Context Navigation

← Previous Post
Next Post →

Setting Orbital Parameters

I've had to do this a couple of times - and thought it might be good to post this here…

Given 2 bodies in orbit about each other - calculate the initial positions and velocities given the semi-major axis, the masses, and the eccentricity.

If we assume the orbit is about the z-axis, and that aphelion occurs when both bodies are on the x-axis then at aphelion the velocities must be in the y-axis and the locations are on the x-axis. This leaves us with 4 unknowns v_1,v_2,r_1,r_2.

The center of mass residing at the origin implies that m_1r_1=m_2r_2

and we have r_1+r_2 = a which can be solved for

r_1=a\frac{m_2}{M}

r_2=a\frac{m_1}{M}

We also must have zero net momentum so

m_1v_1=m_2v_2

which implies that

\frac{v_1}{r_1} = \frac{v_2}{r_2}

Which is consistent with them having the same angular velocity at aphelion.

Now h=\frac{L}{\mu} = \left( \frac{r_1 v_1}{m_2} + \frac{r_2 v_2}{m_1} \right ) M = a \left ( v_1 + v_2 \right )

And we know that twice the entire area of the ellipse is swept out in one orbital period

A=\pi a b = \pi a^2(1-e^2)

T=2 \pi \sqrt{\frac{a^3}{GM}}

we have

h=a \left ( v_1+v_2 \right ) = \frac{2A}{T}=\sqrt{a(1-e^2)GM}

v_1+v_2=\sqrt{\frac{GM(1-e^2)}{a}}

which then gives us

v_1=\frac{\sqrt{\frac{GM(1-e^2)}{a}}}{1+\frac{m_1}{m_2}}

v_2=\frac{\sqrt{\frac{GM(1-e^2)}{a}}}{1+\frac{m_2}{m_1}}

which together with

r_1=a\frac{m_2}{M}

r_2=a\frac{m_1}{M}

define

\mathbf{x_1}=[r_1,0,0] \mathbf{v_1}=[0,v_1,0]

and

\mathbf{x_2}=[-r_2,0,0] \mathbf{v_2}=[0,-v_2,0]

Posted: 11 years ago
Author: Jonathan
Categories: (none)

Comments

No comments.

Download in other formats:

RSS Feed

astrobear

Context Navigation

Setting Orbital Parameters

Comments

Download in other formats: