Blog Week 3 - Orbital Elements and Project Spec

This weeks blog will be relatively short, as I will also be writing a Project Specification, which is due in a week today. Here is an overview of what it will contain:

•   The project specification is a more detailed description of the work to be done. Two sides of A4 should suffice. 
• This should document the initial avenue of research / development which the student follows; indicate any prototype developments which are envisaged; suggest comparative studies of different design approaches etc. The initial project plan should be drawn up after a preliminary investigation of the literature. 
• This deliverable is not formally marked, but it will inform the supervisor’s assessment of conduct. Failure to submit on time (without good reason) will have a negative effect on the conduct grade.

Following my supervisor meeting this week, it became clear that I was looking in the wrong area for the equations I needed to calculate an orbit. What I needed were Orbital Elements. These are 6 elements, some detailed in the diagram below, from this website. 

 

The actual 6 orbital elements are:

Eccentricity (e)
Semimajor Axis (a)
Inclination (i)
Longitude of the ascending node (Ω)
Argument of periapsis ()
Mean anomaly (M)

First, a couple of equations are needed to calculate the planets polar coordinates, (r, v). Kepler's equation allows us to solve the Eccentric anomaly, once we have calculated the Mean anomaly. The two equations are shown below:

 

Once E has been calculated, r and v can be:

I can make use of this data to calculate the Heliocentric Ecliptic coordinates (x,y,z) for a planet, which are an X,Y,Z coordinate system centered on the Sun in which the ecliptic lies in the X-Y plane. The equations to calculate these points are:

Now, what I really need are the Heliocentric Equatorial coordinates (X,Y,Z) for a planet, in which the coordinate system centred on the Sun with the Sun's equator lying on the X-Y plane - the type I will base my system around. The equations to calculate these points are:

 

Onto the most important part of the blog - this data is confirmed in a PDF document on the JPL Solar System Dynamics website (a NASA source). The PDF also lists the Keplerian Elements for the time period 1800 AD - 2050 AD, an invaluable resource for me to be able to calculate the planetary orbits for this project. It should be noted that the website states that the data for these positions is only approximate, but it should be efficient enough for this project.