For hyperbolic orbits, the orbit curve approaches a straight line at great distance from the orbiting centre. The more eccentric the orbit, the greater the angle between these asymptote straight lines.
ϑ
The number of days to complete a full orbit is calculated by dividing the current "Areal Velocity" (or sweep) into the area of the deduced ellipse:

Period (seconds) = π × a × b / Av

where a and b are the major and minor radii of the deduced ellipse in metres.
Av is the Areal Velocity in (metres2 second-1)

The period will be constant while the two body solution of planet and sun is an accurate approximation. In reality, effects from other planets disturb the elliptical orbit, hence why the period and areal velocity may be seen to change. A more accurate orbital period would need to take into account these effects.
The position is calculated from the centre of mass of the two objects, this is most often likely very close to the centre of the largest of the two objects, ie sun. The three numbers are the vector coordinates for the X,Y and Z axis directions. The X/Z plane is designated as the elliptical plane (the common orbital plane for the solar system). The directions of the X and Z axis are arbitary.

The units used are astronomical units (AU) for the primary and secondary bodies, ie sun and planets. One astronomical unit (AU) is defined as the average distance of the Earth from the Sun.

1 AU = 149597870700 metres

For moons we use a Lunar distance (LD), which is defined as the average distance of the Earth's moon (Luna) from the Earth.

1 LD = 384402000 metres = 0.002569569 AU

These Earth-centric units are used for the convenience of being able to conceptualise the distances.
The distance from the centre of mass of the two objects, this is most often likely very close to the centre of the largest of the two objects, ie sun.

The units used are astronomical units (AU) for the primary and secondary bodies, ie sun and planets. One astronomical unit (AU) is defined as the average distance of the Earth from the Sun.

1 AU = 149597870700 metres

For moons we use a Lunar distance (LD), which is defined as the average distance of the Earth's moon (Luna) from the Earth.

1 LD = 384402000 metres = 0.002569569 AU

These Earth-centric units are used for the convenience of being able to conceptualise the distances.
The position angle from the centre of the orbit ellipse - from the main axis (containing the centre of mass). The angle here is always positive between 0 and 180°

This is a dynamically calculated figure derived from the current orbital distance and velocity to deduce the ellipse properties.
The centre of mass of the two bodies is one of the ellipse focal points.

ϑ
The following equations relate the distance to focal point (orbital distance) to the angle and and ellipse radii.

r2 = a2sin2ϑ + b2cos2ϑ

where ϑ is angle from major axis and r is distance from centre

the cosine rule also gives us:

f2 + r2 - 2frcosϑ = d2

where f is the length of the triangle base from the ellpse centre to the focal point, and d is the current distance from focal point (distance form centre of mass) and has been solved from solving the closest approach using the orbital energy.
The closest and furtherest approach are calculated for common centre of mass of the two bodies. With a planet and sun, this is generally very close to the centre of the sun. For Jupiter and the sun, this centre just clears the surface of the sun.

To calculate the approaches, the centre of mass is treated as the foci of the orbit ellipse with a reduced mass of m13/(m1+m2)2 where m2 is the mass (kg) of the body being analyzed and m1 is the mass (kg) of the body it is most effected by gravitationally (ie, the sun).

The orbit where only two bodies are involved obeys two fundamental rules in Newtonian physics: conservation of angular momentum and conservation of energy.
This is represented by the following equations:

V × R × sinϑ = P
E = m2V2/2 - G×Mr×m1/2×R

ϑ
V⃗
R⃗

  • E is the constant energy (kg.m.s-2)
  • V is the velocity of the analyzed object with respect to the centre of mass
  • P is the Areal Velocity (metres2.second-1) - ie angular momentum divided by the mass
  • m1(kg) is the mass of the analyzed body
  • Mr is the reduced mass (kg) described above
  • ϑ is the angle between the velocity tangent and the position from centre of mass position
  • R (metres) is the distance from the centre of mass
  • G is the gravitational constant

At the closest and furtherest approaches, the velocity tangent is orthogonal and sinϑ becomes 1. The two equations reduce to a quadratic:

Er×R2/2 + G×Mr×R/2 - P2/2 = 0 (note: Er = E/m2)

The two solutions for R give the closest and furtherest approaches respectively assuming the body is in orbit.
Kepler's second law of planetary motion states:

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time

The rate at which this line sweeps out area is called the Areal Velocity. This is related to the angular momentum of the orbiting body divided by it's mass. The Areal Velocity can be calculated by:

V × R × sinϑ / 2 = P

  • P is the Areal Velocity (metres2.second-1) - ie angular momentum divided by the mass (and halved)
  • R (metres) is the distance from the centre of mass
  • ϑ is the angle between the velocity tangent and the position from centre of mass position
ϑ
V⃗
R⃗
ΔA
Above, ΔA is the small segment of area swept over a small increment of time Δt.
ΔA/Δt is the Areal Velocity
The major and minor axii of the elipse (A & B), can be deduced from the focii distances derived from the closest and furtherest approach distances.
A
B
f

The foci distance f has the following properties:

f2 + a2 = B2

ie B shares the length of the hypotenuse of the above triangle with side A and length f.
The main axis direction projected onto the x-z plane expressed as an angle from the postive z direction. The main axis direction here is considered to be the vector from the ellipse centre, through the centre of mass to the point of closest approach.
ϑ
The main axis direction expressed as a unit vector (length 1) of the x,y,z axis directions. The main axis direction here is considered to be the vector from the ellipse centre, through the centre of mass to the point of closest approach.
The deviation of the orbital path plane from the x-z plane.

A vector direction orthogonal to the orbit plane is deduced from a vector "cross product" of the velocity and relative position vectors, represented:

V⃗×R⃗ = P⃗

P⃗ is the angular momentum and is considered to have a direction orthogonally towards the observer if the rotation is seen to be anti-clockwise. The position vector R⃗ is from the centre of mass, and the velocity is with respect to the inertial frame of the centre of mass.

The elleptical plane of the solar system can be defined in numerous ways, but here the inclination is simply the deviation from the x-z plane which approximates the ecliptic. The ecliptic is defined by the Earth's orbital plane, but even this varies.
V⃗
R⃗
P⃗
The eccentricity is a way to define the elongation of an ellipse. The smaller the number the more the ellipse is like a circle.

A
B
f

The eccentricity is defined by the ratio of the focus distance divided by the major radius:

E = f / B

  • E is the eccentricity
  • f is the distance from the centre to the focal point
  • B is the major radius