Long-period comets as a tracer of the Oort cloud structure

Abstract

A previous study showed that a fingerprint of the initial shape of synthetic Oort clouds was detectable in the flux of “new” long-period comets. The present study aims to explain in detail how such a fingerprint is propagated by different classes of observable comets to improve the detection of fingerprints. It appears that three main long-term behaviors of observable comets are involved in this propagation: (1) comets that remain frozen during the entire time span and become observable only because of an increase in their orbital energy at the very end of their propagation; (2) comets whose perihelion distance performs an almost complete galactic cycle, while their galactic longitude of the ascending node and cosine of the galactic inclination remain almost constant; (3) comets whose perihelion distance and cosine of the galactic inclination perform a full galactic cycle, while their galactic longitude of the ascending node performs a half a cycle. This investigation allowed us to define four different zones for the previous perihelion distance, in which one or two of the above long-term behaviors dominate. Considering the distribution of the cosine of the ecliptic inclination and the galactic longitude of the ascending node at the previous perihelion distance, for the different zones, several fingerprints of the initial disk shape were highlighted. Such fingerprints appeared to be quite robust since they were still present considering the reconstructed orbital elements, i.e., the elements obtained from the original orbit after a backward propagation over one orbital period considering only the galactic tides.

Appendices

A Forbidden region

The ecliptic coordinates of the Galactic pole in the North Hemisphere are $l=180.02^\circ$ and $b=29.81^\circ$; i.e., the galactic plane is inclined by $\eta = 60.19^\circ$ with respect to the ecliptic plane. The origin of the galactic longitude $\varGamma$ is located toward the present galactic center, with ecliptic coordinates of $l=266.84^\circ$ and $b=-5.54^\circ$. Calling A the ascending node of the ecliptic with respect to the galactic plane, the galactic longitude of A is, thus, $L_\mathrm{{A}}=186.38^\circ$. Consequently, an object with an ecliptic inclination close to $0^\circ$ would have its galactic longitude of the ascending node close to $L_\mathrm{{A}}=186.38^\circ$. Similarly, an object with an ecliptic inclination near $180^\circ$ will have its galactic longitude of the ascending node near $6.38^\circ$ (see Leinert et al. 1998, for more details on the transformations between ecliptic and galactic coordinate systems). More generally, given an ecliptic inclination, not all values of the galactic longitude of the ascending node are allowed.

Fig. 7

Left: position of a prograde orbit with respect to the ecliptic plane and the galactic plane. Right: the same for a retrograde orbit. See text for the meaning of the angles

Considering A as the origin for the ecliptic and galactic longitude, we note ${\tilde{\Omega }}_\mathrm{{E}}$ and ${\tilde{\Omega }}_\mathrm{{G}}$ as the ecliptic and galactic longitude, respectively. Using spherical trigonometry, one can show the following:

$$\begin{aligned} \tan {\tilde{\Omega }}_\mathrm{{G}}=\frac{1}{\cos \eta }\left( \frac{\sin \tilde{\Omega _\mathrm{{E}}}}{\cos \tilde{\Omega _\mathrm{{E}}}+\frac{\tan \eta }{\tan i_\mathrm{{E}}}}\right) . \end{aligned}$$

(1)

If $|\tan i_\mathrm{{E}}| > \tan \eta$, then $\tan {\tilde{\Omega }}_\mathrm{{G}} \rightarrow \pm \infty$ as soon as $\cos \tilde{\Omega _\mathrm{{E}}} \rightarrow -\frac{\tan \eta }{\tan i_\mathrm{{E}}}$. Noting that for $x \rightarrow 0$, $\sin x \sim x$, and $\cos x -1 \sim -x^2/2$, this is also true for $|\tan i_\mathrm{{E}}| = \tan \eta$ (similarly in $\pi$). Consequently, when $i_\mathrm{{E}}\in [\eta , \pi -\eta ]$, ${\tilde{\Omega }}_\mathrm{{G}}$, and thus, $\Omega _\mathrm{{G}}$ can take any value. If $|\tan i_\mathrm{{E}}| < \tan \eta$, then $\tan {\tilde{\Omega }}_\mathrm{{G}}$ is a bound smooth function of ${\tilde{\Omega }}_\mathrm{{E}}$. Differentiating Eq. 1 with respect to ${\tilde{\Omega }}_\mathrm{{E}}$, one finds that extrema are obtained when

$$\begin{aligned} \cos {\tilde{\Omega }}_\mathrm{{E}}=-\frac{\tan i_\mathrm{{E}}}{\tan \eta }. \end{aligned}$$

