A Simple Formula for the Third Integral of Motion of Disk-Crossing Stars in the Galaxy

Abstract

We present an analytical simple formula for an approximated third integral of motion associated with nearly equatorial orbits in the Galaxy:is the vertical amplitude of the orbit at galactocentric distance R and Σ I (R) is the integrated dynamical surface mass density of the disk, a quantity which has recently become measurable. We also suggest that this relation is valid for disk-crossing orbits in a wide variety of axially symmetric galactic models, which range from razor-thin disks to disks with non-negligible thickness, whether or not the system includes bulges and halos. We apply our formalism to a Miyamoto-Nagai model and to a realistic model for the Milky Way. In both cases, the results provide fits for the shape of nearly equatorial orbits which are better than the corresponding curves obtained by the usual adiabatic approximation when the orbits have vertical amplitudes comparable to the disk's scale height. We also discuss the role of this approximate third integral of motion in modified theories of gravity.

Envelopes and vertical amplitudes of disc-crossing orbits

“…1-3). These errors are of the same order of magnitude as the errors appearing in the comparison of formula (3) with flattened Stäckel models (Vieira & Ramos-Caro 2014). Indeed, numerical experiments with Stäckel potentials (the Kuzmin-Kutuzov potential of Dejonghe & de Zeeuw 1988) show that the predictions of Eqs.…”

“…However, it is well known that this approximation has a very limited range of applicability, being valid only for orbits whose vertical excursions are very small (Binney & McMillan 2011). In particular, their vertical amplitudes must be much smaller than the disk thickness (Vieira & Ramos-Caro 2014). Variants of this formula were proposed in order to overcome this problem, as for instance the corrections to the adiabatic approximation of Binney & McMillan (2011).…”

“…which is commonly used to model the disk component of Galaxy mass models (Allen & Santillán 1991;Irrgang et al 2013;Barros et al 2016) and also to analyze the Saggitarius dwarf galaxy debris (Johnston et al 1995;Helmi 2004;Law et al 2009;Ibata et al 2013;Deg & Widrow 2013). The disk's vertical edge is estimated as ζ /a ≈ 3b/a if b ≪ a (Vieira & Ramos-Caro 2014). We analyze various orbits for different values of energy E, angular momentum L z , and initial conditions.…”

“…We presented in this paper an extension of the modeling of the envelopes of disk-crossing orbits on the system's meridional plane. It is based on the disk's surface density distribution, which was previously presented for the cases of off-equatorial orbits around razor-thin disks (Vieira & Ramos-Caro 2015;Vieira & Ramos-Caro 2016) and of orbits in three-dimensional disks with vertical amplitudes comparable to the disk thickness (Vieira & Ramos-Caro 2014). The generalization discussed here encompasses the two known limits for the envelope estimates: The limit of very low vertical amplitudes (usual adiabatic approximation, Eq.…”

“…In spite of the significant advances in recent years, closed-form analytical approximations for the non-classical integral I 3 (and for the corresponding envelope) are still very scarce, and the existent results have limited range of validity. The usual adiabatic approximation based on decoupled radial and vertical motions (Binney & Tremaine 2008), for instance, is only valid very close to the equatorial plane (Binney & McMillan 2011;Vieira & Ramos-Caro 2014). It was recently found, based on the study of orbits in razor-thin disks (Vieira & Ramos-Caro 2015;Vieira & Ramos-Caro 2016), that a simple relation can describe the envelopes Z(R) of disk-crossing orbits whose amplitudes are near the disk's vertical edge (Vieira & Ramos-Caro 2014).…”