The Relative Velocity between Two Confocal Keplerian Orbits

U.S. Naval Observatory, Washington DC

murison@alpheratz.net

April 24, 2007

Abstract We derive analytically the relative velocity vector and relative velocity magnitude between two particles on confocal Keplerian orbits.

Subject headings: celestial mechanics—methods: analytical

The XML version of this document is available on the web at
http://murison.alpheratz.net/papers/TwoBodyDistance/RelativeVelocity.xml

The PDF version of this document is available on the web at
http://murison.alpheratz.net/papers/TwoBodyDistance/RelativeVelocity.pdf

1 Introduction

2 Relative Elements and the Relative Distance Vector

3 Relative Velocity

4 Appendix: Optimized C Code for the Squared Relative Velocity

1 Introduction

Hilton et al. 1996 [1] and Hilton 2002 [2] used a two-body scattering angle as a filter for determining significant asteroid-asteroid close approaches. The scattering angle $ϕ$ in the center-of-mass frame is given by

\tan \frac{ϕ}{2} = \frac{G (m_{1} + m_{2})}{v^{2} b}

(1)

where $m_{1}$ and $m_{2}$ are the asteroid masses, $b$ is the impact parameter, and $v$ is the magnitude of the relative encounter velocity. Thus, the relative encounter speed is important to the observed effects of an encounter deflection. It is possible for a distant encounter at low relative speed to suffer a larger deflection than a closer encounter at higher relative speed. In this Note, we calculate explicitly the relative encounter velocity and its magnitude for two confocal Keplerian orbits, in terms of relative orbital elements ${α, e_{1}, e_{2}, ι, ω, Ω; E_{1}, E_{2}}$ . First, however, we present the relative elements in terms of the orbital elements of the two orbits with respect to some intertial frame, ${a_{k}, e_{k}, ι_{k}, ω_{k}, Ω_{k}, M_{k}}, k = 1, 2$ .

2 Relative Elements and the Relative Distance Vector

Consider the set of relative orbital elements ${α, e_{1}, e_{2}, ι, ω, Ω; E_{1}, E_{2}}$ , where

α = \frac{a_{2}}{a_{1}},

(2)

the relative orientation elements ${ι, ω, Ω}$ are as shown in Figure 1, and $E_{1} (t)$ and $E_{2} (t)$ are the eccentric anomalies of the two bodies. The transformation from separate orbital elements for each orbit to relative elements is given by (Murison and Munteanu 2007 [3])

\begin{matrix} \cos Ω \sin ι & = & \sin ω_{1} \sin ι_{2} \sin Δ Ω - \cos ω_{1} (\sin ι_{1} \cos ι_{2} - \cos ι_{1} \sin ι_{2} \cos Δ Ω) \\ \sin Ω \sin ι & = & \cos ω_{1} \sin ι_{2} \sin Δ Ω + \sin ω_{1} (\sin ι_{1} \cos ι_{2} - \cos ι_{1} \sin ι_{2} \cos Δ Ω) \\ \cos ω \sin ι & = & \sin ω_{2} \sin ι_{1} \sin Δ Ω + \cos ω_{2} (\cos ι_{1} \sin ι_{2} - \sin ι_{1} \cos ι_{2} \cos Δ Ω) \\ \sin ω \sin ι & = & - \cos ω_{2} \sin ι_{1} \sin Δ Ω + \sin ω_{2} (\cos ι_{1} \sin ι_{2} - \sin ι_{1} \cos ι_{2} \cos Δ Ω) \\ \cos ι & = & \cos ι_{1} \cos ι_{2} + \sin ι_{1} \sin ι_{2} \cos Δ Ω \end{matrix}}

(3)

where $Δ Ω = Ω_{2} - Ω_{1}$ . The eccentric anomalies can be calculated from the respective mean anomalies via the Kepler equation,

E (t) - e \sin E (t) = M (t) = n (t - τ) = M_{0} + n (t - t_{0})

(4)

where $M_{0}$ is the mean anomaly at epoch $t = t_{0}$ , $τ$ is a time of pericenter passage, and $n$ is the mean motion $2 π / T$ with $T$ the orbital period. With equations (3) and (4) we may transform from separate orbital elements with respect to some inertial frame at epoch $t_{0}$ to relative elements.

$OrbitalDistance$

Figure 1: Relative orbital elements.