(2)

The extreme values of ${\tilde{\Omega }}_\mathrm{{G}}$ are then the solutions of

$$\begin{aligned} \tan {\tilde{\Omega }}_\mathrm{{G}} = \pm \frac{1}{\cos \eta } \frac{\tan i_\mathrm{{E}}}{\sqrt{\tan ^2 \eta - \tan ^2 i_\mathrm{{E}}}}. \end{aligned}$$

(3)

Let us call ${\tilde{\Omega }}_{\mathrm{{E}}1}$ the solution such that ${\tilde{\Omega }}_{\mathrm{{E}}1} \in [0,\pi /2[$.

As shown in Fig. 7(left), one notes that when $0\le i < \eta$, when ${\tilde{\Omega }}_\mathrm{{E}}=0$ or $\pi$, ${\tilde{\Omega }}_\mathrm{{G}}=0$. Consequently, the range of values spanned by ${\tilde{\Omega }}_\mathrm{{G}}$ is $[-{\tilde{\Omega }}_{\mathrm{{E}}1},{\tilde{\Omega }}_{\mathrm{{E}}1}]$. Further, because $\Omega _\mathrm{{G}}={\tilde{\Omega }}_\mathrm{{G}}+L_\mathrm{{A}}$, the range for $\Omega _\mathrm{{G}}$ is $[L_\mathrm{{A}}-{\tilde{\Omega }}_{\mathrm{{E}}1},L_\mathrm{{A}}+{\tilde{\Omega }}_{\mathrm{{E}}1}]$.

Similarly, when $\pi -\eta \le i < \pi$, as shown in Fig. 7(right), one notes that if ${\tilde{\Omega }}_\mathrm{{E}}=0^\circ$ or $\pi$, then ${\tilde{\Omega }}_\mathrm{{G}}=\pi$. Thus, the range of values spanned by ${\tilde{\Omega }}_\mathrm{{G}}$ is $[\pi -{\tilde{\Omega }}_{\mathrm{{E}}1},\pi +{\tilde{\Omega }}_{\mathrm{{E}}1}]$, and the range for $\Omega _\mathrm{{G}}$ is $[\pi +L_\mathrm{{A}}-{\tilde{\Omega }}_{\mathrm{{E}}1},\pi +L_\mathrm{{A}}+{\tilde{\Omega }}_{\mathrm{{E}}1}]$.

In Fig. 4, the gray areas correspond to the forbidden regions for $\Omega _\mathrm{{G}}$.

When $0 \le i_\mathrm{{E}} \le \eta$, one can compute the extreme value of $i_\mathrm{{E}}$ as a function of ${\tilde{\Omega }}_\mathrm{{G}}=\Omega _\mathrm{{G}}-L_\mathrm{{G}}$. We obtain

$$\begin{aligned} \tan i_\mathrm{{E}}=\frac{\sin \eta \Vert \tan (\Omega _\mathrm{{G}}-L_\mathrm{{G}})\Vert }{\sqrt{1+\tan ^2\eta \tan ^2(\Omega _\mathrm{{G}}-L_\mathrm{{G}})}}. \end{aligned}$$

(4)

B Line $i_\mathrm{{G}}=90^\circ$

When we consider the orbital elements of an observable comet at the perihelion preceding the passage where the comet is observable, it means that we are considering a specific time at which the strength of the action of the Galactic tides is particularly strong because the perihelion distance is quickly decreasing toward the observable region.

It is well known from Heisler and Tremaine (1986) and Matese and Whitman (1992) that the strength of the Galactic tides increases with $\sin i_\mathrm{{G}}$, where $i_\mathrm{{G}}$ is the Galactic inclination. Consequently, our observable comets should have a preference for $\sin i_\mathrm{{G}}$ close to one.

One can easily find, using spherical trigonometry, that $i_\mathrm{{G}}=90^\circ$ implies that:

$$\begin{aligned} \cos i_\mathrm{{E}} = - \cos \, (\Omega _\mathrm{{G}}-6.38^\circ ) \sin \eta . \end{aligned}$$

(5)