An ellipse is defined by its semi-major axis 'a' (half its maximum diameter) and semi-minor axis 'b' (half its minimum diameter).

Ellipses can also be described using ecentricity 'e' which describes the ellipses deviation from circularity. The relationship between these parameters is:

$$b = a \sqrt{1 - e^2}$$

• $e == 0$ is a circle
• $0 < e < 1$ is an ellipse
• $e == 1$ is a parabola
• $e > 1$ is a hyperbola

An ellipse has a minimum radius 'rmin' and a maximum radius 'rmax' as measured from one of its focal points. These can be caclulated using the following equations:

$$rmin = a (1 - e)$$ $$rmax = a (1 + e)$$

Assuming a small body orbiting a much larger body with mass 'm' with no external forces other than gravity 'g', the orbital velocity 'v' can be calculated from its orbital radius 'r' using:

$$v = \sqrt{g m (\frac2r - \frac1a)}$$

The orbital period 'T' can be calculated using:

$$T = 2 \pi \sqrt{\frac{a^3}{g m}}$$

The mean motion 'n' is the angular speed for a body to complete one orbit and can be calculated using:

$$n = \frac{2 \pi}{T}$$

The mean anomaly 'M' is the fraction of an elliptical orbits period that has elapsed since the orbiting body passed minimum radius. At time 't' and where the time the body was at minimum radius is '$t_0$', the mean anomaly is calcualted using:

$$M = n (t - t_0)$$

The eccentric anomaly 'E' is useful to compute the position of a body moving in an orbit. It appears in Keplers equation which is:

$$M = E - e \sin{E}$$

The issue is that there is no closed form solution for calculating E from the above equation. Instead numerical approximation using Newton Raphson Method can be employed:

$$E_{n + 1} = E_n - \frac{(E_n - e \sin{E_n} - M_t)}{(1 - e \cos{E_n})}$$

For most elliptical orbits using an initial value $E_0 = M_t$ is usually sufficient. The approximation should then be repeated until the desired accuracy is reached.

The cartesian position of an obiting body at any time can now be calculated using the equations:

$$x = a (\cos{E} - e)$$ $$y = b \sin{E}$$

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#Physics #Math #Ellipse #Orbit