Lambert Problem: Determining Orbit Semi-Major Axis $a$

[Dustinsfl]

Suppose that an unknown space object is detected and tracked over a short interval of time. Radar measurements give the position vectors as
$$
r_1 = R_{\earth} (0.5\mathbf{i} + 0.6\mathbf{j} + 0.7\mathbf{k}) ,\quad
r_2 = R_{\earth} (0\mathbf{i} + 1.1\mathbf{j} + 0\mathbf{k})
$$
over an elapsed time of 13 minutes. Determine the semi-major axis $a$ of the orbit for this object's trajectory; is it an elliptical, parabolic or hyperbolic?

http://img32.imageshack.us/img32/6995/screenshotfrom201303231.png

\noindent From $\mathbf{r}_1$ and $\mathbf{r}_2$, we can determine $c$ and then $s$.
Additionally, we already have enough to find $\Delta\nu$.
\begin{alignat}{7}
r_1 & = & \quad 6681.96 & \quad &r_2 & = & 7008.1\\
c & = & \sqrt{\lVert \mathbf{r}_2 - \mathbf{r}_1\rVert} & & s & = & \frac{r_1 + r_2 +c}{2}\\
& = & 6339.06 & & & = & 10014.6\\
\Delta t & = & 13\cdot 60 & & \Delta\nu & = & \arccos\left(\frac{\mathbf{r}_1\cdot\mathbf{r}_2}{r_1r_2}\right)\\
& = & 780 & & & = & 55.1048^{\circ}
\end{alignat}
With these relationships, we can solve for $a_m$, $\beta_m$, and then $t_m$. Since $\Delta\nu < \pi$, we can use the exact value of $\beta_m$ that is returned. Moreover, $\alpha_m = \pi$ for the minimum semi-major axis.
\begin{alignat} {7}
a_m & = & \frac{s}{2} & \quad & \beta_m & = & 2\arcsin\left(\sqrt{\frac{s - c}{s}}\right)\\
& = & 5007.28 & & & = & 74.5754^{\circ}
\end{alignat}
Finally, for $t_m$, we need to solve numerically.

In[265]:= Solve[
 Sqrt[\[Mu]]*tm == Sqrt[s^3/8]*(\[Pi] - \[Beta]m + Sin[\[Beta]m]), tm]

Out[265]= {{tm -> 1573.66}}

Now, $\Delta t < t_m$ so $\alpha = \alpha_0$.

I need to find $\alpha$ and $\beta$ in order to solve
$$
\sqrt{\frac{\mu}{a^3}}\Delta t = \alpha - \beta -(\sin(\alpha) - \sin(\beta)).
$$

Formulas I have are
$$
\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{r_1 + r_2 + c}{4a}} = \sqrt{\frac{s}{2a}}
$$
and
$$
\sin\left(\frac{\beta}{2}\right) = \sqrt{\frac{r_1 + r_2 - c}{2a}} = \sqrt{\frac{s-c}{2a}}
$$
but those equations are dependent on $a$ which I need to find.
Also,
$$
\alpha - \beta = 2E_m
$$
$E_m$ is the mean anomlay.

 

[jvicens]

Dear fellow scientist,

Thank you for sharing your calculations and formulas. Based on your findings, it seems that the object's trajectory is an elliptical orbit. This is because the value of $\Delta\nu$ is less than $\pi$, which indicates that the orbit is closed and not open-ended like a parabolic or hyperbolic orbit.

To find the semi-major axis $a$, we can use the formula you mentioned: $\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{s}{2a}}$. Rearranging this equation, we get $a = \frac{s}{2\sin^2\left(\frac{\alpha}{2}\right)}$.

To solve for $\alpha$, we can use the formula $\sqrt{\frac{\mu}{a^3}}\Delta t = \alpha - \beta -(\sin(\alpha) - \sin(\beta))$. Substituting in the values we have, we get $\sqrt{\frac{\mu}{a^3}}\cdot 780 = \alpha - 74.5754^{\circ} -(\sin(\alpha) - \sin(74.5754^{\circ}))$. This equation can be solved numerically to find the value of $\alpha$, which will then give us the value of $a$.

I hope this helps in determining the semi-major axis and confirming that the object's trajectory is indeed an elliptical orbit.