On the stability of isothermal shocks in black hole accretion disks

The paper derives an explicit translation-mode growth/decay rate for isothermal disk shocks, λ0 = [1/(u+0(rs)+u−0(rs))]·[1/rs − 2/(rs − rh)^2 + `^2/r^3s], and shows inner shocks are unstable while outer shocks are stable; this is stated in the main text (eq. 43) and Appendix C (eq. 77) and supported numerically . The candidate solution independently reproduces the same main formula via a concise shock-fitting argument and reaches the same stability conclusion. However, both the paper and the candidate introduce a sign slip when rewriting the result in terms of g(rs) and sech a. The paper defines g(rs) in §3.1.4 as g = −(1/2)[1/rs − 2/(rs − rh)^2 + `^2/r^3s] , but later writes λ0 = −g(rs) sech a = [1/rs − 2/(rs − rh)^2 + `^2/r^3s]/(u+0+u−0) (eq. 77) ; with the given definitions and u+0+u−0 = −2 cosh a, the consistent identity should read λ0 = g(rs) sech a. The candidate repeats this minus sign and also misstates sgn relations between J′, N, and g, though it ultimately lands on the correct stability classification. In short, the central formula (eq. 43) and the stability conclusion are correct in the paper and are independently re-derived by the model, but both contain a minor sign inconsistency in the g–sech rewrite.