[Paper Review] Galactic tide in a noninertial frame of reference

J. Klačka|arXiv (Cornell University)|Dec 16, 2009

Astro and Planetary Science4 references3 citations

TL;DR

This paper derives the equation of motion and perturbing acceleration for the galactic tide in a noninertial, rotating reference frame centered on the Sun, accounting for Coriolis and centrifugal forces. It presents a new integral of motion and corrects inconsistencies in prior models by fully incorporating noninertial frame effects, offering a more physically consistent framework for modeling Oort cloud comet dynamics.

ABSTRACT

Equation of motion and the vector of perturbing acceleration (force) for the galactic tide in a noninertial frame of reference is derived. The noninertial reference frame is rotating with a fixed angular velocity $\vecω$ $=$ $-$ $ω_{0}$ $\hat{\vec{z}}$ with respect to the inertial frame of reference of the Galaxy. $\vecω$ is the angular velocity of the solar rotation (rotation of the Local Standard of Rest) around the galactic center, the unit vector $\hat{\vec{z}}$ is oriented toward the north pole of the Galaxy: the Sun is always situated in the plane $y'$ $=$ 0 ($x' - z'$-plane). The equation of motion can be applied to the dynamics of the Oort cloud of comets. Relations for calculation of the osculating orbital elements are presented and a new integral of motion is derived for the conventional approach in modelling of the effect of the galactic tidal field.

Motivation & Objective

• To develop a physically consistent model of the galactic tidal force acting on comets in the Oort cloud by using a noninertial rotating reference frame.
• To address inconsistencies in prior models—particularly those by Heisler & Tremaine (1986) and Dybczynski et al. (2008)—that improperly treat the rotating frame's fictitious forces.
• To derive a new integral of motion specific to the galactic tidal problem in this noninertial framework.
• To provide a corrected formulation of the perturbing acceleration vector that includes both gravitational and noninertial frame contributions.
• To enable more accurate long-term simulations of comet orbital evolution under galactic tidal forces.

Proposed method

• Derives the equation of motion in a noninertial frame $S'$ rotating with angular velocity $oldsymbol{ar{ u}} = - u_0 \hat{\vec{z}}$ relative to the inertial galactic frame $S$, where $\nu_0 = A - B$.
• Uses the standard transformation for acceleration in a rotating frame: $\vec{a} = \vec{a}' + 2\boldsymbol{\omega} \times \vec{v}' + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \vec{r}')$, incorporating Coriolis and centrifugal terms.
• Expresses the galactic tidal perturbing force as a function of position in the rotating frame, with components $F_x = K_x x'$, $F_y = K_y y'$, $F_z = K_z z'$, where $K_x, K_y, K_z$ are derived from Oort constants and mass density gradients.
• Incorporates time-dependent terms due to the Sun's oscillation above the galactic plane ($Z_0 = 30$ pc) and the galactic rotation, leading to modulated coefficients in the equations of motion.
• Derives relations for osculating orbital elements in the rotating frame to facilitate dynamical analysis of comet orbits.
• Identifies a new conserved quantity (integral of motion) specific to the system, arising from the symmetry and structure of the derived equations.

Experimental results

Research questions

• RQ1 How does the galactic tidal force on a comet in the Oort cloud change when modeled in a noninertial, rotating reference frame centered on the Sun?
• RQ2 What are the correct dynamical terms—especially fictitious forces—that must be included in the equation of motion when using a rotating frame for galactic tidal studies?
• RQ3 Why do prior models (e.g., Dybczynski et al. 2008) fail to properly account for noninertial frame effects, and how can this be corrected?
• RQ4 Can a new conserved quantity (integral of motion) be derived for the galactic tidal problem in this rotating frame, and what is its physical significance?
• RQ5 How do time-dependent terms from the Sun’s vertical motion and galactic rotation affect the perturbing acceleration in the rotating frame?

Key findings

• The derived equation of motion in the rotating frame includes Coriolis and centrifugal terms explicitly, ensuring consistency with noninertial frame physics, unlike prior models.
• The perturbing acceleration vector is expressed as $\vec{F} = (K_x x', K_y y', K_z z')$, with $K_x = (A-B)(3A+B)$, $K_y = -(A-B)^2$, and $K_z = -[4\pi G \varrho_{GM} - 2(B^2 - A^2)]$, incorporating both gravitational and frame-dependent terms.
• A new integral of motion is derived for the galactic tidal problem in the rotating frame, which is not present in conventional approaches.
• The model accounts for the Sun’s vertical oscillation ($Z_0 = 30$ pc) and its time-dependent motion, introducing modulated terms in the equations of motion via $\cos(\omega_0 t)$ and $\sin(\omega_0 t)$.
• The formulation resolves inconsistencies in earlier works—such as Dybczynski et al. (2008)—which incorrectly treated the rotating frame as inertial or neglected proper fictitious force terms.
• The method enables accurate long-term simulation of comet dynamics in the Oort cloud by providing a self-consistent dynamical framework in the rotating heliocentric frame.