Distribution of Planetary Nebulae Perpendicular to the Disk

Abstract

From an analytical solution to the Boltzmann-Poisson equations for a thin, verti­cally isothermal self-gravitating disk with the assumptions that the mass spectrum of stars is expressible as an inverse power-law and the velocity dispersion perpen­dicular to the plane follows a law of the form < V ² _z (m) >= V ² _o for m < m* and < V ² _z (m) >= V ² _o (m * /m)^θ for m ≥ m*, we have obtained a vertical height distribution of planetary nebulae : ${n_{PN}}(z) = \int_m^{{m_u}} {n(m,z)} {T_{PN}}/T(m)dm$, where m _u = 7.0m _⊙, m _l = 1.0m _⊙ and n(m, z) = n(m,o)exp(-ø(z)/ < V_z ²(m) >), ø(z) being the potential at z. Figure 1 shows a normalised height distribution for var­ious values of V _o where we have assumed a Salpeter slope, a ρ(o) = 0.10M _⊙ pc ^-3 and set m* = m _l = 1.0m _⊙. Figure 2 shows filled circles obtained from the ob­servational sample of Daub (1982. ApJ 260, 612) superposed on the theoretical distributions. Although there is agreement for large values of z, closer to the plane the observational sample falls below the theoretical curves. Since the observational sample is size-limited, we may be missing a larger fraction of small nebulae closer to the plane. It is also possible a single value of V _o is not a correct representation of reality as these nebulae originate from stars of greatly differing ages and V _o may have changed over the lifetime of the Galactic disk.