In terms of the relative elements, and with respect to a coordinate frame whose $x$ axis is along the pericenter and $z$ axis is parallel to the angular momentum vector of orbit 1, the relative distance vector is

\frac{1}{a_{1}} \vec{R} (t) = \frac{1}{a_{1}} ({\vec{r}}_{2} - {\vec{r}}_{1}) = α 𝒬 (Ω, ι, ω) \cdot \vec{q} (E_{2}) - \vec{q} (E_{1})

(5)

where in general a Keplerian orbit may be expressed in the form $\vec{r} = a 𝒬 (Ω, ι, ω) \cdot \vec{q} (E)$ , where the vector

\vec{q} (E) = [\begin{matrix} \cos E - e \\ \sqrt{1 - e^{2}} \sin E \\ 0 \end{matrix}]

(6)

specifies the position along the orbit path, and the orbit orientation matrix $𝒬 (Ω, ι, ω)$ is a product of three orthogonal rotations,

𝒬 (Ω, ι, ω) = [\begin{matrix} \cos Ω \cos ω - \sin Ω \sin ω \cos ι & - \cos Ω \sin ω - \sin Ω \cos ω \cos ι & \sin Ω \sin ι \\ \sin Ω \cos ω + \cos Ω \sin ω \cos ι & - \sin Ω \sin ω + \cos Ω \cos ω \cos ι & - \cos Ω \sin ι \\ \sin ω \sin ι & \cos ω \sin ι & \cos ι \end{matrix}]

(7)

This formulation cleanly separates the orbit scale $a$ , the shape of the orbit $e$ plus the position of the particle at a given time $\vec{q}$ , and the orientation of the orbit in space $𝒬 (Ω, ι, ω)$ .

3 Relative Velocity

Differentiation of the relative distance vector yields the relative velocity vector,

\frac{1}{a_{1}} \vec{V} (t) = \frac{1}{a_{1}} \frac{d \vec{R}}{d t} = α 𝒬 (Ω, ι, ω) \cdot \frac{d \vec{q} (E_{2} (t))}{d t} - \frac{d \vec{q} (E_{1} (t))}{d t}

(8)

Now, from the Kepler equation (4), we have

\dot{E} = \frac{n}{1 - e \cos E}

(9)

Define a mass parameter

μ_{k} \equiv G (M + m_{k}), k \in {1, 2}

(10)

where $M$ is the central mass. Since $M ≫ m_{k}$ ,

\frac{μ_{2}}{μ_{1}} = \frac{M + m_{2}}{M + m_{1}} = \frac{1 + ϵ_{2}}{1 + ϵ_{1}} = 1 + ϵ_{2} - ϵ_{1} - ϵ_{1} ϵ_{2} + ϵ_{1}^{2} - \dots

(11)

where $ϵ_{k} = \frac{m_{k}}{M}$ . Thus, for the asteroids $\frac{μ_{2}}{μ_{1}} \approx 1$ and $μ_{1} \approx μ_{2} \approx G M_{⊙} \equiv μ_{⊙}$ . The mean motion and semimajor axis are related by Kepler's third law, $μ = n^{2} a^{3}$ (where, in general, $μ$ is $G$ times the sum of the two masses). Hence, we may generally write

\frac{d \vec{q}}{d t} = \frac{\partial \vec{q}}{\partial E} \dot{E} = \frac{n}{1 - e \cos E} \frac{\partial \vec{q}}{\partial E} = \frac{1}{1 - e \cos E} \sqrt{\frac{μ}{a^{3}}} [\begin{matrix} - \sin E \\ \sqrt{1 - e^{2}} \cos E \\ 0 \end{matrix}]

(12)

Let $β = α - 1 = \frac{a_{2} - a_{1}}{a_{1}}$ . Then the relative velocity, (8), becomes

\sqrt{\frac{a_{1}}{μ_{⊙}}} \vec{V} (t) = \sqrt{\frac{1}{1 + β}} \frac{1}{1 - e_{2} \cos E_{2}} 𝒬 (Ω, ι, ω) \cdot \frac{\partial \vec{q} (E_{2})}{\partial E_{2}} - \frac{1}{1 - e_{1} \cos E_{1}} \frac{\partial \vec{q} (E_{1})}{\partial E_{1}}

(13)

where the velocity is scaled by the velocity of a circular orbit of radius $a_{1}$ ,

n_{1} a_{1} = \sqrt{\frac{μ_{⊙}}{a_{1}}}

(14)

