Proximity Operations About Apophis Through Its 2029 Earth Flyby

Abstract

The dynamics and control of a satellite in proximity to the asteroid Apophis across its Earth close approach in 2029 is evaluated and investigated. First, the feasibility of carrying out close proximity operations about Apophis when in its heliocentric orbit phase is evaluated and shown to be feasible. Then three different types of close proximity motion relative to Apophis are analyzed that will enable a spacecraft to take observations throughout the Earth close approach. These are maintaining a relative orbit that is somewhat distant from Apophis, hovering along the Earth–Apophis line, or maintaining orbit about Apophis through the flyby. Each of these are shown to be feasible, albeit challenging, and some basic aspects of these operations are noted and discussed.

Appendix

The following solution has been developed for relative motion about an eccentric orbit [19, 20]. Here we note that these previous solutions can be extended without loss of generality to the hyperbolic case as well. Define a state vector as ${\bar{\Xi }}_{0} = \left[ {X}_{0}, Y_{0}, Z_{0}, X_{0}', Y_{0}', Z_{0}' \right]$. Then the general orbit solution for linearized motion about an eccentric or hyperbolic orbit can be specified as the linear mapping

$$\begin{aligned} {\bar{\Xi }}_{0}= & {} \Phi (f,f_{o}) {\bar{\Xi }}_{o}, \end{aligned}$$

(24)

where $\Phi \in {\mathbf {R}}^{6\times 6}$ is the state transition matrix for the system. The entries of $\Phi$ can be written out in detail as

$$\begin{aligned} \Phi= & {} \left[ \begin{array}{cccccc} \phi _{XX} &{} \phi _{XY} &{} 0 &{} \phi _{XX'} &{} \phi _{XY'} &{} 0 \\ \phi _{YX} &{} \phi _{YY} &{} 0 &{} \phi _{YX'} &{} \phi _{YY'} &{} 0 \\ 0 &{} 0 &{} \phi _{ZZ} &{} 0 &{} 0 &{} \phi _{ZZ'} \\ \phi _{X'X} &{} \phi _{X'Y} &{} 0 &{} \phi _{X'X'} &{} \phi _{X'Y'} &{} \\ \phi _{Y'X} &{} \phi _{Y'Y} &{} 0 &{} \phi _{Y'X'} &{} \phi _{Y'Y'} &{} 0 \\ 0 &{} 0 &{} \phi _{Z'Z} &{} 0 &{} 0 &{} \phi _{Z'Z'} \end{array} \right] , \end{aligned}$$

(25)

where we inserted zeros in all of the cross coupling terms between the out-of-plane and in-plane terms. The remaining terms are then, taking $f_{o} = 0$,

$$\begin{aligned} \phi _{XX}= & {} \frac{1}{1-e}\left[ 4 + 2e - 3\cos f - 3e\cos ^{2} f - 3 e (2+e)\sin f(1+e\cos f) L \right] , \end{aligned}$$

(26)

$$\begin{aligned} \phi _{XY}= & {} 0, \end{aligned}$$

(27)

$$\begin{aligned} \phi _{XX'}= & {} \frac{1}{1+e} \sin f (1+e\cos f), \end{aligned}$$

(28)

$$\begin{aligned} \phi _{XY'}= & {} \frac{1}{1-e} \left[ 2 + 2e - 2\cos f - 2e\cos ^{2} f - 3 e (1+e)\sin f(1+e\cos f) L \right] , \end{aligned}$$

(29)

$$\begin{aligned} \phi _{YX}= & {} \frac{1}{1-e} \left[ 3\sin f ( 2 + e \cos f) - 3 (2+e)(1+e\cos f)^{2} L \right] , \end{aligned}$$

(30)

$$\begin{aligned} \phi _{YY}= & {} 1, \end{aligned}$$

(31)

$$\begin{aligned} \phi _{YX'}= & {} \frac{1}{1+e} \left[ \cos f (2+e\cos f) - (2+e)\right] , \end{aligned}$$

(32)

$$\begin{aligned} \phi _{YY'}= & {} \frac{1}{1-e}\left[ 2\sin f (2+e\cos f) - 3(1+e)(1+e\cos f)^{2} L \right] , \end{aligned}$$

(33)

$$\begin{aligned} \phi _{X'X}= & {} \frac{1}{1-e}\left[ 3\sin f + 3 e \sin 2f - 3 e (2+e) ( \cos f + e\cos 2f )L - \frac{3e(2+e)\sin f}{1+e\cos f}\right] , \end{aligned}$$

(34)

$$\begin{aligned} \phi _{X'Y}= & {} 0, \end{aligned}$$

(35)

$$\begin{aligned} \phi _{X'X'}= & {} \frac{1}{1+e}\left[ \cos f + e \cos 2f\right] , \end{aligned}$$

(36)

$$\begin{aligned} \phi _{X'Y'}= & {} \frac{1}{1-e} \left[ 2 \sin f + 2 e \sin 2f - 3e (1+e)(\cos f + e\cos 2f)L - \frac{3e(1+e)\sin f}{1+e\cos f}\right] , \end{aligned}$$

(37)

$$\begin{aligned} \phi _{Y'X}= & {} \frac{1}{1-e} \left[ 6\cos f + 3 e \cos 2f - 3(2+e)(1 - 2 e \sin f (1+e\cos f)L) \right] , \end{aligned}$$

(38)

$$\begin{aligned} \phi _{Y'Y}= & {} 0, \end{aligned}$$

(39)

$$\begin{aligned} \phi _{Y'X'}= & {} \frac{-2\sin f(1+e\cos f) }{1+e}, \end{aligned}$$

(40)

$$\begin{aligned} \phi _{Y'Y'}= & {} \frac{1}{1-e} \left[ 4 \cos f + 2 e \cos 2f - 3(1+e)(1 - 2e\sin f(1+e\cos f)L) \right] , \end{aligned}$$

(41)

$$\begin{aligned} \phi _{ZZ}= & {} \cos f, \end{aligned}$$

(42)

$$\begin{aligned} \phi _{ZZ'}= & {} \sin f, \end{aligned}$$

(43)

$$\begin{aligned} \phi _{Z'Z}= & {} -\sin f, \end{aligned}$$

(44)

$$\begin{aligned} \phi _{Z'Z'}= & {} \cos f, \end{aligned}$$

(45)

where we note the function L is defined as

$$\begin{aligned} L(f)= & {} \int \frac{ df}{(1+e\cos f)^{2}} \end{aligned}$$

(46)

$$\begin{aligned}= & {} \sqrt{\frac{\mu }{p^{3}}} t, \end{aligned}$$

(47)

where t is the time from perihelion. Thus we see that L will linearly increase in time, and could lead to an overall secular drift.