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celestial-mechanics:lagrange-coefficients-and-orbit-equation [2016/05/26 00:25] trinh created |
celestial-mechanics:lagrange-coefficients-and-orbit-equation [2016/05/26 00:44] (current) |
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====== Solution of the orbit equation from initial conditions ====== | ====== Solution of the orbit equation from initial conditions ====== | ||

+ | |||

+ | How is the orbit equation solved given initial position and velocity? | ||

+ | |||

+ | First, we create a coordinate system through the perifocal frame. Let $(\bar{x},\bar{y},\bar{z})$ be a new coordinate system where $\bar{x}$ is directed through the periapsis. Let $\bar{y}$ be directed at 90 degrees true anomaly (so in the plane of the orbit), and $\bar{z}$ taken perpendicular to both. | ||

+ | |||

+ | We then have | ||

+ | \begin{gather} | ||

+ | \mb{r} = \bar{x} \hat{p} + \bar{y} \hat{q} \\ | ||

+ | \mb{v} = \dot{\bar{x}} \hat{p} + \dot{\bar{y}}\hat{q} | ||

+ | \end{gather} | ||

+ | for unit vectors in the plane. | ||

+ | |||

+ | It can be shown that, given initial conditions, $\mb{r}_0$ and $\mb{v}_0$, | ||

+ | \begin{gather} | ||

+ | \mb{r} = f \mb{r}_0 + g \mb{v}_0 \\ | ||

+ | \mb{v} = \dot{f} \mb{r}_0 + \dot{g} \mb{v}_0 | ||

+ | \end{gather} | ||

+ | where | ||

+ | \begin{equation} | ||

+ | f = \frac{\bar{x}\dot{\bar{y_0}} - \bar{y}\dot{\bar{x_0}}}{h} \qquad | ||

+ | g = \frac{-\bar{x}\bar{y_0} + \bar{y}\bar{x_0}}{h} | ||

+ | \end{equation} | ||

+ | and $h$ is the constant magnitude of the angular momentum, now related by | ||

+ | \begin{equation} | ||

+ | h = \bar{x_0}\dot{\bar{y_0}} - \bar{y_0}\dot{\bar{x_0}}. | ||

+ | \end{equation} | ||

+ | |||

+ | Basically, after a lot of work, you can show that $f$ and $g$ and their derivatives can instead be written in terms of | ||

+ | \begin{equation} | ||

+ | f = f(\Delta \theta; r, \mb{r}_0, \mb{v}_0) \qquad | ||

+ | g = g(\Delta \theta; r, \mb{r}_0, \mb{v}_0), | ||

+ | \end{equation} | ||

+ | where $r$ is known from the solution of the orbit equation, and $\Delta \theta$ is the desired change in the true anomaly from its initial value. | ||