Orbital Elements
We are essentially done with the problem we started off with. However, there are still some advanced topics left which will simplify our lives, and some setups left to consider. Before that, though, we’re going to discuss orbital elements. These are the elements we need to categorize an orbit, and give us a more general overview of what information we need to fix an orbit. This will be helpful in trying to figure out what kind of shape, orientation, etc. our orbit has. This also serves as a recap of things that we’ve introduced earlier.
Before that, however, let’s have a look over some terminology that we’ll commonly use. You should review positional astronomy (specifically, coordinate systems and the sphere) before moving on, as we’ll use terms from there.
- Reference plane: plane used as a reference to define orbital elements. The reference plane is typically taken to be the ecliptic (the plain containing the orbit) or the equator.
- Reference direction (♈︎): direction in the reference plane used to define the orbital elements. The reference direction is typically taken to be the vernal equinox
- Node: intersection of the orbiting body’s orbital plane with the reference plane.
- Ascending node (☊): point at which the orbiting body crosses the reference plane from below to above.
- Descending node (☋): point at which the orbiting body crosses the reference plane from above to below.
- Periapsis (☉): point at which the orbiting body is closest to the focus of the ellipse. If the primary body is the Sun, this point is called perihelion (☉). If the primary body is Earth, this point is called perigee (☉). If the primary body is the Moon, this point is called perilune (☉).
- Apoapsis (☽): point at which the orbiting body is farthest from the focus of the ellipse. If the primary body is the Sun, this point is called aphelion (☽). If the primary body is Earth, this point is called apogee (☽). If the primary body is the Moon, this point is called apolune (☽).
Orbital elements are the parameters required to uniquely identify a specific orbit.
When we view them from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. A visualisation here certainly helps, so I would recommend playing a bit with the wonderful visuals by Bartosz Ciechanowski.
Anyway, when viewed from a non-inertial frame centered on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. In general, six such parameters are used to describe the orbit when viewed from such a frame.
Two parameter are required to describe the size and shape of the orbit. These are:
- Semi-major axis : half the distance between the apoapsis and periapsis (long axis of the ellipse). This value is positive for elliptical orbits, infinite for parabolic trajectories, and negative for hyperbolic trajectories. This can hinder its usability when working with different types of trajectories, as it is not always nice to work with.
- Eccentricity : shape of the ellipse. It describes how much it deviates from a perfect a circle. An eccentricity of zero describes a perfect circle, values less than 1 describe an ellipse, values greater than 1 describe a hyperbolic trajectory, and a value of exactly 1 describes a parabola.
Three parameters are required to descrive the orientation of the orbit. These are:
- Inclination : vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node. Inclinations from to are typically used to denote retrograde orbits.1
- Longitude of the ascending node : describes the angle from the ascending node of the orbit (☊ in the diagram) to the reference frame’s reference direction (♈︎ in the diagram). This is measured in the reference plane. This is clearly undefined for perfectly coplanar orbits, but is often set to zero instead by convention.
- Argument of periapsis : defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis. This is again, undefined for circular orbits, but is often set to zero instead by convention.
Another alternative parameter commonly used is
- Longitude of periapsis : describes the angle between the reference direction (♈︎) and the periapsis, measured partly in the reference plane and partly in the orbital plane. It is defined as . Unlike the longitude of the ascending node, this value is defined for orbits where the inclination is zero.
One parameter is needed to describe the position of the body around its orbit, and the time at which this occurs. These are:
- Time of periapsis : time at which the orbiting body is at periapsis. This value is not defined for circular orbits, as they do not have a uniquely defined point of periapsis.
Another parameter commonly used is
- Mean Longitude : Mean longitude is similar to mean anomaly, in the sense that it increases linearly with time and does not represent the real angular displacement. Unlike mean anomaly, mean longitude is defined relative to the vernal point, which means it is defined for circular orbits.
Eulerian Angles
The Euler angles are three angles used to describe the orientation of a coordinate system with respect to another fixed coordinate system. The angles , , are the Euler angles characterizing the orientation of the coordinate system. Let , and define the coordinate system of the reference plane, and , and define the coordinate system of the orbital plane. N in the figure denotes the direction of the ascending node, while X is the reference direction.
The transformation between the two coordinate systems is given by the following equations:
and
If the orbiting body is at a distance , having true anamoly , its coordinates are , and . Therefore, the coordinates of the orbiting body in the reference frame are given by
If the reference frame is the ecliplic, the (heliocentric) ecliptic coordinates of the orbiting body are given by
Problems
Assume that at this exact moment a satellite in circular heliocentric orbit is in a solar transit.
- Find the distance of the satellite from earth 65 days later.
- Find the equatorial coordinates of this satellite 140 days later.
You are given that the orbital inclination of the satellite is , orbital radius is , longitude of ascending node is and inclination of the ecliptic relative to the equator is . Assume the orbit of Earth to be circular.
We are given that
Note that since there is no periapsis (the orbit is circular), we can’t determine . Instead, we use the ascending node as a reference point, and set there. The orbital period of the satellite is , which gives the mean motion . Writing eqn 3.4.3 to 3.4.5 at a later time when the mean anamoly (which is equal to the true anamoly) is ,
- The position of Earth at the same time is given by
where a = .
At , . Putting this into the equations, we get
The distance between Earth and the satellite is, therefore,
- The mean anamoly after 140 days is . The position of the satellite at this time is given by
Now, using eq 3.4.6, we can find the ecliptic coordinates of the satellite.
To get the equatorial coordinates from this, we use the following equations
Substituing the values for , and , we get
-
These are orbits where the orbit rotates opposite to the rotation of the primary body, for example a satellite rotating in a direction opposite to Earth’s rotation. ↩︎