Partial Derivatives of Observables with Respect to Two-Body Orbital Elements

U.S. Naval Observatory, Washington, DC

murison@usno.navy.mil

18 December, 2006

Abstract We calculate the partial derivatives of the observable quantities ${α, δ, r, \dot{r}}$ , where $α$ is the right ascension and $δ$ is the declination, with respect to the two-body orbital elements $\{a, e, ι, Ω, ω, E\}$ , where $E$ is the eccentric anomaly. These observables are the only ones available, at least in principle, for the determination of orbital elements of orbiting bodies. We avoid the complexity of expressing the observables directly in terms of orbital elements by instead expressing them in Cartesian coordinates and then invoking the chain rule for derivatives.

Subject headings: celestial mechanics—two-body problem—astrometry—observables

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1 Components of the Two-Body State Vector

1.1 The Position Vector

1.2 Scalar Distance

1.3 Radial Velocity

1.4 Velocity Vector

2 Partial Derivatives of the State Vector

2.1 Position

2.2 Velocity

3 Observables and the State Vector

3.1 Astrometric Angles

3.2 Scalar Distance

3.3 Radial Velocity

4 Partials of the Observables

4.1 Astrometric Angles

4.2 Scalar Distance

4.3 Radial Velocity

4.4 Evaluation of the Partial Derivatives

5 Appendix: Explicit State Vector Derivatives

1 Components of the Two-Body State Vector

1.1 The Position Vector

$The Orbital Elements.$

Figure 1: The orbital elements and angular momentum vector.

Figure 1 shows an orbit with the standard set of orbital elements ${a, e, ι, ω, Ω, θ}$ , where $θ$ is the true anomaly and $\vec{h}$ is the angular momentum (per mass). To reduce expression complexity, we shall use the eccentric anomaly $E$ as the fast angular variable instead of the true anomaly. See Figure 2.

An unperturbed two-body elliptical orbit radius vector can be written

\vec{r} (E) = a 𝒬 (Ω, ι, ω) \cdot \vec{q} (E)

(1)

where $a$ is the semimajor axis, $e$ the orbital eccentricity, $ι$ the orbital inclination, $Ω$ the longitude of the ascending node, and $ω$ the argument of pericenter. The position/shape vector is

\vec{q} (E) = ⟨ \cos E - e, \sqrt{1 - e^{2}} \sin E, 0 ⟩

(2)

and the orbit orientation matrix is

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}]

(3)

$Eccentric and TrueAnomalies$

Figure 2: The geometric relation between true anomaly

θ

and eccentric anomaly

E

where

𝒬 (Ω, ι, ω) = ℛ_{z} (Ω) \cdot ℛ_{x} (ι) \cdot ℛ_{z} (ω)

(4)

is the composition of the rotation matrices

ℛ_{x} (ψ) = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ψ & - \sin ψ \\ 0 & \sin ψ & \cos ψ \end{matrix}]

(5)

ℛ_{z} (ψ) = [\begin{matrix} \cos ψ & - \sin ψ & 0 \\ \sin ψ & \cos ψ & 0 \\ 0 & 0 & 1 \end{matrix}]

(6)

The matrices (5) and (6) individually rotate a vector counterclockwise around the respective x or z coordinate axis of an inertial frame. (Alternatively, one may interpret these as rotating the coordinate frame clockwise about the respective axes.)

The formulation (1) cleanly separates the orbit scale $a$ , the shape of the orbit plus the position of the particle at a given time $\vec{q} (t)$ (with $E (t)$ obtained via the Kepler equation), and the orientation of the orbit in space $𝒬 (Ω, ι, ω)$ .

