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q eff regions near the Sun. In our original paper, we attributed high temperatures and negative heat —uxes at the boundary of the polar coronal hole to ...
THE ASTROPHYSICAL JOURNAL, 564 : 1062È1065, 2002 January 10 ( 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

ERRATUM : ““ SEMIEMPIRICAL TWO-DIMENSIONAL MAGNETOHYDRODYNAMIC MODEL OF THE SOLAR CORONA AND INTERPLANETARY MEDIUM ÏÏ (ApJ, 523, 812 [1999]) EDWARD C. SITTLER, JR., AND MADHULIKA GUHATHAKURTA1 NASA Goddard Space Flight Center, Greenbelt, MD 20771

We report the results of an error in our previous computations, which resulted in quantitatively signiÐcant errors in our r, j maps of e†ective temperature T and e†ective heat Ñux q at the boundaries of the polar coronal hole and equatorial eff eff regions near the Sun. In our original paper, we attributed high temperatures and negative heat Ñuxes at the boundary of the polar coronal hole to inaccuracies of the magnetic Ðeld model. We also suspected inaccuracies in our density model. The error is important only close to the Sun, so we will show only revised Figures 9a, 10a, and 11a from the original paper. We will also show Ðgures giving the fractional error for T and q as a function of r and j. When we correct the problem of a sign error in eff eff our code which computes the radial derivative of the mass density, the above-mentioned anomalies go away, and we have well-behaved solutions. So, the discovery of this error has resolved the problem in our two-dimensional maps of T and q . eff eff Correction.ÈThe error can be traced to the radial derivative of the density, which can be expressed as follows :

C

D C D C D C D

LN (z) LN (z) j2 LN(z) LN (z) cs [ p exp [ \ p ] Lz Lz w2(z) Lz L(z)

j2 j2 Lw(z) ] 2[N (z) [ N (z)] , exp [ cs p w2(z) w3(z) Lz for which z \ 1/r. The error was in the form of a negative sign for the third term on the right-hand side of the above equation, which is shown here in the correct form as having a positive sign. Combined with the directional derivative LN(r, h)/Lh we can compute the component of the spatial derivative of N(r, h) along the magnetic Ðeld direction (h \ n/2 [ j is the colatitude). This is then used in our computation of T via equation (12) eff in our original publication. The errors in T then a†ect our computation of q via equation (15) in our original paper. eff eff Results.ÈIn Figures 1 and 2 we plot the normalized errors of T and q as a function of r, j. The normalized error is given eff eff by the following expressions for T and q , respectively : eff eff 2(T [ T ) 2(q [ q ) new , old . dT \ old dq \ new eff eff (T ] T ) (q ] q ) old new new old If one compares Figure 1 in this paper with Figure 9a in our original paper, one can see that the enhanced temperatures reported in our original paper at the boundary of the polar coronal hole and equatorial regions can be largely explained by the errors displayed in Figure 1. A similar comparison can be made between Figure 2 in this paper and Figure 10a in our original paper, except here the error caused an underestimate of q . eff T and q , respectively ; they should be viewed as the In Figures 3 and 4 we show revised two-dimensional maps of eff in our original paper. As can be seen from replacement for the corresponding Ðgures, Figure 9a and Figure 10a,effrespectively, these Ðgures, there is a lack of enhanced temperatures at the boundary of the polar coronal hole and equatorial regions and a lack of suppressed values of the heat Ñux at corresponding regions. The contours now vary smoothly across the boundary of the polar coronal hole, and none of the anomalies mentioned in our original paper exist now. The absence of the anomalies in the revised calculations reinforces the correctness of the model calculations and the magnetic Ðeld model used. The integrations are performed along the magnetic Ðeld, which is highly divergent at the polar coronal hole boundary (i.e., octupole term dominates near the Sun), and by properly taking into account latitudinal gradients, we obtain solutions which are free of anomalies. For completeness, we show in Figure 5 the revised two-dimensional map of the plasma beta reported in our original paper (Fig. 11a). In the original paper the plasma beta was overestimated by more than a factor of 2 in the equatorial regions near the Sun. Finally, the error reported here is also present in the papers by E. C. Sittler, Jr., & M. Guhathakurta (in AIP Conf. Proc. 471, Solar Wind 9, ed. S. Habbal [New York : AIP, 1999], 401), and M. Guhathakurta & E. C. Sittler, Jr. (in AIP Conf. Proc. 471, Solar Wind 9, ed. S. Habbal [New York : AIP, 1999], 79). The impact on the Sittler & Guhathakurta paper is marginal at best because the impacted region is occupied by a tilted current sheet for which no solution is given because of the presence of closed Ðeld lines. In the paper by Guhathakurta & Sittler, the error shows up for the r \ 2.5R curve in Figure 4, which shows S T . eff

1 Also Physics Department, Catholic University of America, Washington, DC.

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FIG. 1.ÈTwo-dimensional map of dT as a function of r, j inside of 5 solar radii. Color is used to show intensity ; a color bar is used for quantitative eff and model magnetic Ðeld lines. See text for deÐnitions. purposes. We also superpose contours of dT eff

FIG. 2.ÈTwo-dimensional map of dq as a function of r, j inside of 5 solar radii. Color is used to show intensity ; a color bar is used for quantitative purposes. We also superpose contours of eff dq and model magnetic Ðeld lines. See text for deÐnitions. eff

FIG. 3.ÈRevised two-dimensional map of T for Fig. 9a in our original paper eff

FIG. 4.ÈRevised two-dimensional map of q

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eff

for Fig. 10a in our original paper

FIG. 5.ÈRevised two-dimensional map of plasma beta for Fig. 11a in our original paper

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