Erratum: Spectral Lines for Polarization Measurements of the Coronal Magnetic Field. II. Consistent Treatment of the Stokes Vector for Magnetic-dipole Transitions (1999, ApJ, 522, 524)

The expressions 37(b) and 38(b) for the coefficient ${E}_{{{JJ}}_{0}}$ given in the published article are wrong. The general Equation 36(c), however, is correct. In deriving from it the special cases represented by Equations 37(b) and 38(b), a factor 1/4 was accidentally dropped from the second addendum within the square bracket in both equations. We are grateful to Gabriel Dima and Tom Schad for pointing this out (see also Dima & Schad 2020). Additionally, in Equation 37(b), the $(2J+1)$ factor in the denominator of that second addendum should have been $(2J-1)$ instead. Hence, after some straightforward algebraic simplifications, the correct Equations 37(b) and 38(b) become the following,

$$\begin{eqnarray}&&{E}_{J,J+1}=\displaystyle \frac{1}{\sqrt{5}}\,\sqrt{\displaystyle \frac{(2J-1)(2J+3)}{J(J+1)}}\left({\bar{g}}_{{\alpha }_{0}J,{\alpha }_{0}J+1}-\displaystyle \frac{3}{2}\,{g}_{{\alpha }_{0}J+1}\,\displaystyle \frac{J+2}{2J+3}\right),\end{eqnarray} \tag{ 37b }$$

$$\begin{eqnarray}&&{E}_{{J}_{0}+1,{J}_{0}}=\displaystyle \frac{1}{\sqrt{5}}\,\sqrt{\displaystyle \frac{(2{J}_{0}+1)(2{J}_{0}+5)}{({J}_{0}+1)({J}_{0}+2)}}\left({\bar{g}}_{{\alpha }_{0}{J}_{0}+1,{\alpha }_{0}{J}_{0}}-\displaystyle \frac{3}{2}\,{g}_{{\alpha }_{0}{J}_{0}}\,\displaystyle \frac{{J}_{0}}{2{J}_{0}+1}\right).\end{eqnarray} \tag{ 38b }$$

We take the opportunity in this erratum to also provide the general expressions for ${D}_{{{JJ}}_{0}}$ and ${E}_{{{JJ}}_{0}}$, after algebraically reducing the 6-j and 9-j symbols in Equations 36(b) and 36(c):

$$\begin{eqnarray}&&{D}_{{{JJ}}_{0}}=2\sqrt{\displaystyle \frac{2}{5}}\,\displaystyle \frac{{(-1)}^{J-{J}_{0}+1}(2J+1)\sqrt{J(J+1)(2J-1)(2J+3)}}{(J+{J}_{0})(J+{J}_{0}+1)(J+{J}_{0}+2)(J-{J}_{0}+1)!({J}_{0}-J+1)!},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{E}_{{{JJ}}_{0}} & = & \displaystyle \frac{1}{\sqrt{5}}\,\sqrt{\displaystyle \frac{(2J-1)(2J+3)}{J(J+1)}}\left\{{\bar{g}}_{{\alpha }_{0}J,{\alpha }_{0}{J}_{0}}+\displaystyle \frac{3}{4}\,{g}_{{\alpha }_{0}{J}_{0}}\,\displaystyle \frac{{[J(J+1)-{J}_{0}({J}_{0}+1)]}^{2}-5J(J+1)-3{J}_{0}({J}_{0}+1)+6}{(2J-1)(2J+3)}\right\}.\end{array}\end{array}\end{eqnarray} \tag{ 2 }$$

Both expressions are generally valid for all $J,{J}_{0}$ such that $| J-{J}_{0}| \leqslant 1$. The expression for ${E}_{{{JJ}}_{0}}$ is additionally restricted to $J\geqslant 1$. This is not a limitation, however, as levels with $J\lt 1$ cannot be aligned, and therefore the weight of ${E}_{{{JJ}}_{0}}$ in the expression for Stokes V (see Equation 35(c) in the published article) is identically zero in that case.

We note that Equations (1) and (2) are valid also in the case $J={J}_{0}$. Although this condition is typically not met in the emission coronal lines of the M1 type, such a case is also important, since the polarization coefficients ${D}_{{{JJ}}_{0}}$ and ${E}_{{{JJ}}_{0}}$ apply to any dipolar transition between two levels $(\alpha J)$ and $({\alpha }_{0}{J}_{0})$, in the saturation regime of the Hanle effect (see, e.g., Casini et al. 2017, where a general expression analogous to Equation (1) was derived for the coefficient $D(J,{J}_{0})={D}_{{{JJ}}_{0}}/\sqrt{2}$). For the specific case of $J={J}_{0}$, we thus find

$$\begin{eqnarray}&&{D}_{{JJ}}=-\displaystyle \frac{1}{\sqrt{10}}\,\sqrt{\displaystyle \frac{(2J-1)(2J+3)}{J(J+1)}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{E}_{{JJ}} & = & \displaystyle \frac{1}{\sqrt{5}}\,\sqrt{\displaystyle \frac{(2J-1)(2J+3)}{J(J+1)}}\left({\bar{g}}_{\alpha J,{\alpha }_{0}J}-\displaystyle \frac{3}{2}\,{g}_{{\alpha }_{0}J}\right)=-\sqrt{2}\,{D}_{{JJ}}\left({\bar{g}}_{\alpha J,{\alpha }_{0}J}-\displaystyle \frac{3}{2}\,{g}_{{\alpha }_{0}J}\right).\end{array}\end{array}\end{eqnarray} \tag{ 4 }$$

In particular, we note that the polarizability ${D}_{{{JJ}}_{0}}$ is always negative when $J={J}_{0}$ (see, e.g., Landi Degl'Innocenti & Landolfi 2004, Table 10.1).