1.2 Scalar Distance

The matrix $𝒬 (Ω, ι, ω)$ is orthogonal, since it is composed of orthogonal matrices. Hence, $𝒬 {(Ω, ι, ω)}^{⊤} \cdot 𝒬 (Ω, ι, ω) = ℐ$ , where $ℐ$ is the identity matrix. The magnitude of the radius vector can therefore formally be derived from (1) as

\begin{matrix} r^{2} & \equiv & {\vec{r}}^{⊤} \cdot \vec{r} \\ = & a^{2} ({\vec{q}}^{⊤} \cdot 𝒬^{⊤}) \cdot (𝒬 \cdot \vec{q}) \\ = & a^{2} {\vec{q}}^{⊤} \cdot (𝒬^{⊤} \cdot 𝒬) \cdot \vec{q} \\ = & a^{2} ({\vec{q}}^{⊤} \cdot ℐ \cdot \vec{q}) \\ = & a^{2} ({\vec{q}}^{⊤} \cdot \vec{q}) \\ = & a^{2} q^{2} \end{matrix}

(7)

From the definition (2), $q^{2} = {(1 - e \cos E)}^{2}$ . Thus, the scalar distance is

r (E) = \sqrt{{\vec{r} (E)}^{⊤} \cdot \vec{r} (E)} = a \sqrt{{\vec{q} (E)}^{⊤} \cdot \vec{q} (E)} = a (1 - e \cos E)

(8)

1.3 Radial Velocity

The radial velocity is the time derivative of the magnitude of the position vector. If we write the velocity vector in radial and perpendicular parts,

\vec{v} = \dot{r} \hat{r} + r \dot{θ} \hat{θ}

(9)

One immediately sees that the projection onto the line of sight of an observer at the coordinate origin is

\vec{v} \cdot \hat{r} = \dot{r}

(10)

We shall use (8) to calculate $\dot{r}$ . We first need $\dot{E}$ . Thus, take the time derivative of the Kepler equation

E - e \sin E = M = n (t - τ)

(11)

where $n$ is the mean motion and $τ$ is the time of pericenter passage, to get

\dot{E} (1 - e \cos E) = n

(12)

which by (8) is just

\dot{E} = \frac{n a}{r}

(13)

Differentiating (8), we find our result,

\dot{r} = \frac{n a^{2}}{r} e \sin E = \frac{n a}{1 - e \cos E} e \sin E

(14)

1.4 Velocity Vector

The two-body velocity is, by differentiating (1),

\vec{v} = a 𝒬 (Ω, ι, ω) \cdot \frac{\partial \vec{q} (E)}{\partial E} \dot{E}

(15)

From (2), (8), and (13) follows the result

\begin{matrix} \vec{v} & = & \frac{n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot \frac{\partial \vec{q} (E)}{\partial E} \\ = & \frac{n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot [\begin{matrix} - \sin E \\ \sqrt{1 - e^{2}} \cos E \\ 0 \end{matrix}] \end{matrix}

(16)

2 Partial Derivatives of the State Vector

2.1 Position

From (1) it is straightforward to calculate the partial derivatives of the position vector with respect to the orbital parameters $\{a, e, ι, Ω, ω, E\}$ . First, though, recall that $μ = n^{2} a^{3}$ (Kepler's third law), where $μ = G (m_{1} + m_{2})$ conveniently hides most of the physical units. Thus, for the derivative with respect to semimajor axis, we see by the Kepler equation, (11), that the eccentric anomaly is an explicit function of semimajor axis, so that

\frac{\partial E}{\partial a} = - \frac{3 n}{2 r} (t - τ)

(17)

We then find

\begin{matrix} \frac{\partial \vec{r}}{\partial E} & = & a 𝒬 (Ω, ι, ω) \cdot \frac{\partial \vec{q} (E)}{\partial E} \\ = & a 𝒬 (Ω, ι, ω) \cdot ⟨ - \sin E, \sqrt{1 - e^{2}} \cos E, 0 ⟩ \end{matrix}

(18)

\begin{matrix} \frac{\partial \vec{r}}{\partial a} & = & 𝒬 (Ω, ι, ω) \cdot (\vec{q} (E) + a \frac{\partial \vec{q} (E)}{\partial E} \frac{\partial E}{\partial a}) \\ = & \frac{\vec{r}}{a} - \frac{3}{2} \frac{n a}{r} (t - τ) 𝒬 (Ω, ι, ω) \cdot ⟨ - \sin E, \sqrt{1 - e^{2}} \cos E, 0 ⟩ \end{matrix}

