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Scientific paper
Nov 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008apj...687.1461l&link_type=abstract
The Astrophysical Journal, Volume 687, Issue 2, pp. 1461-1463.
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4
Scientific paper
Two errors, that to a degree cancel out, have emerged in the formalism in the original paper. Fundamental conclusions of the paper are slightly changed, and the quantitative agreement of the model with the numerical simulations of Blondin & Mezzacappa (ApJ, 642, 401 [2006]) is improved, especially with regard to predicted growth rates. Equation (2) should have read(ρ+Δρ)[(∂(ur+vr))/∂t+(ur+vr)(∂(ur+vr))/∂r+((uφ+vφ)/rsinθ)(∂(ur+vr))/∂φ-((uφ+vφ)2)/r=-(∂(P+ΔP))/∂r-GM((ρ+Δρ))/(r2), (2)giving a perturbed radial velocity (originally equation 10)vr=(2uφvφ)/(iω'r)-(vθ)/(iω'r)(∂ur)/∂θ-δ/(iω'+urλ)λc2s. (10)Correct equations throughout the rest of the paper are obtained by the replacement GM/r-c2s-->-c2s. I thank an anonymous referee for pointing this out. For reference, the corrected dispersion relation (eq. [22]) is given here, also including the correction of other typographical errors,ω4+ω3[iurs(1/L+5/2r)-i(uri)/aL1/Q(1+βQ)/(1-βQ)-ω2[(l(l+1))/(r2)vsur-(c2s)/aL(1/L+5/2r)-ω2[(c2s)/2(1/L+4γ-5/γ-11/r)(1/L+5/2r)(1+Q(1+βQ)/(1-βQ))+(c2s)/2(1/L+4γ-5/γ-11/r)1/aL(1-Q(1+βQ)/(-βQ))+ω[iurs(1/L+5/2r)l(l+1)(c2s)/(r2)+i(uri)/aLl(l+1)(c2s)/(r2)+i(uri)/aLQ(1+βQ)/(1-βQ)l(l+1)(vsurs)/(r2)+ω(ic2s)/2(1/L+4γ-5/γ-11/r)2[urs(1/L+5/2r)(1-Q(1+βQ)/(1-βQ))+(uri)/aL(1+Q(1+βQ)/(1-βQ))-ω[(c2s)/2aL(1/L+4γ-5/γ-11/r)(1/L+5/2r)(3iurs+(iurs)/Q(1+βQ)/(1-βQ)+iuri+(iuri)/Q(1+βQ)/(1-βQ))-(c2s)/aL(1/L+5/2r)(l(l+1)c2s)/(r2)+(c2s)/2aL(1/L+4γ-5/γ-11/r)(l(l+1)vsurs)/(r2)(1-Q(1+βQ)/(1-βQ))-1/ω(c2s)/2aL(1/L+5/2r)l(l+1)(c2s)/(r2)(1/L+4γ-5/γ-11/r)[-iurs(1+Q(1+βQ)/(1-βQ))-iuri(1-Q(1+βQ)/(1-βQ))-1/ω(l(l+1)vsurs)/(r2)[i(uri)/2aLc2s(1/L+4γ-5/γ-11/r)2(1+Q(1+βQ)/(1-βQ))+i(uri)/aL(l(l+1)c2s)/(r2)=0. (22)
The second issue resides in the value of a. Rather than ``fine-tuning'' to give the best agreement with Blondin & Mezzacappa (ApJ, 642, 401 [2006]), the corrected dispersion relation gives acceptable real frequencies and growth rates for l>=1 for a wide range of a>>1. This corresponds to the perturbation either vanishing or being very small at the inner boundary. For l=0 the expansion in terms of λur/ω is probably inappropriate, and growth rates in the case with advection are unreliable.
As before, we compute growth rates with and without the terms in iurs and iuri in the dispersion relation, interpreting growth with iurs=iuri=0 as being due to a purely acoustic process, while growth requiring nonzero urs and uri must be an advective-acoustic mechanism. Corrected versions of Table 1 and Figure 2 are given for γ=4/3 and a=100. Just as before, at large shock radii we find the advective-acoustic mechanism dominating. However, in a revision to our original paper, this mechanism also dominates at smaller shock radii, and in nonrotating cases we find no purely acoustic instability at all. Corrected results for γ=1.36 have very similar real frequencies, but lower growth rates.
We also give a corrected version of Table 3, for γ=4/3 and a=100. For Ω>0, frequencies ω'=ω+mΩ≶0 should be identified with m=+/-1, from the definition of δ. The corotating mode (m=1 for Ω>0) is enhanced in frequency and growth rate, the counterrotating mode is diminished (m=0 is unchanged), and the conclusion that instability grows fastest in the equatorial plane remains unchanged. As before, in these rotating cases, postshock advection is not necessary for growth, suggesting an acoustic instability.
We also take the opportunity to correct some typographical errors in Appendix B. Equation (B4) should have readδ(ri)=B+exprsriλ+dr+B-exprsriλ-dr=-(vr(ri))/iωaL=-(c2s)/(ω2aL)(B+λ'+exprsriλ+dr+B-λ'-exprsriλ-dr), (B4)leading to a corrected form for equation (B7),βQ=exprirs(λ+-λ-)dr=exprirs(1/L+4γ-5/γ-11/r)Qdr~=((rs)/(ri))[(4γ-6)/(γ-1)Q. (B7)Hence, for 1<γ<1.5, β-->0 as ri/rs-->0, and as γ-->1 for fixed rs/ri.
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