Dispersion velocity revisited

Abstract

The aim of this paper is to provide an analytical tool, which might improve models in which the particle-in-a-box approach has been applied and that may be also used when the thin disk approximation could not be longer appropriate. The dispersion velocity is the root-mean-square planetesimal, asteroid, or Kuiper belt object velocity with respect to the local mean circular orbit. This velocity is a function of the object orbital eccentricity and inclination. We calculate a general expression of the dispersion velocity for the planar case in which the object’s orbit has no inclination with respect to the local mean circular orbit and for the spatial case in which it has an inclined orbit. Our general expression of the square of the dispersion velocity may be expanded around any value of e for the planar and spatial cases, being in space an exact solution of the orbital inclination i. We expanded our expression around $e=0$ with $i=e/2$ to study solid accretion rates and collision probabilities. We find that in the whole range of eccentricities and inclinations, our results are lower than solid accretion rates and collision probabilities computed by using the expressions of the dispersion velocity usually adopted in the literature. We apply our expressions of the square of the dispersion velocity expanded around e=0 and up to sixth order in e in our numerical model of planetary formation with planetesimal fragmentation and in our model of the collisional frequency on large asteroids. Our formalism, although generally giving lower values than previous approximations, validates the formerly used estimates for the applications presented here. In addition, we calculate the statistical velocity dispersion obtaining a straightforward expression as a function of the eccentricity.

Appendices

Jacobi integral

1.1 Planar case

In the frame of the restricted three-body problem (Danby 1992; Kaula 1968) for orbital inclination $i=0^{o}$, the square of the relative velocity $v^{2}_{2Saf}$ in the protoplanet rotational system is given by Safronov (1972):

$$\begin{aligned} v^{2}_{2Saf} = v_{0}^2 \left[ 3 - \left( \frac{1-e^2}{1+ e \cos {\varphi }}\right) - 2 (1 + e \cos {\varphi })^{1/2} \right] . \end{aligned}$$

(59)

We average Eq. (59) over one orbital period as:

$$\begin{aligned} <v^{2}_{2Saf} >= \frac{1}{2 \pi } \int _{0}^{2 \pi }{ (v^{2}_{2Saf})\; d\varphi }. \end{aligned}$$

(60)

Then,

$$\begin{aligned} \langle v^{2}_{2Saf} \rangle = v_{0}^2 \left[ 3 - \left\langle \frac{1-e^2}{1+ e \cos {\varphi }}\right\rangle - 2 \left\langle (1 + e \cos {\varphi })^{1/2}\right\rangle \right] , \end{aligned}$$

(61)

where

$$\begin{aligned} \left\langle \frac{1-e^2}{1 + e \cos {\varphi }}\right\rangle = (1-e^2)^{1/2}. \end{aligned}$$

(62)

Substituting Eqs. (62) and (17) in Eq. (61), we arrive to Eq.(18), i.e., $\langle v^{2}_{2Saf}\rangle$=$v^{2}_{DP}$.

1.2 Spatial case

In a reference system rotating with the protoplanet around the star, the squared relative velocity $v_{3Saf}^2$ of a planetesimal with an eccentric, inclined orbit with respect to the circular orbit of the protoplanet (Danby 1992; Kaula 1968) at the point of intersection of the orbits is given by

$$\begin{aligned} v_{3Saf}^2 = v_0^2 \left[ 3 - \left( \frac{1-e^2}{1+e\cos {\varphi }}\right) - 2 \cos {i} (1+e\cos {\varphi })^{1/2}\right] . \end{aligned}$$

(63)

If $i=0^{o}$, Eq. (63) equals Eq. (59), i.e., $v_{D3Saf}^2 = v_{D2Saf}^2$. Averaging Eq. (63) over one orbital period

$$\begin{aligned} \langle v^{2}_{3Saf} \rangle =v_{0}^2 \left[ 3 - \left\langle \frac{1-e^2}{1+ e \cos {\varphi }}\right\rangle - 2 \cos {i} \left\langle (1+e\cos {\varphi })^{1/2}\right\rangle \right] , \end{aligned}$$

(64)

and substituting Eq. (62) in the second term and Eq. (17) in the third term of the right hand of Eq. (64), we arrive to Eq. (25), i.e., $\langle v^{2}_{3Saf} \rangle =v_{DS}^2$. Note that when $i=0^{o}$, $v_{DS}^2 = v_{DP}^2=\langle v^{2}_{2Saf}\rangle =\langle v^{2}_{3Saf} \rangle$.

Planar case plus the average over a vertical oscillation

The epicyclic approximation (Binney and Tremaine 1987) describes the motion of a particle in the meridional plane, where the particle motion is given by a simple harmonic oscillation in the XY plane plus a simple harmonic oscillation in the vertical direction. In a similar way, for (e, i) $<<$ 1, the square of the standard dispersion velocity in space $v^{2}_{3sdt}$ (the square of Eq. (33), Lissauer and Stewart (1993)) may be obtained as the first term of Eq. (19) plus the contribution of the average over a vertical oscillation.

From Fig.1, the vertical component of the planetesimal position measured with respect to the XY plane is

$$\begin{aligned} z= R \sin {(\omega +\varphi )} \sin {i}. \end{aligned}$$

(65)

Then, deriving Eq. (65)

$$\begin{aligned} {\dot{z}}= {\dot{R}} \sin {(\omega +\varphi )} \sin {i} + R {\dot{\varphi }} \cos {(\omega +\varphi )} \sin {i}. \end{aligned}$$

(66)

Substituting Eqs. (7) and (8), in Eq. (66) we obtain

$$\begin{aligned} {\dot{z}}= & {} \frac{v_{0} \sin {i} }{(1-e^2)^{1/2} (1 + {\tilde{a}})^{1/2}} [(1 + e \cos {\varphi }) \cos {(\omega +\varphi )} \nonumber \\{} & {} + e \sin {\varphi } \sin {(\omega +\varphi )} ]. \end{aligned}$$

(67)

Substituting Eq. (9) in Eq. (67), rising to the square and averaging in the way shown in Eq. (13), we get

$$\begin{aligned} \langle {\dot{z}}^2 \rangle = v_{0}^2 \sin ^2 {i} \left( 1 - \frac{(1-e^2)^{1/2}}{2}\right) . \end{aligned}$$

(68)

Finally, developing Eq. (68) in power series around $(e=0,i=0)$, approximating $\sin {i}$ $\sim$ i, adding the first term of Eq. (19) and keeping terms up to second order in (e,i), we arrive to the square of Eq. (33).