A uniformly valid asymptotic density-potential relation for spiral galaxies

Abstract

An asymptotic relation between density and potential perturbations which is uniformly valid for arbitrary pitch angle β for spiral galaxies with number of spiral arms m ⩾ 2, is given. For small pitch angles (¦β¦⪡ 1) 2 2 The pitch angle will be denoted by β throughout so as to leave i for the imaginary unit. , it is reduced to various known asymptotic relations. When ¦β¦is not small, this new result is suitable for discussing the open spiral waves forming barred spiral galaxies. The final result is given by Eq. (5.40), where \\ ̂ gy 1 and \\ ̂ gs 1 are the complex amplitudes of the potential perturbation ψ 1 and the density perturbation σ 1 defined by Eqs. (5.32) and (5.33), respectively. The region in the (− kr, m) plane where the present result is uniformly valid is given by Eq. (6.1) and shown in Fig. 2. In Eq. (6.1), k is the radial wave number and K is the total wave number. When β = 0, Eq. (5.40) is reduced to Eq. (6.4), an asymptotic relation obtained by Shu (1970). When ¦β¦⪡ 1, Eq. (5.40) is reduced to Eq. (6.4), an equivalent form of Bertin and Mark's result (1979). When β = π 2 , Eq. (5.40) is reduced to Eq. (6.5), an asymptotic relation for the “open limit”. For the middle range of the pitch angle, neither the “tightly wound limit” result for ¦β¦⪡ 1 nor the “open limit” result for β ≈ π 2 is valid, but the present result is still valid.