Erratum: “Potential-density Pairs for a Family of Finite Disks” (2009, ApJ, 693, 1310)

Earl Schulz

doi:10.3847/1538-4357/aab9ba

1. Discussion

Review of the published article found errors that affect Equations (5), (13), (20), (26), and (29). These are purely typographical errors that do not affect the other equations, numerical results, or supporting calculations.

In Equation (5), the operand of the second square root term should be (R² − (α + Iz)²), not (R² + (α − Iz)²). In Equation (13), the denominator of the lead term, not the numerator, should include the term α; and the sign of last term is incorrect. In Equation (20), the leading sign is incorrect. In Equation (26), the first term includes an extra factor of R²; and the sign of the last term is incorrect. In Equation (29), for R > α, the denominator of the lead term should include the term α³, not α.

The corrected equations are as follows:

$\begin{eqnarray}&&{F}_{R,\hom }(R,z;\alpha )=-\displaystyle \frac{\pi \alpha {\sigma }_{0}{\rm{G}}}{R}\left[\displaystyle \frac{\alpha -{\mathtt{I}}\,z}{\sqrt{{R}^{2}-{(\alpha -{\mathtt{I}}z)}^{2}}}+\displaystyle \frac{\alpha +{\mathtt{I}}\,z}{\sqrt{{R}^{2}-{(\alpha +{\mathtt{I}}z)}^{2}}}\right],\end{eqnarray} \tag{ 5 }$

$\begin{eqnarray}{{\rm{\Phi }}}_{\mathrm{Mac}}(R,z;)\alpha & = & \quad -\displaystyle \frac{\pi {\sigma }_{0}{\rm{G}}}{4\alpha }\left[(2{\alpha }^{2}-{R}^{2}+2{z}^{2})\left({\sin }^{-1}\left(\displaystyle \frac{\alpha +{\mathtt{I}}\,z}{R}\right)+{\sin }^{-1}\left(\displaystyle \frac{\alpha -{\mathtt{I}}\,z}{R}\right)\right)\right.\\ & & +\alpha (\sqrt{{R}^{2}-{(\alpha -{\mathtt{I}}z)}^{2}}+\sqrt{{R}^{2}-{(\alpha +{\mathtt{I}}z)}^{2}})\\ & & +3{\mathtt{I}}\,z(\sqrt{{R}^{2}-{(\alpha -{\mathtt{I}}z)}^{2}}-\sqrt{{R}^{2}-{(\alpha +{\mathtt{I}}z)}^{2}})\phantom{\rule{}{3.25ex}}],\end{eqnarray} \tag{ 13 }$

$\begin{eqnarray}&&{F}_{z,\mathrm{Mac}}(0,z;\alpha )=\displaystyle \frac{2\pi {\sigma }_{0}{\rm{G}}}{\alpha }\left[z{\sin }^{-1}\left(\displaystyle \frac{\alpha }{\sqrt{{\alpha }^{2}+{z}^{2}}}\right)-\alpha \,\mathrm{sgn}(z)\right],\end{eqnarray} \tag{ 20 }$

$\begin{eqnarray}{F}_{R,D2}(R,z;\alpha ) & = & -\displaystyle \frac{3\pi {\sigma }_{0}{\rm{G}}}{16R{\alpha }^{3}}\left[2{R}^{2}(4{\alpha }^{2}-3{R}^{2}+12{z}^{2}){\sin }^{-1}\left(\displaystyle \frac{{f}_{1}-{f}_{2}}{2R}\right)\right.\\ & & +\sqrt{2}\alpha (2{\alpha }^{4}-5{\alpha }^{2}{R}^{2}+4{\alpha }^{2}{z}^{2}+3{R}^{4}-25{R}^{2}{z}^{2}+2{z}^{4})\displaystyle \frac{\sqrt{{f}_{1}{f}_{2}-{f}_{3}}}{{f}_{1}{f}_{2}}\\ & & \left.-\sqrt{2}| z| (2{\alpha }^{4}+9{\alpha }^{2}{R}^{2}+4{\alpha }^{2}{z}^{2}-13{R}^{4}-11{R}^{2}{z}^{2}+2{z}^{4})\displaystyle \frac{\sqrt{{f}_{1}{f}_{2}+{f}_{3}}}{{f}_{1}{f}_{2}}\right],\end{eqnarray} \tag{ 26 }$

$\begin{eqnarray}&&{{\rm{\Phi }}}_{D2}(R,0;\alpha )=-\displaystyle \frac{3\pi {\sigma }_{0}{\rm{G}}}{32{\alpha }^{3}}\left[(8{\alpha }^{4}-8{\alpha }^{2}{R}^{2}+3{R}^{4}){\sin }^{-1}\left(\displaystyle \frac{\alpha }{R}\right)+3\alpha (2{\alpha }^{2}-{R}^{2})\sqrt{{R}^{2}-{\alpha }^{2}}\right]\ {\rm{for}}\ R\geqslant \alpha .\end{eqnarray} \tag{ 29b }$

I thank Keith Clemens whose careful work uncovered errors in the published article.

Erratum: "Potential-density Pairs for a Family of Finite Disks" (2009, ApJ, 693, 1310)

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Erratum: "Potential-density Pairs for a Family of Finite Disks" (2009, ApJ, 693, 1310)

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