In component form, eq. (13) is

\begin{matrix} \sqrt{\frac{a_{1}}{μ_{⊙}}} \vec{V} (t) & = & \frac{1}{1 - e_{2} \cos E_{2}} \frac{\vec{B} (E_{1}, E_{2})}{\sqrt{1 + β}} - \frac{1}{1 - e_{1} \cos E_{1}} [\begin{matrix} - \sin E_{1} \\ \sqrt{1 - e_{1}^{2}} \cos E_{1} \\ 0 \end{matrix}] \\ = & \frac{1 - \frac{1}{2} β + \frac{3}{8} β^{2} - \dots}{1 - e_{2} \cos E_{2}} \vec{B} (E_{1}, E_{2}) - \frac{1}{1 - e_{1} \cos E_{1}} [\begin{matrix} - \sin E_{1} \\ \sqrt{1 - e_{1}^{2}} \cos E_{1} \\ 0 \end{matrix}] \end{matrix}

(15)

where

\vec{B} (E_{1}, E_{2}) = [\begin{matrix} C_{2} \sqrt{1 - e_{2}^{2}} \cos E_{2} - C_{4} \sin E_{2} \\ C_{1} \sqrt{1 - e_{2}^{2}} \cos E_{2} - C_{3} \sin E_{2} \\ (\cos ω \sqrt{1 - e_{2}^{2}} \cos E_{2} - \sin ω \sin E_{2}) \sin ι \end{matrix}]

(16)

and

\begin{matrix} C_{1} & = & - \sin Ω \sin ω + \cos Ω \cos ω \cos ι \\ C_{2} & = & - \cos Ω \sin ω - \sin Ω \cos ω \cos ι \\ C_{3} & = & \sin Ω \cos ω + \cos Ω \sin ω \cos ι \\ C_{4} & = & \cos Ω \cos ω - \sin Ω \sin ω \cos ι \end{matrix}}

(17)

Upon squaring (15), we find the relative velocity magnitude,

\frac{a_{1}}{μ_{⊙}} {‖ \vec{V} (t) ‖}^{2} = \frac{1 + e_{1} \cos E_{1}}{1 - e_{1} \cos E_{1}} + \frac{1}{1 + β} \frac{1 + e_{2} \cos E_{2}}{1 - e_{2} \cos E_{2}} - \frac{2}{\sqrt{1 + β}} \frac{g (E_{1}, E_{2})}{(1 - e_{1} \cos E_{1}) (1 - e_{2} \cos E_{2})}

(18)

where

\begin{matrix} g (E_{1}, E_{2}) & = & \sqrt{1 - e_{2}^{2}} \cos E_{2} (C_{1} \sqrt{1 - e_{1}^{2}} \cos E_{1} - C_{2} \sin E_{1}) \\ - \sin E_{2} (C_{3} \sqrt{1 - e_{1}^{2}} \cos E_{1} - C_{4} \sin E_{1}) \end{matrix}

(19)

An optimized C function to calculate eq. (18) is shown in the Appendix (Section 4). For completeness, expanding on $ϵ$ , we have

\begin{matrix} \frac{a_{1}}{μ_{⊙}} {‖ \vec{V} (t) ‖}^{2} & = & \frac{1 + e_{1} \cos E_{1}}{1 - e_{1} \cos E_{1}} + \frac{1 + e_{2} \cos E_{2}}{1 - e_{2} \cos E_{2}} (1 - β + β^{2} - \dots) \\ - \frac{2 g (E_{1}, E_{2})}{(1 - e_{1} \cos E_{1}) (1 - e_{2} \cos E_{2})} (1 - \frac{1}{2} β + \frac{3}{8} β^{2} - \dots) \end{matrix}

(20)

References

[1] Hilton, J.L., Seidelmann, P.K., and Middour, J. (1996), "Prospects for Determining Asteroid Masses", Astronomical Journal 112, 2319-2329.

[2] Hilton, J.L. (2002), "Asteroid Masses and Densities", in Asteroids III, eds. Bottke et al., University of Arizona Press, Tuscon, pp. 103-112.

[3] Murison, M.A., and Munteanu, A. (2007), "On the Distance Function between Two Confocal Keplerian Orbits", submitted to Astronomical Journal.

4 Appendix: Optimized C Code for the Squared Relative Velocity

double vsquared( double E1, double E2, double e1, double e2,