Plotting Two Masses' Orbit Using Ellipse Equation and Initial Conditions

[ice109]

Homework Statement

plot the orbit of two masses using the equation for an ellipse and coordinate system at one of the foci

Masses Initial Positions Initial Velocities
i mi x1 x2 x3 v_1 v_ 2 v_ 3
1 1 0.651 0.585 -0.238 -0.755 -0.828 -0.865 -0.726
2 0.931 -0.096 0.000 0.357 -0.209 0.107 -0.660

Homework Equations

$$\mathbf{r} = \mathbf{r_1} - \mathbf{r_2}$$

$$r=\frac{a\cdot(1-\varepsilon^{2})}{1+\varepsilon\cdot\cos\theta};\,\!$$

$$\varepsilon = \sqrt{1 + \frac{2EL^{2}}{k^{2}\mu}}$$

$$L=\left|\mathbf{L} \right|=\left| \mathbf{r} \times \mathbf{p}\right|$$

$$\mu = \frac{1}{\frac{1}{m_{1}} + \frac{1}{m_{2}}} = \frac{m_{1}m_{2}}{m_{1} + m_{2}}$$

$$E =\frac{\mu \dot{r}^2}{2} -\frac{k}{r} + \frac{L^2}{2\mu r^2}$$

$$k = - G m_1 m_2$$

The Attempt at a Solution

so all i have to do is plot r a function of theta which seems simple enough. the equation for the conic section only has two parameters i don't know E and L. so from what i understand both are constant of motion so i was so i can just evaluate them at the initial conditions plug them into, the formula and plot? am i right?

to check i calculated the eccentricity: 1.11843. could someone check to make that's one should get for the relative orbit of the light one to the heavier one?

 

[ice109]

anyone?