S(r,\phi ) = & \log R(r,\phi ) + \log S(r,\phi -\frac{\pi }{4}). \end{aligned}$$

(40)

As all these functions are periodic in $\phi$, we proceed by looking for a periodic function $\psi (r,\phi ) = \log S(r,\phi )$ satisfying

$$\begin{aligned} \psi (r,\phi )-\psi (r,\phi -\frac{\pi }{4}) = & \log R(r,\phi ) . \end{aligned}$$

(41)

In Fourier space, this equation becomes

$$\begin{aligned} \Psi ^{*}(r,\omega )(1-\mathrm{e}^{-\mathrm{i}\pi \omega /4}) = & \widehat{\log R}^{*}(r,\omega ) \end{aligned}$$

(42)

where the wide hat means that the Fourier transformation is applied to $\log R$ and not to $R$. $\psi (r,\phi )$ and $R(r,\phi )$ being periodic, both sides may be expressed as Fourier series:

$$\begin{aligned} \Psi _{k}(r)(1-\mathrm{e}^{-\mathrm{i}\pi k/4}) = & \widehat{\log R}_{k}(r) \qquad \forall ~ k ~\mbox{in} ~[0,\ldots,N-1], \end{aligned}$$

(43)

where $N$ is the number of angular samples. This differential problem is only partially solvable: indeed, we note that the average value remains undefined, as well as the harmonic terms corresponding to $k \in \{0,8,16,\dots ,8n\}$. In practice, and because of the noise, the determination can well be limited to the first five harmonic terms:

$$\begin{aligned} \Psi _{n}(r) = & \frac{\widehat{\log R}_{k}(r)}{1-\mathrm{e}^{-\mathrm{i}k\pi /4}} \qquad k = \{1,2,3,4,5\}. \end{aligned}$$

The radial variations of the $\Psi _{k}(r)$ coefficients are strongly correlated (thus offering some extra robustness) and a third order polynomial of $r$ is ample.

In practice the missing values in the $(r,\phi )$ frame due to the square format of the initial images poses an additional problem alleviated by limiting the expansion series to the first terms in the outer part of image where data are missing.

The polar form of the function $\log S = \psi (r,\phi )$ is obtained with the truncated Fourier series:

$$\begin{aligned} \psi (r,\phi ) \approx & \sum _{1}^{5} \Psi _{k}(r)\;\mathrm{e}^{- \mathrm{i}2\pi k\phi }. \end{aligned}$$

From $\psi (r,\phi )$ the derivation of $S(x,y)$ is straightforward (Figure 51). We note that the very construction of the function $\psi (r,\phi )$ ensures its smoothness and henceforth that of $S(x,y)$.

Figure 51

The LASCO-C2 correction pattern $S(x,y)$ for the polarized brightness image (left panel) and its histogram (right panel).