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celestial-mechanics:lagrange-coefficients-and-orbit-equation [2016/05/26 01:25]
trinh created 		celestial-mechanics:lagrange-coefficients-and-orbit-equation [2016/05/26 01:44] (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.