Mathematics – General Topology
Scientific paper
Mar 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006apj...640l.107b&link_type=abstract
The Astrophysical Journal, Volume 640, Issue 1, pp. L107-L107.
Mathematics
General Topology
Scientific paper
In the course of performing an extension of this work, we discovered an error in the metric component gtφ. The value of gtφ in equation (3) must be divided by 2, and the corrected equations (3) and (4) are, respectively,Ω=-(gtφ+λgtt)/(gφφ+λgtφ)=((2a/r)-[λ(4a2-r2Δ)/A])/((A/r2)-(2λa/r)), (3)vt=(Ar2Δ)/((1-u2)[A2-4λarA+λ2r2(4a2-r2Δ)])1/2. (4)
The forms of no other equations as displayed in the Letter are affected as they are written in terms of vt, Ω, gtφ, etc. However, the numerical results presented earlier are modified, as are some of our conclusions.
The general topology of the flows remains the same, but there are small changes to the inner and middle sonic point locations for given initial assumptions. For example, in Figure 1, the identical parameters a=0.2, E=1.0003, λ=3.1, and γ=4/3 yield nearly identical curves, but they give new values for ri (=5.0847), rm (=13.564), and ro (=1093.8, which is only slightly different).
The correction to gtφ produces results that are in many respects more desirable than those reported earlier and are summarized here. The replacement Figure 2 shows that a more limited range of λ-values completely spans the full range of positive spin parameters. For counterrotating flows through ro, the variations of Aδ with a are not monotonic; Mδ shows a maximum, and Tδ, ρδ, pδ show minima at a~=0.5. Now we get common [P3 solutions for both a and -a, up to a<~0.57.
For prograde (positive a) flows through ri (relevant during shock formation), the general profiles for the variations of ρδ and pδ with a remain exactly the same as that for flow through ro. Mδ and Tδ also show a similar overall trend for flows through ri and ro. However, for flows through ri, the variations of Mδ and Tδ with a are not monotonic; Mδ has a minimum, and Tδ has a maximum, at some intermediate a-value (near the highest a of their valid multitransonicity range, for fixed E, λ, γ). The comparisons of magnitude for fixed E, λ, γ, a are [Mδ,ρδ](ri)<[Mδ,ρδ](ro) and [Tδ,pδ(ri)>[Tδ,pδ(ro).
We find that for both prograde and retrograde flows, [Tδ,pδ nonlinearly and monotonically correlate with γ, while [Mδ,ρδ] anticorrelate with γ. If one keeps E,γ,|a| fixed but decreases λ (which mimics the introduction of viscosity), [Tδ,ρδ,pδ nonlinearly decrease, and Mδ increases, for prograde flow, whereas the trends of Mδ and Tδ are the opposite for retrograde flow.
Barai Paramita
Das Tapas K.
Wiita Paul J.
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