(19)

\begin{matrix} \frac{\partial \vec{r}}{\partial e} & = & a 𝒬 (Ω, ι, ω) \cdot \frac{\partial \vec{q} (E)}{\partial e} \\ = & - a 𝒬 (Ω, ι, ω) \cdot ⟨ 1, \frac{e \sin E}{\sqrt{1 - e^{2}}}, 0 ⟩ \end{matrix}

(20)

\frac{\partial \vec{r}}{\partial {Ω, ι, ω}} = a \frac{\partial 𝒬 (Ω, ι, ω)}{\partial {Ω, ι, ω}} \cdot \vec{q} (E)

(21)

2.2 Velocity

For the quantity $n a$ that appears in the velocity terms (cf. eq. (16)), we have

n a = \sqrt{\frac{μ}{a}}

(22)

One must be careful to not naively take $\partial (n a) / \partial a = n$ . Right away, something must be wrong with that, since $n a$ is a velocity (the circular velocity of an orbit with radius $a$ ), and physical intuition indicates that an increase in semimajor axis should cause $n a$ to decrease. Instead we must take the derivative of the right-hand side of (22). Hence,

\frac{\partial (n a)}{\partial a} = \frac{\partial}{\partial a} \sqrt{\frac{μ}{a}} = - \frac{1}{2} \sqrt{\frac{μ}{a^{3}}} = - \frac{1}{2} n

(23)

One must also remember the dependence of $E$ on semimajor axis in the derivative of $\vec{v}$ with respect to $a$ (cf. eq. (17)). The partials of the velocity vector (16) then become

\begin{matrix} \frac{\partial \vec{v}}{\partial a} & = & - \frac{1}{2} \frac{n}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot \frac{\partial \vec{q} (E)}{\partial E} + \frac{n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot \frac{\partial^{2} \vec{q} (E)}{{\partial E}^{2}} \frac{\partial E}{\partial a} \\ = & - \frac{1}{2} \frac{\vec{v}}{a} + \frac{3}{2} (t - τ) {(\frac{n a}{r})}^{2} 𝒬 (Ω, ι, ω) \cdot [\begin{matrix} \cos E \\ \sqrt{1 - e^{2}} \sin E \\ 0 \end{matrix}] \end{matrix}

(24)

\begin{matrix} \frac{\partial \vec{v}}{\partial e} & = & \frac{n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot (\frac{\cos E}{1 - e \cos E} \frac{\partial \vec{q} (E)}{\partial E} + \frac{\partial^{2} \vec{q} (E)}{\partial E \partial e}) \\ = & \frac{n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot (\frac{\cos E}{1 - e \cos E} [\begin{matrix} - \sin E \\ \sqrt{1 - e^{2}} \cos E \\ 0 \end{matrix}] + [\begin{matrix} 0 \\ \frac{- e \cos E}{\sqrt{1 - e^{2}}} \\ 0 \end{matrix}]) \end{matrix}

(25)

\begin{matrix} \frac{\partial \vec{v}}{\partial {Ω, ι, ω}} & = & \frac{n a}{1 - e \cos E} \frac{\partial 𝒬 (Ω, ι, ω)}{\partial {Ω, ι, ω}} \cdot \frac{\partial \vec{q} (E)}{\partial E} \\ = & \frac{n a}{1 - e \cos E} \frac{\partial 𝒬 (Ω, ι, ω)}{\partial {Ω, ι, ω}} \cdot ⟨ - \sin E, \sqrt{1 - e^{2}} \cos E, 0 ⟩ \end{matrix}

(26)

\begin{matrix} \frac{\partial \vec{v}}{\partial E} & = & \frac{- n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot (\frac{e \sin E}{1 - e \cos E} \frac{\partial \vec{q} (E)}{\partial E} - \frac{\partial^{2} \vec{q} (E)}{{\partial E}^{2}}) \\ = & \frac{- n a}{1 - e \cos E} 𝒬 (Ω, ι, ω) \cdot (\frac{e \sin E}{1 - e \cos E} [\begin{matrix} - \sin E \\ \sqrt{1 - e^{2}} \cos E \\ 0 \end{matrix}] + [\begin{matrix} \cos E \end{matrix} \end{matrix}