Taking Projectile Motion to make Planetary Trajectories

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10Exahertz
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Hi all I have been trying to find A set of equations that can allow me to map the movement of a planetary body on a polar coordinate sheet (a 2-D Problem). As well as allow me to find out about information such as the perigee and apogee radaii.
So far i have been using Keplers equations and guessing the eccentricity of the orbit. But that is not really a good way of approaching it. However, I couldn't figure out how to do it with Keplers equations so I turned to Projectile Motion. I am currently trying to figure out how to use the basic projectile motion diff eq to map a planets orbital trajectory. Only problem I haven't tried to account for yet is that the normal projectile motion equations seem to assume a flat earth.
I have attached a screenshot of some of my work.
 
Attachments
  • Handwritten notes titled projectile motion equation: r(t) = c0 + r-dot t r-hat + one half r-dot-dot t squared r-hat, with labels r_i and v_i and GM/r^2, then r(t) = r_i + one half GM/r^2 t^2, then u''(phi) and r''(phi) equations with F = GMm/r^2.
    Screenshot_20170719-174058.png
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Discussion

It's pretty easy to apply Newton's gravitational law and the Second law of motion (any co ordinates you want) and do a step by step approach to give an animation. Start with a given velocity and position and apply the gravitational acceleration over a short time Δt. That gives you another position and velocity. The piecewise linear approach will introduce a small error but reducing the step size can reduce it as much as you want.
Re "flat Earth"; the acceleration you need is not constant g (flat Earth) but GM/r2, directed along the radius from the Sun. I did this on a Psion 3a, many years ago with a compiled 'Basic' language. You could leave it running for a long time before the orbit precessed noticeably. It's not a very sophisticated method by you can find perigee and apogee easily enough by finding the turning values of r.
 
Well I created an equation but its implicitly defined. I am not sure if the computer would have a problem with this. But I do considering i would like to be able to map this out without the assistance of a computer.
Here is what I have gotten so far(linked image), but maybe I am overthinking the simplification. Any help in simplifying this so its only dependent on r would be helpful.
 
Attachments
  • Handwritten derivation of a position vector in polar components: p(r, phi, t) = r_i r-hat + (1/2)(GM/r^2) t^2 r-hat + omega r t phi-hat, and below it p(t) = r_i r-hat + (1/2)(GM/r^2) t^2 r-hat + (l/(mu r)) t phi-hat, with phi-dot = l/(mu r^2) = omega and a note reading conservation of angular momentum.
    Screenshot_20170720-165516.png
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