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I have the following system of ODEs used the propagate the trajectory of a particle in free space about a gravitating body:

$$x'(t) = v_{x}(t)$$ $$y'(t) = v_{y}(t)$$ $$v_{x}'(t) = \frac{-G M x(t)}{(x(t)^{2}+y(t)^{2})^{3/2}}$$ $$v_{y}'(t) = \frac{-G M y(t)}{(x(t)^{2}+y(t)^{2})^{3/2}}$$

Where $t$ is time, $x$ and $y$ are the Cartesian coordinate points of the particle, $v_{x}$ and $v_{y}$ are the $x$- and $y$-velocity components of the particle, $G$ is the universal gravitational constant and $M$ is the mass of the gravitating body (for example, Earth). A 3D example of a particle orbiting a larger mass body is shown below:

Example of a particle orbiting a larger body

If I solve this system of ODEs by integrating the equations using, for example, MATLAB's ode45 solver, I get values for $x$, $y$, $v_{x}$ and $v_{y}$ with respect to time $t$. However, I would also like to be able to find values for the radial velocity, $v_{r}$, of the particle with respect to time $t$ (that is, the speed at which the particle moves towards or away from the gravitating body along a radial line), without having to solve the system of ODEs using a polar coordinate formulation.

enter image description here

As such, is there a way to formulate the radial acceleration, $v_{r}'= dv_{r}/dt$ of the particle, using the available Cartesian states $(x(t),y(t),v_{x}(t),v_{y}(t))$, as a 5th equation to be added to the original system of 4 equations, so that when the system is integrated, I receive output for the radial velocity of the particle as well?

So, for example, the new system of equations becomes:

$$x'(t) = v_{x}(t)$$ $$y'(t) = v_{y}(t)$$ $$v_{x}'(t) = \frac{-G M x(t)}{(x(t)^{2}+y(t)^{2})^{3/2}}$$ $$v_{y}'(t) = \frac{-G M y(t)}{(x(t)^{2}+y(t)^{2})^{3/2}}$$ $$v_{r}'(t) = ?$$

Any help would be much appreciated

EDIT: The following formulation given by nnovich-OK is possible in Mathematica using:

$r'(t) = \frac{d}{dt}\sqrt{x(t)^{2}+y(t)^{2}}$

which is shown in the image below for a snippet of Mathematica code that numerically integrates the system of 5 ODEs:

enter image description here

However, I have not been able to get such a formulation to work in MATLAB, and as such I was hoping for a more "traditional" approach without having to take the derivative of the radius with respect to time.

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$v_r$ is tied to $x(t)$ and $y(t)$, which you already found by solving system of ODE: $$ r(t) = \sqrt{x(t)^2 + y(t)^2} $$ $$ v_r (t) = r'(t) = (\sqrt{x(t)^2 + y(t)^2})' $$

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  • $\begingroup$ Thanks nnovich-OK. The form $\frac{d}{dt}\sqrt{x(t)^{2}+y(t)^{2}}$ definitely works in Mathematica, however, I have been unable to get it to work in MATLAB (I'll update my original post to give an example) $\endgroup$ – InquisitiveInquirer Apr 12 '17 at 18:36
  • $\begingroup$ Sadly I'm not familiar with MATLAB. If trouble is complexity of the formula, you can try manually simplified variant: $v_r (t) = \frac{x(t) v_x(t) + y(t) v_y(t)}{\sqrt{x(t)^2+y(t)^2}}$ $\endgroup$ – nnovich-OK Apr 12 '17 at 18:45
  • $\begingroup$ The above equation you just gave for $v_{r}(t)$ matches my data well, thanks very much nnovich-OK, problem solved. $\endgroup$ – InquisitiveInquirer Apr 12 '17 at 18:51

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