The Path of the Wanderers

The Heliocentric Model of the Solar System

While the ancients did not understand the motions of the “wandering stars,” Newton and Kepler made great strides toward understanding them.

Assuming the simple case of a single body, A, orbiting another, B, the two bodies will orbit around a common barycentre that represents the centre of mass of the system. Thus while it is acceptable to say that one body orbits another, it is more physically accurate to say that both bodies are in a mutual orbit about each other.

Assume the existence of two objects, A and B, with masses MA and MB, respectively. Let aA be the distance from A to the barycentre of a system, and let aB be the distance from B to the barycentre. The ratio of aA / aB is determined purely by inverse ratio of the two masses. For example if MA = MB, then aA = aB, and if MA = 3MB, then aA = (1/3)MB. Physically, this is because A is more massive than B, and so it has more gravity and is thus B needs to be farther from the centre of mass to balance the system.

In the latter case, the gravitational acceleration of A on B is also greater, causing B to have a higher velocity than A. Thus despite B having a wide orbit, it will complete the orbit at the same time as A, and thus, the orbital period, P, of the two will be equal. The fact that is true for all values of is important for constraining the orbital period of one object if the other is not visible for whatever reason. The orbital period of both bodies may be defined as

$\displaystyle P = 2\pi \sqrt{a^3 / \mu}$

Where a is the semi-major axis, roughly the average distance between A and B (and is defined as half the length of the orbits longest axis, the “major axis”), and μ is defined as μ = GM, the product of the gravitational constant, G, multiplied by the total masses of the two bodies, M. It is necessary to consider the sum of the masses because the gravitational pull of both objects on each other will contribute to the orbital period. In cases where MA > MB, the contribution of B may be considered negligible and the orbital period determined more or less accurately without taking it into account (for example, calculating the orbital period of Earth about the sun may be done reasonably well without taking Earth’s gravitational pull on the sun into account).

Not all orbits are perfect circles. Some may be eccentric. Mathematically, an orbit should never be thought of as a circle, but rather as an ellipse with the barycentre at one focus. However an orbit with an eccentricity of e = 0 is physically indistinguishable from circular.

For an eccentric orbit (any orbit for which e > 0), there exists a point in the orbit where the bodies are closest to each other, this is referred to as the periapsis, ap. The point in the orbit where the two bodies are most separated is the apoapsis aa, though these terms vary depending on which body is orbited. Aphelion and perihelion are used for bodies orbiting the sun, and apastron and periastron are used for bodies orbiting other stars.

If e = 0, the orbit may be considered circular. If 0 < e < 1, then the orbit is eccentric. If e = 1, then the orbit is parabolic, and not bound to the orbited body. If e > 1, then the orbit is hyperbolic and is not bound to the orbited body. In the case of e ≥ 1, there exists a periapsis, but no apoapsis.

Regardless of the eccentricity of an orbit, the orbital period will remain constant for a constant value of a. However the orbital velocity will change in such a way so that a line drawn from the planet to the star will cover a constant amount of area per unit time. As such, a planet will orbit faster at periastron, and slower at apastron.

Conveniently, the eccentricity of a planet around the barycenter will be equal to the eccentricity of the star’s orbit around the barycentre. Note that if a body in a circular orbit accelerates (be it by its own power like a spacecraft, or by a gravitational pull from another body), its orbit will become more eccentric.

Inclination for the planets in our solar system is defined as the angle between the plane of the planet’s orbit and the plane of Earth’s orbit, which defines the “ecliptic.” However this is invalid for other planetary systems for which Earth is not presently orbiting to provide such a reference. As such, we use a different reference plane upon which the sky is. We view the sky as if in the centre of an all-encompassing sphere. When looking at a planetary system from the viewpoint of Earth, we may consider the background sky to be a perpendicular plane, and determine the inclination of an orbit as the angle between that body’s orbital plane and the plane of the sky.

Inclination of a planetary system

Thus an edge-on orbit around some distant star would be perpendicular to the plane of the sky, and would be considered to have an inclination of 90°, and face-on orbits are parallel to the plane of the sky and have an inclination of 0° or 180°. As with the orbital period and eccentricity, both the planet and the star will have an equal inclination (and indeed their orbital planes are identical).

One may rotate an orbit about the orbited body within the orbit plane, such that the rotation axis is perpendicular to the orbit plane. This is called the longitude of perihelion, ω, because it rotates the perihelion longitudinally around the orbit, and is measured in degrees. For an orbit with e = 0, this does not make a noticeable change in the orbit, and in such cases, it is customary to set ω = 0°. The reference angle for ω is the line between the observer and the star. There is a 180° difference between ωplanet and ωstar.

The longitude of the ascending node, Ω, is the angle around the reference plane where the orbital plane of the planet in an inclined orbit crosses the reference plane. For the specific case of the ascending node, the planet rises above the reference plane (more northward). For the specific case of when the planet dips below the reference plane, this is called the longitude of the descending node. From visual observation alone, it is impossible to discern the ascending node from the descending node for a distant planetary system. The value for Ω is equal for both the planet and the star. For non-inclined orbits we let Ω = 0°.

The mean anomaly of an orbit, M, is the position of the planet around the orbit, in degrees, and is unrelated to the shape of the orbit. It is typically given with an ephemeris, or a date (or more scientifically, ‘epoch’) for which the value of M is true. There is a 180° offset between the values of M for the star and the planet for the same epoch.

The elements of an orbit